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FQXi FORUM
January 22, 2019
Does universe has any dimension or it is dimension-less?
In the universe we can observe only distances not dimensions. Thee dimensional Euclid space, four dimensional Riemann space and multidimensional geometrical spaces are merely mathematical models and not physical realities.
As in the universe we can observe only distances we can only observe motion. Time as a clock run and different geometries are man inventions with which he describes motion i.e material change in the universe. Universe itself is dimension-less and time-less. Space-time is a math model only and has no correspondence in physical universe.
A see more adequate picture of the universe in three-dimensional objects of different sizes from Planck size above. This picture is more elegant and less complicated as a picture of n-dimensional physical objects coupled together as a Russian Matjuskas.
Whatever model we take, we have to be aware it is only a picture. Conscious observer is aware of that.
yours amrit
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In the universe we can observe only distances not dimensions. Thee dimensional Euclid space, four dimensional Riemann space and multidimensional geometrical spaces are merely mathematical models and not physical realities.
As in the universe we can observe only distances we can only observe motion. Time as a clock run and different geometries are man inventions with which he describes motion i.e material change in the universe. Universe itself is dimension-less and time-less. Space-time is a math model only and has no correspondence in physical universe.
A see more adequate picture of the universe in three-dimensional objects of different sizes from Planck size above. This picture is more elegant and less complicated as a picture of n-dimensional physical objects coupled together as a Russian Matjuskas.
Whatever model we take, we have to be aware it is only a picture. Conscious observer is aware of that.
yours amrit
PS
Regarding the title here, mathematic is not an illusion. Mathematic is a projection of consciousness into human mind. Logic is based on neuronal dynamics that corresponds fundamental physical properties and dynamics of the universe. Logic and mathematic are realizations – manifestations of consciousness in the rational part of human mind.
“Illusion” is thinking that mathematical objects as space-time, gravitational waves, hypothetical particle “chronos” and some others are physical realities. “Illusion” is created by not clearly distinguishing between “scientific picture” and universe itself.
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Regarding the title here, mathematic is not an illusion. Mathematic is a projection of consciousness into human mind. Logic is based on neuronal dynamics that corresponds fundamental physical properties and dynamics of the universe. Logic and mathematic are realizations – manifestations of consciousness in the rational part of human mind.
“Illusion” is thinking that mathematical objects as space-time, gravitational waves, hypothetical particle “chronos” and some others are physical realities. “Illusion” is created by not clearly distinguishing between “scientific picture” and universe itself.
Topos theory or topoi describes sets of sheaves which have some categorical equivalence by functors. So what Isham and Doring are setting up is a system where one observer will detect things under one algebraic variety, while in general observations can occur under a whole set of such varieties. A formalism of quantum gravity may well have this sort of feature. My general comment is that this tends to appeal to abstractions as a way of doing physics. What is likely to change how we think about physics is some physical principle, or some new manner of thinking where obstructions to the quantization of gravity are removed. The formalism which emerges from this could then involve this sort of category theory.
Cheers LC
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Cheers LC
The previous posting "topos or not topos",I recall reading and trying to get around the complex notions and ideas/ what I found is that the macro system can be thought as being "True" whilst the quantum system being thought of as "False". The bigger the system the more real or relative it becomes, and conversely the more micro a system becomes, the more trouble one has in retaining relative properties such as position ie.
For all intensive purpose, the undetectableness of a system is related to it's "falseness"? one can be mathematical certain of existing within relative 4-D space-time, but one cannot transpose the mathematics below quantum scales with exact certainties.
Ultimately one can coclude this:What is True on large scales, may not be proveable "True" on small scales, and what is "False" on large scales may not be proveable on small scales?
Due the exactness of relativity, the opposing systems such as string theory may, by its very nature be only identified by its "falseness"?..if stringtheory was "true" then relativity must be false..at least by the insights of topos theory, as I understand it that is !
best p.v
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For all intensive purpose, the undetectableness of a system is related to it's "falseness"? one can be mathematical certain of existing within relative 4-D space-time, but one cannot transpose the mathematics below quantum scales with exact certainties.
Ultimately one can coclude this:What is True on large scales, may not be proveable "True" on small scales, and what is "False" on large scales may not be proveable on small scales?
Due the exactness of relativity, the opposing systems such as string theory may, by its very nature be only identified by its "falseness"?..if stringtheory was "true" then relativity must be false..at least by the insights of topos theory, as I understand it that is !
best p.v
Stringtheory is a beautiful theory, because of it's inability to be mathematically proven ?
It sounds almost as if these words are ringing out from a distant past philosophy, maybe one of Jung?
very interesting idea's
p.v
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It sounds almost as if these words are ringing out from a distant past philosophy, maybe one of Jung?
very interesting idea's
p.v
How to prove string theory in one easy lesson.
Add both sides of the blackboard to get string theory to balance.
and add the five string theories.
1/3 APPLE+ 1/3 ORANGE+ 1/3 ORANGE= 1 APPLE/ORANGE.
Add the equatons three at a time selecting the equations intelligently to be added.
Then keep adding until you have just one equation for all string theory.
Then put this equation to the test................
in the real world.
Steve
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Add both sides of the blackboard to get string theory to balance.
and add the five string theories.
1/3 APPLE+ 1/3 ORANGE+ 1/3 ORANGE= 1 APPLE/ORANGE.
Add the equatons three at a time selecting the equations intelligently to be added.
Then keep adding until you have just one equation for all string theory.
Then put this equation to the test................
in the real world.
Steve
I have a tangential interest in this. A quantum error correction code exists on the 24 + n dimensional spacetime, where n = 2 on the light cone gauge of the n = 3 Jordan exceptional algebra. The 24 -dimensions describes a Leech lattice system, and the additional n dimensions are a projective variety. This projective variety defines another covering quantum code called a Goppa code. This has topoi mathematics, since projective varieties define sheaves, and the coding structure is a categorical system on sheaves --- similar to the Grothendieck system or etale topos.
However, before blasting away with that I am trying to establish a physical reasoning for this. In some ways this is harder than the mathematics. It centers around the obstruction to the existence of a continuous time operator in quantum mechanics.
In what I am proposing string theory does play a role. The 26 dimensions above pertain to the 26-dimensional bosonic string. I don’t think the problem with string theory is mathematically proving things. The mathematics can be worked out. The problem is really with direct empirical observations. Topos theory, if that is called to really bear on physics problems, will indicate how certain theoretical observations (what an observer detects) varies in ways which appear radically different, but which ultimately have some categorical equivalency.
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However, before blasting away with that I am trying to establish a physical reasoning for this. In some ways this is harder than the mathematics. It centers around the obstruction to the existence of a continuous time operator in quantum mechanics.
In what I am proposing string theory does play a role. The 26 dimensions above pertain to the 26-dimensional bosonic string. I don’t think the problem with string theory is mathematically proving things. The mathematics can be worked out. The problem is really with direct empirical observations. Topos theory, if that is called to really bear on physics problems, will indicate how certain theoretical observations (what an observer detects) varies in ways which appear radically different, but which ultimately have some categorical equivalency.
Hermann Weyl once commented:
'While topology has succeeded fairly well in mastering continuity, we do not yet understand the inner meaning of the restriction to differentiable manifolds. Perhaps one day physics will be able to discard it. '
We are now ready to do that. In fact it has been done and the results are so amazing that most theoretical physicists cannot even recognize or understand the successful completion of Einstein's 3-part relativity project: [1] Special Relativity (relativity of S-T for inertial frames); [2] General Relativity (relativity for inertial + accelerated frames); [3] discrete conformal relativity (discrete relativity of scale).
Welcome to the 21st century,
RLO
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'While topology has succeeded fairly well in mastering continuity, we do not yet understand the inner meaning of the restriction to differentiable manifolds. Perhaps one day physics will be able to discard it. '
We are now ready to do that. In fact it has been done and the results are so amazing that most theoretical physicists cannot even recognize or understand the successful completion of Einstein's 3-part relativity project: [1] Special Relativity (relativity of S-T for inertial frames); [2] General Relativity (relativity for inertial + accelerated frames); [3] discrete conformal relativity (discrete relativity of scale).
Welcome to the 21st century,
RLO
Lawrence@:A quantum error correction code exists on the 24 + n dimensional spacetime, where n = 2 on the light cone gauge of the n = 3 Jordan exceptional algebra. The 24 -dimensions describes a Leech lattice system, and the additional n dimensions are a projective variety. This projective variety defines another covering quantum code called a Goppa code. This has topoi mathematics, since...
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There once was a man named Kaku,
Who imagined himself a guru.
While hunting hidden worlds, alas,
He stuck his head far up his ass,
And declared: "That world is made of poo!"
Gratis,
RLO
www.amherst.edu/~rloldershaw
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Who imagined himself a guru.
While hunting hidden worlds, alas,
He stuck his head far up his ass,
And declared: "That world is made of poo!"
Gratis,
RLO
www.amherst.edu/~rloldershaw
The relationship between the 26 and 10-11 dimensional cases is subtle. The bosonic string has Virasoro cancellation of central anomaly for D = 26, and for supersymmetric theory it is 11 dimensional. The 11 dimensional case involves M^4xS^7, where the S^7, or its related S^6 under the light cone gauge, contains the Calabi-Yau data under compactification. The 7-sphere here plays a central role in this matter, for it defines a holonomy which is crucial in defining a cubic action. I will say that this is holographic, and there are ways of working a matrix theory this way. The best approach is the Jordan exceptional algebra. This is the algebra of the octonions, which extends the E_8 octonions into a triality. The J^2(O) = R (+) V [(+) = oplus] is of the form
[equation]J^2(O) = \left(\matrix{z_0 & {\cal O}\cr
{\bar{\cal O}} & z_1}\right)[/equation]
which is extended to the J^3(O) matrix algebra. The triality in J^3(O) includes an E_8 matrix of vectors and two spinor matrices. These do correspond to a Feynman diagram where a vector (boson) decays into a spinor and its conjugate. The Feynman diagram is a three-way spoke, with V ~ O going in and O’ and O” going out. The J^3(O) is R (+) J^2(O) (+) θ (+) bar-θ,
[equation]J^3(O) = \left(\matrix{z_1 & {\cal O}_0 & {\bar{\cal O}}_2\cr
{\bar{\cal O}}_0 & z_2 & {\cal O}_1\cr
{\cal O}_2 & {\bar{\cal O}}_1 & z_0}\right)[/equation]
for the spinorial (fermionic fields) octonions in O and O”. The vector term given by the ocotinion O, V ~ J^2(O) decays into V --> ψ + bar-ψ. This is a manifestation of the automorphism of G_2, and is an elementary Feynman diagram for a supersymmetric gauge interaction. So the two spinorial octonions are the O^2 or two E_8’s which you refer to.
The space here is 27 dimensional. The O’s are each 8-dimensional, which gives a general span of 3x8 + 3 = 27 dimensional. The above triality condition, along with some anomaly cancellations of vertex algebras, defines a space of reduced dimension of 8 + 3 = 11 dimensional. On the light cone frame (infinite momentum frame) the space in 27 dimensions is reduced to 26 dimensions and the 11 dimensional space to 10. These are the corresponding bosonic string Lorentizian spacetime and the supersymmetric space of supergravity respectively. The diagonal elements of this matrix define a Chern-Simons lagrangian of scalar terms for x_i --> p_i + A_i (the cubic nature of this is apparent) and a general Lagrangian defined as the determinant of J^3(O) under all triality transformations defines a cubic action. This then determines an equivalency between a field theory term and a dual boundary field. This field ~ boundary of dual field is a cornerstone of AdS.CFT. The AdS/CFT can be found for the case where two scalars in J^3(O) define timelike directions.
Cheers LC
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[equation]J^2(O) = \left(\matrix{z_0 & {\cal O}\cr
{\bar{\cal O}} & z_1}\right)[/equation]
which is extended to the J^3(O) matrix algebra. The triality in J^3(O) includes an E_8 matrix of vectors and two spinor matrices. These do correspond to a Feynman diagram where a vector (boson) decays into a spinor and its conjugate. The Feynman diagram is a three-way spoke, with V ~ O going in and O’ and O” going out. The J^3(O) is R (+) J^2(O) (+) θ (+) bar-θ,
[equation]J^3(O) = \left(\matrix{z_1 & {\cal O}_0 & {\bar{\cal O}}_2\cr
{\bar{\cal O}}_0 & z_2 & {\cal O}_1\cr
{\cal O}_2 & {\bar{\cal O}}_1 & z_0}\right)[/equation]
for the spinorial (fermionic fields) octonions in O and O”. The vector term given by the ocotinion O, V ~ J^2(O) decays into V --> ψ + bar-ψ. This is a manifestation of the automorphism of G_2, and is an elementary Feynman diagram for a supersymmetric gauge interaction. So the two spinorial octonions are the O^2 or two E_8’s which you refer to.
The space here is 27 dimensional. The O’s are each 8-dimensional, which gives a general span of 3x8 + 3 = 27 dimensional. The above triality condition, along with some anomaly cancellations of vertex algebras, defines a space of reduced dimension of 8 + 3 = 11 dimensional. On the light cone frame (infinite momentum frame) the space in 27 dimensions is reduced to 26 dimensions and the 11 dimensional space to 10. These are the corresponding bosonic string Lorentizian spacetime and the supersymmetric space of supergravity respectively. The diagonal elements of this matrix define a Chern-Simons lagrangian of scalar terms for x_i --> p_i + A_i (the cubic nature of this is apparent) and a general Lagrangian defined as the determinant of J^3(O) under all triality transformations defines a cubic action. This then determines an equivalency between a field theory term and a dual boundary field. This field ~ boundary of dual field is a cornerstone of AdS.CFT. The AdS/CFT can be found for the case where two scalars in J^3(O) define timelike directions.
Cheers LC
Oops~ I copied part of the post in the name box! Yikes, that looks crazy!
Cheers LC
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Cheers LC
All matter and constituent particles are continuously changing position and thus form within space, and all particles are in continuous motion, all things are in a continuous state of becoming. I have described the necessary energy for universal motion and development of structure from the complexity of the medium as the Universal potential energy difference between the two states of existence,...
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I would like a more radical return to really sound mathematics instead of mysticism. Isn't set theory based topology unable to cut the real line into two symmetrical parts?
Let me comment on this:
"Isham and colleagues have identified a topos in which quantum theory appears to make logical sense—as long as you embrace a new type of logic, in which "true" and "false" are no longer your only options. There are now multiple shades in between."
Cantor's naive set theory deliberately ignored so called 4th logical option. Isn't it justified to ask for a honest while less exciting solution instead?
"In a sense, using a topos is changing the whole of mathematics," says Döring.
How can the application of a branch of a mathematics that is based on a - as I found out illfounded - theory change the whole of mathematics to the better?
Incidentally to me, the words "in some sense" tend to hide a vague speculation.
Eckard Blumschein
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Let me comment on this:
"Isham and colleagues have identified a topos in which quantum theory appears to make logical sense—as long as you embrace a new type of logic, in which "true" and "false" are no longer your only options. There are now multiple shades in between."
Cantor's naive set theory deliberately ignored so called 4th logical option. Isn't it justified to ask for a honest while less exciting solution instead?
"In a sense, using a topos is changing the whole of mathematics," says Döring.
How can the application of a branch of a mathematics that is based on a - as I found out illfounded - theory change the whole of mathematics to the better?
Incidentally to me, the words "in some sense" tend to hide a vague speculation.
Eckard Blumschein
I said "I do not know whether the new mathematics proposed in the article is actually required." That was perhaps a little too diplomatic to convey my genuine opinion.
I am sure that this mathematics is not necessary to explain the physics that is observed. Quaternion mathematics will do the job nicely, in my opinion. The quaternion mathematical approach has to be the way forward as it is the logical way to model the universe, that answers the foundational questions and overcomes the paradoxes. Doug Sweetster seems to be doing some good work here.
Doug Sweetster's maths
Not being a mathematician I would not like to analyse or criticize his work. It would only serve to show my own limited ability. Though I would be interested in the opinions of other mathematicians. Just to get a measure of how well he is doing. I am really optimistic about this approach. If there are "issues" that is still OK.
As Feyman said at the start of his Noble prize lecture. "We have a habit in writing articles published in scientific journals to make the work as finished as possible, to cover all the tracks, to not worry about the blind alleys or to describe how you had the wrong idea first, and so on. So there isn't any place to publish, in a dignified manner, what you actually did in order to get to do the work, although, there has been in these days, some interest in this kind of thing."
I think the "in a dignified manner" is important as it is about reputation. I do not know who said it first (it may have been Lord Snowdon or Lord Lichfield) but there is a saying that "the secret of being a good photographer is never to show people your bad photographs" or words to that effect. I would dare to say that all good scientists and good photographers also have a lot of work that they are not proud of because they do not attain the fully formed idea or perfect work from the outset every time.
With regard to the topos mathematics. I do not think it is without any value just because it is unnecessary within physics. I categorise it as interesting, unknown potential, not immediately useful.
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I am sure that this mathematics is not necessary to explain the physics that is observed. Quaternion mathematics will do the job nicely, in my opinion. The quaternion mathematical approach has to be the way forward as it is the logical way to model the universe, that answers the foundational questions and overcomes the paradoxes. Doug Sweetster seems to be doing some good work here.
Doug Sweetster's maths
Not being a mathematician I would not like to analyse or criticize his work. It would only serve to show my own limited ability. Though I would be interested in the opinions of other mathematicians. Just to get a measure of how well he is doing. I am really optimistic about this approach. If there are "issues" that is still OK.
As Feyman said at the start of his Noble prize lecture. "We have a habit in writing articles published in scientific journals to make the work as finished as possible, to cover all the tracks, to not worry about the blind alleys or to describe how you had the wrong idea first, and so on. So there isn't any place to publish, in a dignified manner, what you actually did in order to get to do the work, although, there has been in these days, some interest in this kind of thing."
I think the "in a dignified manner" is important as it is about reputation. I do not know who said it first (it may have been Lord Snowdon or Lord Lichfield) but there is a saying that "the secret of being a good photographer is never to show people your bad photographs" or words to that effect. I would dare to say that all good scientists and good photographers also have a lot of work that they are not proud of because they do not attain the fully formed idea or perfect work from the outset every time.
With regard to the topos mathematics. I do not think it is without any value just because it is unnecessary within physics. I categorise it as interesting, unknown potential, not immediately useful.
Further links to Doug Sweetster's work.
Einstein's vision I: Classical unified field equations for gravity and electromagnetism using Riemannian quaternions by Doug Sweetster
New Quaternion Math Leads to New Reasons Why Physics Work, you tube video
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Einstein's vision I: Classical unified field equations for gravity and electromagnetism using Riemannian quaternions by Doug Sweetster
New Quaternion Math Leads to New Reasons Why Physics Work, you tube video
The process of cutting a line is the Dedekind cut.
Topos theory does not involve Cantor's transfinite numbers. Topos theory does invoke the Zariski topology, or systems of projective and algebraic varieties which are in general non-Hausdorff. The only connection with set theory is with foundational issues that are purely mathematical. Most people working on this or related issues of sheaf theory in algebraic geometry are not focused particularly on set theoretic underpinnings.
Sweetser's GEM theory!? I have encountered this guy on other blog or forum sites. First off, if you want to study quaternions, then study Clifford algebras. My comment on Sweetser's stuff is that he is, as I recall, building up a theory of gravity based on electromagnetism, or the quaternionic formulism of EM going back to Maxwell. He has been plugging on this for years, and the whole thing keeps snowballing into an endless tangle and nest of complicated formulae. I and others would find difficulties with his ideas and he would keep patching it up, and the whole thing kept evolving into this increasingly ponderous web of complexity. I don't think his theory beyond the second order in post-Newtonian gravity can be correct.
Cheers LC
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Topos theory does not involve Cantor's transfinite numbers. Topos theory does invoke the Zariski topology, or systems of projective and algebraic varieties which are in general non-Hausdorff. The only connection with set theory is with foundational issues that are purely mathematical. Most people working on this or related issues of sheaf theory in algebraic geometry are not focused particularly on set theoretic underpinnings.
Sweetser's GEM theory!? I have encountered this guy on other blog or forum sites. First off, if you want to study quaternions, then study Clifford algebras. My comment on Sweetser's stuff is that he is, as I recall, building up a theory of gravity based on electromagnetism, or the quaternionic formulism of EM going back to Maxwell. He has been plugging on this for years, and the whole thing keeps snowballing into an endless tangle and nest of complicated formulae. I and others would find difficulties with his ideas and he would keep patching it up, and the whole thing kept evolving into this increasingly ponderous web of complexity. I don't think his theory beyond the second order in post-Newtonian gravity can be correct.
Cheers LC
I would just like to explain how the ideas I have posted here tie in with my previous posts on subjective reality.
I said "The objects potentially observed states of being can be imagined existing sequentially as spatial variations along the 4th dimension not as multiple possibilities occupying the same 3D space. Though the imagined other possibilities have no existence outside of the imagination when the particular state at the moment of observation is determined. They were only potential candidates for the prize of being in existence at the moment of observation."
Rather than moment of observation I should have said at that particular spatial configuration where the observer or apparatus detected the information concerning the particle. That spatial configuration and condition of particle is just one in a sequence passed through. The information gathered about the particle is extracted from objective reality and is formed into a subjective reality. It is wrongly assumed that that information gathered on the particle is in some way more real than the other possibilities, that might have been detected if the moment (spatial configuration) of detection had been different. Which is to say it was undertaken when there was a different spatial configuration and condition of particle. The different possibilities having had their real existence along the 4th spatial dimension as the particle took its particular path through quaternion space. The particle if undisturbed would also not remain in the condition that it would have been detected in, if that detection was carried out.
Subjective reality freezes the condition of the particle to identify a single reality within 3D space rather than giving it an extra degree of spatio-energetic freedom in which to exist. It does not have to be this or that it can be both but not within the same moment or exact same spatial configuration. It is an elusive dynamic entity, never ceasing in its change of quaternion spatial position. Its identity formed from the sequence, that is the spatio-energetic change, rather than being fixed in a singular identifiable identity within 3D space that is unchanging.
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I said "The objects potentially observed states of being can be imagined existing sequentially as spatial variations along the 4th dimension not as multiple possibilities occupying the same 3D space. Though the imagined other possibilities have no existence outside of the imagination when the particular state at the moment of observation is determined. They were only potential candidates for the prize of being in existence at the moment of observation."
Rather than moment of observation I should have said at that particular spatial configuration where the observer or apparatus detected the information concerning the particle. That spatial configuration and condition of particle is just one in a sequence passed through. The information gathered about the particle is extracted from objective reality and is formed into a subjective reality. It is wrongly assumed that that information gathered on the particle is in some way more real than the other possibilities, that might have been detected if the moment (spatial configuration) of detection had been different. Which is to say it was undertaken when there was a different spatial configuration and condition of particle. The different possibilities having had their real existence along the 4th spatial dimension as the particle took its particular path through quaternion space. The particle if undisturbed would also not remain in the condition that it would have been detected in, if that detection was carried out.
Subjective reality freezes the condition of the particle to identify a single reality within 3D space rather than giving it an extra degree of spatio-energetic freedom in which to exist. It does not have to be this or that it can be both but not within the same moment or exact same spatial configuration. It is an elusive dynamic entity, never ceasing in its change of quaternion spatial position. Its identity formed from the sequence, that is the spatio-energetic change, rather than being fixed in a singular identifiable identity within 3D space that is unchanging.
The relevance of my previous post to the article is that, in my opinion, it should not take a -complete- change in the way mathematics is done to explain scientific observations. There has already been considerable evidence that the universe can be described using mathematic, giving good correspondence. Given a workable interpretation of what is being observed, intelligent and careful development and application of currently known mathematical technique should suffice.
Doug Sweetser did say that, (according to his calculations) gravity could be regarded as a potential or a metric. That does correspond with what I have been saying about how gravity can be understood. Though, not being a mathematician, I can not realistically give a well considered opinion on how well the mathematics was performed. I am grateful for Lawrences's opinion but would also like to hear from other mathematicians regarding Doug Sweetster's work
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Doug Sweetser did say that, (according to his calculations) gravity could be regarded as a potential or a metric. That does correspond with what I have been saying about how gravity can be understood. Though, not being a mathematician, I can not realistically give a well considered opinion on how well the mathematics was performed. I am grateful for Lawrences's opinion but would also like to hear from other mathematicians regarding Doug Sweetster's work
Sweetser's idea involve these very long and complicated equations. As time went on in following his reasoning they grew more complicated, with specially define product rules and all sorts of things. As I remember some of these product rules were not mathematically consistent. I am not going to pour through his stuff to ferret this out again, unless there comes some trumpet blast from major figures in physics and a growing chorus all giving support for his theory. I sort of doubt that will happen.
As for mathematics in general, I don't believe in crafting physics entirely around a mathematical system. New physical principles have to be advanced as the primary motivator, which then require some mathematical system of description. On the other hand, as physics advances it will most likely require ever more advanced (or better put abstract) mathematics. So a good grounding in higher mathematics makes for a better tool box.
Cheers LC
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As for mathematics in general, I don't believe in crafting physics entirely around a mathematical system. New physical principles have to be advanced as the primary motivator, which then require some mathematical system of description. On the other hand, as physics advances it will most likely require ever more advanced (or better put abstract) mathematics. So a good grounding in higher mathematics makes for a better tool box.
Cheers LC
Lawrence you said "New physical principles have to be advanced as the primary motivator, which then require some mathematical system of description."
The following all apply to a quaternion energy-space not space-time.
All matter has continuous aforeward change of position along the scalar dimension.(Necessary for gravity and arrow of time.)
All subatomic particles have...
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The following all apply to a quaternion energy-space not space-time.
All matter has continuous aforeward change of position along the scalar dimension.(Necessary for gravity and arrow of time.)
All subatomic particles have...
view entire post
No. Dedekind`s cut does not symmetrically divide the line of numbers into two equal parts. Such division without remaining neutral element is impossible for integer and rational numbers while obviously possible for a continuum every part of which has parts, endlessly. Dedekind explained in §4 "Creation of Irrational Numbers" how his cut CREATES a number. He and Cantor abandoned the old notion of continuum.
Lawrence Crowell wrote: Topos theory does not involve Cantor's transfinite numbers. Topos theory does invoke the Zariski topology, or systems of projective and algebraic varieties which are in general non-Hausdorff. The only connection with set theory is with foundational issues that are purely mathematical. Most people working on this or related issues of sheaf theory in algebraic geometry are not focused particularly on set theoretic underpinnings.
I noticed constructivists who tried a topos without the law of excluded middle.
I see the problem deeper hidden within the limitation of the notion number.
Eckard Blumschein
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Lawrence Crowell wrote: Topos theory does not involve Cantor's transfinite numbers. Topos theory does invoke the Zariski topology, or systems of projective and algebraic varieties which are in general non-Hausdorff. The only connection with set theory is with foundational issues that are purely mathematical. Most people working on this or related issues of sheaf theory in algebraic geometry are not focused particularly on set theoretic underpinnings.
I noticed constructivists who tried a topos without the law of excluded middle.
I see the problem deeper hidden within the limitation of the notion number.
Eckard Blumschein
Eckard Blumschein,
Please forgive the baby steps and baby talk that are necessary for me to take.
Firstly I like the idea of open sets very much because when thinking of particles in continuous motion within quaternion space it is never possible to give a location of a particle as it is forever changing. So it is not really appropriate to talk of distance because of the difficulty identifying a location and because the particle can move freely in 4 dimensions. Distance isn't as straight forward as the line between two points. However if it is squiggling about in 4 dimensional space then that squiggle is an open set. (I was wondering how a squiggle of 4 dimensional change in spatial position could be described.) Keeping track of where the elusive dynamic entity might be in its squiggle also sounds jolly useful.So if I am understanding the idea of sheaves correctly then that is good too.
A problem with the notion of number is interesting. If I use the number 1 to represent a singular object, that number stays with the object. However from one configuration of the contents of space to the next the object is changed. Its position in quaternion space has altered. It has moved along the 4th spatial dimension and this gives rotation within 3D space. All of the subatomic particles within the object will have changed spatial position. Depending upon the object there may have been chemical change within the structure giving a different spatial configuration of the constituent matter. The surrounding medium will have changed in position.
I keep thinking about the animation of the gimbal rings Wikipedia gimbal lock
The point being that 1 is not really just 1. It is a whole sequence of alternative 1s. All may seem to paradoxically exist within 3D space because we do not appreciated that the space and the object are in a continuous process of change with 4 degrees of freedom. So it is not exactly the same 3D space or object. Even though we have attached 1 to the object and it is considered to be the same 1. When an observation is made it is as if the handle of the fruit machine has been pulled. The gimbals rotate with 4 different degrees of freedom and at the precise moment of detection one 1 from the whole range of possibilities is selected as the observed existential reality. Quantum physicists would say the wave function has collapsed.
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Please forgive the baby steps and baby talk that are necessary for me to take.
Firstly I like the idea of open sets very much because when thinking of particles in continuous motion within quaternion space it is never possible to give a location of a particle as it is forever changing. So it is not really appropriate to talk of distance because of the difficulty identifying a location and because the particle can move freely in 4 dimensions. Distance isn't as straight forward as the line between two points. However if it is squiggling about in 4 dimensional space then that squiggle is an open set. (I was wondering how a squiggle of 4 dimensional change in spatial position could be described.) Keeping track of where the elusive dynamic entity might be in its squiggle also sounds jolly useful.So if I am understanding the idea of sheaves correctly then that is good too.
A problem with the notion of number is interesting. If I use the number 1 to represent a singular object, that number stays with the object. However from one configuration of the contents of space to the next the object is changed. Its position in quaternion space has altered. It has moved along the 4th spatial dimension and this gives rotation within 3D space. All of the subatomic particles within the object will have changed spatial position. Depending upon the object there may have been chemical change within the structure giving a different spatial configuration of the constituent matter. The surrounding medium will have changed in position.
I keep thinking about the animation of the gimbal rings Wikipedia gimbal lock
The point being that 1 is not really just 1. It is a whole sequence of alternative 1s. All may seem to paradoxically exist within 3D space because we do not appreciated that the space and the object are in a continuous process of change with 4 degrees of freedom. So it is not exactly the same 3D space or object. Even though we have attached 1 to the object and it is considered to be the same 1. When an observation is made it is as if the handle of the fruit machine has been pulled. The gimbals rotate with 4 different degrees of freedom and at the precise moment of detection one 1 from the whole range of possibilities is selected as the observed existential reality. Quantum physicists would say the wave function has collapsed.
Giving the identity of 1 to any object or particle is problematic. The only unchangeable thing about object or particle is that it has continuous change of quaternion spatial position and thus possesses energy. That gives it existence.
For a particle its location is changing, direction of spin can change ,charge can change, rotation can change. How then is it truely compatible with identity as a singular thing? A macroscopic object is also continuously changing although this is not generally perceived, except perhaps with regard to biological organisations of matter.
We may designate names for particular stages of organisation such as foetus, baby, toddler, child, teenager, young adult, adult , senior or elder. They are obviously not all the same spatial organisation of matter but may still be regarded as 1. The recognition of spatial difference in organisation could be extended so it is acknowledged that day to day the 1 has changed and is not the same 1 as yesterday or even minute to minute or second to second. That is not how we think about the macroscopic world and the objects we identify as singular and name.
It is therefore perhaps the concept of 1 itself that needs revising to acknowledge that 1 is more that just 1 but many different 1s.
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For a particle its location is changing, direction of spin can change ,charge can change, rotation can change. How then is it truely compatible with identity as a singular thing? A macroscopic object is also continuously changing although this is not generally perceived, except perhaps with regard to biological organisations of matter.
We may designate names for particular stages of organisation such as foetus, baby, toddler, child, teenager, young adult, adult , senior or elder. They are obviously not all the same spatial organisation of matter but may still be regarded as 1. The recognition of spatial difference in organisation could be extended so it is acknowledged that day to day the 1 has changed and is not the same 1 as yesterday or even minute to minute or second to second. That is not how we think about the macroscopic world and the objects we identify as singular and name.
It is therefore perhaps the concept of 1 itself that needs revising to acknowledge that 1 is more that just 1 but many different 1s.
The explanations I have given in my previous 2 posts are compatible with the scientific idea of particles existing as a probability until "wave function collapse" but explains mechanically why that should be so. It unites the probabilistic nature of the sub atomic realm with the apparently definitive nature of the macroscopic realm of everyday experience.
Thinking gimbal rings again. Wave function can then be viewed as description of the alterations in orientation of change of spatial position generated from continuous change with 4 degrees of freedom. (It can be hypothesised that spin is the oscillation or rotation along the 4th dimension as it is not rotation within 3D space.)
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Thinking gimbal rings again. Wave function can then be viewed as description of the alterations in orientation of change of spatial position generated from continuous change with 4 degrees of freedom. (It can be hypothesised that spin is the oscillation or rotation along the 4th dimension as it is not rotation within 3D space.)
Dear Georgina, you say: Giving the identity of 1 to any object or particle is problematic. The only unchangeable thing about object or particle is that it has continuous change of quaternion spatial position and thus possesses energy. That gives it existence.
Yes, "motion" and "energy" are two empirical fundaments of physics. All rest is a description.
yours amrit
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Yes, "motion" and "energy" are two empirical fundaments of physics. All rest is a description.
yours amrit
Eckard,
Cutting the reals does involve a Dedekind cut. As for cutting the reals into two equal parts, that gets one into the matter of adding infinities. To be honest I leave the matter of axiomatic set theory to people who work on that. A mathematician friend of mine calls set theory "set on your ass theory," jokingly to indicate how the subject has tangential contact with other more realistic mathematics.
Topos theory might be compared to what happens in quantum mechanics. QM does admit the overlap of states which correspond to physically distinct outcomes under a measurement. In topos theory sheaves may have completely different pre-S structures or different algebraic varieites --- elliptic curves, projective structures and so forth. Yet a functor between two sheaves remove these distinctions, so what might be seen as contradictory sets can then obtains as categoriies.
Cheers LC
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Cutting the reals does involve a Dedekind cut. As for cutting the reals into two equal parts, that gets one into the matter of adding infinities. To be honest I leave the matter of axiomatic set theory to people who work on that. A mathematician friend of mine calls set theory "set on your ass theory," jokingly to indicate how the subject has tangential contact with other more realistic mathematics.
Topos theory might be compared to what happens in quantum mechanics. QM does admit the overlap of states which correspond to physically distinct outcomes under a measurement. In topos theory sheaves may have completely different pre-S structures or different algebraic varieites --- elliptic curves, projective structures and so forth. Yet a functor between two sheaves remove these distinctions, so what might be seen as contradictory sets can then obtains as categoriies.
Cheers LC
Lawrence,
while I enjoy "set on your arse theory" I consider Buridan's ass something important in physics. The matter has far reaching consequences and deserves more clarification than I am able to provide this evening. Please find a bit food for thought in topic 527.
If we agree that real numbers are different from rational ones then Georgina is wrong: The distinction between open and closed sets is invalid for really real numbers. oo + oo = oo at least according to the really great ones like Galilei.
What about topos theory, shouldn't we consult Grothendieck himself?
Sincerely,
Eckard
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while I enjoy "set on your arse theory" I consider Buridan's ass something important in physics. The matter has far reaching consequences and deserves more clarification than I am able to provide this evening. Please find a bit food for thought in topic 527.
If we agree that real numbers are different from rational ones then Georgina is wrong: The distinction between open and closed sets is invalid for really real numbers. oo + oo = oo at least according to the really great ones like Galilei.
What about topos theory, shouldn't we consult Grothendieck himself?
Sincerely,
Eckard
Amrit,
thank you for the reply.
The singular thing identified as 1 changes and so its description can change. The 1 thing is not the identical 1 thing from spatial configuration to spatial configuration (moment to moment if using time as proxy). It is possible to notice the change in the sub atomic particle because it is such a minimal "thing" but continuous change is also occurring in the macroscopic realm. However because macroscopic objects are such large collections of particles in complex organisation the subtle but ceaseless change is not readily apparent.
This is the elusive connection between the behaviour observed at the smallest scale with the macroscopic scale. We only perceive macroscopic objects as unchanging and singular in their identity because mostly the change is subtle and therefore overlooked. Occurring on a scale too small to perceive or too slowly to be apparent to our unaided senses. Such as the erosion of a rock. (We also do not perceive the continuous change in quaternion spatial position of all objects, even those considered stationary.) There is a difference between subjective reality of experience that tells us most macroscopic objects are singular and unchanging in identity and objective reality that exists outside of that experience in which everything at every scale is continuously changing.
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thank you for the reply.
The singular thing identified as 1 changes and so its description can change. The 1 thing is not the identical 1 thing from spatial configuration to spatial configuration (moment to moment if using time as proxy). It is possible to notice the change in the sub atomic particle because it is such a minimal "thing" but continuous change is also occurring in the macroscopic realm. However because macroscopic objects are such large collections of particles in complex organisation the subtle but ceaseless change is not readily apparent.
This is the elusive connection between the behaviour observed at the smallest scale with the macroscopic scale. We only perceive macroscopic objects as unchanging and singular in their identity because mostly the change is subtle and therefore overlooked. Occurring on a scale too small to perceive or too slowly to be apparent to our unaided senses. Such as the erosion of a rock. (We also do not perceive the continuous change in quaternion spatial position of all objects, even those considered stationary.) There is a difference between subjective reality of experience that tells us most macroscopic objects are singular and unchanging in identity and objective reality that exists outside of that experience in which everything at every scale is continuously changing.
A Dedekind cut of the reals gives (∞, 0) [0, ∞) or (∞, 0] (0, ∞), which are not exactly equal. I am not sure this has great impact on physics, unless there is a delta function set at 0. A course in real analysis covers these matters pretty extensively.
Grothendieck disappeared sometime back in the 1980s, and is suspected of living in S. France somewhere. The man is a bit of a subject from “A Beautiful Mind.” During the Vietnam war he conducted lectures on algebraic geometry in the forests outside Hanoi as B-52s were bombing the place. I am not sure if he had much of an audience. In fact as I understand he is not a citizen of any country. He was fairly notorious for strange behavior. I think getting a hold of the man is a bit like trying to put a neutrino in a box. Connes’ book on noncommutative geometry covers some aspects of Grothendieck’s fibration system.
Cheers LC
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Grothendieck disappeared sometime back in the 1980s, and is suspected of living in S. France somewhere. The man is a bit of a subject from “A Beautiful Mind.” During the Vietnam war he conducted lectures on algebraic geometry in the forests outside Hanoi as B-52s were bombing the place. I am not sure if he had much of an audience. In fact as I understand he is not a citizen of any country. He was fairly notorious for strange behavior. I think getting a hold of the man is a bit like trying to put a neutrino in a box. Connes’ book on noncommutative geometry covers some aspects of Grothendieck’s fibration system.
Cheers LC
Eckard Blumschein,
You said "If we agree that real numbers are different from rational ones then Georgina is wrong: The distinction between open and closed sets is invalid for really real numbers. oo + oo = oo at least according to the really great ones like Galilei."
Please could you elaborate on why this should be so? I know nothing about this topos mathematics other than what I have read on this site and Wikipedia. The open set just sounded a good description that might fit with what I am trying to explain, to the best of my limited ability.
From Wikipedia..
A set U is open if any point x in U can be moved by a small amount in any direction and still be in the set U. The notion of an open set provides a fundamental way to speak of nearness of points in a topological space, without explicitly having a concept of distance defined. Concepts that use notions of nearness, such as the continuity of functions, can be translated into the language open sets.
I was thinking that U could be that thing that we consider a particle and x could be the elusive dynamic entity squiggling about within set U with 4 degrees of freedom. So the set is made from all the squiggling. I am not a mathematician, so please forgive my language.
I am contemplating something that exists only because of its change in position in 4 dimensions. A component of that change in position (4th dimensional change) is not directly countable and could be envisioned being along a real number line. (Real numbers being used to measure continuous quantities.) It can be hypothesised that this is the spin and aforeward progression along the 4th dimension of a particle (except for antimatter which may progress aftward). The entity itself is counted as 1 although its continuous change means that it is not the identical 1 but a series of alternative 1s.
Its existence in our universe depends upon its 4th dimensional change in position, a real number continuous change.It can not be directly measured, we can only use time as a proxy and approximation. The 1 that is observed "frozen in an instant of existence" is then a function of being considered singular entity in 3D space denoted by the rational number 1 and having an unmeasurable real number change in position along the 4th spatio-energetic dimension. What does that make it? The entity can be considered as having both a rational component, if its changing is ignored and it is considered just 1 entity, and a real component because of its continuous change.
I think the interesting problem is with saying it is 1 when it is just a "snapshot" of the manifestation of something that is in a process of continuous change.
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You said "If we agree that real numbers are different from rational ones then Georgina is wrong: The distinction between open and closed sets is invalid for really real numbers. oo + oo = oo at least according to the really great ones like Galilei."
Please could you elaborate on why this should be so? I know nothing about this topos mathematics other than what I have read on this site and Wikipedia. The open set just sounded a good description that might fit with what I am trying to explain, to the best of my limited ability.
From Wikipedia..
A set U is open if any point x in U can be moved by a small amount in any direction and still be in the set U. The notion of an open set provides a fundamental way to speak of nearness of points in a topological space, without explicitly having a concept of distance defined. Concepts that use notions of nearness, such as the continuity of functions, can be translated into the language open sets.
I was thinking that U could be that thing that we consider a particle and x could be the elusive dynamic entity squiggling about within set U with 4 degrees of freedom. So the set is made from all the squiggling. I am not a mathematician, so please forgive my language.
I am contemplating something that exists only because of its change in position in 4 dimensions. A component of that change in position (4th dimensional change) is not directly countable and could be envisioned being along a real number line. (Real numbers being used to measure continuous quantities.) It can be hypothesised that this is the spin and aforeward progression along the 4th dimension of a particle (except for antimatter which may progress aftward). The entity itself is counted as 1 although its continuous change means that it is not the identical 1 but a series of alternative 1s.
Its existence in our universe depends upon its 4th dimensional change in position, a real number continuous change.It can not be directly measured, we can only use time as a proxy and approximation. The 1 that is observed "frozen in an instant of existence" is then a function of being considered singular entity in 3D space denoted by the rational number 1 and having an unmeasurable real number change in position along the 4th spatio-energetic dimension. What does that make it? The entity can be considered as having both a rational component, if its changing is ignored and it is considered just 1 entity, and a real component because of its continuous change.
I think the interesting problem is with saying it is 1 when it is just a "snapshot" of the manifestation of something that is in a process of continuous change.
Last post was me. These numbers are annoying me.
The rational number assigned to an entity is just an integer quantity. Giving how many of it there are. But a physical entity is not just an integer quantity. It must also be seen to be a continuous process of change that gives it existence. So it must therefore be represented by a series of real numbers, if it is to be properly comprehended, whilst still retaining its integer assignment.
I am not saying that the rational integers are different from the real numbers. The integers must form part of the full set of real numbers.
It is how they are applied within physics to an entity that is problematic when the real number (4th dimensional) change is not considered. There is then a series of the same integer applying to different spatial manefestations of an entity. Which is correct because there can still the same quantity of it, whatever it has become.
1 ice cube, 1 puddle, 1 vapour cloud. Or 1 electron, 1 positron or 1 infant, 1 child, 1 adult. Same quantity but also changing form.Bizarrely this means if 1 sheep turns into 1 banana it is still OK because there is still only 1 entity. There is still only 1 kind of 1, but a series of different manifestations of the 1 thing.I know I'm repeating myself and probably getting unnecessarily concerned about how this is described numerically but I want to be sure I have understood and clarified it reasonably.
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The rational number assigned to an entity is just an integer quantity. Giving how many of it there are. But a physical entity is not just an integer quantity. It must also be seen to be a continuous process of change that gives it existence. So it must therefore be represented by a series of real numbers, if it is to be properly comprehended, whilst still retaining its integer assignment.
I am not saying that the rational integers are different from the real numbers. The integers must form part of the full set of real numbers.
It is how they are applied within physics to an entity that is problematic when the real number (4th dimensional) change is not considered. There is then a series of the same integer applying to different spatial manefestations of an entity. Which is correct because there can still the same quantity of it, whatever it has become.
1 ice cube, 1 puddle, 1 vapour cloud. Or 1 electron, 1 positron or 1 infant, 1 child, 1 adult. Same quantity but also changing form.Bizarrely this means if 1 sheep turns into 1 banana it is still OK because there is still only 1 entity. There is still only 1 kind of 1, but a series of different manifestations of the 1 thing.I know I'm repeating myself and probably getting unnecessarily concerned about how this is described numerically but I want to be sure I have understood and clarified it reasonably.
Rational numbers are the ratio of two integers r = a/b. Now there are numbers which we call irrational, for they can't be expressed as such. A simple example is sqrt{2}. Assume it is a rational number sqrt{2} = a/b. Now this ratio must be something other than 2^n/2^m, for otherwise we would have sqrt{2} is some multiple of 2 or 1/2. Therefore b must be an odd number. Now we then have
sqrt{2}^2 = (a/b)^2,
or a^2 = 2b^2. Therefore a^2 is even and then must also be a. So a is a number a = 2c, and
2b^2 = 4c^2 ==> c^2 = b^2/2
so that b^2 is clearly even as well, and so them must be b. Yet b must be odd by above, so this is a contradiction. Therefore sqrt{2} can’t be expressed as the ratio of two integers.
The great majority of numbers are irrational, and in fact they can’t be counted. Galois algebra is a system for defining roots of polynomial equations, which are irrational, but the procedure describes a countably infinite set – not all of them. There are in the set of irrational numbers transcendental numbers, such as π. These numbers are not algebraic and they exist in a set which is not countably infinite. This then gets into the matter of infinite cardinalities, which in set theory results in the Cantor description of transfinite numbers and the continuum. The continuum hypothesis is a lynchpin conjecture on the Zermelo-Fraenkel set theory. Bernays and Cohen showed by using Godel’s theorems that the continuum hypothesis is consistent with ZF, but not provable.
What you describe with open sets is a way of setting up Hausdorff point set topology. There are non-Hausdorff topologies as well, such as the moduli space for general relativity or Zariski topology that topos theory depends on. I am not sure how far to go in describing this stuff. This gets into advanced undergraduate to first year graduate school mathematics.
Cheers LC
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sqrt{2}^2 = (a/b)^2,
or a^2 = 2b^2. Therefore a^2 is even and then must also be a. So a is a number a = 2c, and
2b^2 = 4c^2 ==> c^2 = b^2/2
so that b^2 is clearly even as well, and so them must be b. Yet b must be odd by above, so this is a contradiction. Therefore sqrt{2} can’t be expressed as the ratio of two integers.
The great majority of numbers are irrational, and in fact they can’t be counted. Galois algebra is a system for defining roots of polynomial equations, which are irrational, but the procedure describes a countably infinite set – not all of them. There are in the set of irrational numbers transcendental numbers, such as π. These numbers are not algebraic and they exist in a set which is not countably infinite. This then gets into the matter of infinite cardinalities, which in set theory results in the Cantor description of transfinite numbers and the continuum. The continuum hypothesis is a lynchpin conjecture on the Zermelo-Fraenkel set theory. Bernays and Cohen showed by using Godel’s theorems that the continuum hypothesis is consistent with ZF, but not provable.
What you describe with open sets is a way of setting up Hausdorff point set topology. There are non-Hausdorff topologies as well, such as the moduli space for general relativity or Zariski topology that topos theory depends on. I am not sure how far to go in describing this stuff. This gets into advanced undergraduate to first year graduate school mathematics.
Cheers LC
Lawrence and Georgina,
Continuity vs. discreteness is an important issue in physics. Common sense distinguishes between what is considered as a number of countable indivisible items and what is modeled as an endlessly divisible and therefore uncountable liquid. Peirce defined a continuum as something every part of which has parts. Euclid called a point something that does not have parts....
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Continuity vs. discreteness is an important issue in physics. Common sense distinguishes between what is considered as a number of countable indivisible items and what is modeled as an endlessly divisible and therefore uncountable liquid. Peirce defined a continuum as something every part of which has parts. Euclid called a point something that does not have parts....
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I think you might want to read up on real variables, or take a course on the subject. With an MS in math I took a graduate level course, but frankly the subject is not exactly my cup of tea in mathematics. I find it tedious and not terribly interesting. Yet, issues of open and closed sets are well defined, including lim_[sup} and lim_{inf} of sets which approach sets (both open and closed) in certain Cauchy-like sequences. This is fairly classical mathematics, mid to late 19th century stuff. It all leads to generalized theory of integration such as Fatou's theorem and the Lebesgue integral.
With physics information is discrete. Even in classical mechanics. For a wave a discontinuities in a wave front will define a phase velocity for the flow of information. We might think of this with regards to Dirac's comb, or a discrete set of pulses. What is continuous does not convey information. This geometric aspect of theory permits us to work with differential and integral equations. This is IMO a great confusion with the Planck scale. This is the scale where the area of a black hole horizon is determined by a deBroglie wave, leading to L_p = sqrt{Għ/c^3} as some smallest region where physics obtains. Many people think this is some type of discretizing of spacetime. This is wrong, for all it tells us is the minimal scale where we can measure information about physics.
I will largely avoid the Zermelo-Fraenkel set theory issues. The subject captured my attention a number of years ago, and went through a lot of this. I can’t really regurgitate much of it on a short blog post. I will say that largely I found the whole thing consistent and that is worked. I did not find it of much value for physics, so I have largely not studied this subject in over 10 years. The set-theory mavens did come up with an interesting proof in the 80s on the set of homeomorphic 4-manifolds and 7-spheres that are not diffeomorphic. So the subject is not without merits. These guys also work to make tests on how certain areas of mathematics have axiomatic structure consistent with ZF set theory.
Lawrence B. Crowell
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With physics information is discrete. Even in classical mechanics. For a wave a discontinuities in a wave front will define a phase velocity for the flow of information. We might think of this with regards to Dirac's comb, or a discrete set of pulses. What is continuous does not convey information. This geometric aspect of theory permits us to work with differential and integral equations. This is IMO a great confusion with the Planck scale. This is the scale where the area of a black hole horizon is determined by a deBroglie wave, leading to L_p = sqrt{Għ/c^3} as some smallest region where physics obtains. Many people think this is some type of discretizing of spacetime. This is wrong, for all it tells us is the minimal scale where we can measure information about physics.
I will largely avoid the Zermelo-Fraenkel set theory issues. The subject captured my attention a number of years ago, and went through a lot of this. I can’t really regurgitate much of it on a short blog post. I will say that largely I found the whole thing consistent and that is worked. I did not find it of much value for physics, so I have largely not studied this subject in over 10 years. The set-theory mavens did come up with an interesting proof in the 80s on the set of homeomorphic 4-manifolds and 7-spheres that are not diffeomorphic. So the subject is not without merits. These guys also work to make tests on how certain areas of mathematics have axiomatic structure consistent with ZF set theory.
Lawrence B. Crowell
Lawrence and Eckard,
Thank you for your explanations.I probably should think more about numbers.
It seems to me that the 4th spatio-energetic dimension must qualify as really representable by real numbers because there is no way to count or measure along it but if it is spatial then it must have distances. We can only use a continuous measurement undertaken in 3D space to represent that measurement (time).
I also think that even the concept of a point is problematic because to identify it the universe must be artificially frozen. If not only energy but spatial change is conserved then the whole universe is engaged in continuous change of quaternion spatial position. So a single point does not have an unchanging position in quaternion space. So the concept of a point is a mathematical abstraction rather than an accurate representation of physical reality.
At the very least any point with existence rather than a purely mathematical abstraction is moving along the 4th spatial dimension and therefore is rotating in 3D space. If it is a point on a piece of matter such as a page there is the rotation of the earth, orbit of the sun and rotation of the galaxy to consider. Those gimbals turn and the point, although it can be identified as 1 single point in space, also has a whole sequence of different spatial position due to 4 degrees of freedom of movement within space, which are unobserved and unmeasurable and it would seem to me must be considered as representable by real numbers.
Eckard Blumschein You said "Euclid called a point something that does not have parts." But is that really so if the point also ought to be described by a real sequence? So there is it seems to me a difference between the integer 1 and a single existential entity and between a point in mathematics and a point within quaternion energy-space or if you must think in terms of time within space-time. There is a difference between physics and mathematics.
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Thank you for your explanations.I probably should think more about numbers.
It seems to me that the 4th spatio-energetic dimension must qualify as really representable by real numbers because there is no way to count or measure along it but if it is spatial then it must have distances. We can only use a continuous measurement undertaken in 3D space to represent that measurement (time).
I also think that even the concept of a point is problematic because to identify it the universe must be artificially frozen. If not only energy but spatial change is conserved then the whole universe is engaged in continuous change of quaternion spatial position. So a single point does not have an unchanging position in quaternion space. So the concept of a point is a mathematical abstraction rather than an accurate representation of physical reality.
At the very least any point with existence rather than a purely mathematical abstraction is moving along the 4th spatial dimension and therefore is rotating in 3D space. If it is a point on a piece of matter such as a page there is the rotation of the earth, orbit of the sun and rotation of the galaxy to consider. Those gimbals turn and the point, although it can be identified as 1 single point in space, also has a whole sequence of different spatial position due to 4 degrees of freedom of movement within space, which are unobserved and unmeasurable and it would seem to me must be considered as representable by real numbers.
Eckard Blumschein You said "Euclid called a point something that does not have parts." But is that really so if the point also ought to be described by a real sequence? So there is it seems to me a difference between the integer 1 and a single existential entity and between a point in mathematics and a point within quaternion energy-space or if you must think in terms of time within space-time. There is a difference between physics and mathematics.
Eckard, you said "Euclid called a point something that does not have parts. Accordingly, genuine continua like a volume, an area, or a line can just be approximated by a finite number of points, no matter how densely the points are thought to be arranged. Any finite set of points is zero-dimensional."
In existential physics however it would seem that no point can be zero dimensional because of the continuous change of spatial position of that point, with 4 degrees of freedom.(Even though it may appear stationary) Therefore no set of points within existential physics can be zero dimensional.
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In existential physics however it would seem that no point can be zero dimensional because of the continuous change of spatial position of that point, with 4 degrees of freedom.(Even though it may appear stationary) Therefore no set of points within existential physics can be zero dimensional.
I am only thinking about what numbers -are- because it seems relevant to the current contemplation of how they are used for modelling in physics. I have no love for them nor them for me. (I am now re-acquainted with the various terms for different kinds of numbers, thank you.) It seems to me, following on from my previous posts, that the only way to accurately model even a point or single entity is to use a series of quaternion numbers.
Eckard you said "Of course, only approximations to such really real numbers can be numerically handled. On the other hand, several apparent problems including Schroedinger's cat vanish when we do not forget that mathematics is always treating the reals as if they were rationals. Weyl called them aptly a sauce."
I think that is really interesting. I like the analogy of sauce very much too.
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Eckard you said "Of course, only approximations to such really real numbers can be numerically handled. On the other hand, several apparent problems including Schroedinger's cat vanish when we do not forget that mathematics is always treating the reals as if they were rationals. Weyl called them aptly a sauce."
I think that is really interesting. I like the analogy of sauce very much too.
An existential point rather than a mathematical abstract point must be seen to have some duration in time however minuscule.If it has no duration it has no existence. So it can not be zero dimensional. There has to be at the very least 1 dimension of change pertaining to that existential point giving rise to the perceived existence in time. That being 4th dimensional change. An existential point also can not be stationary in quaternion space unless it is perceived to be moving extremely quickly because all of the matter that we perceive to be stationary is actually moving very fast through space. The existential point will have to move very fast to be stationary in objective rather than subjective reality. So the point will have to have perceived velocity within 3D space, which is a quaternion change in position. That velocity being made up of a spatial change in position within 3D space and a change in position along the 4th spatio-energetic dimension.
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The minimum description of an existential point or entity must be two different quaternions. The entity or point must change position in quaternion energy-space (space-time if you really must) in order to have existence. It must also have spatial existence however minimal in all dimensions. This makes an existential point or entity different from a mathematical point or integer which just does not and can not exist outside of abstract mathematics.In my opinion. So quaternion mathematics seems, even from this minute analysis, a superior language for describing the working of the universe.
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In this linkhydrogen atom one can see that the Proton is the point, discrete and seperate, from the Electron, which is continuous and, faip, it's wavelike properties make it infinity like ? So what "seperates" the point from the continuum?
The Proton is 3-D pointlike sitting within a 2-D continuous fieldlike Electron?
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The Proton is 3-D pointlike sitting within a 2-D continuous fieldlike Electron?
Anonymous,
It would be nice to know who you are and who you are addressing. Then I could decide whether it is appropriate for me to reply or whether you want to talk to someone else. I can only say what I think and it would not be a conventional mainstream physics answer.Nice pictures by the way.
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It would be nice to know who you are and who you are addressing. Then I could decide whether it is appropriate for me to reply or whether you want to talk to someone else. I can only say what I think and it would not be a conventional mainstream physics answer.Nice pictures by the way.
Whether or not there is a minimum difference in change of quaternion position that should form the minimum difference in quaternion frame used to model it ie whether there is a jump from position to position or continuous flow can not be known. Attempts to determine this experimentally will,as Eckard pointed out, be constrained by the limit of measurement that is possible. It will therefore depend upon the model that is chosen to represent the change. For a particle it could be assumed that one complete spin oscillation would represent 1 change of quaternion position although smaller distances being fractions of oscillations could be imagined.
Change in 4th dimensional position is not directly measurable. It is reasonable to assume that that dimension should comprise the full set of real numbers,(including those fractions without finite answer). Which is the same as for other physical processes.
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Change in 4th dimensional position is not directly measurable. It is reasonable to assume that that dimension should comprise the full set of real numbers,(including those fractions without finite answer). Which is the same as for other physical processes.
oops..last "anon" was me!
A 3-D point thus has a definate boundary, it is seperate, almost an individual system. 2-D entities, are continuous, like the Electron Wave around the Proton, it is deemed continuous. But when Electrons move across to other Protons, they must be 3-D, or particle-like?
The Particle Wave Duality should really be a dimensional transport action, for the Electrons, Photons and Proton? The Electron shells are transitional points between 3-D and 2-D space's,area's ?
There is more interesting aspects for seperated and continuous boundaries.
best p.v
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A 3-D point thus has a definate boundary, it is seperate, almost an individual system. 2-D entities, are continuous, like the Electron Wave around the Proton, it is deemed continuous. But when Electrons move across to other Protons, they must be 3-D, or particle-like?
The Particle Wave Duality should really be a dimensional transport action, for the Electrons, Photons and Proton? The Electron shells are transitional points between 3-D and 2-D space's,area's ?
There is more interesting aspects for seperated and continuous boundaries.
best p.v
The magical Electron/..take a Universe size box containing a Hydrogen Atom and a single Proton, eventually these two particles will commute to a cal area, there the Electron will transpose across to the single Proton, and then back to it's original source. Now waht is interesting is the discontinuous and discreteness of it's relative positions, it leaves one Proton as an ejected Photon, and arrive's at the next Proton as a Photon of certain energy.
It seems to be that a standing Electron wave around a Proton, is really eqivelent to an ejected Photon, constrained by the dimensionality volume it occupies, pointlike but not actually a 3-Dimensional entity, until it attaches itself to the Proton? Of course Electron shells are altered vastly by the introduction of excess Protons, ie more than two-particles!
Now whilst I do not contend any greater dimensions other than 3-d2-d or 1-d, i do accept that the 11-D or even 26-D models must have their own magical transformational energies, all totally hidden of course by the very nature of our mathematical dimensional illusionary exixtence?
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It seems to be that a standing Electron wave around a Proton, is really eqivelent to an ejected Photon, constrained by the dimensionality volume it occupies, pointlike but not actually a 3-Dimensional entity, until it attaches itself to the Proton? Of course Electron shells are altered vastly by the introduction of excess Protons, ie more than two-particles!
Now whilst I do not contend any greater dimensions other than 3-d2-d or 1-d, i do accept that the 11-D or even 26-D models must have their own magical transformational energies, all totally hidden of course by the very nature of our mathematical dimensional illusionary exixtence?
Hi Georgina, you wrote: There is a difference between subjective reality of experience that tells us most macroscopic objects are singular and unchanging in identity and objective reality that exists outside of that experience in which everything at every scale is continuously changing.
Conscious observer experiences universe in an objective way. He experiences what senses perceive.
In today scientific experience between perception and experience there is a mind elaboration. Because of that this experience is "rational" and not "objective".
Conscious observer and his direct and objective experience opens new dimensions in physics. We can see how much rational mind elaboration influences our experience of the universe.
Time is a first subject I work on from the point of Conscious observer.
yours amrit
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Conscious observer experiences universe in an objective way. He experiences what senses perceive.
In today scientific experience between perception and experience there is a mind elaboration. Because of that this experience is "rational" and not "objective".
Conscious observer and his direct and objective experience opens new dimensions in physics. We can see how much rational mind elaboration influences our experience of the universe.
Time is a first subject I work on from the point of Conscious observer.
yours amrit
Hi Paul,
In answer to your question, firstly the proton is far more massive than an electron and therefore is a far greater disturbance of the medium of space. Greater mass gives greater inertia which is the energy required to change trajectory through quaternion space. So it is less inclined to move than an electron. The centre of the proton is the exit point for the 4th dimensional axis...
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In answer to your question, firstly the proton is far more massive than an electron and therefore is a far greater disturbance of the medium of space. Greater mass gives greater inertia which is the energy required to change trajectory through quaternion space. So it is less inclined to move than an electron. The centre of the proton is the exit point for the 4th dimensional axis...
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Amrit ,
I think the reason we appear to be not in complete agreement here is because you and I are not using the term objective in the same way. I think you mean by "objective", dealing purely with that which is observed without interpretation or model. To use a photographic analogy you are using the raw file. The problem with a raw file is that it contains too much data to efficiently handle and use for other purposes.
Rendering the data into a more compact and manageable format means that some data is lost but the remaining data can be manipulated and used far more easily. That is the purpose of the explanatory model.To make the infinitely complicated comprehensible. Yes, it is rational.It is not possible to map the position of every sub atomic particle in the universe and note every minute energy change relating to each spatial change in position. That is the objective reality that I think you refer to. The "raw file" of the universe is too immense.
Quaternion mathematics is used to model all sorts of complex systems such as hydrological or meteorological systems. It does not map the position of every molecule but uses the rotations of bodies of substance to give an approximation that allows comprehension of the processes that are occurring. That is what I propose is used throughout science where a process is under investigation.
When I use the term objective reality I mean that underlying reality that exists without the observer. It can not be known by experimental observation because as soon as there is observation a subjective reality is formed. The data observed has to be selected and processed by the brain or device for the observation to exist. The data is changed by the process of observation although the objective reality itself is unchanged. (Perhaps I need to think of a new term to avoid this ambiguity of meaning that you have highlighted.) The Prime reality interface between objective and subjective reality is the limit of scientific method. Objective reality is inaccessible to that method and can only be modelled.
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I think the reason we appear to be not in complete agreement here is because you and I are not using the term objective in the same way. I think you mean by "objective", dealing purely with that which is observed without interpretation or model. To use a photographic analogy you are using the raw file. The problem with a raw file is that it contains too much data to efficiently handle and use for other purposes.
Rendering the data into a more compact and manageable format means that some data is lost but the remaining data can be manipulated and used far more easily. That is the purpose of the explanatory model.To make the infinitely complicated comprehensible. Yes, it is rational.It is not possible to map the position of every sub atomic particle in the universe and note every minute energy change relating to each spatial change in position. That is the objective reality that I think you refer to. The "raw file" of the universe is too immense.
Quaternion mathematics is used to model all sorts of complex systems such as hydrological or meteorological systems. It does not map the position of every molecule but uses the rotations of bodies of substance to give an approximation that allows comprehension of the processes that are occurring. That is what I propose is used throughout science where a process is under investigation.
When I use the term objective reality I mean that underlying reality that exists without the observer. It can not be known by experimental observation because as soon as there is observation a subjective reality is formed. The data observed has to be selected and processed by the brain or device for the observation to exist. The data is changed by the process of observation although the objective reality itself is unchanged. (Perhaps I need to think of a new term to avoid this ambiguity of meaning that you have highlighted.) The Prime reality interface between objective and subjective reality is the limit of scientific method. Objective reality is inaccessible to that method and can only be modelled.
Amrit, I did not clarify. Your objective reality is my raw subjective reality.
That raw subjective reality has been obtained by selection of input by sense organs (or internally generated) and by activity of unconscious brain processes. It is closer to the raw objective reality than subjective reality formed by active conscious analysis of data or application of a scientific model to data but is still not (my definition of) objective reality itself.
You are not further rendering that raw subjective reality by thought or active investigation.
However If you are alive and conscious your brain is working either on external input or internal input. In a meditative state you may be able to minimise this activity but not stop it. The experience of a meditative state is not (my definition of) objective reality itself but an altered state of brain function and so altered consciousness. You can only become one with (my definition of) objective reality when you are dead. That is not to dispute the spiritual value and mental and physical health benefits of meditation.
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That raw subjective reality has been obtained by selection of input by sense organs (or internally generated) and by activity of unconscious brain processes. It is closer to the raw objective reality than subjective reality formed by active conscious analysis of data or application of a scientific model to data but is still not (my definition of) objective reality itself.
You are not further rendering that raw subjective reality by thought or active investigation.
However If you are alive and conscious your brain is working either on external input or internal input. In a meditative state you may be able to minimise this activity but not stop it. The experience of a meditative state is not (my definition of) objective reality itself but an altered state of brain function and so altered consciousness. You can only become one with (my definition of) objective reality when you are dead. That is not to dispute the spiritual value and mental and physical health benefits of meditation.
Hi again Amrit,
I understand your objection and upon further consideration I see that I was not precise enough.
I wrote "There is a difference between subjective reality of experience that tells us most macroscopic objects are singular and unchanging in identity and objective reality that exists outside of that experience in which everything at every scale is continuously changing."
I should have said "......in which, according to this explanatory model, everything at every scale is continuously changing."
That would have shown that this was a model of objective reality rather than objective reality itself. Which as I have said on numerous occasions can not be known by any direct means. I apologise for not making this clear.
The fact that it is only a model of objective reality does in fact mean that is just another subjective reality. However it is a logical explanatory representation of objective reality based on a self consistent model. Rather than spiritual acceptance of ultimate unknowability. Both can be correct. The ultimate "raw file" of the universe is incomprehensible because the amount and complexity of data can not be handled by a human mind. It has to be reduced to a comprehensible approximation and model for logical understanding of its function and for use as a predictive scientific framework.
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I understand your objection and upon further consideration I see that I was not precise enough.
I wrote "There is a difference between subjective reality of experience that tells us most macroscopic objects are singular and unchanging in identity and objective reality that exists outside of that experience in which everything at every scale is continuously changing."
I should have said "......in which, according to this explanatory model, everything at every scale is continuously changing."
That would have shown that this was a model of objective reality rather than objective reality itself. Which as I have said on numerous occasions can not be known by any direct means. I apologise for not making this clear.
The fact that it is only a model of objective reality does in fact mean that is just another subjective reality. However it is a logical explanatory representation of objective reality based on a self consistent model. Rather than spiritual acceptance of ultimate unknowability. Both can be correct. The ultimate "raw file" of the universe is incomprehensible because the amount and complexity of data can not be handled by a human mind. It has to be reduced to a comprehensible approximation and model for logical understanding of its function and for use as a predictive scientific framework.
amrit,
I do also understand that one can choose to ignore the complexity and ultimate un-knowability and rather just accept that everything is just one. This viewpoint is still compatible with the model I have been proposing. As every energy change is conserved within quaternion space and every change in spatial position is conserved, in that it causes another spatial change in position, then everything is connected within energy-space. There are spatial and energetic boundaries to those things we regard as objects but there is also continuous change in those objects and continuity between everything. One might say the "one entity" is continuously flowing and changing in quaternion space, manifesting different forms, which we identify, but it is still the "one entity".
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I do also understand that one can choose to ignore the complexity and ultimate un-knowability and rather just accept that everything is just one. This viewpoint is still compatible with the model I have been proposing. As every energy change is conserved within quaternion space and every change in spatial position is conserved, in that it causes another spatial change in position, then everything is connected within energy-space. There are spatial and energetic boundaries to those things we regard as objects but there is also continuous change in those objects and continuity between everything. One might say the "one entity" is continuously flowing and changing in quaternion space, manifesting different forms, which we identify, but it is still the "one entity".
Yes Georgina,
there is only one energy in the universe taking different forms. My research is how to build an adequate picture of this permanent change. One is clear. Flow of change is timeless, has no duration. Duration comes into existence when we measure energy flow with clocks.
yours amrit, Eternity is NOW.
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there is only one energy in the universe taking different forms. My research is how to build an adequate picture of this permanent change. One is clear. Flow of change is timeless, has no duration. Duration comes into existence when we measure energy flow with clocks.
yours amrit, Eternity is NOW.
Paul,
just to clarify a couple of points from my reply to your question.
When I said that a proton is actually a 4D object like all 3D objects, I was attempting to say that -all objects we regard as 3D with in 3D space are actually quaternion objects within quaternion energy-space. They are certainly not not 4 vector dimensional objects with in space-time. I hope this clarification avoids a misunderstanding of the meaning I was trying to convey.
I also said that an electron has 4 degrees of freedom. I should have perhaps said that an electron within an atom, as it can be regarded as moving within a 2D plane (which must actually be a 3D structure, but very thin, within 3D space, and therefore also ultimately quaternion in structure) is not moving significantly within 1 of the 3 vector spatial dimension and therefore is only using 3 of the 4 available degrees of freedom. That does not mean it can not use that other vector spatial degree of freedom if disturbed from its stable position by a photon of disturbance.
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just to clarify a couple of points from my reply to your question.
When I said that a proton is actually a 4D object like all 3D objects, I was attempting to say that -all objects we regard as 3D with in 3D space are actually quaternion objects within quaternion energy-space. They are certainly not not 4 vector dimensional objects with in space-time. I hope this clarification avoids a misunderstanding of the meaning I was trying to convey.
I also said that an electron has 4 degrees of freedom. I should have perhaps said that an electron within an atom, as it can be regarded as moving within a 2D plane (which must actually be a 3D structure, but very thin, within 3D space, and therefore also ultimately quaternion in structure) is not moving significantly within 1 of the 3 vector spatial dimension and therefore is only using 3 of the 4 available degrees of freedom. That does not mean it can not use that other vector spatial degree of freedom if disturbed from its stable position by a photon of disturbance.
Amrit ,
I'm glad that we do seem to share similar understanding with regard to the ultimate nature of the universe and the question of time. Even though we may express our understanding differently.
You said "there is only one energy in the universe taking different forms. My research is how to build an adequate picture of this permanent change."
Herein lies the problem. "The Tao that can be expressed is not the eternal Tao." Tao te ching, Lao Tzu
As soon as one tries to describe or model the objective reality, it becomes a subjective reality fashioned by the working of the mind rather than the raw existence itself. I must accept that any model is just a model and can only ever be a model. A representation, not the "ultimate entity" itself. The questions I now seek to answer are... How good a likeness does a particular model provide and can it be improved in any way to correspond more closely to observation and experience?
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I'm glad that we do seem to share similar understanding with regard to the ultimate nature of the universe and the question of time. Even though we may express our understanding differently.
You said "there is only one energy in the universe taking different forms. My research is how to build an adequate picture of this permanent change."
Herein lies the problem. "The Tao that can be expressed is not the eternal Tao." Tao te ching, Lao Tzu
As soon as one tries to describe or model the objective reality, it becomes a subjective reality fashioned by the working of the mind rather than the raw existence itself. I must accept that any model is just a model and can only ever be a model. A representation, not the "ultimate entity" itself. The questions I now seek to answer are... How good a likeness does a particular model provide and can it be improved in any way to correspond more closely to observation and experience?
Dear Georgina,
for 30 years I watch my mind and I can tell you there is a lot of ideas in scientific mind that have no correspondence with "objective" reality.
"Scientific picture" of the world can be sharpened only by awakening of the observer. This is where physics meets pure meditation, pure witnessing of the mind. I live in timeless universe that is real, objective. New definition of physical time as "Physical time is run of clocks in space" is the first step into physics that will reach to the ultimate peacks of the human understanding and knowing.
yours sincerely, amrit
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for 30 years I watch my mind and I can tell you there is a lot of ideas in scientific mind that have no correspondence with "objective" reality.
"Scientific picture" of the world can be sharpened only by awakening of the observer. This is where physics meets pure meditation, pure witnessing of the mind. I live in timeless universe that is real, objective. New definition of physical time as "Physical time is run of clocks in space" is the first step into physics that will reach to the ultimate peacks of the human understanding and knowing.
yours sincerely, amrit
Amrit,
I am not sure that the experience of meditation is necessary for the improvement of physics. The meditative state does give insight into an altered perception of reality. One can realise that experience is not itself the underlying reality. However it does not in itself give rise to explanation of either the everyday experience, which is a biologically generated simulation of...
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I am not sure that the experience of meditation is necessary for the improvement of physics. The meditative state does give insight into an altered perception of reality. One can realise that experience is not itself the underlying reality. However it does not in itself give rise to explanation of either the everyday experience, which is a biologically generated simulation of...
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Anyone who thinks that the raw objective existence or reality can be known by the human mind should perhaps meditate on the matter before posting.
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Dear Georgina
A key question of cosmology is:
"Does universe run in time or is time running in the universe?"
By watching, witnessing the mind one discovers
that time is running in the universe as a run of clocks.
Universe itself is timeless, universe is NOW.
yours amrit
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A key question of cosmology is:
"Does universe run in time or is time running in the universe?"
By watching, witnessing the mind one discovers
that time is running in the universe as a run of clocks.
Universe itself is timeless, universe is NOW.
yours amrit
Amrit ,
as you know I do not disagree with that. A superior model can be constructed without time as a foundational element but with the subjective experience of time produced by the process of continuous change, IMO. This allows a number of foundational questions to be answered.
The fact that you have come to this realisation via meditation rather than intuition in an alert everyday state of mind or logical deduction does not mean that the meditative trance is a superior method of investigation of the universe. It is just a different use of the brain that can give a different experience.
Freed from external stimuli, and the chatter and interference of the logical mind one can, perhaps, experience the perspective of the intuitive, comparative often unconscious mind that does not measure and does not use time. We have two hemispheres of the brain that analyse the stimuli received in different ways. One is not correct and the other incorrect they are just different interpretations of the same data. A model that gives both perspectives, time not being foundational but time being experienced seems to be a good representation complying with the perspective of both cerebral hemispheres.
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as you know I do not disagree with that. A superior model can be constructed without time as a foundational element but with the subjective experience of time produced by the process of continuous change, IMO. This allows a number of foundational questions to be answered.
The fact that you have come to this realisation via meditation rather than intuition in an alert everyday state of mind or logical deduction does not mean that the meditative trance is a superior method of investigation of the universe. It is just a different use of the brain that can give a different experience.
Freed from external stimuli, and the chatter and interference of the logical mind one can, perhaps, experience the perspective of the intuitive, comparative often unconscious mind that does not measure and does not use time. We have two hemispheres of the brain that analyse the stimuli received in different ways. One is not correct and the other incorrect they are just different interpretations of the same data. A model that gives both perspectives, time not being foundational but time being experienced seems to be a good representation complying with the perspective of both cerebral hemispheres.
If an existential entity, rather than mathematical abstract, must be defined by at least two different quaternions,( because it is -change- in quaternion spatio-energetic position that gives existence to every entity), this will solve Zeno of Elea's arrow paradox.
The arrow never is at rest in a (subjectively experienced) instant of time. The arrow is defined by and only exists because of its change in position. Rather than ever having a static position.
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The arrow never is at rest in a (subjectively experienced) instant of time. The arrow is defined by and only exists because of its change in position. Rather than ever having a static position.
Georgina
arrow move in space only and not in time. There is no paradox here. Paradox arises because one think arrov moves in time.
yours amrit
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arrow move in space only and not in time. There is no paradox here. Paradox arises because one think arrov moves in time.
yours amrit
FQXi might be the best place to ask for basics of mathematics that are more appropriate for physics, not at all tedious as compared with Lawrence's set on the ass theory. In physics, continuity is an issue. Its definition by Peirce "something every part of which has parts" contradicts to Euclid's definition of a point: "something that does not have parts".
What is wrong for instance with a Dedekind cut? Already §2 of Stetigkeit und Irrationale Zahlen: "Vergleichung der rationalen Zahlen mit den Punkten einer geraden Linie" (Interpretation of rational numbers as points of a straight line).
Actually, it does nor matter what we count. We always start with one basic measure, for instance a unit length. Basic measures are assumed positive of course and exactly given without any doubt. The integer numbers of properties along a street start with number one and are actually finite while potentially infinite (Archimedes axiom). Nobody imagines property 1 as a point. It extends from the beginning of 1 to its border to 2. So the integer number n primarily represents an interval and not the belonging endpoint of it. Counting needs intervals. If a natural number is missing, the the line is discontinuous.
A piece of an interval can relate to unity in either a countable or an uncountable manner. Ever a countable one can be as small as we like. The same piece that is not a countable part with respect to one can be countable with respect to a basis that is not rationally connected to the measure one.
Nonlinear tunctions could be understood in terms of "real intervalls", which should be called irreal intervals because they are of size zero. It makes no sense to declare irreal intervals open or closed because they are naked without any size that could be added or removed.
Mathematics would perhaps no longer need a lot of boring courses in real analysis and other futile trouble. Physics could loose illusions too.
If this suggestion of mine is correct, then sets of points seem to be inappropriate. I cannot imagine that this idea is new and would like to know how it was refuted.
Eckard
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What is wrong for instance with a Dedekind cut? Already §2 of Stetigkeit und Irrationale Zahlen: "Vergleichung der rationalen Zahlen mit den Punkten einer geraden Linie" (Interpretation of rational numbers as points of a straight line).
Actually, it does nor matter what we count. We always start with one basic measure, for instance a unit length. Basic measures are assumed positive of course and exactly given without any doubt. The integer numbers of properties along a street start with number one and are actually finite while potentially infinite (Archimedes axiom). Nobody imagines property 1 as a point. It extends from the beginning of 1 to its border to 2. So the integer number n primarily represents an interval and not the belonging endpoint of it. Counting needs intervals. If a natural number is missing, the the line is discontinuous.
A piece of an interval can relate to unity in either a countable or an uncountable manner. Ever a countable one can be as small as we like. The same piece that is not a countable part with respect to one can be countable with respect to a basis that is not rationally connected to the measure one.
Nonlinear tunctions could be understood in terms of "real intervalls", which should be called irreal intervals because they are of size zero. It makes no sense to declare irreal intervals open or closed because they are naked without any size that could be added or removed.
Mathematics would perhaps no longer need a lot of boring courses in real analysis and other futile trouble. Physics could loose illusions too.
If this suggestion of mine is correct, then sets of points seem to be inappropriate. I cannot imagine that this idea is new and would like to know how it was refuted.
Eckard
Georgina, unconscious observer experiences timeless space through inner time as "present moment". Conscious observer is aware of inner time. He experiences timeless space as "eternal ever lasting present moment". In cosmic space is always NOW, present. Arrov moves always in timeless space, means in present.
Paradox has arised with preposition that in present moment arrow is still, that arrow moves from the past to the future. This is pure illusion. Past and future belongs to the mind, in the universe there is only NOW - present. Wrong prepositio has created a mental paradox that does not exist in physical reality.
yours amrit
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Paradox has arised with preposition that in present moment arrow is still, that arrow moves from the past to the future. This is pure illusion. Past and future belongs to the mind, in the universe there is only NOW - present. Wrong prepositio has created a mental paradox that does not exist in physical reality.
yours amrit
Amrit,
I agree that the arrow is not actually moving in time. It can be modelled as moving through quaternion energy-space. I am aware that you dislike the use of dimensions for description. However if all descriptive and measurement terms are removed, it is then not possible to describe or measure anything and science can not get very far at all. Using the quaternion spatio-energetic description, time is not a dimension and the spacial element can be regarded as the means to measure change rather than being fundamental change itself. So, if space is also removed from consideration all then that is left is the un-measured energy (equivalent to that spatial change of position that is being ignored).
The point I was making was that the arrow is the change in position. That change in position being the energy that is the arrow itself. According to this model, objects only exist because of their quaternion change in position which gives them mass energy.If the object is observed to be moving through 3D space, that change in position is a part of the physical identity of the object. An existential object, rather than mathematical abstract entity, must be defined by at least two quaternions. That definition includes the element of change within it. (Meaning the arrow is not a collection of mathematical points that must be considered stationary). This solves the paradox.
Although integers and points can be used for physical description it is my opinion that they are part of an abstract language that does not relate directly to the existential reality of physics. Therefore producing abstract outcomes rather than outcomes that relate more directly to that which is observed. That is not to say that non quaternion mathematics is not useful.
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I agree that the arrow is not actually moving in time. It can be modelled as moving through quaternion energy-space. I am aware that you dislike the use of dimensions for description. However if all descriptive and measurement terms are removed, it is then not possible to describe or measure anything and science can not get very far at all. Using the quaternion spatio-energetic description, time is not a dimension and the spacial element can be regarded as the means to measure change rather than being fundamental change itself. So, if space is also removed from consideration all then that is left is the un-measured energy (equivalent to that spatial change of position that is being ignored).
The point I was making was that the arrow is the change in position. That change in position being the energy that is the arrow itself. According to this model, objects only exist because of their quaternion change in position which gives them mass energy.If the object is observed to be moving through 3D space, that change in position is a part of the physical identity of the object. An existential object, rather than mathematical abstract entity, must be defined by at least two quaternions. That definition includes the element of change within it. (Meaning the arrow is not a collection of mathematical points that must be considered stationary). This solves the paradox.
Although integers and points can be used for physical description it is my opinion that they are part of an abstract language that does not relate directly to the existential reality of physics. Therefore producing abstract outcomes rather than outcomes that relate more directly to that which is observed. That is not to say that non quaternion mathematics is not useful.
Amrit and Georgina,
Your positions remind me of Einstein's helpless admission concerning the NOW and also of mathematical evidence for existence, e.g. of god and the fundamental theorem of analysis.
Does the real projective line provide evidence for the existence of a point plus-minus infinity?
Amrit denies the existence of past and future. What does existence mean to him?
I prefer to rather consider two mutually excluding views: Backward and forward.
Any actual process unites an unlimited diversity of influences from its past. They are real in the sense they left traces and cannot be changed any more.
Any prediction, plan, or preparation is limited to the expected future and always uncertain to some extend with respect to its agreement with what will become reality. There is no measurable future. Even in bird's brain, future is a mental construct.
Abstraction from reality removes the restriction of processes to the past and of predictions to the future. Mathematics is not an illusion but a tool. There are a few very basic illusions in mathematics and scores of illusionary applications of mathematics in physics.
Eckard
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Your positions remind me of Einstein's helpless admission concerning the NOW and also of mathematical evidence for existence, e.g. of god and the fundamental theorem of analysis.
Does the real projective line provide evidence for the existence of a point plus-minus infinity?
Amrit denies the existence of past and future. What does existence mean to him?
I prefer to rather consider two mutually excluding views: Backward and forward.
Any actual process unites an unlimited diversity of influences from its past. They are real in the sense they left traces and cannot be changed any more.
Any prediction, plan, or preparation is limited to the expected future and always uncertain to some extend with respect to its agreement with what will become reality. There is no measurable future. Even in bird's brain, future is a mental construct.
Abstraction from reality removes the restriction of processes to the past and of predictions to the future. Mathematics is not an illusion but a tool. There are a few very basic illusions in mathematics and scores of illusionary applications of mathematics in physics.
Eckard
Eckard,
You ask "Does the real projective line provide evidence for the existence of a point plus-minus infinity?" Why do you ask? I'm not sure if that is a rhetorical question or statement to make "a point" or an actual question. I am not a mathematician. Though tempted to answer I can see myself falling into an unforeseen well of abstract mathematical theory and reasoning. I don't want to go there.
I am in agreement that there are signs of those influences encountered during the continuous process of change, that can be observed in behaviour and or physical structure, although there is no existential past "out there". The past can however form a part of mathematical analysis. I also agree that the future does not exist but can be imagined. It too can form part of mathematical analysis.
These important differences between mathematics and existential physics need to be borne in mind, rather than assuming that that which exists in the mathematics is physically realistic.
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You ask "Does the real projective line provide evidence for the existence of a point plus-minus infinity?" Why do you ask? I'm not sure if that is a rhetorical question or statement to make "a point" or an actual question. I am not a mathematician. Though tempted to answer I can see myself falling into an unforeseen well of abstract mathematical theory and reasoning. I don't want to go there.
I am in agreement that there are signs of those influences encountered during the continuous process of change, that can be observed in behaviour and or physical structure, although there is no existential past "out there". The past can however form a part of mathematical analysis. I also agree that the future does not exist but can be imagined. It too can form part of mathematical analysis.
These important differences between mathematics and existential physics need to be borne in mind, rather than assuming that that which exists in the mathematics is physically realistic.
Eckard we do not have any experimental data past or future exist as a physical reality.
Past and future belongs to the mind, energy change run in timeless space.
Universe is ageless. "Before" exist only in a sense of numerical order measured with clocks - natural as Earth rotation or made one.
Universe is NOW.
yours amrit
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Past and future belongs to the mind, energy change run in timeless space.
Universe is ageless. "Before" exist only in a sense of numerical order measured with clocks - natural as Earth rotation or made one.
Universe is NOW.
yours amrit
Eckard,
"NOW" is not problematic in the model I am proposing. It is an individual subjective experience that is different for each observer. Each experienced present moment is formed from a patchwork of input arriving at the sense organs and filtered by the type and threshold of those senses and individual brain processing. Which may include such factors as state of alertness, alcohol,drug or caffeine consumption, training to attend to particular stimuli, number of other competing stimuli to be dealt with etc. etc. Further filtering occurs so unimportant or non urgent stimuli are not passed to the conscious mind, although attended to by the sub conscious mind. Association between memory is incorporated so that the input can be comprehended in the light of previous experience and learning and the result is experienced by the conscious mind.
The input has various sources varying in spatial distance from the observer. Possibly varying from millimetres to light years. However it is all "stitched together" by the brain to give a present or now experience of the conscious mind. So there is no universal present, it is all personal subjective experience of input from sources that may be considered to be well spread over time, if using a space-time model.There is no existential past, present or future. The present is an experience, part of the biologically generated subjective reality. It is not an external physical absolute or a foundational element giving rise to objective reality.
There is energy change, producing those forms that we identify, which can be modelled for comprehension within a quaternion framework of 4 spatio-energetic dimensions, without time. Within that structure the processes of chaos and complexity can (and I anticipate will eventually) be modelled and shown to gives rise to the universe that we observe via self assembly. IMO.
Whether an individual wishes to relate to a scientific explanatory model as description of their god is a matter of personal choice. The model just is what it is. What people choose to do with it outside of physics or how they wish to relate to it is a matter for the disciplines of behavioural psychology, sociology and theology to explore not physics.
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"NOW" is not problematic in the model I am proposing. It is an individual subjective experience that is different for each observer. Each experienced present moment is formed from a patchwork of input arriving at the sense organs and filtered by the type and threshold of those senses and individual brain processing. Which may include such factors as state of alertness, alcohol,drug or caffeine consumption, training to attend to particular stimuli, number of other competing stimuli to be dealt with etc. etc. Further filtering occurs so unimportant or non urgent stimuli are not passed to the conscious mind, although attended to by the sub conscious mind. Association between memory is incorporated so that the input can be comprehended in the light of previous experience and learning and the result is experienced by the conscious mind.
The input has various sources varying in spatial distance from the observer. Possibly varying from millimetres to light years. However it is all "stitched together" by the brain to give a present or now experience of the conscious mind. So there is no universal present, it is all personal subjective experience of input from sources that may be considered to be well spread over time, if using a space-time model.There is no existential past, present or future. The present is an experience, part of the biologically generated subjective reality. It is not an external physical absolute or a foundational element giving rise to objective reality.
There is energy change, producing those forms that we identify, which can be modelled for comprehension within a quaternion framework of 4 spatio-energetic dimensions, without time. Within that structure the processes of chaos and complexity can (and I anticipate will eventually) be modelled and shown to gives rise to the universe that we observe via self assembly. IMO.
Whether an individual wishes to relate to a scientific explanatory model as description of their god is a matter of personal choice. The model just is what it is. What people choose to do with it outside of physics or how they wish to relate to it is a matter for the disciplines of behavioural psychology, sociology and theology to explore not physics.
I think the main point was that physics interfaces with biology (at the sense organs). I have called this the Prime Reality Interface. We -experience- the universe from the biological side of that interface. Which includes the experience of time.
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Issues of "epsilons and deltas" in mathematics pertain to the consistency of the mathematics, which might be employed in physics. I Don't tend to worry too much about whether a real line can be dissected in some manner, such as a Dekekind cut, when it comes to directly physical implications. Again these are matters which concern the consistency of the mathematics employed.
The mathematical world which pertains to geometry is pretty tied to point-set theory, which deeper down is tied to the ZF set theory. I don't think this is likely to change in the future. The issues involved were inspired by calculus and the nature of infinitesimals (what Newton called fluxions) and the exact meaning of the limit.
Cheers LC
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The mathematical world which pertains to geometry is pretty tied to point-set theory, which deeper down is tied to the ZF set theory. I don't think this is likely to change in the future. The issues involved were inspired by calculus and the nature of infinitesimals (what Newton called fluxions) and the exact meaning of the limit.
Cheers LC
A problem with geometry is that it is static. However it seems that change is a most fundamental feature of the universe and the particles and matter within it. That is not reflected by the static nature of geometry, which can at best only give a series of static snap shots of 3D space. A block universe does not appear to move or change. That does not mean that geometry should not be used but its artificial rigidity should be borne in mind when interpreting its meaning.
Change is incorporated into the mathematics of quaternions which can give a sequence of rotations used to build up 3D structure. Chirality or handedness is seen in some inorganic substances and biological molecules. I think that it is not unreasonable to hypothesise that the nature of the particular rotation could lead to this chirality of structure. Which may be due to the particular shape of a structure under construction, giving drag or balance factors, that leads to a particular orientation of rotation in quaternion space.
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Change is incorporated into the mathematics of quaternions which can give a sequence of rotations used to build up 3D structure. Chirality or handedness is seen in some inorganic substances and biological molecules. I think that it is not unreasonable to hypothesise that the nature of the particular rotation could lead to this chirality of structure. Which may be due to the particular shape of a structure under construction, giving drag or balance factors, that leads to a particular orientation of rotation in quaternion space.
Georgina, you say:
"NOW" is not problematic in the model I am proposing. It is an individual subjective experience that is different for each observer.
I would no agree, NOW is not subjective in a sense of "personal", NOW is objective in a sense that observer experiences what senses perceive.
In all people on this planet and all beings in the universe observer is the same, identical: observer is consciousness itself.
yours amrit
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"NOW" is not problematic in the model I am proposing. It is an individual subjective experience that is different for each observer.
I would no agree, NOW is not subjective in a sense of "personal", NOW is objective in a sense that observer experiences what senses perceive.
In all people on this planet and all beings in the universe observer is the same, identical: observer is consciousness itself.
yours amrit
PS, read my article on file attached
attachments: 1_Bridging_Observer_and_Observed__amrit_sorli.pdf
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attachments: 1_Bridging_Observer_and_Observed__amrit_sorli.pdf
Amrit,
We will have to disagree then. Your usage of the term objective is not the same as mine. What the senses perceive is a filtered sub set of the data available in the universe which is further filtered and modified by the brain. The subject experiencing the product of that filtering and processed data is not directly experiencing the universe but a biological simulation. I call that biological simulation subjective reality. Every individual biological entity will be experiencing its own subjective reality interpretation of the external universe.Which includes an individual perception of a present. You are not a part of my present and I am not a part of yours.
Philosophically it may be argued that each individual is a part of a single consciousness but unaware of that. It may be the interpretation of the right hemisphere in a meditative state, which tends not to reduce and classify but thinks holistically, bringing things together and comparing. That is not the experience of alert mind in a usual situation.
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We will have to disagree then. Your usage of the term objective is not the same as mine. What the senses perceive is a filtered sub set of the data available in the universe which is further filtered and modified by the brain. The subject experiencing the product of that filtering and processed data is not directly experiencing the universe but a biological simulation. I call that biological simulation subjective reality. Every individual biological entity will be experiencing its own subjective reality interpretation of the external universe.Which includes an individual perception of a present. You are not a part of my present and I am not a part of yours.
Philosophically it may be argued that each individual is a part of a single consciousness but unaware of that. It may be the interpretation of the right hemisphere in a meditative state, which tends not to reduce and classify but thinks holistically, bringing things together and comparing. That is not the experience of alert mind in a usual situation.
Amrit,
When I said "That is not the experience of alert mind in a usual situation." I meant that the usual experience is not to "feel" that all life has one consciousness.
In the alert usual frame of mind objects and individuals are considered separate in both substance and consciousness. This is the left brain functioning as it should to allow us to make sense of the array of sensory input that arrives at the brain as flows of electrical impulses. Differentiating it into different and distinct sources.
The left brain is not correct and the right brain incorrect or vice versa. Both are interpretations of the data performed differently. I do not think that the right hemisphere perspective is more real or correct. A model that is able to fit with the interpretation of both cerebral hemispheres is in my opinion better than one that only fits with the interpretation of one of the two hemispheres.
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When I said "That is not the experience of alert mind in a usual situation." I meant that the usual experience is not to "feel" that all life has one consciousness.
In the alert usual frame of mind objects and individuals are considered separate in both substance and consciousness. This is the left brain functioning as it should to allow us to make sense of the array of sensory input that arrives at the brain as flows of electrical impulses. Differentiating it into different and distinct sources.
The left brain is not correct and the right brain incorrect or vice versa. Both are interpretations of the data performed differently. I do not think that the right hemisphere perspective is more real or correct. A model that is able to fit with the interpretation of both cerebral hemispheres is in my opinion better than one that only fits with the interpretation of one of the two hemispheres.
Lawrence,
Both the closed interval [0,a] and the open interval(0,a) are deemed to have the same Lebesgue (Lebeg) measure. Do we need non-measurable Vitali sets? Do we need all the confusion due to sets of points at all? I would be interested in letters by Baire, Borel, or Lebeg on Cantor's sets.
The boundary between the property of mine and the property of my neighbor does neither...
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Both the closed interval [0,a] and the open interval(0,a) are deemed to have the same Lebesgue (Lebeg) measure. Do we need non-measurable Vitali sets? Do we need all the confusion due to sets of points at all? I would be interested in letters by Baire, Borel, or Lebeg on Cantor's sets.
The boundary between the property of mine and the property of my neighbor does neither...
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The point-set topological approach to mathematics is so far the most consistent approach taken. There have been alternatives proposed, but they run into other difficulties. As for integrating from -∞ to ∞, breaking this up would change the answer if there is a delta function at zero. If the integral is set up between (-∞, o^-) and ()^+, ∞) that would ignore the delta function. There are theorems about this for general functions instead of distributions, and the result is in general there is no difference.
It is not hard to show there is a difference between countable infinity and uncountable infinity. The Cantor diagonalization “slash” method illustrates there exists sets which have an infinite cardinality which can’t be mapped to a countably infinite set. This leads to the power set generalization or the continuum hypothesis אּ_1 = 2^{אּ_0}, which in turn leads to higher aleph values for transfinite numbers. The continuum hypothesis has been found to be consistent in ZF set theory by Bernays and Cohen, but not provable. This was a set theoretic approach to using Godel’s theorem.
I don’t know if any of that stuff has much use in physics. It is pretty abstract stuff which tell us nothing about physical quantities. As for real analysis, I took a graduate course in it. It is not exactly my cup of tea. I prefer group theory, algebra, differential geometry and topology, and am much more prepared to make intelligent comments on that. Yet the subject of real variables does get into general integration theory, and integration over function of L^p spaces. For p = 2 this includes Banach and Hilbert spaces.
As a general comment, mathematics as known is pretty solidly founded with lots of theorems which render these systems consistent. The point-set topological approach to these foundations is pretty much bedrock stuff. People have looked into alternatives, but nothing has worked out with the degree of consistency and power.
Cheers LC
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It is not hard to show there is a difference between countable infinity and uncountable infinity. The Cantor diagonalization “slash” method illustrates there exists sets which have an infinite cardinality which can’t be mapped to a countably infinite set. This leads to the power set generalization or the continuum hypothesis אּ_1 = 2^{אּ_0}, which in turn leads to higher aleph values for transfinite numbers. The continuum hypothesis has been found to be consistent in ZF set theory by Bernays and Cohen, but not provable. This was a set theoretic approach to using Godel’s theorem.
I don’t know if any of that stuff has much use in physics. It is pretty abstract stuff which tell us nothing about physical quantities. As for real analysis, I took a graduate course in it. It is not exactly my cup of tea. I prefer group theory, algebra, differential geometry and topology, and am much more prepared to make intelligent comments on that. Yet the subject of real variables does get into general integration theory, and integration over function of L^p spaces. For p = 2 this includes Banach and Hilbert spaces.
As a general comment, mathematics as known is pretty solidly founded with lots of theorems which render these systems consistent. The point-set topological approach to these foundations is pretty much bedrock stuff. People have looked into alternatives, but nothing has worked out with the degree of consistency and power.
Cheers LC
Georgina you say: The subject experiencing the product of that filtering and processed data is not directly experiencing the universe but a biological simulation.
Myself I'm pragmatic, I use my rational mind as a tool when I need it.
In the direct experience of the universe there is Oneness.
yours amrit
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Myself I'm pragmatic, I use my rational mind as a tool when I need it.
In the direct experience of the universe there is Oneness.
yours amrit
Lawrence,
You wrote: “As for integrating from -∞ to ∞, breaking this up would change the answer if there is a delta function at zero. If the integral is set up between (-∞, o^-) and ()^+, ∞) that would ignore the delta function.”
I do not stick on arbitrarily chosen definitions when I understand the essence. My honest main concern is to get rid of...
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You wrote: “As for integrating from -∞ to ∞, breaking this up would change the answer if there is a delta function at zero. If the integral is set up between (-∞, o^-) and ()^+, ∞) that would ignore the delta function.”
I do not stick on arbitrarily chosen definitions when I understand the essence. My honest main concern is to get rid of...
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@Eckard,
I will have to get back to this. To be honest I suspect you are drumming up a problem or controversy where none exists, or the answers or theorems exist.
LC
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I will have to get back to this. To be honest I suspect you are drumming up a problem or controversy where none exists, or the answers or theorems exist.
LC
Lawrence,
Zermelo saved Cantor's naive set theory in 1904 when Julius Koenig objected against well-ordering of the real numbers by fabricating the axiom of choice. In ZFC stands Z for Zermelo and C for choice. Lebeg's measure does not need the axiom of choice.
Zermelo's 1908 improved proof of the well-ordering is based on exhaustion. Zermelo ignored that the original meaning infinity cannot be exhausted.
By the way, Dedekind explained in his letter to Weber on Jan. 24, 1888 why the irrational numbers are not immediately identical with his cuts.
The only miracle to me is why not even Poincaré, who called Cantor someone who spoils the young generation, Borel who rejected the axiom of choice, and Lebec were not encouraged enough as to not consider measure belonging to sets but the other way round consider measures the primary objects in mathematics and physics. Maybe, the proponents of Cantor's untenable naive definition of a set were just too strong and those like Kronecker and Brouwer too weak.
Regards,
Eckard
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Zermelo saved Cantor's naive set theory in 1904 when Julius Koenig objected against well-ordering of the real numbers by fabricating the axiom of choice. In ZFC stands Z for Zermelo and C for choice. Lebeg's measure does not need the axiom of choice.
Zermelo's 1908 improved proof of the well-ordering is based on exhaustion. Zermelo ignored that the original meaning infinity cannot be exhausted.
By the way, Dedekind explained in his letter to Weber on Jan. 24, 1888 why the irrational numbers are not immediately identical with his cuts.
The only miracle to me is why not even Poincaré, who called Cantor someone who spoils the young generation, Borel who rejected the axiom of choice, and Lebec were not encouraged enough as to not consider measure belonging to sets but the other way round consider measures the primary objects in mathematics and physics. Maybe, the proponents of Cantor's untenable naive definition of a set were just too strong and those like Kronecker and Brouwer too weak.
Regards,
Eckard
The axiom of choice leads to some curious results, such as the Banach-Tarski decomposition of a sphere into an unmeasurable set and its recomposition into two spheres. This theorem was suggested as a basis for particle physics and a parton model approach to QCD and energy scaling. I am not so sure about that.
Cantor's transfinite numbers remained in the hinderland of set theory for some time. Yet Bernays and Cohen found in the 1960s that the continuum system Cantor proposed was consistent in ZF, but not provable. So the Cantor transfinite set system moved into the domain of established mathematics. Set theory gets even stranger, with least inaccessible cardinal numbers by Uman which are infintely greater than the aleph numbers, even אּ_ω or אּ_∞. These are in a class even further up the transfinite ladder, and there are a couple of set thoeretic infinite systems far beyond that. Of course for me I can't entertain these ideas that much. These don't appear to be effective for mathematical physics. So as far as I see much of this is a sort of mathematical playground that has little bearing on the natural world --- at least for now.
Cheers LC
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Cantor's transfinite numbers remained in the hinderland of set theory for some time. Yet Bernays and Cohen found in the 1960s that the continuum system Cantor proposed was consistent in ZF, but not provable. So the Cantor transfinite set system moved into the domain of established mathematics. Set theory gets even stranger, with least inaccessible cardinal numbers by Uman which are infintely greater than the aleph numbers, even אּ_ω or אּ_∞. These are in a class even further up the transfinite ladder, and there are a couple of set thoeretic infinite systems far beyond that. Of course for me I can't entertain these ideas that much. These don't appear to be effective for mathematical physics. So as far as I see much of this is a sort of mathematical playground that has little bearing on the natural world --- at least for now.
Cheers LC
describes reality as a nested density fluctuations of infinitelly dense particle field. Does some connection to topos theory exist there? I don't see any yet.
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Absolutely fascinating. Even though I'm convinced that we are decades from having a real breakthrough in quantum mechanics I wish I had the capability to understand a brain storm between these two fascinating minds.
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Kerr solution: J = aGM^2/c
m(n) = [n]^1/2 [constant], i.e., sqrt[n] [constant]
where: a = 1/n and
constant = corrected Planck mass = 674 Mev
-n----n]^1/2[constant]----Empirical mass---Agreement
1/36------112.3------muon 105.7------------94.0 %
1/25------134.8------pion 134.98-----------99.9 %
1/2--------476.6-----kaon 497.7-------------95.8 %
3/4--------583.7-----eta 547.8--------------93.4%
1----------674---------Planck mass-------- -----
2----------953.2-------proton 938-------------98.3 %
2----------953.2-------neutron939.2?--------98.5%
2----------953.2-------eta' 958--------------99.5 %
3--------1167.4-------Lambda 1115.7------95.4 %
3--------1167.4-------Sigma 1192----------97.9 %
4--------1348.0-------Xi 1314.8------------97.5 %
5--------1507.1-------N ~ 1450------------96.1 %
6--------1651---------Omega 1672.5-------98.7 %
7--------1783---------TAU 1784.1---------99.95%
8--------1906.3-------D 1864.-------------97.8 %
10------2131.4-------D(s) 2112.2-----------99.1 %
12------2334.8-------Lam(c)2284.9---------97.8%
Well, that is the 16 most common and stable of the
particles observed, with the exception of the electron
which has n = 1/(1319)^2 and I want to study that a
bit more. Maybe only a full K-N solution will suffice here.
My argument is that this high degree of ordering
demands an explanation. The fact that it was achieved
with the admittedly very approximate Kerr solution
makes things even more interesting. The fact that
Discrete Scale Relativity is definitively required to
determine the crucial value of the corrected Planck
mass should be fully appreciated.
Barking dogs may now start barking.
Scientists will undoubtedly start thinking.
Happy Winter Solstice [33rd anniversary of DSR]
Robert L. Oldershaw
www.amherst.edu/~rloldershaw
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m(n) = [n]^1/2 [constant], i.e., sqrt[n] [constant]
where: a = 1/n and
constant = corrected Planck mass = 674 Mev
-n----n]^1/2[constant]----Empirical mass---Agreement
1/36------112.3------muon 105.7------------94.0 %
1/25------134.8------pion 134.98-----------99.9 %
1/2--------476.6-----kaon 497.7-------------95.8 %
3/4--------583.7-----eta 547.8--------------93.4%
1----------674---------Planck mass-------- -----
2----------953.2-------proton 938-------------98.3 %
2----------953.2-------neutron939.2?--------98.5%
2----------953.2-------eta' 958--------------99.5 %
3--------1167.4-------Lambda 1115.7------95.4 %
3--------1167.4-------Sigma 1192----------97.9 %
4--------1348.0-------Xi 1314.8------------97.5 %
5--------1507.1-------N ~ 1450------------96.1 %
6--------1651---------Omega 1672.5-------98.7 %
7--------1783---------TAU 1784.1---------99.95%
8--------1906.3-------D 1864.-------------97.8 %
10------2131.4-------D(s) 2112.2-----------99.1 %
12------2334.8-------Lam(c)2284.9---------97.8%
Well, that is the 16 most common and stable of the
particles observed, with the exception of the electron
which has n = 1/(1319)^2 and I want to study that a
bit more. Maybe only a full K-N solution will suffice here.
My argument is that this high degree of ordering
demands an explanation. The fact that it was achieved
with the admittedly very approximate Kerr solution
makes things even more interesting. The fact that
Discrete Scale Relativity is definitively required to
determine the crucial value of the corrected Planck
mass should be fully appreciated.
Barking dogs may now start barking.
Scientists will undoubtedly start thinking.
Happy Winter Solstice [33rd anniversary of DSR]
Robert L. Oldershaw
www.amherst.edu/~rloldershaw
This would have been more interesting if you described their ideas more and spent less time extolling the virtues of collaboration.
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Lawrence,
Let's not forget that I am claiming having unveiled a basic mistake: It is natural to count measures, not points. Several oddities of set theory and related topology can be ascribed to this mistake.
You wrote: "The axiom of choice leads to some curious results, such as the Banach-Tarski decomposition of a sphere into an unmeasurable set and its recomposition into two spheres."
-- This is well known. It is also no secret that many mathematicians distrust the AC.
You wrote: "This theorem was suggested as a basis for particle physics and a parton model approach to QCD and energy scaling. I am not so sure about that."
-- I am not interested in such suggestions as long as they seem to be unfounded. I cannot even decipher what QCD stands for. I would read QED quantum electrodynamics.
You wrote: "Cantor's transfinite numbers remained in the hinderland of set theory for some time."
-- Can you please tell me the meaning of hinderland, a word missing in my dictionaries. Hinterland in German might have a different meaning.
You wrote: "Yet Bernays and Cohen found in the 1960s that the continuum system Cantor proposed was consistent in ZF, but not provable."
-- I was only aware of Cohen "The Independence of Continuum Hypothesis" PNAS 1963.
You wrote: "So the Cantor transfinite set system moved into the domain of established mathematics."
-- I prefer positive evidence as do positivists. The existence of god has been mathematically proven while the opposite seems to be impossible. ;-)
I see Cantor's evidence just delusion.
You wrote: "Set theory gets even stranger, with least inaccessible cardinal numbers by Uman which are infintely greater than the aleph numbers, even אּ_ω or אּ_∞. These are in a class even further up the transfinite ladder, and there are a couple of set thoeretic infinite systems far beyond that. Of course for me I can't entertain these ideas that much. These don't appear to be effective for mathematical physics. So as far as I see much of this is a sort of mathematical playground that has little bearing on the natural world --- at least for now."
-- Uman is not known to me. Replace little by no, and I will agree. I appreciate having learned the apt word playground from you.
Regards,
Eckard
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Let's not forget that I am claiming having unveiled a basic mistake: It is natural to count measures, not points. Several oddities of set theory and related topology can be ascribed to this mistake.
You wrote: "The axiom of choice leads to some curious results, such as the Banach-Tarski decomposition of a sphere into an unmeasurable set and its recomposition into two spheres."
-- This is well known. It is also no secret that many mathematicians distrust the AC.
You wrote: "This theorem was suggested as a basis for particle physics and a parton model approach to QCD and energy scaling. I am not so sure about that."
-- I am not interested in such suggestions as long as they seem to be unfounded. I cannot even decipher what QCD stands for. I would read QED quantum electrodynamics.
You wrote: "Cantor's transfinite numbers remained in the hinderland of set theory for some time."
-- Can you please tell me the meaning of hinderland, a word missing in my dictionaries. Hinterland in German might have a different meaning.
You wrote: "Yet Bernays and Cohen found in the 1960s that the continuum system Cantor proposed was consistent in ZF, but not provable."
-- I was only aware of Cohen "The Independence of Continuum Hypothesis" PNAS 1963.
You wrote: "So the Cantor transfinite set system moved into the domain of established mathematics."
-- I prefer positive evidence as do positivists. The existence of god has been mathematically proven while the opposite seems to be impossible. ;-)
I see Cantor's evidence just delusion.
You wrote: "Set theory gets even stranger, with least inaccessible cardinal numbers by Uman which are infintely greater than the aleph numbers, even אּ_ω or אּ_∞. These are in a class even further up the transfinite ladder, and there are a couple of set thoeretic infinite systems far beyond that. Of course for me I can't entertain these ideas that much. These don't appear to be effective for mathematical physics. So as far as I see much of this is a sort of mathematical playground that has little bearing on the natural world --- at least for now."
-- Uman is not known to me. Replace little by no, and I will agree. I appreciate having learned the apt word playground from you.
Regards,
Eckard
I think we are close to being able to work out a problem here with SU(4). I wrote on The Beautiful Truth on SU(4) and SU(3)xU(1). This seems to imply QCD in a parton-like theory with Bjorken scaling is a low energy stringy theory that is holographically dual to an AdS_3 (or at higher energy AdS_4) spacetime physics. In this way we might work how hadron scattering is dual to holographic fields of a supergravity multiplet.
We could start working this up after the New Years I think. We will need to communicate off the FQXI website here.
Today is the solstice, so happy Yule and ring those solstice bells. This is the season we all light up candles and faralitos --- in our modern age electric lights, in keeping with an ancient tradition of trying to bring the light back. Hannukah is past, so the light of the Menorah are no more. The Romans had Sol Invictus and Saturnalia, and the Christian celebration of Jesus’ birth is to carry the idea of light entering back into the world.
Cheers LC
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We could start working this up after the New Years I think. We will need to communicate off the FQXI website here.
Today is the solstice, so happy Yule and ring those solstice bells. This is the season we all light up candles and faralitos --- in our modern age electric lights, in keeping with an ancient tradition of trying to bring the light back. Hannukah is past, so the light of the Menorah are no more. The Romans had Sol Invictus and Saturnalia, and the Christian celebration of Jesus’ birth is to carry the idea of light entering back into the world.
Cheers LC
I pasted in the above post into the wrong FQXI page.
I don't think that Cantor's work is a delusion, though I am not in a position to rigorously defend it. The problem is that I am not a set theorist particularly, though I have a tangential acquaintance with it. There are arguments over the role of the AC. I had some years ago an interesting converstation with Chaitan over this. The mathematics of transfinite mathematics does have it origin in the dichotomy between countable and uncountable infinity. There is little mathematical debate over the existence of that.
The issue with physics, where it might have some bearing, is that physical objects occur in discrete packets. We measure things in clumps, where even quantum measurements involve a spot on a photoplate or an electronic click that registers a bit of information. So we actually measure things which are not continuous in a strict mathematical sense. These are particles or dynamical quantities. Yet we are faced with some curious issues with physics that dates back to F = ma. There you have a dynamical entity (force) equal to a kinematical quantity (mass) times a geometric quantity (acceleration). The geometric stuff involves relationships between things we measure. It is in this domain that we have continuous quantities. It is also here that the foundations of mathematics enter into physics. Most physicists, myself included, might only appeal to some point-set theory, such as compactness or paracompactness and so forth. If one were so game they might delve into deeper set theoretic issues. However, most physicists don't go there.
The loop quantum gravity crowd tends to see physical spacetime as a discrete system. However, the recent measurement by Fermi spacecraft on the simultaneous arrival of photons of widely different wavelength from billions of light years shoots down a major prediction of LQG.
Cheers LC
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I don't think that Cantor's work is a delusion, though I am not in a position to rigorously defend it. The problem is that I am not a set theorist particularly, though I have a tangential acquaintance with it. There are arguments over the role of the AC. I had some years ago an interesting converstation with Chaitan over this. The mathematics of transfinite mathematics does have it origin in the dichotomy between countable and uncountable infinity. There is little mathematical debate over the existence of that.
The issue with physics, where it might have some bearing, is that physical objects occur in discrete packets. We measure things in clumps, where even quantum measurements involve a spot on a photoplate or an electronic click that registers a bit of information. So we actually measure things which are not continuous in a strict mathematical sense. These are particles or dynamical quantities. Yet we are faced with some curious issues with physics that dates back to F = ma. There you have a dynamical entity (force) equal to a kinematical quantity (mass) times a geometric quantity (acceleration). The geometric stuff involves relationships between things we measure. It is in this domain that we have continuous quantities. It is also here that the foundations of mathematics enter into physics. Most physicists, myself included, might only appeal to some point-set theory, such as compactness or paracompactness and so forth. If one were so game they might delve into deeper set theoretic issues. However, most physicists don't go there.
The loop quantum gravity crowd tends to see physical spacetime as a discrete system. However, the recent measurement by Fermi spacecraft on the simultaneous arrival of photons of widely different wavelength from billions of light years shoots down a major prediction of LQG.
Cheers LC
Lawrence,
I would appreciate explanations why in the end Dedekind equated numbers with points. I myself found out so far:
Homer used the word pempazein (to five) for counting. Obviously counting was performed at the five fingers of a hand. The Greeks used
alpha, beta, gamma, ..., eta instead of 1,´2, 3,... , 8,
iota, kappa, lambda,... pi, instead of 10, 20, 30, ..., 80
rho, sigma, tau, ... omega, instead of 100, 200, 300, ..., 800
Vau = 6, Koppa = 90, and Sampi = 900
Aritoteles quoted the Phythagorean Eurytos who defined the unity (monav) as a point without position and accordingly a point as a unity that has a position.
Accordingly I tend to blame the intention of Eurytos to play with figurated numbers for the perhaps worst mistake in mathematics.
I will later deal with your defense of Cantor's naivity. Alfred Nobel did perhaps know why he decide, let be no price for mathematics. He did not like
Mittag-Leffler who supported Cantor.
Regards,
Eckard
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I would appreciate explanations why in the end Dedekind equated numbers with points. I myself found out so far:
Homer used the word pempazein (to five) for counting. Obviously counting was performed at the five fingers of a hand. The Greeks used
alpha, beta, gamma, ..., eta instead of 1,´2, 3,... , 8,
iota, kappa, lambda,... pi, instead of 10, 20, 30, ..., 80
rho, sigma, tau, ... omega, instead of 100, 200, 300, ..., 800
Vau = 6, Koppa = 90, and Sampi = 900
Aritoteles quoted the Phythagorean Eurytos who defined the unity (monav) as a point without position and accordingly a point as a unity that has a position.
Accordingly I tend to blame the intention of Eurytos to play with figurated numbers for the perhaps worst mistake in mathematics.
I will later deal with your defense of Cantor's naivity. Alfred Nobel did perhaps know why he decide, let be no price for mathematics. He did not like
Mittag-Leffler who supported Cantor.
Regards,
Eckard
The assignment of points with numbers goes back a long way. From meter stick lengths to coordinate grids on maps that is what we do. So the issue of assigning a real number of every point on a line, or pairs and n-tuples on planes or higher dimensional space goes back to some pretty classical ideas. Calculus is based ultimately on the idea that an infinitesimal constructed from a limit computes something, or numbers are assigned to points.
This procedure, which has practical uses, has lead to various questions about the foundations of mathematics. In the 20th century this lead to Cantor's pointing out how countable infinity was less than a continuum infinity, Godel's discovery that axiomatic systems are incomplete with statements that are true and unprovable, and Cohen's fusion of these two in his demonstration the continuum hypothesis is consistent in ZF, but not provable.
Cheers LC
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This procedure, which has practical uses, has lead to various questions about the foundations of mathematics. In the 20th century this lead to Cantor's pointing out how countable infinity was less than a continuum infinity, Godel's discovery that axiomatic systems are incomplete with statements that are true and unprovable, and Cohen's fusion of these two in his demonstration the continuum hypothesis is consistent in ZF, but not provable.
Cheers LC
Lawrence,
Let me reveal mistakes step by step. You wrote:
"The assignment of points with numbers goes back a long way. From meter stick lengths to coordinate grids on maps that is what we do. So the issue of assigning a real number of every point on a line, or pairs and n-tuples on planes or higher dimensional space goes back to some pretty classical ideas."
-- The primary meaning of numbers is still rudimentary to be seen in the Roman numbers: I, II, III, etc. Numbers are based on the choice of a unity "one". The next steps were repetitious recognition of this unity. If the unity is a length, then any decent number is also a length. In other words, the primary and therefore correct meaning of a number is a measure. Points do not have a measure. A number indicates how often the unity is repeated or split. Any piece of an ideal one-dimensional meter stick is still to be thought to be one-dimensional. The measure is given by two points: at zero and at the end. For instance, the number 35 is only correct if we start count at zero an not at 100cm. This clarification seem to be trivial. However, it avoids a lot of wrong consequences.
Regards,
Eckard
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Let me reveal mistakes step by step. You wrote:
"The assignment of points with numbers goes back a long way. From meter stick lengths to coordinate grids on maps that is what we do. So the issue of assigning a real number of every point on a line, or pairs and n-tuples on planes or higher dimensional space goes back to some pretty classical ideas."
-- The primary meaning of numbers is still rudimentary to be seen in the Roman numbers: I, II, III, etc. Numbers are based on the choice of a unity "one". The next steps were repetitious recognition of this unity. If the unity is a length, then any decent number is also a length. In other words, the primary and therefore correct meaning of a number is a measure. Points do not have a measure. A number indicates how often the unity is repeated or split. Any piece of an ideal one-dimensional meter stick is still to be thought to be one-dimensional. The measure is given by two points: at zero and at the end. For instance, the number 35 is only correct if we start count at zero an not at 100cm. This clarification seem to be trivial. However, it avoids a lot of wrong consequences.
Regards,
Eckard
As I see it you are calling into question the whole development of mathematics going back to calculus. Of course I can't write up a review of classical mathematics leading up to point-set topology and later foundational aspects. I am not sure where your departure lies. I would imagine that few physicists would cry foul over a rejection of Cantor transfinite set theory, and many mathematician might be unperturbed. However, if you point of departure reaches further into established or classical mathematics I suspect your objections will gather a smaller audience.
I am at best a sort of meatball mathematician, but mathematics involves relationships between structures which are consistent. That is the primary requirement. The idea of limits in calculus, with infinitesimal lengths and the like, might sound strange from one perspective. However, from a formal perspective these systems work. That is the primary requirement for any successful mathematical system.
Cheers LC
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I am at best a sort of meatball mathematician, but mathematics involves relationships between structures which are consistent. That is the primary requirement. The idea of limits in calculus, with infinitesimal lengths and the like, might sound strange from one perspective. However, from a formal perspective these systems work. That is the primary requirement for any successful mathematical system.
Cheers LC
Lawrence,
Why do you think the interpretation of numbers as measures instead of points is not agreeable with calculus?
According to
www.classicpersuasion.org/pw/burnet/egp.htm?chapter=2
"the Euclidean representation of numbers as lines was adopted to avoid the difficulties raised by the discovery of irrational quantities". Euclid is not only famous for qed but also for his book "Elements" which was printed in 1500 editions and there are uncounted handwritten copies. Who abandoned his notion of number and why? Was it at the time of Descartes?
Regards,
Eckard
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Why do you think the interpretation of numbers as measures instead of points is not agreeable with calculus?
According to
www.classicpersuasion.org/pw/burnet/egp.htm?chapter=2
"the Euclidean representation of numbers as lines was adopted to avoid the difficulties raised by the discovery of irrational quantities". Euclid is not only famous for qed but also for his book "Elements" which was printed in 1500 editions and there are uncounted handwritten copies. Who abandoned his notion of number and why? Was it at the time of Descartes?
Regards,
Eckard
I was not clear on your objection I think. Calculus is based on measure theory, at least integration is largely based on it. To be honest I am a bit unclear on what you see as so troubling. The real line is the set of all real numbers, which exist at all points on the line.
Cheer LC
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Cheer LC
Lawrence,
I found four conditions for a non-principal ultrafilter in non-standard mathematics:
1 modus ponens, 2 closed under intersection, and 3 dichotomy. While 1+2 define a filter and 1+2+3 define an ultrafilter, I am mainly interested in 4 non-principality, i.e., adding or deleting a finite munber of elements to/from A does not affect whether A is in p.
The latter well agrees with my understanding of really real numbers in contrast to rational ones. In other words, a single number or even a finite number of real numbers can be neglected when considered submerged into the genuine continuum of really real numbers. Accordingly, I prefer |sign(-->0)|=1, not =0.
I know: It is absolutely uncommon to write -->0 or in the general case -->x for the argument of a function. When we write f(x) we assume x subject to trichotomy: having neighbors, which are either smaller or equal or larger.
Now I will tell you what makes the difference when we do not consider points but measures: In any case including integration and its inverse the arrow of measure has always the same direction wrt zero. Then there is not f(x=0) but always only a f(x-->0) with a direction according to |x|>0. The point x=0 has no measure and no bearing. Absolute exactness is a fiction that contradicts to the fiction of absolute continuity.
While the replacement of points by positive or negative distances from zero will have diverse implications for the issue continuum vs. discrete approximation, I felt in particular urged to remove some illogicalities that hindered me to both "correctly" and reasonably perform a restriction to IR^+ and gave rise to trouble with delta(0) and the lower limit of integration in case of unilateral Laplace transform.
In contrast to points, measures are not zero-dimensional.
Cantor's point sets do strictly speaking NOT constitute a genuine (Peirce) and physically relevant continuum because he excluded what he called the infinitum absolutum. He claimed that there are cases of an infinitum creatum sive transfinitum in nature. They aren't.
After Abel called mathematics a mess, an increasing crowd of mathematicians strove for rigorous formalisms. Many were even ready to accept Cantor's seemingly correct proof for a more than countably infinite number of real numbers. Cantor enchanted the experts by keeping fix all of infinitely much of numbers!
Do not take me wrong. I do not deny that irrational numbers are uncountable, i.e. outside the rationals, and we may consider them to constitute the Peirce continuum. However, they must not quantitatively compared with the rationals. Infinity is strictly speaking an absolute property that cannot be increased. Infinitesimals are elements for linear approximation that are well described by means of epsilontics and notions like "small of higher order", or "as if they were infinite".
Regards,
Eckard
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I found four conditions for a non-principal ultrafilter in non-standard mathematics:
1 modus ponens, 2 closed under intersection, and 3 dichotomy. While 1+2 define a filter and 1+2+3 define an ultrafilter, I am mainly interested in 4 non-principality, i.e., adding or deleting a finite munber of elements to/from A does not affect whether A is in p.
The latter well agrees with my understanding of really real numbers in contrast to rational ones. In other words, a single number or even a finite number of real numbers can be neglected when considered submerged into the genuine continuum of really real numbers. Accordingly, I prefer |sign(-->0)|=1, not =0.
I know: It is absolutely uncommon to write -->0 or in the general case -->x for the argument of a function. When we write f(x) we assume x subject to trichotomy: having neighbors, which are either smaller or equal or larger.
Now I will tell you what makes the difference when we do not consider points but measures: In any case including integration and its inverse the arrow of measure has always the same direction wrt zero. Then there is not f(x=0) but always only a f(x-->0) with a direction according to |x|>0. The point x=0 has no measure and no bearing. Absolute exactness is a fiction that contradicts to the fiction of absolute continuity.
While the replacement of points by positive or negative distances from zero will have diverse implications for the issue continuum vs. discrete approximation, I felt in particular urged to remove some illogicalities that hindered me to both "correctly" and reasonably perform a restriction to IR^+ and gave rise to trouble with delta(0) and the lower limit of integration in case of unilateral Laplace transform.
In contrast to points, measures are not zero-dimensional.
Cantor's point sets do strictly speaking NOT constitute a genuine (Peirce) and physically relevant continuum because he excluded what he called the infinitum absolutum. He claimed that there are cases of an infinitum creatum sive transfinitum in nature. They aren't.
After Abel called mathematics a mess, an increasing crowd of mathematicians strove for rigorous formalisms. Many were even ready to accept Cantor's seemingly correct proof for a more than countably infinite number of real numbers. Cantor enchanted the experts by keeping fix all of infinitely much of numbers!
Do not take me wrong. I do not deny that irrational numbers are uncountable, i.e. outside the rationals, and we may consider them to constitute the Peirce continuum. However, they must not quantitatively compared with the rationals. Infinity is strictly speaking an absolute property that cannot be increased. Infinitesimals are elements for linear approximation that are well described by means of epsilontics and notions like "small of higher order", or "as if they were infinite".
Regards,
Eckard
Analysis uses measures because there is no "machine" which can compute or enumerate all the elements of the reals. Yet measure theory is still based on point set topology.
It has been a long time since I concerned myself with these issues. Going back to the late 19th and early 20th century history of mathematics it is clear there were considerable debates over this. I would tend to say that unless somebody comes up with a new system of mathematics that the current system will remain. I think it would require a huge abstracted idea to reframe these foundations of mathematics.
Cheers LC
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It has been a long time since I concerned myself with these issues. Going back to the late 19th and early 20th century history of mathematics it is clear there were considerable debates over this. I would tend to say that unless somebody comes up with a new system of mathematics that the current system will remain. I think it would require a huge abstracted idea to reframe these foundations of mathematics.
Cheers LC
That type of relationship (between Döring and Isham) is what I've been looking for my entire career. Unfortunately, I'm already a tenured department chair which severely limits my freedoms. Congrats to Döring on finding such a thing when he did.
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You're a "chair"?
That wiould explain the dead wood, but how in the world can a chair type a message on a keyboard?
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That wiould explain the dead wood, but how in the world can a chair type a message on a keyboard?
This statement:"According to quantum mechanics, you cannot ask what the properties of a quantum particle are before it is measured. Prior to observation, the particle exists in a superposition of multiple contradictory states."
There’s no simple yes/no or true/false answer to questions about the state of the particle." is a puzzling statement?..isn't a "measure" actually allready asking the !question" ? To measure something is to ask and thus locate a particle's properties? Being measured is axtually a default knowing "before" question?
Reducing observation down to simplistic "yes/no" questions is not a viable process, as stated in the article, but what if we looked at a reverse perspective?..for instance, any macro entity such as an observer, when trying to locate a micro entity/paticle is actually looking from large-scale to small-scale, so what about the micro particle looking outwards..any quantum actually "know" it is being measured by default, a quantum will "know" about a MACRO entity long before the entity has located the quantum!
If one tries to locate a needle in a haystack, one could find it by chance quite quickly, or it may take an eternity, something large trying to locate something small, by its very nature has a lot of variables. Now reverse the "logic" process, what if the needle could relay its knowledge of the "seeker2, the macro entity trying to locate it, if it could signal to the observer, here I am!, look over here!...then there would be interesting alterations to the Macro to Quantum, information exchange domains?
The very fact that a Macro observer moves the Quantum needle from location to location, a brownian motion like effect by default? The needle is no observer, it has no knowledge of hide'n'seek, even though it is being moved by the act of observer trying to locate it, the observation is only one-way.
Now logical sense may have limitations, by the process of cross scale domains?..would it make sense to walk into a haysack barn and proclaim:Needle, Iam not looking for you... the questioning seems to have relevance?
Who can doubt that actually looking for an actual sowing needle for instance, causes the sowing needle to be NOT located with ease, yet as soon as one gives up and turns to walk away, the needle appears under your foot..OOWWCHH!
It is not that trees in a forest do not exist or fall over when nobody is measuring them, its the facat the quastioning itself, appears to constrain the actual outcome of probable answers"
I do really admire the ones who take the challenge to unravel the cpmplex observer/observation_measured/measure fields, quite interesting.
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There’s no simple yes/no or true/false answer to questions about the state of the particle." is a puzzling statement?..isn't a "measure" actually allready asking the !question" ? To measure something is to ask and thus locate a particle's properties? Being measured is axtually a default knowing "before" question?
Reducing observation down to simplistic "yes/no" questions is not a viable process, as stated in the article, but what if we looked at a reverse perspective?..for instance, any macro entity such as an observer, when trying to locate a micro entity/paticle is actually looking from large-scale to small-scale, so what about the micro particle looking outwards..any quantum actually "know" it is being measured by default, a quantum will "know" about a MACRO entity long before the entity has located the quantum!
If one tries to locate a needle in a haystack, one could find it by chance quite quickly, or it may take an eternity, something large trying to locate something small, by its very nature has a lot of variables. Now reverse the "logic" process, what if the needle could relay its knowledge of the "seeker2, the macro entity trying to locate it, if it could signal to the observer, here I am!, look over here!...then there would be interesting alterations to the Macro to Quantum, information exchange domains?
The very fact that a Macro observer moves the Quantum needle from location to location, a brownian motion like effect by default? The needle is no observer, it has no knowledge of hide'n'seek, even though it is being moved by the act of observer trying to locate it, the observation is only one-way.
Now logical sense may have limitations, by the process of cross scale domains?..would it make sense to walk into a haysack barn and proclaim:Needle, Iam not looking for you... the questioning seems to have relevance?
Who can doubt that actually looking for an actual sowing needle for instance, causes the sowing needle to be NOT located with ease, yet as soon as one gives up and turns to walk away, the needle appears under your foot..OOWWCHH!
It is not that trees in a forest do not exist or fall over when nobody is measuring them, its the facat the quastioning itself, appears to constrain the actual outcome of probable answers"
I do really admire the ones who take the challenge to unravel the cpmplex observer/observation_measured/measure fields, quite interesting.
Lawrence,
Hopefully you will agree that Leshan, Döring, Paul Valetta and someone whose essay is characterized by frequent use of "in some sense but" might simply be too old as to follow the logics of Euclid and any critically thinking child: Point, line, etc. are different from reality in that being infinitely small or thin.
I consider the idea that points are like pebbles or like symmetrically arranged patterns on a dice a cancer to be removed from mathematics. Let's admire the wisdom of Alexandria which was summarized by Euclid: A point is what does not have parts.
The usual notion of a cut means to separate something by placing a knife between parts. Dedekind's cut is not in position to separate two continua from each other. Accordingly we have to re-understand the notion singularity and should be very skeptical concerning claimed singularities in physics. This does not mean they are useless. I enjoy using them as I am using points and lines, which strictly speaking do not exist either.
I see children, very young physicists, and to some extent mathematically proficient engineers like Wolfgang Mueckenheim who is a dean in Augsburg and a professor who lectures mathematics best suited to contribute to an overdue correction of some basics of mathematics. Do not get me wrong. I do not entirely share his views. Old chairs shy back from the huge heap of rubble already feared by Fraenkel.
Regards,
Eckard
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Hopefully you will agree that Leshan, Döring, Paul Valetta and someone whose essay is characterized by frequent use of "in some sense but" might simply be too old as to follow the logics of Euclid and any critically thinking child: Point, line, etc. are different from reality in that being infinitely small or thin.
I consider the idea that points are like pebbles or like symmetrically arranged patterns on a dice a cancer to be removed from mathematics. Let's admire the wisdom of Alexandria which was summarized by Euclid: A point is what does not have parts.
The usual notion of a cut means to separate something by placing a knife between parts. Dedekind's cut is not in position to separate two continua from each other. Accordingly we have to re-understand the notion singularity and should be very skeptical concerning claimed singularities in physics. This does not mean they are useless. I enjoy using them as I am using points and lines, which strictly speaking do not exist either.
I see children, very young physicists, and to some extent mathematically proficient engineers like Wolfgang Mueckenheim who is a dean in Augsburg and a professor who lectures mathematics best suited to contribute to an overdue correction of some basics of mathematics. Do not get me wrong. I do not entirely share his views. Old chairs shy back from the huge heap of rubble already feared by Fraenkel.
Regards,
Eckard
Lawrence,
When you reminded of controversy on set theory you certainly meant how Cantor managed to get published with support by Weierstrass and Hurwitz and despite of Kronecker's rejection. His populist attitude was widely appreciated: "The essence of mathematics is just its freedom." While there was not really a need for a set theory, Cantor's stunning proofs were seemingly correct. Kronecker gave up and died. In 1884, Cantor got insane for the first time because he failed to provide an already announced proof of his CH. Hilbert declared in 1900 the CH a most important problem. In this case and also in argumentations by Zermelo and by Fraenkel, sets of points were not at all put in question. From the very beginning in 1872 Cantor agreed with Dedekind and perhaps virtually all other mathematicians on the fallacious idea that there must be more real numbers as compared to the rational ones because the latter are a subset of the former.
My knowledge of the controversy is mainly based on original papers and Fraenkel's 1923 book. Christian Betsch who earned a 1,000,000.00 Deutschmark price, which was a fortune after inflation, also did not deal with the question whether numbers are points or measures.
Having looked in vain for any possibility to justify Cantor's naive cardinality, I see Galilei's and Euclid's insights still valid and a huge crowd of mediocre mathematicians busy with a tempest in a teapot for more than a hundred years.
The late Fraenkel cautiously admitted that set theory merely made mathematics more "interesting". I would rather say more bizarre.
Let me stress that I am not an antisemit. Cantor's opponent Kronecker was a wealthy Jew while Cantor himself got a Catholic education from his mother. Stupid Germans said one has to be Jewish as to understand the set theory. Wouldn't this possibly mean that Jews can in general be mislead more easily? The list of Hurwitz, Hadamard, Bernstein, Mittag, Schloemilch, Hessenberg, Hausdorff, v. Neumann, Bernays, Goedel, Fraenkel, Grothendieck, Robinson, Cohen, Levi, etc. is not less impressive than the list of those who Cantor himself called opponents of his transfinite numbers including Cauchy, Galilei, Gauss, Hegel, v. Helmholtz, Kant, Kronecker, Leibniz, and Newton.
Fraenkel himself stated that there were only a very few who resisted: Poincaré, Brouwer, and I would like to add Kolmogoroff, Heyting, Lorenzen, Bishop.
However, science is not a democracy, and it must not be attributed to belief. What ultimately counts are only solid arguments, experimental verification and valuable applications in practice.
Regards,
Eckard
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When you reminded of controversy on set theory you certainly meant how Cantor managed to get published with support by Weierstrass and Hurwitz and despite of Kronecker's rejection. His populist attitude was widely appreciated: "The essence of mathematics is just its freedom." While there was not really a need for a set theory, Cantor's stunning proofs were seemingly correct. Kronecker gave up and died. In 1884, Cantor got insane for the first time because he failed to provide an already announced proof of his CH. Hilbert declared in 1900 the CH a most important problem. In this case and also in argumentations by Zermelo and by Fraenkel, sets of points were not at all put in question. From the very beginning in 1872 Cantor agreed with Dedekind and perhaps virtually all other mathematicians on the fallacious idea that there must be more real numbers as compared to the rational ones because the latter are a subset of the former.
My knowledge of the controversy is mainly based on original papers and Fraenkel's 1923 book. Christian Betsch who earned a 1,000,000.00 Deutschmark price, which was a fortune after inflation, also did not deal with the question whether numbers are points or measures.
Having looked in vain for any possibility to justify Cantor's naive cardinality, I see Galilei's and Euclid's insights still valid and a huge crowd of mediocre mathematicians busy with a tempest in a teapot for more than a hundred years.
The late Fraenkel cautiously admitted that set theory merely made mathematics more "interesting". I would rather say more bizarre.
Let me stress that I am not an antisemit. Cantor's opponent Kronecker was a wealthy Jew while Cantor himself got a Catholic education from his mother. Stupid Germans said one has to be Jewish as to understand the set theory. Wouldn't this possibly mean that Jews can in general be mislead more easily? The list of Hurwitz, Hadamard, Bernstein, Mittag, Schloemilch, Hessenberg, Hausdorff, v. Neumann, Bernays, Goedel, Fraenkel, Grothendieck, Robinson, Cohen, Levi, etc. is not less impressive than the list of those who Cantor himself called opponents of his transfinite numbers including Cauchy, Galilei, Gauss, Hegel, v. Helmholtz, Kant, Kronecker, Leibniz, and Newton.
Fraenkel himself stated that there were only a very few who resisted: Poincaré, Brouwer, and I would like to add Kolmogoroff, Heyting, Lorenzen, Bishop.
However, science is not a democracy, and it must not be attributed to belief. What ultimately counts are only solid arguments, experimental verification and valuable applications in practice.
Regards,
Eckard
Singularities in physics usually indicate a breakdown of a a physical principle. Singularities in general relativity tell us some more global principle is involved, such as quantization.
One problem I have with commenting on set theory is that I am not sufficiently educated in the topic to make much contribution to any such discussion. Mathematics builds up model systems, and one can build systems which avoid some of these curious issues of seemingly paradoxical cardinalities. The mathematicians do though like to work with the most general or powerful (powerful in being able to prove the widest array of theorems) system possible. We have a similar situation in physics with string theory, which makes a vast array of "predictions" that are not all realistic, but which is able to address certain physics problems better than the alternatives. ZFC theory does seem to fit the bill for mathematicians who work on these issues.
Cheers LC
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One problem I have with commenting on set theory is that I am not sufficiently educated in the topic to make much contribution to any such discussion. Mathematics builds up model systems, and one can build systems which avoid some of these curious issues of seemingly paradoxical cardinalities. The mathematicians do though like to work with the most general or powerful (powerful in being able to prove the widest array of theorems) system possible. We have a similar situation in physics with string theory, which makes a vast array of "predictions" that are not all realistic, but which is able to address certain physics problems better than the alternatives. ZFC theory does seem to fit the bill for mathematicians who work on these issues.
Cheers LC
Inevitably it seems, when physicists get stuck for an answer, the fault must lie in the mathematics. To me, that is sort of like blaming one's inability to tell a story on one's lack of vocabulary--and then blaming the audience for not understanding one's newly-invented terms when one decides to tell the story anyway.
Even in the days before mathematics and physics were truly separate disciplines, new mathematical methods came under suspicion by those seeking more intuitive, even "obvious" and simpler ways of describing perceived reality. The application of zero as a number, complex analysis, Newton's fluxions, Riemannian geometry ... examples abound for mathematical theories, methods and techniques that resisted physical application until forced by necessity.
One needn't worry about replacing mathematics with another system. It will replace itself. The most social and adaptable of the arts, mathematical language evolves just as natural language evolves--to meet the demands of describing new experiences. One need be reminded that almost all that we know of objective reality, however, is counterintuitive, not simple nor obvious until incorporated fully into the culture. And why should we be concerned about closing the gap between language and meaning? That gap is where understanding lives, the place where we find objective knowledge, and where we find the most interesting questions to ask of the "subtle, but not malicious" reality that we share.
Tom
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Even in the days before mathematics and physics were truly separate disciplines, new mathematical methods came under suspicion by those seeking more intuitive, even "obvious" and simpler ways of describing perceived reality. The application of zero as a number, complex analysis, Newton's fluxions, Riemannian geometry ... examples abound for mathematical theories, methods and techniques that resisted physical application until forced by necessity.
One needn't worry about replacing mathematics with another system. It will replace itself. The most social and adaptable of the arts, mathematical language evolves just as natural language evolves--to meet the demands of describing new experiences. One need be reminded that almost all that we know of objective reality, however, is counterintuitive, not simple nor obvious until incorporated fully into the culture. And why should we be concerned about closing the gap between language and meaning? That gap is where understanding lives, the place where we find objective knowledge, and where we find the most interesting questions to ask of the "subtle, but not malicious" reality that we share.
Tom
Tom and Lawrence,
Given, my - as I am claiming - not entirely unsubstantiated suspicion that SUSY is based on a mistake will not be refuted by experiments with LHC or the like. Might I then hope for contributing to a more humble readiness to put at least the worst and not even positivist speculations in question? Zeh (4th ed. , p. 130) admits that there are reasons why white holes are unrealistic. I doubt that past, future, naked and other physical singularities must be taken as reality.
I looked for the origin and early meaning of the notion singularity at the time of Bernoulli and Euler and found it closely related to what I suspect. Let me explain it in the language of a boy who knows a ball and the globe. Is the middle point something special? No. It just belongs to the ideal equal distance from surface.
Triviality stems from three ways. The model has a node of ramification. Two streets in reality do not.
Seemingly, a theoretical physicist should reach the age of Methusalem as to have a chance for getting educated enough for answering my questions in a manner that does nobody understand. I see already Lawrence overqualified in mathematics.
Again, I consider my reproach worth: It is more reasonable to consider numbers as measures and not as points. I just found the notorious lack of clarity also in a conference paper by Jeffrey, Labahn, et al. of Wateloo on integration of signum, piecewise and related functions. Do we really need true lies, differently defined infinities, differently defined signum functions, etc.?
Indeed, reality and appropriate mathematics are not malicious. At the moment, Wikipedia seems to include a huge collection of mistake-related twists.
I recall that a textbook for engineers showed a function |sign(x)| like this
1 ____ ____ as if it intended to illustrate Lipshitz continuity.
0 V
Regards,
Eckard
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Given, my - as I am claiming - not entirely unsubstantiated suspicion that SUSY is based on a mistake will not be refuted by experiments with LHC or the like. Might I then hope for contributing to a more humble readiness to put at least the worst and not even positivist speculations in question? Zeh (4th ed. , p. 130) admits that there are reasons why white holes are unrealistic. I doubt that past, future, naked and other physical singularities must be taken as reality.
I looked for the origin and early meaning of the notion singularity at the time of Bernoulli and Euler and found it closely related to what I suspect. Let me explain it in the language of a boy who knows a ball and the globe. Is the middle point something special? No. It just belongs to the ideal equal distance from surface.
Triviality stems from three ways. The model has a node of ramification. Two streets in reality do not.
Seemingly, a theoretical physicist should reach the age of Methusalem as to have a chance for getting educated enough for answering my questions in a manner that does nobody understand. I see already Lawrence overqualified in mathematics.
Again, I consider my reproach worth: It is more reasonable to consider numbers as measures and not as points. I just found the notorious lack of clarity also in a conference paper by Jeffrey, Labahn, et al. of Wateloo on integration of signum, piecewise and related functions. Do we really need true lies, differently defined infinities, differently defined signum functions, etc.?
Indeed, reality and appropriate mathematics are not malicious. At the moment, Wikipedia seems to include a huge collection of mistake-related twists.
I recall that a textbook for engineers showed a function |sign(x)| like this
1 ____ ____ as if it intended to illustrate Lipshitz continuity.
0 V
Regards,
Eckard
Usually in the case of physics the fault of a theory lies not in the mathematics, but in the postulates of theory which result in some obstruction. This occurred with electromagnetism and mechanics, which Einstein first saw when he imagined himself on a reference frame moving with and electromagnetic wave. Similarly Planck realized that a problem with getting the spectra of a black body right was due to the assumption of having harmonic oscillators with a continuum of energy. By removing this obstruction then the apparent paradox is removed. From a mathematical perspective this does usually mean the adoption of a more general or powerful mathematical system for theoretical formulation.
When it comes to set theory that is pretty far removed from physics by and large. It is sometimes called metamathematics, for it concerns itself with the foundations of mathematical thought and how mathematical models are consistent. As I indicated I have really a tangential knowledge of this subject, so I am probably the last person to work up a more general foundation which could replace ZFC, or contribute much in such an effort.
This is not directly related to the issue of Topos theory, which is a categorical system of sheaves. There the type of algebraic geometry one works with is dependent upon the category (functor system etc) employed. This gives rise to this idea that mathematics in certain setting can "Lie."
Cheers LC
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When it comes to set theory that is pretty far removed from physics by and large. It is sometimes called metamathematics, for it concerns itself with the foundations of mathematical thought and how mathematical models are consistent. As I indicated I have really a tangential knowledge of this subject, so I am probably the last person to work up a more general foundation which could replace ZFC, or contribute much in such an effort.
This is not directly related to the issue of Topos theory, which is a categorical system of sheaves. There the type of algebraic geometry one works with is dependent upon the category (functor system etc) employed. This gives rise to this idea that mathematics in certain setting can "Lie."
Cheers LC
Yes, usually the fault lies not in the mathematics, and I would like to add that the words like theory, theorem, theology, theocracy, Theodor, etc. relate to theo=deo=deus=Zeus=god.
Maybe, for this or a similar reason Euler did not call his work a theory of ships?
You Lawrence uttered the common opinion:If there are obstructions, then the mathematics must be made more general. I strongly object in cases where I consider a theory already as too general.
Is it justified to use future time for an analysis of what relates to a process that is finished? No.
Does it make sense to allow for a diversity of arbitrary to choose definitions e.g. in connection with a function at a discontinuity and its value(s) there?
No.
Does the complex representation of a function that has measurable values only for positive arguments really provide an additional degree of freedom? No.
Isn't it a at least extremely risky to speculatively interpret without justification as real not just the obviously matching solution of an equation that fits to some extent to the physical reality but all of it, e.g. all Schwarzschild solutions?
I consider Gantor's utterly naive theory an exception, something really wrong and in the word ueberabzaehlbar reminding of Nietzsche's Uebermensch. Cantor admitted having got it directly from god.
So far I do not yet understand why Euclid's correct attribution of a numbers to the measure of a pieces of a line was forgotten. Maybe, the Romans were in mathematics the true "Vandals".
Regards,
Eckard
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Maybe, for this or a similar reason Euler did not call his work a theory of ships?
You Lawrence uttered the common opinion:If there are obstructions, then the mathematics must be made more general. I strongly object in cases where I consider a theory already as too general.
Is it justified to use future time for an analysis of what relates to a process that is finished? No.
Does it make sense to allow for a diversity of arbitrary to choose definitions e.g. in connection with a function at a discontinuity and its value(s) there?
No.
Does the complex representation of a function that has measurable values only for positive arguments really provide an additional degree of freedom? No.
Isn't it a at least extremely risky to speculatively interpret without justification as real not just the obviously matching solution of an equation that fits to some extent to the physical reality but all of it, e.g. all Schwarzschild solutions?
I consider Gantor's utterly naive theory an exception, something really wrong and in the word ueberabzaehlbar reminding of Nietzsche's Uebermensch. Cantor admitted having got it directly from god.
So far I do not yet understand why Euclid's correct attribution of a numbers to the measure of a pieces of a line was forgotten. Maybe, the Romans were in mathematics the true "Vandals".
Regards,
Eckard
As a layman, I am surprised:
http://anubis.dkuug.dk/JTC1/SC2/WG2/docs/n2708.pdf
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http://anubis.dkuug.dk/JTC1/SC2/WG2/docs/n2708.pdf
Let me explain why I consider it relevant for the relationship between mathematics including topos theory and physics that some Greeks did already operate with zero.
Most likely those like Platon (427-348) and Pythagoras (570-500) are to be blamed for the attitude of those mathematicians not to accept zero as a number,
negative numbers, etc., who expected that the world is based on laws of mathematics and anything is number.
I first looked at Descartes for several reasons:
- He referred to the rigorous work by Euclid.
- He used the understanding of numbers like points on a line.
- He prepared the invention of calculus by Fermat, Newton, and Leibniz.
The sloppy understanding of numbers like points on a line was, however, perhaps ubiquitous already before Descartes. Albert of Saxony, a rector of the university of Paris, founder of the university of Vienna and finally Bishop of Halberstadt was influenced by Buridan and Ockham. He compiled the knowledge of his time. He wrote: "There is no simple concept of a point, a vacuum, or the infinite, and although imaginary hypotheses provide an interesting detour, physics must in the end provide an account of the natural order of things."
Did he not know Euclid's uncorrectable definition of a point?
He wrote: "If A smaller B then there exists a quantity C such that A smaller B smaller C."
I suspect that the strive for mathematical strength was not agreeable with philosophical points of view as mentioned above. It was in this sense a strength of the middle ages to loose rigorosity.
Dedekind's attitude was then also opposite to Plato's. We may better understand the history of mathematics and physics when we are focusing on the question how the scientists tried to cope with unavoidable pragmatism and the need to accept physical restrictions to the much wider mathematical possibilities. This should give rise for a scrutiny of a large part of modern, in the sense of utterly speculative, theories.
Eckard
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Most likely those like Platon (427-348) and Pythagoras (570-500) are to be blamed for the attitude of those mathematicians not to accept zero as a number,
negative numbers, etc., who expected that the world is based on laws of mathematics and anything is number.
I first looked at Descartes for several reasons:
- He referred to the rigorous work by Euclid.
- He used the understanding of numbers like points on a line.
- He prepared the invention of calculus by Fermat, Newton, and Leibniz.
The sloppy understanding of numbers like points on a line was, however, perhaps ubiquitous already before Descartes. Albert of Saxony, a rector of the university of Paris, founder of the university of Vienna and finally Bishop of Halberstadt was influenced by Buridan and Ockham. He compiled the knowledge of his time. He wrote: "There is no simple concept of a point, a vacuum, or the infinite, and although imaginary hypotheses provide an interesting detour, physics must in the end provide an account of the natural order of things."
Did he not know Euclid's uncorrectable definition of a point?
He wrote: "If A smaller B then there exists a quantity C such that A smaller B smaller C."
I suspect that the strive for mathematical strength was not agreeable with philosophical points of view as mentioned above. It was in this sense a strength of the middle ages to loose rigorosity.
Dedekind's attitude was then also opposite to Plato's. We may better understand the history of mathematics and physics when we are focusing on the question how the scientists tried to cope with unavoidable pragmatism and the need to accept physical restrictions to the much wider mathematical possibilities. This should give rise for a scrutiny of a large part of modern, in the sense of utterly speculative, theories.
Eckard
Having dealt a little with ancient and medieval mathematics, I changed my attitude towards Dedekind. His somewhat bewildering claim to create a new number was indeed a brave antithesis to Kronecker who considered the natural numbers as made by god.
Medieval mathematicians often equated the number one with god. "I am the Lord your God. You must not have other gods beside me."
The rational numbers are based on the primary measure 1. Irrational numbers may also be considered numbers if one admits different measures as the basis, e.g. the diameter of a circle or the diagonal of a square.
What about Cantor's idea to count in excess of infinity, I maintain my objections. Cardinalities like aleph_2 etc. are nonsense. While any irrational number is not exactly addressable with reference to the basic measure 1, there are infinitely many "gods" in the sense of - as Peirce understood the mere possibility of - infinitely many basic measures that would allow their own system of counting. The genuine continuum could be imagined to consist of all of them together. One must, however, not hope for unification of all "gods" in a manner other than calculation as if the reals were rationals. It is reasonable and sometimes necessary to accept that continuity and the measure-one based numbers exclude and mutually complement each other.
Eckard
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Medieval mathematicians often equated the number one with god. "I am the Lord your God. You must not have other gods beside me."
The rational numbers are based on the primary measure 1. Irrational numbers may also be considered numbers if one admits different measures as the basis, e.g. the diameter of a circle or the diagonal of a square.
What about Cantor's idea to count in excess of infinity, I maintain my objections. Cardinalities like aleph_2 etc. are nonsense. While any irrational number is not exactly addressable with reference to the basic measure 1, there are infinitely many "gods" in the sense of - as Peirce understood the mere possibility of - infinitely many basic measures that would allow their own system of counting. The genuine continuum could be imagined to consist of all of them together. One must, however, not hope for unification of all "gods" in a manner other than calculation as if the reals were rationals. It is reasonable and sometimes necessary to accept that continuity and the measure-one based numbers exclude and mutually complement each other.
Eckard
truth of the universe is not in numbers but in proper understending of time
yours amrit
attachments: Linear_Time_PastPresentFutuure_is_based_on_Neuronal_Dynamics_Physical_Time_is_based_on_Motion.pdf
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yours amrit
attachments: Linear_Time_PastPresentFutuure_is_based_on_Neuronal_Dynamics_Physical_Time_is_based_on_Motion.pdf
There is a whole lot of machinery involving n-tupes of points on a manifold or the real line. In particular metric geometry is founded on this. The role of measures come with the integration of functions over the reals, or some manifold in L^p spaces. This is particularly of importantce when the function has some strange support, such as "continuous almost everywhere." I don't think that the point on a space perspective is inconsistent with the concept of a measure.
Cheers LC
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Cheers LC
I did not yet manage reading Euclid`s book VII with his definition of a number. So I only guess from secondary literature that already he go it correctly: The primary basis of counting is the number one. It denotes an extension, in particular a length, i.e. the distance between two points. Countable are repetitions and splits of it also in combination, i.e. one-based numbers that we may obtain with the four basic operations addition, multiplication, division, and subtraction. Seemingly these measures are not much different from points because we tacitly assume that they all refer to the point zero, called the neutral element of addition and the point one, the neutral element of multiplication. As there is no limit to the amount of countable, i.e. rational points, there is also no limit to the amount of countable measures. Accordingly, there are uncountable, i.e. irrational measures.
What justifies my obstinacy? While the addition of dimensionless points does not yield for instance a one-dimensional line, measures do have a priori the dimension of their basic element. We need not the unrealistic fiction "all of infinitely many" for getting a different quality by resorting to an unimaginably large quantity. Infinity was originally and is according to all reasonable logic still simply the property to have no limit. It is not a quantity even if god-related speculations led to absurd rather than merely abstruse twists.
Is this arguing perhaps correct but pointless? No. The reason for me to deal with this topic emerged from practice. I am still not yet in position to overlook all consequences. The point on a space concept follows with some caveats from the measure concept. EPR and the question whether or not a single point does exist in a continuum prove pointless. It does not matter that Buridan's ass seems to be first mentioned not by Buridan himself but by Pierre Bale (1647-1706). It nicely describes the problem. If 0 is a measure then I consider |sign(0)|=1 correct. The constructivists/intuitionists suffered from troublesome attempts to remedy the consequences of basing mathematics on sets of points.
Strange constructs like Weierstrass's monster and Cantor's dust were interesting from mathematical playground point of view. Did they have implications in physics?
Eckard
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What justifies my obstinacy? While the addition of dimensionless points does not yield for instance a one-dimensional line, measures do have a priori the dimension of their basic element. We need not the unrealistic fiction "all of infinitely many" for getting a different quality by resorting to an unimaginably large quantity. Infinity was originally and is according to all reasonable logic still simply the property to have no limit. It is not a quantity even if god-related speculations led to absurd rather than merely abstruse twists.
Is this arguing perhaps correct but pointless? No. The reason for me to deal with this topic emerged from practice. I am still not yet in position to overlook all consequences. The point on a space concept follows with some caveats from the measure concept. EPR and the question whether or not a single point does exist in a continuum prove pointless. It does not matter that Buridan's ass seems to be first mentioned not by Buridan himself but by Pierre Bale (1647-1706). It nicely describes the problem. If 0 is a measure then I consider |sign(0)|=1 correct. The constructivists/intuitionists suffered from troublesome attempts to remedy the consequences of basing mathematics on sets of points.
Strange constructs like Weierstrass's monster and Cantor's dust were interesting from mathematical playground point of view. Did they have implications in physics?
Eckard
I can only say that point-set topology used to underlie differential geometry and algebraic geometry has yet to find successful rivals. I have not studied the history of this in more recent times, but there have been attempts to construct alternatives in the 20th century, but these seemed to lead to greater difficulties or are not sufficiently powerful to build rich mathematical structures.
The Wierstrass monster set is used in electrical engineering. You might have noticed how cell phones don't have an antenna sticking out. The problem back then was how to get an antenna that received on a wide bandwidth. The solution was to construct a chip with an epitaxial form of the monster set. This permitted a wide range of EM reception, and the antenna sticking out of the phone is removed. As for Cantor dust, that is what an energy surface or torus of dynamics turns into when "punctured." So the Cantor dust is something which is a backbone of chaotic dynamics.
Cheers LC
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The Wierstrass monster set is used in electrical engineering. You might have noticed how cell phones don't have an antenna sticking out. The problem back then was how to get an antenna that received on a wide bandwidth. The solution was to construct a chip with an epitaxial form of the monster set. This permitted a wide range of EM reception, and the antenna sticking out of the phone is removed. As for Cantor dust, that is what an energy surface or torus of dynamics turns into when "punctured." So the Cantor dust is something which is a backbone of chaotic dynamics.
Cheers LC
Lawrence,
Weierstrass was not recognized until he shocked the mathematic community with a monster function that was already found by the theologian Bolzano. Weierstrass claimed that this function is everywhere continuous and nowhere differentiable. To my knowledge, he did not explain why: This “pathological” behavior is bound to a fictitious completeness: The function is a sum of an actually infinite amount of terms. Even the best approximation with as many terms as you like behaves as expected.
You mentioned fractal antennas as an application. They are not only rather dissimilar to Weierstrass’s monster but they are fundamentally different from it as is anything that belongs to reality. It is impossible to build antennas that do not have an upper limit of frequency. That’s why I consider the belief elusive that Weierstrass’ monster, some admittedly nice work using infinity by Reuben Goodstein, and the like not immediately relevant for reality. Incidentally, already the simple circle and all belonging harmonic functions that are thought ranging from minus infinity to plus infinity do not exactly describe something in reality.
I fear, you will not continue discussion because I and you cannot convince each other. Let me confirm your argument that alternatives to set theory based on points were not successful so far.
I would appreciate any hint to some attempt to resume what perhaps already did Euclid: Consider the notion of number based on the primary measure one, i.e. counting with respect to the distance between two points one of which is the neutral point of addition/subtraction, the other one is the neutral point of multiplication/division. I am pretty sure that you thought of the few constructivists, in particular Brouwer, Weyl, Heyting, and Kolgomoroff. To my knowledge, they tried to create an alternative set theory also based on points. The only ones who were close to my suggestion seem to be Baire, Borel and Lebesgue.
Constructivism was perhaps prompted by some unrealistic aspects of formalism while it proved obviously less efficient. I hope, choosing the measure as the primary notion of number will lead to a more satisfactory result: both realistic and efficient. After I will have found Euclid’s notion of a number in book IIV, I would call it neo-Euclidean. Cantor’s dust has measure zero, regardless whether or not a close approximation might be valuable.
Attribution of always just one number y to a number x is anyway no must for engineers, who are familiar with hysteresis. Purification of mathematics from illusions will be a good basis for the purification of physics from much more phantasm.
Regards,
Eckard
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Weierstrass was not recognized until he shocked the mathematic community with a monster function that was already found by the theologian Bolzano. Weierstrass claimed that this function is everywhere continuous and nowhere differentiable. To my knowledge, he did not explain why: This “pathological” behavior is bound to a fictitious completeness: The function is a sum of an actually infinite amount of terms. Even the best approximation with as many terms as you like behaves as expected.
You mentioned fractal antennas as an application. They are not only rather dissimilar to Weierstrass’s monster but they are fundamentally different from it as is anything that belongs to reality. It is impossible to build antennas that do not have an upper limit of frequency. That’s why I consider the belief elusive that Weierstrass’ monster, some admittedly nice work using infinity by Reuben Goodstein, and the like not immediately relevant for reality. Incidentally, already the simple circle and all belonging harmonic functions that are thought ranging from minus infinity to plus infinity do not exactly describe something in reality.
I fear, you will not continue discussion because I and you cannot convince each other. Let me confirm your argument that alternatives to set theory based on points were not successful so far.
I would appreciate any hint to some attempt to resume what perhaps already did Euclid: Consider the notion of number based on the primary measure one, i.e. counting with respect to the distance between two points one of which is the neutral point of addition/subtraction, the other one is the neutral point of multiplication/division. I am pretty sure that you thought of the few constructivists, in particular Brouwer, Weyl, Heyting, and Kolgomoroff. To my knowledge, they tried to create an alternative set theory also based on points. The only ones who were close to my suggestion seem to be Baire, Borel and Lebesgue.
Constructivism was perhaps prompted by some unrealistic aspects of formalism while it proved obviously less efficient. I hope, choosing the measure as the primary notion of number will lead to a more satisfactory result: both realistic and efficient. After I will have found Euclid’s notion of a number in book IIV, I would call it neo-Euclidean. Cantor’s dust has measure zero, regardless whether or not a close approximation might be valuable.
Attribution of always just one number y to a number x is anyway no must for engineers, who are familiar with hysteresis. Purification of mathematics from illusions will be a good basis for the purification of physics from much more phantasm.
Regards,
Eckard
Dear Everyone,
You have to see this paper:
http://www.scribd.com/doc/24674634/The-Sixth-Problem
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You have to see this paper:
http://www.scribd.com/doc/24674634/The-Sixth-Problem
Of course the application of the monster group as a fractal antenna must have a cut off. so it is a finite element approximation. Similarly numerical representations of fractal cut off the iterations as well. More later.
Cheers LC
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Cheers LC
Greetings,
I just discovered this topic earlier today, but feel I have to weigh in now that I've read through the comments. I have found the discussion between Eckard and Lawrence interesting, also earlier comments by Georgina, so I'll try to weave that into my remarks about the initial topic. I find the work of Isham and Döring quite promising - from what little I know of it - though I...
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I just discovered this topic earlier today, but feel I have to weigh in now that I've read through the comments. I have found the discussion between Eckard and Lawrence interesting, also earlier comments by Georgina, so I'll try to weave that into my remarks about the initial topic. I find the work of Isham and Döring quite promising - from what little I know of it - though I...
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Dear Jonathan,
Thank you for extracting most important questions from in what I disagree with common theory.
Meanwhile I found Book 7 of Euclid's Elements at
http://aleph0.clarku.edu/~djoyce/Java/elements/bookVII/boo
kVII.html
Definition 1:
A unit is that by virtue of which each of the things that exist is called one.
Definition 2:
A number is a multitude composed of units.
Compare this with the wrong notion of numbers like points on the real line.
Compare it also with other definitions and notions of numbers, e.g.:
1) Pythagoras: Anything is number
2) Albert of Saxony: the number of points of a wooden bar
3) Kronecker: Natural numbers were made by god …
4) Dedekind: Numbers are creation of the human mind …
5) Peano’s axioms
6) Frege’s logical definition of numbers
7) Hilbert’s brutal number axioms I-IV
8) Weyl: continuous sauce between Euclidean numbers
9) Russell: number = class of all classes that are similar to a given class
10) v. Neumann: start with the empty set
I conclude: Numbers should no longer be considered like points but as already Euclid correctly understood like measures. The word incommensurable does not exclude that a real number can be a measure.
Regards,
Eckard
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Thank you for extracting most important questions from in what I disagree with common theory.
Meanwhile I found Book 7 of Euclid's Elements at
http://aleph0.clarku.edu/~djoyce/Java/elements/bookVII/boo
kVII.html
Definition 1:
A unit is that by virtue of which each of the things that exist is called one.
Definition 2:
A number is a multitude composed of units.
Compare this with the wrong notion of numbers like points on the real line.
Compare it also with other definitions and notions of numbers, e.g.:
1) Pythagoras: Anything is number
2) Albert of Saxony: the number of points of a wooden bar
3) Kronecker: Natural numbers were made by god …
4) Dedekind: Numbers are creation of the human mind …
5) Peano’s axioms
6) Frege’s logical definition of numbers
7) Hilbert’s brutal number axioms I-IV
8) Weyl: continuous sauce between Euclidean numbers
9) Russell: number = class of all classes that are similar to a given class
10) v. Neumann: start with the empty set
I conclude: Numbers should no longer be considered like points but as already Euclid correctly understood like measures. The word incommensurable does not exclude that a real number can be a measure.
Regards,
Eckard
Dear Jonathan,
Thank you for extracting in what I disagree with common theory.
Meanwhile I found Book 7 of Euclid's Elements at
http://aleph0.clarku.edu/~djoyce/Java/elements/bookVII/boo
kVII.html
Definition 1:
A unit is that by virtue of which each of the things that exist is called one.
Definition 2:
A number is a multitude composed of units.
Compare this with the wrong notion of numbers like points on the real line.
Compare it also with other definitions and notions of numbers, e.g.:
1) Pythagoras: Anything is number
2) Albert of Saxony: The numbers a wooden bar consists of
3) Kronecker: The natural numbers were made by god ...
4) Dedekind: Numbers are creations of the human mind ...
5) Frege's logical definition
6) Peano's axioms
7) Russell:... the class of all classes that are similar to a given class.
8) Hilbert's brutal axioms I-IV
9) Weyl's continuous sauce between Euclid's numbers
10) v. Neumann's start at the empty set
We should admit that numbers correspond to measures instead of points. Why not allow a measure to be strictly speaking incommensurable? Incommensurable does not mean not measurable at all.
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Thank you for extracting in what I disagree with common theory.
Meanwhile I found Book 7 of Euclid's Elements at
http://aleph0.clarku.edu/~djoyce/Java/elements/bookVII/boo
kVII.html
Definition 1:
A unit is that by virtue of which each of the things that exist is called one.
Definition 2:
A number is a multitude composed of units.
Compare this with the wrong notion of numbers like points on the real line.
Compare it also with other definitions and notions of numbers, e.g.:
1) Pythagoras: Anything is number
2) Albert of Saxony: The numbers a wooden bar consists of
3) Kronecker: The natural numbers were made by god ...
4) Dedekind: Numbers are creations of the human mind ...
5) Frege's logical definition
6) Peano's axioms
7) Russell:... the class of all classes that are similar to a given class.
8) Hilbert's brutal axioms I-IV
9) Weyl's continuous sauce between Euclid's numbers
10) v. Neumann's start at the empty set
We should admit that numbers correspond to measures instead of points. Why not allow a measure to be strictly speaking incommensurable? Incommensurable does not mean not measurable at all.
Well said Eckard,
I like the idea of starting from nothing, when talking about number. There is an increase in comprehension when children are taught to count 0,1,2.. instead of 1,2,3... But when I was that age, the latter formalism was the most common or conventional. If we take a constructivist view of this matter, it forces us to find a minimal set of assumptions which allow counting...
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I like the idea of starting from nothing, when talking about number. There is an increase in comprehension when children are taught to count 0,1,2.. instead of 1,2,3... But when I was that age, the latter formalism was the most common or conventional. If we take a constructivist view of this matter, it forces us to find a minimal set of assumptions which allow counting...
view entire post
For Georgina,
I had not been previously aware of Sweetser's work, but if I had been I would likely have given it a mention in my contest essay where I explicitly talk about energy-space as a universal dynamic worthy of consideration. And I love the fact that Quaternions offer significant improvement and simplification over the multi-vector approach - for various types of calculation within Engineering, Computer Graphics, and Physics. Nonetheless, I agree with Lawrence for the most part, in the assessment that Sweetser attempts to make Quaternions do things they don't do so well - making some of his theory a bit over complicated.
My thought is that we want to use the right Math for the job. And it is evident that Quaternion Math was slighted, once multi-vector analysis came on the scene, because there are things it really does better - or handles more easily and simply. I had the idea that Real numbers are good for Classical mechanics and conventional Gravity, Complex numbers are required for describing Wavelike phenomena and Electromagnetism, Quaternions are useful when we include Spin or Rotation - which should mean the Weak Force would be modeled well, and Octonions would then be required where there are still more degrees of freedom (or dimensions) as in the sub-atomic realm - and for the Strong force.
I will have to look more deeply into what Sweetser is saying, however, as I think there will be a few gems.
All the Best,
Jonathan
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I had not been previously aware of Sweetser's work, but if I had been I would likely have given it a mention in my contest essay where I explicitly talk about energy-space as a universal dynamic worthy of consideration. And I love the fact that Quaternions offer significant improvement and simplification over the multi-vector approach - for various types of calculation within Engineering, Computer Graphics, and Physics. Nonetheless, I agree with Lawrence for the most part, in the assessment that Sweetser attempts to make Quaternions do things they don't do so well - making some of his theory a bit over complicated.
My thought is that we want to use the right Math for the job. And it is evident that Quaternion Math was slighted, once multi-vector analysis came on the scene, because there are things it really does better - or handles more easily and simply. I had the idea that Real numbers are good for Classical mechanics and conventional Gravity, Complex numbers are required for describing Wavelike phenomena and Electromagnetism, Quaternions are useful when we include Spin or Rotation - which should mean the Weak Force would be modeled well, and Octonions would then be required where there are still more degrees of freedom (or dimensions) as in the sub-atomic realm - and for the Strong force.
I will have to look more deeply into what Sweetser is saying, however, as I think there will be a few gems.
All the Best,
Jonathan
Jonathan,
Do we really "need to revamp Math", i.e. to try and improve it and hide its faults? To me the verb revamp shows disapproval. It is definitely not just better but as I am convinced necessary to honestly reveal the faults like sets of points instead of measures, aleph_2, and all the other ridiculous nonsense. Set theory based topology is not even in position to separate a piece into two equal halves. No add on will ever remedy what is foundational wrong. I hope FQXi is the place where such attitude will be taken seriously. I do not have decades to wait and wait for the LHC and other effort if it is impossible to find SUSY.
What about my criticism of too much abstraction, I prefer the most adequate descriptions like e.g. measured data belonging to only positive elapsed time for many reasons. Well, it will hurt those who enjoy speculating almost without limit.
I will search for Rudy Rucker.
Problems with
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Do we really "need to revamp Math", i.e. to try and improve it and hide its faults? To me the verb revamp shows disapproval. It is definitely not just better but as I am convinced necessary to honestly reveal the faults like sets of points instead of measures, aleph_2, and all the other ridiculous nonsense. Set theory based topology is not even in position to separate a piece into two equal halves. No add on will ever remedy what is foundational wrong. I hope FQXi is the place where such attitude will be taken seriously. I do not have decades to wait and wait for the LHC and other effort if it is impossible to find SUSY.
What about my criticism of too much abstraction, I prefer the most adequate descriptions like e.g. measured data belonging to only positive elapsed time for many reasons. Well, it will hurt those who enjoy speculating almost without limit.
I will search for Rudy Rucker.
Problems with
Jonathan,
I looked at what I found about the man who is a descendant of Hegel. No, I am not interested in science fiction, set theory and other mysticism.
I prefer Archimedes, Euclid, Galilei, and Ren.
Eckard
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I looked at what I found about the man who is a descendant of Hegel. No, I am not interested in science fiction, set theory and other mysticism.
I prefer Archimedes, Euclid, Galilei, and Ren.
Eckard
Thank you Eckard,
For what it is worth, I greatly appreciated the link to Euclid's Elements, once I corrected the erroneous capital J. It is welcome that we can have this discussion, despite our difference of opinion on the value of Rucker's 'mysticism.'
As to whether Math can or should be revamped, I think there is at least a matter of its being somewhat inconsistent in usage of terms or definitions of concepts for various branches. But Isham's contention that Math needs to be replaced or re-worked entirely is suspect. It does seem like grandstanding.
Nonetheless; I continue to believe we will see a ripple effect across the whole of Mathematics, which results from insights coming out of category theory, topos theory, and other related work. If nothing else, it has promise for showing how the various kinds of Math are functionally related, and this itself is exciting!
All the Best,
Jonathan
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For what it is worth, I greatly appreciated the link to Euclid's Elements, once I corrected the erroneous capital J. It is welcome that we can have this discussion, despite our difference of opinion on the value of Rucker's 'mysticism.'
As to whether Math can or should be revamped, I think there is at least a matter of its being somewhat inconsistent in usage of terms or definitions of concepts for various branches. But Isham's contention that Math needs to be replaced or re-worked entirely is suspect. It does seem like grandstanding.
Nonetheless; I continue to believe we will see a ripple effect across the whole of Mathematics, which results from insights coming out of category theory, topos theory, and other related work. If nothing else, it has promise for showing how the various kinds of Math are functionally related, and this itself is exciting!
All the Best,
Jonathan
Dear Jonathan,
When I asked why not considering measures instead of points basic to numbers, Terence Tao reacted as follows: He deleted my request from his public discussion and pointed to a lesson on fundamentals by his colleague Jim Ralston and Folland's book on Real Analysis.
I did not yet ask someone else for the permission to make his reply to essentially the same question of mine public. However, his more proficient reply caused me to ask myself why ancient mathematics was not able to develop further after Euclid.
I still do not agree on that Euclid's unity-based notion of number is too restricted. I rather consider it the only correct while open for extension basis of mathematics. I am not an admirer of Euclid as a genius but I consider him someone who summarized the best knowledge of ancient mathematics. Remarkably, he did not include the Pythagorean idea of numbers like pattern pebbles to be recognized.
Later more.
Regards,
Eckard
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When I asked why not considering measures instead of points basic to numbers, Terence Tao reacted as follows: He deleted my request from his public discussion and pointed to a lesson on fundamentals by his colleague Jim Ralston and Folland's book on Real Analysis.
I did not yet ask someone else for the permission to make his reply to essentially the same question of mine public. However, his more proficient reply caused me to ask myself why ancient mathematics was not able to develop further after Euclid.
I still do not agree on that Euclid's unity-based notion of number is too restricted. I rather consider it the only correct while open for extension basis of mathematics. I am not an admirer of Euclid as a genius but I consider him someone who summarized the best knowledge of ancient mathematics. Remarkably, he did not include the Pythagorean idea of numbers like pattern pebbles to be recognized.
Later more.
Regards,
Eckard
Once again, Euclid was obviously correct when he based the notion number on the notion unit alias element.
Of course, we may extend his definition 2.
Of course, we may count points. However, the notion number line is somewhat misleading. According to a contribution in Mathforum@Drexel: Historia-Mathematica there is evidence for the use of number line about 5000 years ago in Egypt.
Everybody interprets the number X attributed to a point P on the number line as a measure alias relation to the measure one, namely the distance between a reference point zero and a particular point P in relation to the unit 1, regardless whether this bracket is a larger or a smaller measure than the reference unit 1.
Any measure is based on comparison. Any process of counting is based on its unit one as a reference measure. We may add in the sense of combining measures that refer to different units of reference, e.g. circumference c and diameter d of a circle, so called incommensurable measures, which were well known to Euclid. Since there is no restriction to the resolution of rational measures(= rational numbers), it is even possible, in principle, to decide whether any d-based number is smaller or larger than c.
Could we modify Dedekind's cut as a "bra-cut"? I think so. Let's ask for consequences. If we understood it as a measure alias bracket, a positive number would always be given as the limit from the left. The "number" zero has the measure zero, i.e., it quite naturally requires particular care instead of arbitrarily remedying bans.
Perhaps all my primary objections against present mathematics would disappear. Mathematics would not suffer but perhaps benefit if loosing sets that are no measures. Admittedly, replacing the understanding of numbers as sets of points by the original Euclidean understanding as measures would give rise to, let's say, scrutinize Cantor's naive beliefs and his belonging putative evidences for good. Having carefully read Fraenkel 1923, I consider them untenable anyway.
I also dealt with the somewhat related question why was nearly a seeming standstill in the development of mathematics during about a millennium between Euclid (325-275) and his translator Johannes Campanus of Novarra (1260). My conclusions are not yet reliable.
Eckard
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Of course, we may extend his definition 2.
Of course, we may count points. However, the notion number line is somewhat misleading. According to a contribution in Mathforum@Drexel: Historia-Mathematica there is evidence for the use of number line about 5000 years ago in Egypt.
Everybody interprets the number X attributed to a point P on the number line as a measure alias relation to the measure one, namely the distance between a reference point zero and a particular point P in relation to the unit 1, regardless whether this bracket is a larger or a smaller measure than the reference unit 1.
Any measure is based on comparison. Any process of counting is based on its unit one as a reference measure. We may add in the sense of combining measures that refer to different units of reference, e.g. circumference c and diameter d of a circle, so called incommensurable measures, which were well known to Euclid. Since there is no restriction to the resolution of rational measures(= rational numbers), it is even possible, in principle, to decide whether any d-based number is smaller or larger than c.
Could we modify Dedekind's cut as a "bra-cut"? I think so. Let's ask for consequences. If we understood it as a measure alias bracket, a positive number would always be given as the limit from the left. The "number" zero has the measure zero, i.e., it quite naturally requires particular care instead of arbitrarily remedying bans.
Perhaps all my primary objections against present mathematics would disappear. Mathematics would not suffer but perhaps benefit if loosing sets that are no measures. Admittedly, replacing the understanding of numbers as sets of points by the original Euclidean understanding as measures would give rise to, let's say, scrutinize Cantor's naive beliefs and his belonging putative evidences for good. Having carefully read Fraenkel 1923, I consider them untenable anyway.
I also dealt with the somewhat related question why was nearly a seeming standstill in the development of mathematics during about a millennium between Euclid (325-275) and his translator Johannes Campanus of Novarra (1260). My conclusions are not yet reliable.
Eckard
The timespan between -275 and +1260 is 1535 years.
A rapid development of mathematics begun not before stimulation in particular by Columbus, Copernicus, Kepler, and Galilei when printed books were available in the 17th century. Calculus has roots already published by Fermat in 1629. Calculus was not based on points. In his Geometria indivisibilus continuorum, Bonaventura Cavalieri wrote in 1635 correctly: Indivisible of lines are lines (not points).
In the middle ages, after the crusaders conquered Byzantium in 1204, contributions to mathematics and philosophy came often from bishops and from the first universities:
Campanus, see above
William of Ockham (1300-1349)
Johannes Buridan, U of Paris (Buridan's ass is to be found much later)
Albert of Saxony (1316-1390)
Nicole of Oresme (1323-1382):
graphic representation of functions, rotation of earth
Nicolaus Cusanus (1401-1464): endless universe without a center
Francoise Vieta (1540-1603): infinite product
Eckard
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A rapid development of mathematics begun not before stimulation in particular by Columbus, Copernicus, Kepler, and Galilei when printed books were available in the 17th century. Calculus has roots already published by Fermat in 1629. Calculus was not based on points. In his Geometria indivisibilus continuorum, Bonaventura Cavalieri wrote in 1635 correctly: Indivisible of lines are lines (not points).
In the middle ages, after the crusaders conquered Byzantium in 1204, contributions to mathematics and philosophy came often from bishops and from the first universities:
Campanus, see above
William of Ockham (1300-1349)
Johannes Buridan, U of Paris (Buridan's ass is to be found much later)
Albert of Saxony (1316-1390)
Nicole of Oresme (1323-1382):
graphic representation of functions, rotation of earth
Nicolaus Cusanus (1401-1464): endless universe without a center
Francoise Vieta (1540-1603): infinite product
Eckard
Hello again,
I thank you Eckard for making me think. What you are saying makes a lot of sense, but the focus seems to be on a historical progression. While I feel there is an evolution of Mathematics, and that discovering some higher Math depends upon the conceptual basis resulting from earlier work, it is amazing what has been lost and found again - over the years. Take Archimedes, for example, who is now known to have developed the rudiments of integral calculus. Then it wasn't re-discovered until many years later. And, as you seem to be implying, mathematicians have also picked up a lot of baggage along the way or made things over-complicated sometimes.
I argue that there is a more sensible way to relate to all of this, by the conceptual relationship of ideas resulting from their innate dependencies. That is; from a constructivist and process-theoretic view, all possible operations have a basis. Set theory is founded on the concept of interiority/exteriority. Without topological (hard) distinctions to define boundaries, the ideas of Set theory are not possible to construct, in their conventional form. But, if we generalize on the interiority/exteriority concept a bit - we get near/far or proximal/distal, which takes us into the realm of measurement and/or geometry. So there is a certain inter-relatedness to conceptual bases of ideas.
If I am not mistaken, the ideas found in category theory allow us to construct such mappings, or to discover the ways in which mathematical frameworks and concepts are related. If they are right, Isham and Döring will discover that a unique topos characterizes the view and structure of various 'positions' within the subject of Math, and can potentially show a kind of relativity between logical frameworks - upleveling the idea that there is only one correct logic, and replacing that with a concept more like the idea of a Multiverse in Physics.
Now; the fact the possibility exists doesn't mean they are right, but the territory they are exploring is probably useful regardless.
All the Best,
Jonathan
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I thank you Eckard for making me think. What you are saying makes a lot of sense, but the focus seems to be on a historical progression. While I feel there is an evolution of Mathematics, and that discovering some higher Math depends upon the conceptual basis resulting from earlier work, it is amazing what has been lost and found again - over the years. Take Archimedes, for example, who is now known to have developed the rudiments of integral calculus. Then it wasn't re-discovered until many years later. And, as you seem to be implying, mathematicians have also picked up a lot of baggage along the way or made things over-complicated sometimes.
I argue that there is a more sensible way to relate to all of this, by the conceptual relationship of ideas resulting from their innate dependencies. That is; from a constructivist and process-theoretic view, all possible operations have a basis. Set theory is founded on the concept of interiority/exteriority. Without topological (hard) distinctions to define boundaries, the ideas of Set theory are not possible to construct, in their conventional form. But, if we generalize on the interiority/exteriority concept a bit - we get near/far or proximal/distal, which takes us into the realm of measurement and/or geometry. So there is a certain inter-relatedness to conceptual bases of ideas.
If I am not mistaken, the ideas found in category theory allow us to construct such mappings, or to discover the ways in which mathematical frameworks and concepts are related. If they are right, Isham and Döring will discover that a unique topos characterizes the view and structure of various 'positions' within the subject of Math, and can potentially show a kind of relativity between logical frameworks - upleveling the idea that there is only one correct logic, and replacing that with a concept more like the idea of a Multiverse in Physics.
Now; the fact the possibility exists doesn't mean they are right, but the territory they are exploring is probably useful regardless.
All the Best,
Jonathan
Jonathan,
Let me try and say it as simply as possible: Any measure, no matter whether length, area, wight, or something else can be represented by an abstract quantum called number. As soon as we have chosen a unit measure one, we may imagine this number a multiple/part of a continuum that can be repeated and split without limitation.
The unit has two ideal ends: zero and one, the neutrals of addition and of division. Unfortunately, mathematics calls these ideal points elements. This led to a nonsensical distinction between open and closed measures. Recall Euclid and Peirce: A point has no parts, but each part of a continuum has parts. If we admit the ideal infinite accuracy of "real" numbers then there is no difference between open and closed. The measure x of concern extends from zero to the limit from the left: ]=(=[0, x[. An adjacent measure from x (limit from the right) to y (limit from the left) does not let a gap. The gap has the measure zero.
I prefer |sign(0)|=1, not 0.
In physics we learned where is one body there must not be an other one.
Points without the belonging reference to zero are no correct correlates of numbers.
Regards,
Eckard
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Let me try and say it as simply as possible: Any measure, no matter whether length, area, wight, or something else can be represented by an abstract quantum called number. As soon as we have chosen a unit measure one, we may imagine this number a multiple/part of a continuum that can be repeated and split without limitation.
The unit has two ideal ends: zero and one, the neutrals of addition and of division. Unfortunately, mathematics calls these ideal points elements. This led to a nonsensical distinction between open and closed measures. Recall Euclid and Peirce: A point has no parts, but each part of a continuum has parts. If we admit the ideal infinite accuracy of "real" numbers then there is no difference between open and closed. The measure x of concern extends from zero to the limit from the left: ]=(=[0, x[. An adjacent measure from x (limit from the right) to y (limit from the left) does not let a gap. The gap has the measure zero.
I prefer |sign(0)|=1, not 0.
In physics we learned where is one body there must not be an other one.
Points without the belonging reference to zero are no correct correlates of numbers.
Regards,
Eckard
Let me add what nonsense results from Dedekind's imprecise and intentional style of thinking: I found Wikipedia/complex numbers: "stress and rotate points". I would like calling the complex numbers, which are represented in complex plane, phasors or maybe vectors rather than points. Points cannot be manipulated at all because they do not have parts.
Eckard Blumschein
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Eckard Blumschein
Having read some books on history of mathematics, I feel obliged to blame already Gauss 1831 rather than his pupil Dedekind 1872 for giving rise to the still ongoing confusion between phasor and point with far reaching implications.
On the other hand, Gauss used to utter realistic views, e.g.: "We must humbly acknowledge that if number is only a product of our minds, space has a reality even outside our minds, to which we a priori cannot completely prescribe its laws."
Eckard
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On the other hand, Gauss used to utter realistic views, e.g.: "We must humbly acknowledge that if number is only a product of our minds, space has a reality even outside our minds, to which we a priori cannot completely prescribe its laws."
Eckard
I'm going to have to get better at take-aways before I can understand these comments.
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http://steve-jeffreystheoryofeverything.blogspot.com/2010/02
/jeffreys-theory-of-everything.html
----- Original Message -----
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/jeffreys-theory-of-everything.html
----- Original Message -----
Add sting theory so that you have both sides of the blackboard balanced as it was originally.
And then add the physics equations 1 ODD+ 1 EVEN= 2 ODD.
And 1 ODD+ 1 ODD= 2 EVEN and 2 ODD+ 2 EVEN= 4 EVEN.
By adding the string theory equations at random in a spreadhseet..
THIS IS A TURING MACHINE AND WORTHY OF THE PRIZE.
It can print continuusly forever until you say stop.
And can come up with a new E=MC^2 every day.
Steve
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And then add the physics equations 1 ODD+ 1 EVEN= 2 ODD.
And 1 ODD+ 1 ODD= 2 EVEN and 2 ODD+ 2 EVEN= 4 EVEN.
By adding the string theory equations at random in a spreadhseet..
THIS IS A TURING MACHINE AND WORTHY OF THE PRIZE.
It can print continuusly forever until you say stop.
And can come up with a new E=MC^2 every day.
Steve
Dear Isham and Doring,
Please see the mathematical equation representing our self and the universe we live in.
zero = i = infinity .
If 0 x 0 = 0 is true, then 0 / 0 = 0 is also true
If 0 x 1 = 0 is true, then 0 / 0 = 1 is also true
If 0 x 2 = 0 is true, then 0 / 0 = 2 is also true
If 0 x i = 0 is true, then 0 / 0 = i is also true
If 0 x ~ = 0 is true, then 0 / 0 = ~ is also true
It seems that mathematics, the universal language, is also pointing to the absolute truth that 0 = 1 = 2 = i = ~, where "i" can be any number from zero "0" to infinity "~". We have been looking at only first half of the if true statements in the relative world. As we can see it is not complete with out the then true statements whic are equally true. As all numbers are equal mathematically, so is all creation equal "absolutely".
This proves that 0 = i = ~ or in words "absolutely" nothing = "relatively" everything or everything is absolutely equal. Singularity is not only relative infinity but also absolute equality. There is only one singularity or infinity in the relativistic universe and there is only singularity or equality in the absolute universe and we are all in it.
Love,
Sridattadev.
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Please see the mathematical equation representing our self and the universe we live in.
zero = i = infinity .
If 0 x 0 = 0 is true, then 0 / 0 = 0 is also true
If 0 x 1 = 0 is true, then 0 / 0 = 1 is also true
If 0 x 2 = 0 is true, then 0 / 0 = 2 is also true
If 0 x i = 0 is true, then 0 / 0 = i is also true
If 0 x ~ = 0 is true, then 0 / 0 = ~ is also true
It seems that mathematics, the universal language, is also pointing to the absolute truth that 0 = 1 = 2 = i = ~, where "i" can be any number from zero "0" to infinity "~". We have been looking at only first half of the if true statements in the relative world. As we can see it is not complete with out the then true statements whic are equally true. As all numbers are equal mathematically, so is all creation equal "absolutely".
This proves that 0 = i = ~ or in words "absolutely" nothing = "relatively" everything or everything is absolutely equal. Singularity is not only relative infinity but also absolute equality. There is only one singularity or infinity in the relativistic universe and there is only singularity or equality in the absolute universe and we are all in it.
Love,
Sridattadev.
there is only 1 true number and it is 1
all other numbers are tricks to make the equations meet observations
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all other numbers are tricks to make the equations meet observations
CHRIS ISHAM'S VERY INTERESTING TOPOS-THEORY MATHEMATIZATION OF PHYSICS MAY HAVE ORIGINATED WHEN HE DROVE EDWARD SIEGEL[J. NONCRYSTALINE SOLIDS 40, 453(1980)], ONCE AT QUEEN MARY COLLEGE/UNIVERSITY OF LONDON AND INPE/CNPq, SAO JOSE DOS CAMPOS, BRAZIL, FROM IMPERIAL COLLEGE TO A CONFERENCE AT OXFORD CIRCA 1990 AND SEEMS TO FOLLOW ON EDWARD MACKINNON'S CALL FOR AND JACK AND IAN STEWART'S[THE...
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attachments: 2_FULL_PAPER_COMPLEX_QUANTUM-STATISTICS_IN_FRACTAL-DIMENSIONS.pdf, STRATIFIED_FUZZYICS__HoT_HIERARCHY-of-THINKING_from_ASKING_ORTHOGONAL_QUESTIONS_WHAT_WHEN__WHERE__WHY.doc
Tom,
Do you agree with Feynman? I know your great love for the mathematical...
“Physics is not mathematics, and mathematics is not physics. One helps the other. But in physics you have to have an understanding of the connection of words with the real world. It is necessary at the end to translate what you have figured out into English, into the world, into the blocks of copper and...
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Do you agree with Feynman? I know your great love for the mathematical...
“Physics is not mathematics, and mathematics is not physics. One helps the other. But in physics you have to have an understanding of the connection of words with the real world. It is necessary at the end to translate what you have figured out into English, into the world, into the blocks of copper and...
view entire post
Hi Akinbo,
Sure, I agree. I have said repeatedly that unless it can be shown that a mathematical model is independent of its physical result (i.e., that meaning is independent of language), there can be no demonstrated correspondence between the theory and the physics, and hence no rational -- no truly objective -- theory.
The main way in which special and general relativity differs from quantum mechanics is that relativity is written in mathematical language that allows exact solutions to the equations that incorporate the theory (this is true of all classical physics). The reason that quantum theory is subject to the many interpretations that we hear, is that it is not based in equations that incorporate the theory, it is based in the phenomenology of quantum mechanics -- as Feynman said, it's all about explaining "the experiment with the two holes."
Quantum theorists from Bohr, Feynman, Bell and to the present day, have struggled to develop from scratch a quantum foundations theory independent of the experiment; they have not been successful. The tendency among theorists now is to simply accept that nature does not obey a strict mathematical structure, that events are random at foundation.
Just to add, the reason that I don't often respond any more to Pentcho or to Peter J, is that they have a conviction that relativity can -- like quantum mechanics -- be "interpreted" starting with physical results rather than from what the theory says. Nothing could be further from the truth, and such discussions go nowhere. Rational theories are true by correspondence, not by interpretation.
Best,
Tom
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Sure, I agree. I have said repeatedly that unless it can be shown that a mathematical model is independent of its physical result (i.e., that meaning is independent of language), there can be no demonstrated correspondence between the theory and the physics, and hence no rational -- no truly objective -- theory.
The main way in which special and general relativity differs from quantum mechanics is that relativity is written in mathematical language that allows exact solutions to the equations that incorporate the theory (this is true of all classical physics). The reason that quantum theory is subject to the many interpretations that we hear, is that it is not based in equations that incorporate the theory, it is based in the phenomenology of quantum mechanics -- as Feynman said, it's all about explaining "the experiment with the two holes."
Quantum theorists from Bohr, Feynman, Bell and to the present day, have struggled to develop from scratch a quantum foundations theory independent of the experiment; they have not been successful. The tendency among theorists now is to simply accept that nature does not obey a strict mathematical structure, that events are random at foundation.
Just to add, the reason that I don't often respond any more to Pentcho or to Peter J, is that they have a conviction that relativity can -- like quantum mechanics -- be "interpreted" starting with physical results rather than from what the theory says. Nothing could be further from the truth, and such discussions go nowhere. Rational theories are true by correspondence, not by interpretation.
Best,
Tom
Thanks Tom. In your short reply 'truth' appears at least thrice. Now are there two truths, one mathematical and one physical? If there are two types of truth, which one are you looking for?
Mathematically, if something arrives later than it used to under a given condition, it can be interpreted that clocks run slower or time is dilated OR the speed of propagation is reduced/ affected. All are mathematically equivalent.
But in physics, they may not be all equivalent. Some of the mathematically correct choice of interpretation may lead to riddles, paradoxes and absurdities. In that case, the philosopher is likely to prefer the interpretation free of these. What is mathematically false, cannot be physically correct. What is physically false, cannot be mathematically correct.
Your point is however important. As someone said, you serve as a Quality Controller.
Regards,
Akinbo
*Regarding my statement in italics, a line having length and of zero width is physically impossible, same with a surface of zero thickness. Mathematicians should therefore take a look at this conflict.
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Mathematically, if something arrives later than it used to under a given condition, it can be interpreted that clocks run slower or time is dilated OR the speed of propagation is reduced/ affected. All are mathematically equivalent.
But in physics, they may not be all equivalent. Some of the mathematically correct choice of interpretation may lead to riddles, paradoxes and absurdities. In that case, the philosopher is likely to prefer the interpretation free of these. What is mathematically false, cannot be physically correct. What is physically false, cannot be mathematically correct.
Your point is however important. As someone said, you serve as a Quality Controller.
Regards,
Akinbo
*Regarding my statement in italics, a line having length and of zero width is physically impossible, same with a surface of zero thickness. Mathematicians should therefore take a look at this conflict.
I used the word truth only once, Akinbo, and not in the context you ascribe to it. My usage of "truly" means "really," and the use of "true statement" means "logically coherent." In the final case, my use of the word truth means "facts." I am not concerned with truth as you use it -- my concern is for rational, objective knowledge.
You write: "Mathematically, if something arrives later than it used to under a given condition, it can be interpreted that clocks run slower or time is dilated OR the speed of propagation is reduced/ affected. All are mathematically equivalent."
No. You are assuming an ideal clock that doesn't exist.
"What is mathematically false, cannot be physically correct. What is physically false, cannot be mathematically correct."
Nothing is ever physically false. There are only physically false interpretations of phenomena. That is why a mathematically complete theory that corresponds with the physical phenomena it incorporates and describes, is not open to interpretation.
"*Regarding my statement in italics, a line having length and of zero width is physically impossible, same with a surface of zero thickness. Mathematicians should therefore take a look at this conflict."
There is no conflict. The ideal line is a metric of 1 dimension, and the ideal surface is a plane of 2 dimensions. Your confusion results from thinking that because we live in 3 dimensions we cannot describe existence in any other terms. Language itself, however, is dimensionless.
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You write: "Mathematically, if something arrives later than it used to under a given condition, it can be interpreted that clocks run slower or time is dilated OR the speed of propagation is reduced/ affected. All are mathematically equivalent."
No. You are assuming an ideal clock that doesn't exist.
"What is mathematically false, cannot be physically correct. What is physically false, cannot be mathematically correct."
Nothing is ever physically false. There are only physically false interpretations of phenomena. That is why a mathematically complete theory that corresponds with the physical phenomena it incorporates and describes, is not open to interpretation.
"*Regarding my statement in italics, a line having length and of zero width is physically impossible, same with a surface of zero thickness. Mathematicians should therefore take a look at this conflict."
There is no conflict. The ideal line is a metric of 1 dimension, and the ideal surface is a plane of 2 dimensions. Your confusion results from thinking that because we live in 3 dimensions we cannot describe existence in any other terms. Language itself, however, is dimensionless.
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