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FQXi FORUM

February 28, 2021

CATEGORY:
Trick or Truth Essay Contest (2015)
[back]

TOPIC: The Deeper Roles of Mathematics in Physical Laws by Kevin H Knuth [refresh]

TOPIC: The Deeper Roles of Mathematics in Physical Laws by Kevin H Knuth [refresh]

Many have wondered how mathematics, which appears to be the result of both human creativity and human discovery, can possibly exhibit the degree of success and seemingly-universal applicability to quantifying the physical world as exemplified by the laws of physics. In this essay, I claim that much of the utility of mathematics arises from our choice of description of the physical world coupled with our desire to quantify it. This will be demonstrated in a practical sense by considering one of the most fundamental concepts of mathematics: additivity. This example will be used to show how many physical laws can be derived as constraint equations enforcing relevant symmetries in a sense that is far more fundamental than commonly appreciated.

Kevin Knuth is an Associate Professor in the Departments of Physics and Informatics at the University at Albany. He is Editor-in-Chief of the journal Entropy, and is the co-founder and President of the robotics company Autonomous Exploration Inc. He has 20 years of experience in applying Bayesian and maximum entropy methods to the design of machine learning algorithms for data analysis applied to the physical sciences. His current research interests include the foundations of physics, autonomous robotics, and the search for and characterization of extrasolar planets.

Kevin,

The two roles, symmetries and calculation, and your class examples help provide more clarity to your points. I wonder about your opinion of equations derived through educated guesses and/or trial-and-error. Such derivative functions must always be suspect in modeling and subjected to testing and peer review. Look at BICEP2. Your conclusive remarks seem to make this point when subscribing to a deeper understanding of math roles in physics regarding order, symmetries, and effectiveness.

Enjoyed your essay, feeling that your cautions are well-founded and your examples and descriptions were a great aid to clarity.

Jim

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The two roles, symmetries and calculation, and your class examples help provide more clarity to your points. I wonder about your opinion of equations derived through educated guesses and/or trial-and-error. Such derivative functions must always be suspect in modeling and subjected to testing and peer review. Look at BICEP2. Your conclusive remarks seem to make this point when subscribing to a deeper understanding of math roles in physics regarding order, symmetries, and effectiveness.

Enjoyed your essay, feeling that your cautions are well-founded and your examples and descriptions were a great aid to clarity.

Jim

report post as inappropriate

Dear Jim

Thank you for your kind and thoughtful comments. I am very glad that you found my examples and descriptions to be a "great aid to clarity." I felt that it was risky to include such detail in an essay. However, without concrete examples, I feared that the ideas presented in the essay would be perceived as mere speculation rather than something that was born of insight and could be backed up with meaningful and relevant examples.

As for equations arrived at through educated guesses or trial-and-error, this is difficult. In any initial exploration, this is really all that one has. Much of physics was developed this way. But once one better understands the interconnections, one can begin to put the pieces of the puzzle together with greater insight. To some degree, this is where we are now in the history of physics. But it is unclear what aspects of physics are derivable from deeper principles and which are contingent (accidental or perhaps even dictated by decree). The distribution of the cosmic microwave background (which you note) is very likely contingent, just as is the current relative positions of South America and Africa. While some things are just not derivable, some are.

Thank you again.

Kevin

Thank you for your kind and thoughtful comments. I am very glad that you found my examples and descriptions to be a "great aid to clarity." I felt that it was risky to include such detail in an essay. However, without concrete examples, I feared that the ideas presented in the essay would be perceived as mere speculation rather than something that was born of insight and could be backed up with meaningful and relevant examples.

As for equations arrived at through educated guesses or trial-and-error, this is difficult. In any initial exploration, this is really all that one has. Much of physics was developed this way. But once one better understands the interconnections, one can begin to put the pieces of the puzzle together with greater insight. To some degree, this is where we are now in the history of physics. But it is unclear what aspects of physics are derivable from deeper principles and which are contingent (accidental or perhaps even dictated by decree). The distribution of the cosmic microwave background (which you note) is very likely contingent, just as is the current relative positions of South America and Africa. While some things are just not derivable, some are.

Thank you again.

Kevin

Hi Kevin,

As I said in my e-mail to you, this is one of the best things you've written. I think it is a fantastic essay and captures some really deep insights.

Cheers,

Ian

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As I said in my e-mail to you, this is one of the best things you've written. I think it is a fantastic essay and captures some really deep insights.

Cheers,

Ian

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Dear Ian

Thank you for your high praise. I really appreciate it.

As I noted in my email reply to you, these are questions that I have had for a long time (many of us have) and regarding the specific problem of additivity, I was pleased to have answered it myself (back in 2003) when studying various derivations of probability theory related to Cox and Jaynes. It became clear to me...

view entire post

Thank you for your high praise. I really appreciate it.

As I noted in my email reply to you, these are questions that I have had for a long time (many of us have) and regarding the specific problem of additivity, I was pleased to have answered it myself (back in 2003) when studying various derivations of probability theory related to Cox and Jaynes. It became clear to me...

view entire post

Dear Kevin,

You have a fascinating and brilliant essay deriving the `ubiquity of addtivity' from the underlying symmetries of commutativity and associativity, and from ordering. It was a pleasure reading the essay and we will need to read it again for better grasp.

We are not clear though, about what stance you finally adopt with regard to the relation between symmetries and physical laws of motion. We are accustomed to relating symmetries to conservation laws, but perhaps not in full generality to the equations of motion and the force laws themselves [say Newton's second law, and his inverse square law of gravitation]. Would you say that eventually we must understand force laws also from symmetries: where does one draw the line - how much from symmetry, and how much from experiment? For instance, it is not obvious how symmetry can dictate that equations of motion be first order or second order in time.

Also, are you suggesting that in general one should seek to derive mathematical axioms [say those of Euclidean geometry] from symmetries?

Questions apart, it was fun reading your very well-written essay.

Regards,

Anshu, Tejinder

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You have a fascinating and brilliant essay deriving the `ubiquity of addtivity' from the underlying symmetries of commutativity and associativity, and from ordering. It was a pleasure reading the essay and we will need to read it again for better grasp.

We are not clear though, about what stance you finally adopt with regard to the relation between symmetries and physical laws of motion. We are accustomed to relating symmetries to conservation laws, but perhaps not in full generality to the equations of motion and the force laws themselves [say Newton's second law, and his inverse square law of gravitation]. Would you say that eventually we must understand force laws also from symmetries: where does one draw the line - how much from symmetry, and how much from experiment? For instance, it is not obvious how symmetry can dictate that equations of motion be first order or second order in time.

Also, are you suggesting that in general one should seek to derive mathematical axioms [say those of Euclidean geometry] from symmetries?

Questions apart, it was fun reading your very well-written essay.

Regards,

Anshu, Tejinder

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Dear Tejinder

Thank you for your very generous comments and questions. I appreciate the opportunity to answer them.

There is much more that I would have liked to have said about the role that these subtle symmetries play in physical law. Your focus on the physical laws of motion is a very interesting example, and you are right that I did not speak specifically to that. I decided...

view entire post

Thank you for your very generous comments and questions. I appreciate the opportunity to answer them.

There is much more that I would have liked to have said about the role that these subtle symmetries play in physical law. Your focus on the physical laws of motion is a very interesting example, and you are right that I did not speak specifically to that. I decided...

view entire post

Thanks Kevin, for your clear and precise reply, which I agree with. I do believe this is path breaking work and I hope to make time to read and understand the papers you have referred to.

This is probably already a direction contained in some of your works: it will be great to understand how quantum mechanics might originate from symmetries, in the sense in which you approach the problem. I believe that a successful application of the symmetry idea ought to yield not quantum theory, but a more general theory to which quantum mechanics is an approximation, because by itself quantum theory suffers from various shortcomings including the measurement problem. [I am reminded of Stephen Adler's work on Trace Dynamics, described in his book `Quantum theory as an emergent phenomenon', where quantum theory is approximately emergent as an equilibrium statistical thermodynamics of an underlying mechanical theory of non-commuting matrices. Fluctuations about equilibrium modify quantum theory in a desirable way]. I will be much interested in knowing your views on the symmetries - quantum theory connection.

Regards,

Tejinder

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This is probably already a direction contained in some of your works: it will be great to understand how quantum mechanics might originate from symmetries, in the sense in which you approach the problem. I believe that a successful application of the symmetry idea ought to yield not quantum theory, but a more general theory to which quantum mechanics is an approximation, because by itself quantum theory suffers from various shortcomings including the measurement problem. [I am reminded of Stephen Adler's work on Trace Dynamics, described in his book `Quantum theory as an emergent phenomenon', where quantum theory is approximately emergent as an equilibrium statistical thermodynamics of an underlying mechanical theory of non-commuting matrices. Fluctuations about equilibrium modify quantum theory in a desirable way]. I will be much interested in knowing your views on the symmetries - quantum theory connection.

Regards,

Tejinder

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Dear Kevin Knuth,

Thanks for your essay heavily focused on symmetry. May I recommend another that you might like: Aldo Filomeno's essay focused on gauge symmetry.

You look at symmetry and additivity. I begin with logic structures and quickly lead to addition structures, which leads to the concept of distance and identity.

And yes, it was somewhat surprising that one can derive the Feynman rules for combining quantum amplitudes by relying on symmetries. One learns a lot in these FQXi contests!

Early in your essay you state that "*as a physicist interested in the foundations, assumptions cause me concern*."

They cause me concern too, and I have been concerned for example that Bell's assumption of a constant field leads to a null experimental result, an obvious contradiction before one even gets into any other analysis. My conclusion is that Bell relied on an oversimplified assumption. The interesting thing is that a local model based on a more realistic assumption actually delivers the quantum correlation unless one throws away measurement information. I invite you to read my essay and I welcome your feedback.

Best,

Edwin Eugene Klingman

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Thanks for your essay heavily focused on symmetry. May I recommend another that you might like: Aldo Filomeno's essay focused on gauge symmetry.

You look at symmetry and additivity. I begin with logic structures and quickly lead to addition structures, which leads to the concept of distance and identity.

And yes, it was somewhat surprising that one can derive the Feynman rules for combining quantum amplitudes by relying on symmetries. One learns a lot in these FQXi contests!

Early in your essay you state that "

They cause me concern too, and I have been concerned for example that Bell's assumption of a constant field leads to a null experimental result, an obvious contradiction before one even gets into any other analysis. My conclusion is that Bell relied on an oversimplified assumption. The interesting thing is that a local model based on a more realistic assumption actually delivers the quantum correlation unless one throws away measurement information. I invite you to read my essay and I welcome your feedback.

Best,

Edwin Eugene Klingman

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And how would you classify the symmetry of activities of a regular tetrahedron with an inscribed sphere? Seems to me to be a "duality" with connections. see topic #2408.

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Dear Kevin,

Your essay is really fantastic and inspiring. You ask very important questions that I have not asked myself. Below I try to comment main aspects of your essay.

The aspect related to quantification.

I start from your example: “…2+1 defines 3 […] this is an axiom of measure theory, which means that it is assumed.” Obviously we can easily find examples where...

view entire post

Your essay is really fantastic and inspiring. You ask very important questions that I have not asked myself. Below I try to comment main aspects of your essay.

The aspect related to quantification.

I start from your example: “…2+1 defines 3 […] this is an axiom of measure theory, which means that it is assumed.” Obviously we can easily find examples where...

view entire post

report post as inappropriate

Dear Jacek

Thank you for your questions and comments.

Your first question relates to closure, which I addressed in the essay. In my example of combining a cup of water, a napkin and a lighted match, the order in which you combine them matters if you choose a description where you count the things: water, napkin, match. However, on combination, these things are not closed. You get...

view entire post

Thank you for your questions and comments.

Your first question relates to closure, which I addressed in the essay. In my example of combining a cup of water, a napkin and a lighted match, the order in which you combine them matters if you choose a description where you count the things: water, napkin, match. However, on combination, these things are not closed. You get...

view entire post

Dear Kevin,

Thank you for your clear and well thought out essay. I liked your perspective, "*..there are two distinct aspects to the role that mathematics plays. The first aspect is related to ordering and associated symmetries, and the second aspect is related to quantification and the equations that enable one to quantify things.*" and I think it provides a basis for considerable thought and discussion.

Your suggestion that we "*step back and release ourselves from familiarity and consider order and symmetry to be fundamental, then we see these equations as rules to constrain our artificial quantifications in accordance with the underlying order and symmetries of our chosen descriptions.*" highlights the importance of underlying order and symmetries. I especially agree with the comment "*quantum mechanics is not a generalized probability theory any more than information theory, geometry, and number theory are generalized probability theories*"

I have used groups and symmetries as a way of visualizing the fundamental particles on the standard model in my essay here. I feel this in many ways is an example of your first aspect of order and symmetries. The second aspect of quantification and equations confirm the proper ordering and symmetries and allow for the predictive power so important to physics.

All the best regarding your essay, it made an enjoyable and thoughtful read.

Regards,

Ed Unverricht

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Thank you for your clear and well thought out essay. I liked your perspective, "

Your suggestion that we "

I have used groups and symmetries as a way of visualizing the fundamental particles on the standard model in my essay here. I feel this in many ways is an example of your first aspect of order and symmetries. The second aspect of quantification and equations confirm the proper ordering and symmetries and allow for the predictive power so important to physics.

All the best regarding your essay, it made an enjoyable and thoughtful read.

Regards,

Ed Unverricht

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Dear Kevin,

You have focused on how mathematics, which appears to be the result of both human creativity and human discovery, can possibly exhibit the degree of success and seemingly-universal applicability to quantifying the physical world as exemplified by the laws of physics.

You have also taken your thought in line with Hamming.

You are trying to explain which one is more...

view entire post

You have focused on how mathematics, which appears to be the result of both human creativity and human discovery, can possibly exhibit the degree of success and seemingly-universal applicability to quantifying the physical world as exemplified by the laws of physics.

You have also taken your thought in line with Hamming.

You are trying to explain which one is more...

view entire post

report post as inappropriate

Dear Pankaj

Thank you for your comments.

I am sorry if you misunderstood my intention.

I did not try to argue which was more fundamental: physics or mathematics. Physics is a description of the natural world. Mathematics is a language that describes things in two very different ways: relationships such as symmetries and quantification with numbers.

I worked to show, explicitly, that the concept of consistent quantification constrained by symmetries leads to specific mathematical laws. (in the example given, it was additivity)

You seem to be claiming that quantum mechanics may not be governed by symmetries and thus not quantifiable via mathematics. Don't give up too fast! We have shown that the Feynman rules for manipulating quantum amplitudes is indeed derivable from fundamental symmetries:

Origin of complex quantum amplitudes and Feynman’s rules

Philip Goyal, Kevin H. Knuth, and John Skilling

Phys. Rev. A 81, 022109 – Published 11 February 2010

http://journals.aps.org/pra/abstract/10.1103/PhysRevA.81

.022109

and

Quantum Theory and Probability Theory: Their Relationship and Origin in Symmetry

Philip Goyal, Kevin H. Knuth

Symmetry 2011, 3(2), 171-206; doi:10.3390/sym3020171

http://www.mdpi.com/2073-8994/3/2/171

T

hese are proofs---not arguments.

Indeed QM *can* be derived and from consistent quantification constrained by symmetries!

Thank you for your comments.

I am sorry if you misunderstood my intention.

I did not try to argue which was more fundamental: physics or mathematics. Physics is a description of the natural world. Mathematics is a language that describes things in two very different ways: relationships such as symmetries and quantification with numbers.

I worked to show, explicitly, that the concept of consistent quantification constrained by symmetries leads to specific mathematical laws. (in the example given, it was additivity)

You seem to be claiming that quantum mechanics may not be governed by symmetries and thus not quantifiable via mathematics. Don't give up too fast! We have shown that the Feynman rules for manipulating quantum amplitudes is indeed derivable from fundamental symmetries:

Origin of complex quantum amplitudes and Feynman’s rules

Philip Goyal, Kevin H. Knuth, and John Skilling

Phys. Rev. A 81, 022109 – Published 11 February 2010

http://journals.aps.org/pra/abstract/10.1103/PhysRevA.81

.022109

and

Quantum Theory and Probability Theory: Their Relationship and Origin in Symmetry

Philip Goyal, Kevin H. Knuth

Symmetry 2011, 3(2), 171-206; doi:10.3390/sym3020171

http://www.mdpi.com/2073-8994/3/2/171

T

hese are proofs---not arguments.

Indeed QM *can* be derived and from consistent quantification constrained by symmetries!

Dear Kevin H. Knuth,

I found your essay interesting. Your remark about addition is thought provoking. Your reductionist approach, I think, coincides with that of mine. I feel that the only law in mathematics is the law of addition; it is not an axiom, but a rule, the eternal rule that cannot be violated even by an omnipotent creator. The rest of the mathematical structures are axiomatic, where axioms can be regarded as properties that we arbitrarily assign. I have been thinking in that direction for some time, but not reached a final conclusion.

In my opinion there are no physical laws; the physical world has no laws of its own. It has only some basic properties. The rules applicable to it are that of mathematics. Or simply, 'the properties are physical' and 'the laws are mathematical'. Why is it so? The changes in the physical world happen by way of motion, the rules of which are mathematical. So a changing world follows mathematical rules. I request you to go through my essay: A physicalist interpretation of the relation between Physics and Mathematics

If it seems interesting, kindly visit my site: finitenesstheory.com, where I propose a theory of everything based on fundamental properties and consequent emergent strucures.

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I found your essay interesting. Your remark about addition is thought provoking. Your reductionist approach, I think, coincides with that of mine. I feel that the only law in mathematics is the law of addition; it is not an axiom, but a rule, the eternal rule that cannot be violated even by an omnipotent creator. The rest of the mathematical structures are axiomatic, where axioms can be regarded as properties that we arbitrarily assign. I have been thinking in that direction for some time, but not reached a final conclusion.

In my opinion there are no physical laws; the physical world has no laws of its own. It has only some basic properties. The rules applicable to it are that of mathematics. Or simply, 'the properties are physical' and 'the laws are mathematical'. Why is it so? The changes in the physical world happen by way of motion, the rules of which are mathematical. So a changing world follows mathematical rules. I request you to go through my essay: A physicalist interpretation of the relation between Physics and Mathematics

If it seems interesting, kindly visit my site: finitenesstheory.com, where I propose a theory of everything based on fundamental properties and consequent emergent strucures.

report post as inappropriate

Dear Jose

Thank you for your comments.

You state that

"Your reductionist approach, I think, coincides with that of mine. I feel that the only law in mathematics is the law of addition; it is not an axiom, but a rule, the eternal rule that cannot be violated even by an omnipotent creator."

I agree and disagree.

I do not think that addition is the only law. Additivity (or any invertible transform of additivity) arises from ordering, closure, commutativity and associativity. I agree that even an omnipotent being could break it, unless of course, he or she or it broke one of those conditions. Then it would be broken. But one could not keep those conditions and break additivity.

I believe that there are fundamental symmetries and relationships and that these place strong constraints on any attempt to consistently quantify phenomena. These constraints give rise to constraint equations (enforcing these basic properties) which we interpret as physical laws.

Thank you for your comments.

You state that

"Your reductionist approach, I think, coincides with that of mine. I feel that the only law in mathematics is the law of addition; it is not an axiom, but a rule, the eternal rule that cannot be violated even by an omnipotent creator."

I agree and disagree.

I do not think that addition is the only law. Additivity (or any invertible transform of additivity) arises from ordering, closure, commutativity and associativity. I agree that even an omnipotent being could break it, unless of course, he or she or it broke one of those conditions. Then it would be broken. But one could not keep those conditions and break additivity.

I believe that there are fundamental symmetries and relationships and that these place strong constraints on any attempt to consistently quantify phenomena. These constraints give rise to constraint equations (enforcing these basic properties) which we interpret as physical laws.

Dear Kevin,

It’s true that Mathematics has deeper roles in Physical Laws, while we consider that Symmetry and Asymmetry are the basic kind of abstracts for both Physical and Mathematical entities, whereas quantization limits the value with 1 and returns 0 on absence of a quanta, while not an abstract.

Thus, Mysterious connection between Physics and Mathematics begins with Three-dimensional Structure formation on the Universe, by the transformation from pre-existence.

With best wishes,

Jayakar

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It’s true that Mathematics has deeper roles in Physical Laws, while we consider that Symmetry and Asymmetry are the basic kind of abstracts for both Physical and Mathematical entities, whereas quantization limits the value with 1 and returns 0 on absence of a quanta, while not an abstract.

Thus, Mysterious connection between Physics and Mathematics begins with Three-dimensional Structure formation on the Universe, by the transformation from pre-existence.

With best wishes,

Jayakar

report post as inappropriate

Dear Jayakar

Thank you for your comments.

I am not sure what you mean when you say that

"Symmetry and Asymmetry are the basic kind of abstracts for both Physical and Mathematical entities"

Physical things sometimes have symmetries. This simply means that one can make a correspondence between some aspect of one physical system and some aspect of a kind of transform of that physical system.

You also note that

"quantization limits the value with 1 and returns 0 on absence of a quanta"

This is an example of a quantification scheme mapping a 0 to one state and a 1 to another.

Basically, what I argue (and explicitly demonstrate) in my essay is that the act of quantification (assigning numbers to states or things or systems or aspects of systems) is often constrained by symmetries and that these constraints result in constraint equations, which can be interpreted as physical laws.

I hope that this helps explain my stance.

Thank you for your comments.

I am not sure what you mean when you say that

"Symmetry and Asymmetry are the basic kind of abstracts for both Physical and Mathematical entities"

Physical things sometimes have symmetries. This simply means that one can make a correspondence between some aspect of one physical system and some aspect of a kind of transform of that physical system.

You also note that

"quantization limits the value with 1 and returns 0 on absence of a quanta"

This is an example of a quantification scheme mapping a 0 to one state and a 1 to another.

Basically, what I argue (and explicitly demonstrate) in my essay is that the act of quantification (assigning numbers to states or things or systems or aspects of systems) is often constrained by symmetries and that these constraints result in constraint equations, which can be interpreted as physical laws.

I hope that this helps explain my stance.

Hi Kevin,

What happens if there is a union ( or battle ) of Knuth and t'Hooft...

From the ground up, such as cellular automata do not seem to present the idea of "selection criteria" so that there is therein an "observer" to make choices as you might describe.

Either something has to "evolve" in that soup to "see" what you're saying, or it has to be totally inherent from the get-go, by definition.

Would you see his efforts as yet another expression of a "measure" still standing on these same principles ? Or as something yet more fundamental and there is some "gap" between his bits flipping and the ability for set theory stuff to exist at all - "what observer/criteria" - in that universe... when do the eyes happen... How...

Did that make sense ? This is a hard one to frame, I've never asked it before.

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What happens if there is a union ( or battle ) of Knuth and t'Hooft...

From the ground up, such as cellular automata do not seem to present the idea of "selection criteria" so that there is therein an "observer" to make choices as you might describe.

Either something has to "evolve" in that soup to "see" what you're saying, or it has to be totally inherent from the get-go, by definition.

Would you see his efforts as yet another expression of a "measure" still standing on these same principles ? Or as something yet more fundamental and there is some "gap" between his bits flipping and the ability for set theory stuff to exist at all - "what observer/criteria" - in that universe... when do the eyes happen... How...

Did that make sense ? This is a hard one to frame, I've never asked it before.

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Dear Allen

Thank you for your comments!

You write:

"What happens if there is a union ( or battle ) of Knuth and t'Hooft..."

I am not sure what a union would look like, but if it were a battle, he would probably simply beat me silly. ;)

I am sorry that your question is not making much sense to me. However, I get the feeling that you are asking something rather deep that has not quite settled in your mind.

Perhaps you can try to rephrase it?

I would be very interested in understanding what you are considering.

My guess is that you are worried about my reliance on an observer that actively quantifies aspects of the universe, and reconciling that with a view of the universe as some kind of cellular automaton that simply marches forward according to some set of "bit flipping" rules.

Is something like this the questions?

Thank you for your comments!

You write:

"What happens if there is a union ( or battle ) of Knuth and t'Hooft..."

I am not sure what a union would look like, but if it were a battle, he would probably simply beat me silly. ;)

I am sorry that your question is not making much sense to me. However, I get the feeling that you are asking something rather deep that has not quite settled in your mind.

Perhaps you can try to rephrase it?

I would be very interested in understanding what you are considering.

My guess is that you are worried about my reliance on an observer that actively quantifies aspects of the universe, and reconciling that with a view of the universe as some kind of cellular automaton that simply marches forward according to some set of "bit flipping" rules.

Is something like this the questions?

Kevin, we meet again. I do like your essay, however, I am not certain I understand what you mean by this: “One can now see that ordering, commutativity and associativity underlie a class of universal phenomena. I will next discuss how this leads to mathematics which gives rise to physical laws with a degree of universal applicability.”

Do these underlie physics or are they simply a means of encoding what “Laws Describe”? An empirically familiar regularity is that some collections of objects (pencils, pennies, rocks, etc.) are invariant under regrouping: a law. What is meant is that regrouping creates things that are the same but not the same: the law of the included middle. They are different in that the grouping is different but quantitatively they are the same. Consider these three principles: 1/ a + b = b + a, 2/ (a +b) + c = a + (b + c) & 3/ a(b + c) = ab + ac. 2 & 3 state regrouping principles. However 1 is a definition of identity as orientation is not encoded symbolically. In other words it doesn’t make any difference which group is label a or b. A mathematical interpretation of these symbols is entirely different as the intent is to reason about symbol construction with a serial ordered set of symbols; a + b = c.

It seems to me that set theory is when objects may not be invariant under regrouping but there is a way of determining the variance. Thus I found it odd that you would start out with set theory. Am I missing something, as it seems to me that you started out assuming the law: invariance under regrouping, with your reference to the union of sets. Have you not assumed that arbitrary members are not lost upon being joined?

Cheers

Rob MacDuff

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Do these underlie physics or are they simply a means of encoding what “Laws Describe”? An empirically familiar regularity is that some collections of objects (pencils, pennies, rocks, etc.) are invariant under regrouping: a law. What is meant is that regrouping creates things that are the same but not the same: the law of the included middle. They are different in that the grouping is different but quantitatively they are the same. Consider these three principles: 1/ a + b = b + a, 2/ (a +b) + c = a + (b + c) & 3/ a(b + c) = ab + ac. 2 & 3 state regrouping principles. However 1 is a definition of identity as orientation is not encoded symbolically. In other words it doesn’t make any difference which group is label a or b. A mathematical interpretation of these symbols is entirely different as the intent is to reason about symbol construction with a serial ordered set of symbols; a + b = c.

It seems to me that set theory is when objects may not be invariant under regrouping but there is a way of determining the variance. Thus I found it odd that you would start out with set theory. Am I missing something, as it seems to me that you started out assuming the law: invariance under regrouping, with your reference to the union of sets. Have you not assumed that arbitrary members are not lost upon being joined?

Cheers

Rob MacDuff

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Dear Rob

Thank you for your comments and questions.

I am not completely sure that I understand what you find confusing. Somehow, we must be talking past one another.

Let me give this a try...

I do not really talk about regrouping, which I feel has connotations of grouping objects to form different objects.

When I discussed combining objects, I spent a little time discussing closure, which is the idea that on combining sets of objects, the objects do not change or combine or annihilate or anything like that.

I do talk about sets, but one only needs to consider that we are considering sets of things that we want to describe quantitatively. That is, we will map sets of things to real numbers.

The point is that if you have sets of things that you want to quantify, and these things obey closure on combination, they can be ordered in some way, and they obey both commutativity and associativity on combination, then the only consistent means that one can conceive of to quantify such sets of objects is via some invertible transform of additivity. This is why adding the quantities of things when you combine them works (as long as you have closure, they can be ordered and as long as the combining rule obeys commutativity and associativity).

Does this help?

Kevin

Thank you for your comments and questions.

I am not completely sure that I understand what you find confusing. Somehow, we must be talking past one another.

Let me give this a try...

I do not really talk about regrouping, which I feel has connotations of grouping objects to form different objects.

When I discussed combining objects, I spent a little time discussing closure, which is the idea that on combining sets of objects, the objects do not change or combine or annihilate or anything like that.

I do talk about sets, but one only needs to consider that we are considering sets of things that we want to describe quantitatively. That is, we will map sets of things to real numbers.

The point is that if you have sets of things that you want to quantify, and these things obey closure on combination, they can be ordered in some way, and they obey both commutativity and associativity on combination, then the only consistent means that one can conceive of to quantify such sets of objects is via some invertible transform of additivity. This is why adding the quantities of things when you combine them works (as long as you have closure, they can be ordered and as long as the combining rule obeys commutativity and associativity).

Does this help?

Kevin

Dear Kevin Knuth

Your presentation is really clear and fundamental. It seems to me, that people, which try to answers some questions from childhood, they present these answers more clearly.

I am interested, how the principle of your derivation si connected with derivation of Russel, that 1+1=2? His derivation is a lot of longer.

I think that physical world is more precisely a consequnce of informatics, that of mathematics, because it can be simulated by a binary computer. Do you think that this binary computer is enough to say that number are consequence of logic?

I think that mathematics and logic are more consequence of physics than oppositely, (naturalism, Smolin). Mathematics is a language where physics is described in short as Torsten Asselmeyer-Maluga in FQXi: '' Without abstraction, our species with a limited brain is unable to reflect the world.'' As a consquence I suppose that fundamental physics can be described still more abstractly, thus quantum gravity should be desribed on a t-shrirt.

Do you think that counting (of sheeps, rocks, apples in essay of Leifer) arises from physical world and is a consequence of physical world, not Platonic world?

My essay

Best regards,

Janko Kokosar

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Your presentation is really clear and fundamental. It seems to me, that people, which try to answers some questions from childhood, they present these answers more clearly.

I am interested, how the principle of your derivation si connected with derivation of Russel, that 1+1=2? His derivation is a lot of longer.

I think that physical world is more precisely a consequnce of informatics, that of mathematics, because it can be simulated by a binary computer. Do you think that this binary computer is enough to say that number are consequence of logic?

I think that mathematics and logic are more consequence of physics than oppositely, (naturalism, Smolin). Mathematics is a language where physics is described in short as Torsten Asselmeyer-Maluga in FQXi: '' Without abstraction, our species with a limited brain is unable to reflect the world.'' As a consquence I suppose that fundamental physics can be described still more abstractly, thus quantum gravity should be desribed on a t-shrirt.

Do you think that counting (of sheeps, rocks, apples in essay of Leifer) arises from physical world and is a consequence of physical world, not Platonic world?

My essay

Best regards,

Janko Kokosar

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Dear Kevin.

It was a real pleasure to read your essay, and I fully agree with your main idea we can summarize in the following terms: (i) There are mathematical edifices being interpretations of set theory specified by group theory, knowing that group theory by definition covers all aspects of symmetry, and (ii) physical laws stricto sensu are experience oriented interpretations of...

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It was a real pleasure to read your essay, and I fully agree with your main idea we can summarize in the following terms: (i) There are mathematical edifices being interpretations of set theory specified by group theory, knowing that group theory by definition covers all aspects of symmetry, and (ii) physical laws stricto sensu are experience oriented interpretations of...

view entire post

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Dear Kevin,

I like how you identify ordering, the ubiquity of additivity, and symmetry, as fundamental concepts allowing us to quantify and to abstract mathematical structures from the physical world. I think your essay is compelling and well written, and sheds light on the unifying vision driving your research work from other of your articles.

Best wishes,

Cristi Stoica

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I like how you identify ordering, the ubiquity of additivity, and symmetry, as fundamental concepts allowing us to quantify and to abstract mathematical structures from the physical world. I think your essay is compelling and well written, and sheds light on the unifying vision driving your research work from other of your articles.

Best wishes,

Cristi Stoica

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Dear Kevin,

Your question

“To what degree are physical laws derivable and to what degree are they accidental, contingent or decreed by Mother Nature?”

inclines me to ask in return, what do you mean by "Mother Nature", who is able to decree the laws?

I hope you would agree that the entire set of laws cannot be "derivable", since when all of them are taken together, there are no laws anymore to derive anything from.

As far as the degree of chance, our essay refutes the option of them being*fully* accidental. That is because their selection from an ensemble of all mathematically possible sets of laws (=full-blown multiverse of Tegmark) by the weak anthropic principle is insufficient.

Regards,

Alexey Burov.

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Your question

“To what degree are physical laws derivable and to what degree are they accidental, contingent or decreed by Mother Nature?”

inclines me to ask in return, what do you mean by "Mother Nature", who is able to decree the laws?

I hope you would agree that the entire set of laws cannot be "derivable", since when all of them are taken together, there are no laws anymore to derive anything from.

As far as the degree of chance, our essay refutes the option of them being

Regards,

Alexey Burov.

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Thank you for an interesting essay.

Your fundamental question about additivity is nice.

Martin Gardner wrote “. . . if two dinosaurs met two other dinosaurs in a clearing there would have been four there even if no humans were around to observe them. The equation 2 + 2 = 4 is a timeless truth.”

Your writing on symmetry is very relevant to my paper. Have a look.

All the best,

Noson Yanofsky

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Your fundamental question about additivity is nice.

Martin Gardner wrote “. . . if two dinosaurs met two other dinosaurs in a clearing there would have been four there even if no humans were around to observe them. The equation 2 + 2 = 4 is a timeless truth.”

Your writing on symmetry is very relevant to my paper. Have a look.

All the best,

Noson Yanofsky

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Dear Professor Knuth,

Your essay is excellent and completely addresses this forum topic. I like how you differentiated between the order/symmetry aspects of mathematics and the quantification aspects of mathematics, and how you profoundly connected fundamental concepts of additivity, symmetry, and order. Your salient thesis of "much of the utility of mathematics arises from our choice of description of the physical world coupled with our desire to quantify it" is a hallmark of my essay, which also looks at the dynamic constraints physical explanation and mathematical representation have on one another. You clearly and rigorously explained how certain laws may be derived as constraint equations enforcing relevant symmetries, and I really appreciated the examples given. The three slit problem you explained succinctly and lucidly, and that's especially revealing how you can derive Feynman Rules via symmetries. I would like to check out your papers on deriving the math of flat spacetime and the quantum symmetrization postulate. Thanks too for the technical endnotes which were very resourceful. Furthermore, the dialogue you had with Tejinder Singh (as well as other comments here) which further developed the connection to physical understanding, was very enlightening. I instantly give this the highest rating.

Please take a moment to review and rate my essay as well.

Thanks, Steve Sax

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Your essay is excellent and completely addresses this forum topic. I like how you differentiated between the order/symmetry aspects of mathematics and the quantification aspects of mathematics, and how you profoundly connected fundamental concepts of additivity, symmetry, and order. Your salient thesis of "much of the utility of mathematics arises from our choice of description of the physical world coupled with our desire to quantify it" is a hallmark of my essay, which also looks at the dynamic constraints physical explanation and mathematical representation have on one another. You clearly and rigorously explained how certain laws may be derived as constraint equations enforcing relevant symmetries, and I really appreciated the examples given. The three slit problem you explained succinctly and lucidly, and that's especially revealing how you can derive Feynman Rules via symmetries. I would like to check out your papers on deriving the math of flat spacetime and the quantum symmetrization postulate. Thanks too for the technical endnotes which were very resourceful. Furthermore, the dialogue you had with Tejinder Singh (as well as other comments here) which further developed the connection to physical understanding, was very enlightening. I instantly give this the highest rating.

Please take a moment to review and rate my essay as well.

Thanks, Steve Sax

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Kevin,

As time grows short, I am revisiting essays I’ve read to assure I’ve rated them. I find that I rated yours on 3/7, rating it as one I could immediately relate to. I hope you get a chance to look at mine: http://fqxi.org/community/forum/topic/2345

Jim

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As time grows short, I am revisiting essays I’ve read to assure I’ve rated them. I find that I rated yours on 3/7, rating it as one I could immediately relate to. I hope you get a chance to look at mine: http://fqxi.org/community/forum/topic/2345

Jim

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Dear Kevin,

Congrats on a brilliant essay. Early in your essay, is the quote, “Familiarity breeds the illusion of understanding”. We are more familiar with pencils, pennies, rocks, sticks, candy, monkeys, planets and stars, all of which are visible than we are with electrons, quarks, virtual particles, etc. I hope you agree on this?

Now your very interesting poser, "“Why is it that when I take two pencils and add one pencil, I always get three pencils? And when I take two pennies and add one penny, I always get three pennies, and so on with rocks and sticks and candy and monkeys and planets and stars. Is this true by definition as in 2+1 defines 3? Or is it an experimental fact so that at some point in the distant past this observation needed to be verified again and again?”, and "Is it not remarkable that 6 sheep plus 7 sheep make 13 sheep; that 6 stones plus 7 stones make 13 stones?", applies to those things we can see. An unstated underlying assumption of 6 + 7 = 13 or 1 + 2 = 3 is that what exists cannot perish. This is the Parmenidean curse I discuss in my essay.

But suppose what exists can perish? Will 6 + 7 still equal 13?

It is certainly easier for a single fundamental particle to perish than for a sheep made up of billions of such particles to do so. Therefore, adding 6 fundamental particles to 7, may not sum up to 13. But because the perishing of a fundamental particle in a sheep does not change the concept of a sheep, although not a certainty, it is more likely that 6 sheep plus 7 sheep has a higher probability of being 13. Certainly, by far more likely than would be the case for a fundamental particle.

All the best in the competition. If you have the time, you may take a look at my essay where I elaborate more on this.

Regards,

Akinbo

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Congrats on a brilliant essay. Early in your essay, is the quote, “Familiarity breeds the illusion of understanding”. We are more familiar with pencils, pennies, rocks, sticks, candy, monkeys, planets and stars, all of which are visible than we are with electrons, quarks, virtual particles, etc. I hope you agree on this?

Now your very interesting poser, "“Why is it that when I take two pencils and add one pencil, I always get three pencils? And when I take two pennies and add one penny, I always get three pennies, and so on with rocks and sticks and candy and monkeys and planets and stars. Is this true by definition as in 2+1 defines 3? Or is it an experimental fact so that at some point in the distant past this observation needed to be verified again and again?”, and "Is it not remarkable that 6 sheep plus 7 sheep make 13 sheep; that 6 stones plus 7 stones make 13 stones?", applies to those things we can see. An unstated underlying assumption of 6 + 7 = 13 or 1 + 2 = 3 is that what exists cannot perish. This is the Parmenidean curse I discuss in my essay.

But suppose what exists can perish? Will 6 + 7 still equal 13?

It is certainly easier for a single fundamental particle to perish than for a sheep made up of billions of such particles to do so. Therefore, adding 6 fundamental particles to 7, may not sum up to 13. But because the perishing of a fundamental particle in a sheep does not change the concept of a sheep, although not a certainty, it is more likely that 6 sheep plus 7 sheep has a higher probability of being 13. Certainly, by far more likely than would be the case for a fundamental particle.

All the best in the competition. If you have the time, you may take a look at my essay where I elaborate more on this.

Regards,

Akinbo

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Dear Akinbo

Thank you for your kind words.

Your comments focus on the idea that some things can perish and others cannot, and that this is central to the fact that 6 + 7 = 13.

I think that this is absolutely correct. In my essay, I call this concept closure and introduce the very same problem by combing a lighted match, a paper napkin and a glass of water. The order in which you combine these objects matter because if you combine the lighted match with the paper napkin, both will perish; whereas if you combine the lighted match with the glass of water they do not.

However, if you consider the constituent atoms, you have closure since atoms joined with atoms leads to collections of atoms.

So closure is the additional important concept here, along with ordering, commutativity and associativity.

All the best

Kevin

Thank you for your kind words.

Your comments focus on the idea that some things can perish and others cannot, and that this is central to the fact that 6 + 7 = 13.

I think that this is absolutely correct. In my essay, I call this concept closure and introduce the very same problem by combing a lighted match, a paper napkin and a glass of water. The order in which you combine these objects matter because if you combine the lighted match with the paper napkin, both will perish; whereas if you combine the lighted match with the glass of water they do not.

However, if you consider the constituent atoms, you have closure since atoms joined with atoms leads to collections of atoms.

So closure is the additional important concept here, along with ordering, commutativity and associativity.

All the best

Kevin

Dear Kevin,

I think you've really nailed the topic, uplifting and inspiring me after reading so much poor and flawed analysis and philosophizing. Your essay presently lies just above mine and is well worth it's place (as I think does mine of course!)

I don't even have any questions to ask you, but hope we might discuss the 'application' of your work in terms of the important consistent insights I hope mine offers if you manage to read it.

Congratulations for a brilliant job.

Peter

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I think you've really nailed the topic, uplifting and inspiring me after reading so much poor and flawed analysis and philosophizing. Your essay presently lies just above mine and is well worth it's place (as I think does mine of course!)

I don't even have any questions to ask you, but hope we might discuss the 'application' of your work in terms of the important consistent insights I hope mine offers if you manage to read it.

Congratulations for a brilliant job.

Peter

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Kevin,

Thanks for your kind comments on my essay, and links. I've now managed the arXiv paper but the 2nd link is dead, could you please re-post, direct or here with the 'link help page' above.

I found the arXiv paper excellent and, as you say, giving a core relationship the sound mathematical formalism required. I checked on our 'It from bit' conversation and we similarly agreed. I've pushed on with many other aspects since then. I don't know if you did go back to my (2010/11) '2020 Vision' essay which identifies some. (I've also identified ex NASA Edward Dowdyes 'Extinction Shift' hypothesis from 1996 as an early version of the same basic dynamic.)

I hope you've found some time to explore my papers too. This arXiv paper also gives a broadish overview, http://arxiv.org/abs/1307.7163 and you should also find the (HJ published) consequential cyclic cosmology paper interesting. A_CYCLIC_MODEL_OF_GALAXY_EVOLUTION_WITH_BARS I do hope we can collaborate on development and presentation as you're far better placed to make headway against the entrenched mainstream direction than John and I are. (Ted Dowdye seems a poor communicator and a loner, the curse of advancing understanding!)

Peter

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Thanks for your kind comments on my essay, and links. I've now managed the arXiv paper but the 2nd link is dead, could you please re-post, direct or here with the 'link help page' above.

I found the arXiv paper excellent and, as you say, giving a core relationship the sound mathematical formalism required. I checked on our 'It from bit' conversation and we similarly agreed. I've pushed on with many other aspects since then. I don't know if you did go back to my (2010/11) '2020 Vision' essay which identifies some. (I've also identified ex NASA Edward Dowdyes 'Extinction Shift' hypothesis from 1996 as an early version of the same basic dynamic.)

I hope you've found some time to explore my papers too. This arXiv paper also gives a broadish overview, http://arxiv.org/abs/1307.7163 and you should also find the (HJ published) consequential cyclic cosmology paper interesting. A_CYCLIC_MODEL_OF_GALAXY_EVOLUTION_WITH_BARS I do hope we can collaborate on development and presentation as you're far better placed to make headway against the entrenched mainstream direction than John and I are. (Ted Dowdye seems a poor communicator and a loner, the curse of advancing understanding!)

Peter

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Hi Peter

Thanks for the links to your papers.

My paper on emergent spacetime is here at JMP, 2014, 55(11), p.112501.

http://scitation.aip.org/content/aip/journal/jmp/55

/11/10.1063/1.4899081

But you can also get it on the arxiv at:

http://arxiv.org/abs/1209.0881

My Contemporary Physics paper, which addresses both spacetime and fermion physics is on the arxiv as well:

http://arxiv.org/abs/1310.1667

Cheers

Kevin

Thanks for the links to your papers.

My paper on emergent spacetime is here at JMP, 2014, 55(11), p.112501.

http://scitation.aip.org/content/aip/journal/jmp/55

/11/10.1063/1.4899081

But you can also get it on the arxiv at:

http://arxiv.org/abs/1209.0881

My Contemporary Physics paper, which addresses both spacetime and fermion physics is on the arxiv as well:

http://arxiv.org/abs/1310.1667

Cheers

Kevin

Kevin,

I have reread your fascinating essay. The question of whether 1 + 2 = 3 is experimental is both profound and provocative. One answer is that for concrete representations of the problem it is empirical, but there is a mathematical abstraction that serves as a model which can be derived from axioms. With the abstract model in hand we just need to check which physical situations comply with the model. This of course just raises deeper questions about how we know the axioms and whether they were invented to fit the physics or they exist in their own right.

I agree very much about the importance of symmetry. After all without temporal and spatial symmetry we could not form any fixed physical laws. I like the way you see symmetries in algebraic principles. I think that this is where they must emerge from

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I have reread your fascinating essay. The question of whether 1 + 2 = 3 is experimental is both profound and provocative. One answer is that for concrete representations of the problem it is empirical, but there is a mathematical abstraction that serves as a model which can be derived from axioms. With the abstract model in hand we just need to check which physical situations comply with the model. This of course just raises deeper questions about how we know the axioms and whether they were invented to fit the physics or they exist in their own right.

I agree very much about the importance of symmetry. After all without temporal and spatial symmetry we could not form any fixed physical laws. I like the way you see symmetries in algebraic principles. I think that this is where they must emerge from

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Dear Kevin,

I very much enjoyed your essay. Your argument strikes me as Kantian- you are arguing for innate categories of perception (mathematics) projected onto reality (physics). Have you thought about things explicitly in these terms? How would you compare your argument to the philosophy of Kant?

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I very much enjoyed your essay. Your argument strikes me as Kantian- you are arguing for innate categories of perception (mathematics) projected onto reality (physics). Have you thought about things explicitly in these terms? How would you compare your argument to the philosophy of Kant?

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