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**Torsten Asselmeyer-Maluga**: *on* 4/21/15 at 11:04am UTC, wrote Dear Armin, thanks for reading my essay and for the comment. You are...

**Torsten Asselmeyer-Maluga**: *on* 4/21/15 at 10:46am UTC, wrote Dear Akinbo, thanks for reading my essay and the comment. In principle, I...

**Torsten Asselmeyer-Maluga**: *on* 4/21/15 at 7:26am UTC, wrote Dear Steve, thanks for the comments and for reading my essay. As you...

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FQXi FORUM

August 12, 2022

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

TOPIC: Mathematics as unifying force for science by Torsten Asselmeyer-Maluga [refresh]

TOPIC: Mathematics as unifying force for science by Torsten Asselmeyer-Maluga [refresh]

In this essay I will discuss the relation between mathematics (in short: math) and physics. Starting with a historical review, the close relation between math and physics is rooted in forecast of experiments in physics and engineering. Then math is simple a tool to tackle these problems. But math and physics changed by a cultural change of our thinking. Therefore, a more global view to problems was created leading to the consideration of general, abstract structures in math and physics as well. In particular, it was the need to understand invisible things like atoms or fields in physics. But math and physics met at this higher level again. In this essay I will also discuss the question why math was created. I see the roots in the requirement for abstraction necessary for a species with limited brain. But math is also limited as discussed by Gödel and Turing. The development of new math is a creative process which is bounded to our brain. So, I disagree with Plato: there is no independent world of ideas. Finally I will discuss the unifying power of math for all science in the future.

I'm a researcher at the German Aerospace Center with widespreaded interests. My current work is in direction of quantum gravity and cosmology. There, I used mathematical methods from topology to understand quantum gravity. In particular, exotic smoothness structures of the spacetime is my main research topic.

I like your section: "MATHEMATICAL STRUCTURES FOR A QUALITATIVE UNDERSTANDING OF SCIENCES"

Do you think that one can "forecast" the future accurately without mathematics?

Good essay.

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Do you think that one can "forecast" the future accurately without mathematics?

Good essay.

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Dear Torsten Asselmeyer-Maluga,

Love it when you talk about models "Physics describes the dynamics of simple objects. Even at the beginning of physics, there were simple models which served as universal models to understand the underlying processes. The harmonic oscillator is one model which is used in mechanics (pendulum), thermodynamics (heat bath), electrodynamics (oscillator of Hertz) and quantum physics (harmonic oscillator, free field quantization in quantum field theory)."

And I feel point 3 of your conclusion is very important "Math changed around 200 years ago and in particular after the Second World War. Now it is the science of structures like groups, modules, categories etc. instead of elegant calculations...". The importance of more complex structures leads to whole new understanding of physics and better predictive power.

Good work, enjoyed the read.

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Love it when you talk about models "Physics describes the dynamics of simple objects. Even at the beginning of physics, there were simple models which served as universal models to understand the underlying processes. The harmonic oscillator is one model which is used in mechanics (pendulum), thermodynamics (heat bath), electrodynamics (oscillator of Hertz) and quantum physics (harmonic oscillator, free field quantization in quantum field theory)."

And I feel point 3 of your conclusion is very important "Math changed around 200 years ago and in particular after the Second World War. Now it is the science of structures like groups, modules, categories etc. instead of elegant calculations...". The importance of more complex structures leads to whole new understanding of physics and better predictive power.

Good work, enjoyed the read.

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Torsten,

Thanks for the encouraging word on my essay. I am going to try to read yours sometime today. Your topic appears related to mine in some ways.

Cheers LC

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Thanks for the encouraging word on my essay. I am going to try to read yours sometime today. Your topic appears related to mine in some ways.

Cheers LC

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There is the prospect that the universe in effect computes things. The universe as a quantum compter, an idea I am partially supportive of, would tell us the universe involves states that are the logical outcomes of certain elementary computations. I am interested in 4-qubit entanglements of 8-qubit systems that are E8. The structure of four manifolds involves a construction with Plucker coordinates and the E8 Cartan matrix. This seems to imply, though I have not seen it in the literature, that for 8 qubits there is not the same SLOCC system based on the Kostant=Sekiguchi theorem. However, I suspect that the structure of 4-spaces might hold the key for something analogous to KS theorem and the structure of 2-3 (GHZ) entanglements that are constructed from G_{abcd}. If the universe has this sort of discrete structure via computation, then it makes some sense to say the universe is in some ways a "machine" that functions by mathematics.

Your essay makes a nice overview of the subject. You avoid some of the metaphysical aspects of this question, such as Platonism. I mention this somewhat briefly, but put this in a certain metaphysical category.

Cheers LC

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Your essay makes a nice overview of the subject. You avoid some of the metaphysical aspects of this question, such as Platonism. I mention this somewhat briefly, but put this in a certain metaphysical category.

Cheers LC

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Lawrence,

Interesting idea to use 4-manifold topology to say something about qubit entanglement. As far as I understand your approach, you have the E8 manifold in mind (which is not smooth). I have to think about it....

In my essay I also made some remarks about the Platonism (but more implicitely). I mostly agree with Gödel: the numbers is (God-)given but the rest belongs to us. I cannot imagine that we only discover mathematics. Even model theory showed us that mathematics is in principle free to use any logical system (but it should agree with the structure of our mind).

Torsten

Interesting idea to use 4-manifold topology to say something about qubit entanglement. As far as I understand your approach, you have the E8 manifold in mind (which is not smooth). I have to think about it....

In my essay I also made some remarks about the Platonism (but more implicitely). I mostly agree with Gödel: the numbers is (God-)given but the rest belongs to us. I cannot imagine that we only discover mathematics. Even model theory showed us that mathematics is in principle free to use any logical system (but it should agree with the structure of our mind).

Torsten

The smooth 4-manifold is dual to the quantized discrete manifold. This duality is STU, or in the case of T duality with R --- > a/R, the large R involved with measuring spacetime across billions of lightyears is dual to a measurement near the Planck scale. Alaine Connes [\link] has found an interesting structure that has an 8-fold structure to it, similar to Bott periodicity, that also looks a lot like quantized geometry. This appears to be similar to the idea of quantum foam, or where a superstring is wildly fluctuating as a graviton.

The E8 manifold is a four manifolds with intersection form given by the Plucker coordinates and E8 Cartan matrix. This forms the "exotic" 4-manifolds that have E8 lattice or group structure. This all again seems to point to some strange relationship between continuous and discontinuous structures. As I think that geometry is a manifestation of quantum entanglement I think it means an observer with access to quantum bits in a large number of entanglements would observe geometry in effect 'breaking down." The STU dual to that would be the observer that does not have access to this level of quantum information, but observes physics commensurate with a smooth realization of geometry.

LC

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The E8 manifold is a four manifolds with intersection form given by the Plucker coordinates and E8 Cartan matrix. This forms the "exotic" 4-manifolds that have E8 lattice or group structure. This all again seems to point to some strange relationship between continuous and discontinuous structures. As I think that geometry is a manifestation of quantum entanglement I think it means an observer with access to quantum bits in a large number of entanglements would observe geometry in effect 'breaking down." The STU dual to that would be the observer that does not have access to this level of quantum information, but observes physics commensurate with a smooth realization of geometry.

LC

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I wrote the following in my essay blog:

In the end there is a bit of a duality here, or a dialectic of sorts. I think that what is measured in physics is discrete. We measure certain observables that have finite values, and quantum physics in particular bears this out pretty seriously. The continuum aspects to physics is pretty much a mathematical issue. Experimental data does not have...

view entire post

In the end there is a bit of a duality here, or a dialectic of sorts. I think that what is measured in physics is discrete. We measure certain observables that have finite values, and quantum physics in particular bears this out pretty seriously. The continuum aspects to physics is pretty much a mathematical issue. Experimental data does not have...

view entire post

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Lawrence,

Ok I see the point. Of course the outcome of experiments is not a real number but as you also point out, one has problems to confirm the discrete structure of spacetime.

I see one reason in the underlying topological nature of physics. You also discussed it in your essay. I will illustrate it in a an example:

If two curves intersect then we measure the number of intersections (a discrete number, gauge or diffeomorphism invariant) but in most cases we are not interested in the coordinates of the intersection. Even sometimes we have problem to determine the coordinate system.

I see the measurement values in physics in this fashion. But then one has a dichotomy between discrete (number of intersections) and continuous. The measured values are in principle discrete but you need the continuum to express the probabilites of quantum mechanics.

I don't see any contradiction in this picture. Of course you will never measure that spacetime has a continuum structure but you can measure a discrete structure. And as you correctly point out: every experiment failed up to now.

In principle I agree with you very much. In particular I like your body-and-soul picture

Best

Torsten

Ok I see the point. Of course the outcome of experiments is not a real number but as you also point out, one has problems to confirm the discrete structure of spacetime.

I see one reason in the underlying topological nature of physics. You also discussed it in your essay. I will illustrate it in a an example:

If two curves intersect then we measure the number of intersections (a discrete number, gauge or diffeomorphism invariant) but in most cases we are not interested in the coordinates of the intersection. Even sometimes we have problem to determine the coordinate system.

I see the measurement values in physics in this fashion. But then one has a dichotomy between discrete (number of intersections) and continuous. The measured values are in principle discrete but you need the continuum to express the probabilites of quantum mechanics.

I don't see any contradiction in this picture. Of course you will never measure that spacetime has a continuum structure but you can measure a discrete structure. And as you correctly point out: every experiment failed up to now.

In principle I agree with you very much. In particular I like your body-and-soul picture

Best

Torsten

I wrote my essay with a certain perspective in mind. I am not locked into any particular perspective on this matter. That spacetime is continuous and smooth seems very much in line with the NASA Fermi and ESA Integral measurements, which involves a large ruler, or equivalently nearly zero energy, measurement of spacetime. Conversely a high transverse momentum measurement would result in chaotic...

view entire post

view entire post

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Lawrence,

thanks for the answer. It is very interesting and I agree with it. Yes I know the foam agrument was always used to introduce the discrete spacetime structure.

I also agree with you about the chaotic nature of quantum fluctuations. Currently I work on a geometric understanding of this point using McMUllen's Field medal work about complex dynamics, fractals and hyperbolic 3-manifolds.

So at first, thanks for the discussion

It help me to understand your point better

Best

Torsten

thanks for the answer. It is very interesting and I agree with it. Yes I know the foam agrument was always used to introduce the discrete spacetime structure.

I also agree with you about the chaotic nature of quantum fluctuations. Currently I work on a geometric understanding of this point using McMUllen's Field medal work about complex dynamics, fractals and hyperbolic 3-manifolds.

So at first, thanks for the discussion

It help me to understand your point better

Best

Torsten

Hi Torsten,

It was a great pleasure to see you here again. I have nothing to add to your essay’s conclusions.

I know and appreciate your other publications referring exotic smoothness structures of the spacetime. In your last one: “How to include fermions into General relativity by exotic smoothness” you wonder: “But then one has the problem to represent QFT by geometric...

view entire post

It was a great pleasure to see you here again. I have nothing to add to your essay’s conclusions.

I know and appreciate your other publications referring exotic smoothness structures of the spacetime. In your last one: “How to include fermions into General relativity by exotic smoothness” you wonder: “But then one has the problem to represent QFT by geometric...

view entire post

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Hi Jacek,

at first I skimed over your essay and found it interesting. Actually I used the exceptional geometries (NIL, SOLV, SL2) in my work. In a previous paper I described the interaction with them (see the paper). The idea is simple: the connection between the knot complements (=fermions) are torus bundles and there are only 3 types of torus bundles which can be identified with interactions (weak, strong and EM). In this paper you will also find the identification of bosons to the geometries.

Currently I think about the SL2 geometry and the Ricci flow.In the past I thought that the Ricci flow is something to do with the measurment process in quantum mechanics but now I changed my mind. I have to go over your essay more carefully. Thanks for your words.

Best Torsten

at first I skimed over your essay and found it interesting. Actually I used the exceptional geometries (NIL, SOLV, SL2) in my work. In a previous paper I described the interaction with them (see the paper). The idea is simple: the connection between the knot complements (=fermions) are torus bundles and there are only 3 types of torus bundles which can be identified with interactions (weak, strong and EM). In this paper you will also find the identification of bosons to the geometries.

Currently I think about the SL2 geometry and the Ricci flow.In the past I thought that the Ricci flow is something to do with the measurment process in quantum mechanics but now I changed my mind. I have to go over your essay more carefully. Thanks for your words.

Best Torsten

Hi Torsten,

Could you please give me the link described above as "(see the paper)" as it does not work and I am really interested in.

Thanks,

Jacek

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Could you please give me the link described above as "(see the paper)" as it does not work and I am really interested in.

Thanks,

Jacek

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Hi Jacek,

try http://arxiv.org/abs/1006.2230 and download the PDF (it is the published version). Secion 8 discussed the gauge group and explains the geometry.

Unfortunately, I see now that I don't mention the geometries (I think the referee don't want to see them and ask to remove them).

Here it is:

finite order = E3 (Eucledian)

Dehn twist = NIL

Anosov = SOLV

Best Torsten

try http://arxiv.org/abs/1006.2230 and download the PDF (it is the published version). Secion 8 discussed the gauge group and explains the geometry.

Unfortunately, I see now that I don't mention the geometries (I think the referee don't want to see them and ask to remove them).

Here it is:

finite order = E3 (Eucledian)

Dehn twist = NIL

Anosov = SOLV

Best Torsten

Dear Torsten,

Great essay! I learned from you. Very enjoyable reading, brief and yet packed with information. You covered Turing, Plato and many others. I also covered Turing and Plato. I also covered number theory briefly. I also concur with Pythagoras that "all things are numbers" and in KQID, it is Einstein complex coordinates Ψ(iτLx,y,z, Lm). However, I started with a premise that everything, yes everything is infinite qbit(00, 1, -1) or Qbit(00, +, -). The infinite contains both finites and infinites. Similarly, finite contains infinites. Because both are governed by infinite law(KQID). Finite law cannot govern infinite entity like the Qbit. Furthermore, as you pointed out everything including the Qbit or our Creator is evolving. Please review and comment on my essay.

Best wishes for the contest and I vote your essay highly,

Leo KoGuan

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Great essay! I learned from you. Very enjoyable reading, brief and yet packed with information. You covered Turing, Plato and many others. I also covered Turing and Plato. I also covered number theory briefly. I also concur with Pythagoras that "all things are numbers" and in KQID, it is Einstein complex coordinates Ψ(iτLx,y,z, Lm). However, I started with a premise that everything, yes everything is infinite qbit(00, 1, -1) or Qbit(00, +, -). The infinite contains both finites and infinites. Similarly, finite contains infinites. Because both are governed by infinite law(KQID). Finite law cannot govern infinite entity like the Qbit. Furthermore, as you pointed out everything including the Qbit or our Creator is evolving. Please review and comment on my essay.

Best wishes for the contest and I vote your essay highly,

Leo KoGuan

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Dear Torsten,

You took the risk to go outside your main domain of expertise and I admire you for that. I intend to give you more comments in a next post.

Meanwhile, you mention the use of dessins d'enfants in your work. I am eager to know if it is related to your remarkable papers on exotic smoothness.

Cheers,

Michel

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You took the risk to go outside your main domain of expertise and I admire you for that. I intend to give you more comments in a next post.

Meanwhile, you mention the use of dessins d'enfants in your work. I am eager to know if it is related to your remarkable papers on exotic smoothness.

Cheers,

Michel

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Dear Michel,

in the last two years I went more deeply in hyperbolic geometric (hyperbolic 3-manifolds). Then I found many interesting relations to finite groups (of course much of it is also covered by a book of Kapovich "Hyperbolic 3-manifolds and discrete groups"). Together with my coauthor Jerzy, we calculated the partition function of a certain quantum field theory and found quasimodular behavior. Then we started to go into it more deeply and again found interesting relations to finite groups (Fuchsian groups). Then we managed to find a folaition of an exotic R^4 and this foliation is given by tessalation of a hyperbolic disk. Here, I found also your picture.

Your essay opened my eyes and it was like a missing link to fulfill another goal of us: to get a geometric description of quantum mechanics (right along your way).

For my there are many really deep thoughts in your essay and I certainly need moer time to grasp them.

Very good work,

Excited greetings

Torsten

in the last two years I went more deeply in hyperbolic geometric (hyperbolic 3-manifolds). Then I found many interesting relations to finite groups (of course much of it is also covered by a book of Kapovich "Hyperbolic 3-manifolds and discrete groups"). Together with my coauthor Jerzy, we calculated the partition function of a certain quantum field theory and found quasimodular behavior. Then we started to go into it more deeply and again found interesting relations to finite groups (Fuchsian groups). Then we managed to find a folaition of an exotic R^4 and this foliation is given by tessalation of a hyperbolic disk. Here, I found also your picture.

Your essay opened my eyes and it was like a missing link to fulfill another goal of us: to get a geometric description of quantum mechanics (right along your way).

For my there are many really deep thoughts in your essay and I certainly need moer time to grasp them.

Very good work,

Excited greetings

Torsten

Dear Torsten,

I read your essay and I like it very much. As you commented on my wall, we are in "boring agreement" :) Yes, I feel the same as you that "Mathematics (in short: math) is not only driven by logic and formal systems of axioms but rather by intuition and creativity." And I agree with the idea that mathematics = understanding structures to forecast the future. Your essay is filled with interesting historical information which exemplifies your point of view and is instructive in the same time.

Best wishes,

Cristi Stoica

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I read your essay and I like it very much. As you commented on my wall, we are in "boring agreement" :) Yes, I feel the same as you that "Mathematics (in short: math) is not only driven by logic and formal systems of axioms but rather by intuition and creativity." And I agree with the idea that mathematics = understanding structures to forecast the future. Your essay is filled with interesting historical information which exemplifies your point of view and is instructive in the same time.

Best wishes,

Cristi Stoica

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Dear Torsten Aßelmeyer-Maluga,

After I realized that the letter ö in Schrödinger is correctly written, I tried the letter ß in Aßelmeyer.

You wrote to LC: "I mostly agree with Gödel: the numbers is (God-)given but the rest belongs to us. I cannot imagine that we only discover mathematics."

Kronecker referred to the natural numbers. I am not familiar with Gödel. What did he mean with "the numbers"? Did he include G. Cantor's transfinite numbers too?

I should avoid hurting the feelings of almost all mathematicians who firmly believe in set theory. However, when Cantor claimed having got CH directly from God, I don't believe this.

Hopefully we can agree on that alephs in excess of aleph_1 didn't find any application in science.

Regards,

Eckard

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After I realized that the letter ö in Schrödinger is correctly written, I tried the letter ß in Aßelmeyer.

You wrote to LC: "I mostly agree with Gödel: the numbers is (God-)given but the rest belongs to us. I cannot imagine that we only discover mathematics."

Kronecker referred to the natural numbers. I am not familiar with Gödel. What did he mean with "the numbers"? Did he include G. Cantor's transfinite numbers too?

I should avoid hurting the feelings of almost all mathematicians who firmly believe in set theory. However, when Cantor claimed having got CH directly from God, I don't believe this.

Hopefully we can agree on that alephs in excess of aleph_1 didn't find any application in science.

Regards,

Eckard

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Dear Eckard,

at first thanks for the correct spelling of my last name. It is absolutely unusual to spell Asselmeyer like Aßelmeyer. Secondly my first name has also a misspelling. Thorsten is correct (Thor from the nordish god of thunder, sten measn stone -> Thunderstone, the stone that makes the thunder).

But now toyour question (or statement): You are right I used implicitely a quote from Kronecker. But Gödel also thought in that direction: the natural numbers were discovered but all the rest is made from us and is not given in some 'world of ideas' (Platon).

From the experimental point of view, you are absolutely right: there are only countable numbers to express the measured values. But the continuum is at least good as a model.

Regards,

Torsten

at first thanks for the correct spelling of my last name. It is absolutely unusual to spell Asselmeyer like Aßelmeyer. Secondly my first name has also a misspelling. Thorsten is correct (Thor from the nordish god of thunder, sten measn stone -> Thunderstone, the stone that makes the thunder).

But now toyour question (or statement): You are right I used implicitely a quote from Kronecker. But Gödel also thought in that direction: the natural numbers were discovered but all the rest is made from us and is not given in some 'world of ideas' (Platon).

From the experimental point of view, you are absolutely right: there are only countable numbers to express the measured values. But the continuum is at least good as a model.

Regards,

Torsten

Dear Torsten,

You didn't confirm agreeing agree on that alephs in excess of aleph_1 didn't find any application in science.

What about non-Dedekind but Euclidean (Maudlin's) numbers?

Regards,

Eckard

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You didn't confirm agreeing agree on that alephs in excess of aleph_1 didn't find any application in science.

What about non-Dedekind but Euclidean (Maudlin's) numbers?

Regards,

Eckard

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Dear Torsten (or Thor stone),

Your essay is a very good survey of the comparative recent history of maths and physics and how one arrived at a " cultural change in our thinking" needed by our specie to adapt the environnement. You are clearly closer to Darwin than Plato and Tegmark.

Can you explain your strange conclusion that "the relation to physics is mainly caused by the simple calculable problems in physics" that seems to contradict your main thesis?

From the Clay Institute's official problem description of Yang-Mills theory by Arthur Jaffe and Edward Witten:

“ [...] one does not yet have a mathematically complete example of a quantum gauge theory in four-dimensional space-time, nor even a precise definition of quantum gauge theory in four dimensions. Will this change in the 21st century? We hope so! ”.

Reading your papers like

http://arxiv.org/pdf/1006.2230v6.pdf

I understand that the theory of four-manifolds including the exotic geometries has something to say. This pefectly fits our topic.

Best.

Michel

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Your essay is a very good survey of the comparative recent history of maths and physics and how one arrived at a " cultural change in our thinking" needed by our specie to adapt the environnement. You are clearly closer to Darwin than Plato and Tegmark.

Can you explain your strange conclusion that "the relation to physics is mainly caused by the simple calculable problems in physics" that seems to contradict your main thesis?

From the Clay Institute's official problem description of Yang-Mills theory by Arthur Jaffe and Edward Witten:

“ [...] one does not yet have a mathematically complete example of a quantum gauge theory in four-dimensional space-time, nor even a precise definition of quantum gauge theory in four dimensions. Will this change in the 21st century? We hope so! ”.

Reading your papers like

http://arxiv.org/pdf/1006.2230v6.pdf

I understand that the theory of four-manifolds including the exotic geometries has something to say. This pefectly fits our topic.

Best.

Michel

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Dear Michel,

thanks for your words.

You are absolutely right, this conclusion is strange. Actually I used the wrong tense and interschange math and physics. The corect statement is:

"the relation to math was mainly caused by the simple calculable problems in physics"

I think then it made more sense.

Thanks for the quote. Yes it is my intention. Our new paper about foliations of exotic R^4 gives also a relation to quantum field theory (we found a factor III_1 algebra which is typical for a QFT)

My remarks about dessins d'enfants were a little bit cryptic. A central point in the construction of the foliation is the embedding of a tree in a hyperbolic disk (here one has a Belyi pair i.e. a polynomial). A central point in the 4-manifold theory is the infinite tree giving a Casson handle. Of course one has finite subtrees. Here comes the dessins d'enfants into play: the embedding of these finite trees are described by this structure.

Currently we try to relate this Casson handle to Connes-Kreimer renormalization theory. If our feeling is true then the action of the absolute Galois group (central for the dessins d'enfants) must be related to the cosmic Galois group.

Of course the whole approach must be related to the interpretation of quantum mechanics too. Even in your essay you presented this relation. Certainly I have to go more deeply into your ideas.

Best

Torsten

thanks for your words.

You are absolutely right, this conclusion is strange. Actually I used the wrong tense and interschange math and physics. The corect statement is:

"the relation to math was mainly caused by the simple calculable problems in physics"

I think then it made more sense.

Thanks for the quote. Yes it is my intention. Our new paper about foliations of exotic R^4 gives also a relation to quantum field theory (we found a factor III_1 algebra which is typical for a QFT)

My remarks about dessins d'enfants were a little bit cryptic. A central point in the construction of the foliation is the embedding of a tree in a hyperbolic disk (here one has a Belyi pair i.e. a polynomial). A central point in the 4-manifold theory is the infinite tree giving a Casson handle. Of course one has finite subtrees. Here comes the dessins d'enfants into play: the embedding of these finite trees are described by this structure.

Currently we try to relate this Casson handle to Connes-Kreimer renormalization theory. If our feeling is true then the action of the absolute Galois group (central for the dessins d'enfants) must be related to the cosmic Galois group.

Of course the whole approach must be related to the interpretation of quantum mechanics too. Even in your essay you presented this relation. Certainly I have to go more deeply into your ideas.

Best

Torsten

Dear Torsten,

Thanks to you I discovered the exotic world of manifolds. I fully agree that mathematics is the driving force for science as you perfectly showed. We have much to share in the near future and I intend to work hard in this direction. My rate this year is eight. New questions to you in preparation.

Best,

Michel

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Thanks to you I discovered the exotic world of manifolds. I fully agree that mathematics is the driving force for science as you perfectly showed. We have much to share in the near future and I intend to work hard in this direction. My rate this year is eight. New questions to you in preparation.

Best,

Michel

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Thank you so much Torsten,

I started to read your book

http://www.maths.ed.ac.uk/~aar/papers/exoticsmooth.pdf

I am also doing mathematical experiments on 3-manifolds

http://magma.maths.usyd.edu.au/magma/handbook/tex

t/742

Another mathematical result of interest

"that every finitely presented group can be realized as the fundamental group of a 4-manifold"

http://mathoverflow.net/questions/30238/constructing-4-manif

olds-with-fundamental-group-with-a-given-presentation

Of course, I just enter your field that I consider a pandora's box.

Best,

Michel

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I started to read your book

http://www.maths.ed.ac.uk/~aar/papers/exoticsmooth.pdf

I am also doing mathematical experiments on 3-manifolds

http://magma.maths.usyd.edu.au/magma/handbook/tex

t/742

Another mathematical result of interest

"that every finitely presented group can be realized as the fundamental group of a 4-manifold"

http://mathoverflow.net/questions/30238/constructing-4-manif

olds-with-fundamental-group-with-a-given-presentation

Of course, I just enter your field that I consider a pandora's box.

Best,

Michel

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Michel,

thanks for your interest in exotic smoothness.

In the last years we found some interesting relations to quantum mechanics for understanding decoherence or what is a quantum state geometrically.

You follow me on ResearchGate and find all relevant papers there. One interesting result for you could be: a quantum state is a wild embedding (see Alexanders horned sphere or Fox wild knot) and we showed that a quantization of tame embedding (a usual embedding) is a wild embdding.

This result is of curse connected to exotic smoothness: conider an exotic S^3xR then a S^3 insider of this space must be a wild embedded S^3.

Currently I try to understand quantum mechanics from this point of view.

Best

Torsten

thanks for your interest in exotic smoothness.

In the last years we found some interesting relations to quantum mechanics for understanding decoherence or what is a quantum state geometrically.

You follow me on ResearchGate and find all relevant papers there. One interesting result for you could be: a quantum state is a wild embedding (see Alexanders horned sphere or Fox wild knot) and we showed that a quantization of tame embedding (a usual embedding) is a wild embdding.

This result is of curse connected to exotic smoothness: conider an exotic S^3xR then a S^3 insider of this space must be a wild embedded S^3.

Currently I try to understand quantum mechanics from this point of view.

Best

Torsten

Torsten,

Very readable essay that cogently presents your 5 basic ideas. The integral function of math I see as connected with computers and modeling, augmenting those mental weaknesses we have and requiring peer review (BICEP2, for example) to get it right.That math is a unifying force for all sciences I see and relate to the new field of quantum biology, DNA studies, and the LHC.

Many of your ideas I mention but cite more of the pragmatics and less of the integral connections you represent.

Your essay traces well the historical to the modern. Because our length is limited, you didn't seem to have time for the quantum needs and connections in physics.

Thanks for giving us the opportunity to share your views.

Jim

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Very readable essay that cogently presents your 5 basic ideas. The integral function of math I see as connected with computers and modeling, augmenting those mental weaknesses we have and requiring peer review (BICEP2, for example) to get it right.That math is a unifying force for all sciences I see and relate to the new field of quantum biology, DNA studies, and the LHC.

Many of your ideas I mention but cite more of the pragmatics and less of the integral connections you represent.

Your essay traces well the historical to the modern. Because our length is limited, you didn't seem to have time for the quantum needs and connections in physics.

Thanks for giving us the opportunity to share your views.

Jim

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Jim,

Thanks for your words (and rating?). Your essay is also on my reading list.

Certainly more later

Torsten

Thanks for your words (and rating?). Your essay is also on my reading list.

Certainly more later

Torsten

Jim,

thanks for writing this essay. It contains a lot of ideas and conclusions to agree with. As you know from ym essay, I'm really interesting into the relation between the disciplines like biology, sociology, physics, math etc. Your essay covered all these question.

It reminds me on a discussion with a biophysicist about consciousness and quantum mechanics. New experiments seem to imply that quantum mechanics is needed to get consciousness and higher brain functions. You explained it also at the example of birds finding their route.

Therefore you will also get a high rate from me.

Best

Torsten

thanks for writing this essay. It contains a lot of ideas and conclusions to agree with. As you know from ym essay, I'm really interesting into the relation between the disciplines like biology, sociology, physics, math etc. Your essay covered all these question.

It reminds me on a discussion with a biophysicist about consciousness and quantum mechanics. New experiments seem to imply that quantum mechanics is needed to get consciousness and higher brain functions. You explained it also at the example of birds finding their route.

Therefore you will also get a high rate from me.

Best

Torsten

Torsten,

Thanks for taking the time to read my essay and for your kind remarks.

Jim

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Thanks for taking the time to read my essay and for your kind remarks.

Jim

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You wrote:

"... numbers as an abstract count of objects was the beginning. ... But math is in particular a relational theory. Let us consider Euclid’s geometry. One needs some obvious basic objects like point, line or surface which is not defined. Then the axioms are given by the relation between the three objects (like: the intersection between two lines is a point). In principle all axiom systems are of this kind."

Euclid's math was built directly on modeling structure in the world of phenomena, and therefore has phenomena as it's "referent". The same cannot be said for much of math that comes since, although it certainly has been adapted (with great effort and creativity) to the task of modeling phenomena.

Before you can claim otherwise, can you answer the question: what is a number?

Also, Euclid's axioms and postulates have the quality of encoding the law-like behavior of phenomena. Does that get carried forward into any subsequent math?

You might want to check out the entry "The Mathematics of Science" by Robert MacDuff.

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"... numbers as an abstract count of objects was the beginning. ... But math is in particular a relational theory. Let us consider Euclid’s geometry. One needs some obvious basic objects like point, line or surface which is not defined. Then the axioms are given by the relation between the three objects (like: the intersection between two lines is a point). In principle all axiom systems are of this kind."

Euclid's math was built directly on modeling structure in the world of phenomena, and therefore has phenomena as it's "referent". The same cannot be said for much of math that comes since, although it certainly has been adapted (with great effort and creativity) to the task of modeling phenomena.

Before you can claim otherwise, can you answer the question: what is a number?

Also, Euclid's axioms and postulates have the quality of encoding the law-like behavior of phenomena. Does that get carried forward into any subsequent math?

You might want to check out the entry "The Mathematics of Science" by Robert MacDuff.

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What is a number? Honestly, I don't know. Counting of objacts in reference to a numer ia an abstract process. Human done it but it don't answers this question.

Here I can answer woth Kronecker: the natural numbers are made by God. But all the rest belongs to Humans.

You are also right, also Euklids geometry contains terms like line point etc. which cannot be defined or explained. The same is true in set theory.

I will have a look into the essay of Robert MacDuff.

Best

Torsten

Here I can answer woth Kronecker: the natural numbers are made by God. But all the rest belongs to Humans.

You are also right, also Euklids geometry contains terms like line point etc. which cannot be defined or explained. The same is true in set theory.

I will have a look into the essay of Robert MacDuff.

Best

Torsten

Dear Torsten,

Please find Euclid's famous, plausible, and compelling definition of a point as "something that has no parts" via Ref. 1 of my essay. Euclid summarized the still useful definitions and axioms of ancient mathematics.

Naive point set theory was logically untenable and therefore substituted by competing among each other i.e. rather arbitrarily chosen systems of axioms (NF, ZF, ZFC, NGB, ...) that were fabricated with the only intention to avoid paradoxes, cf. Fraenkel 1923 and 1984.

Regards,

Eckard

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Please find Euclid's famous, plausible, and compelling definition of a point as "something that has no parts" via Ref. 1 of my essay. Euclid summarized the still useful definitions and axioms of ancient mathematics.

Naive point set theory was logically untenable and therefore substituted by competing among each other i.e. rather arbitrarily chosen systems of axioms (NF, ZF, ZFC, NGB, ...) that were fabricated with the only intention to avoid paradoxes, cf. Fraenkel 1923 and 1984.

Regards,

Eckard

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Dear Torsten Asselmeyer-Maluga

You wrote very exhaustive presentation of mathematics in physics. At this search it is also importantly to find the most precise words, which describe our intuition. One good example of your precise words is: ''Without abstraction, our species with a limited brain is unable to reflect the world.'' Thus math is a process of abstraction. Thus, my conclusion is that the essence of math in pyhsics is to be abstract and simple as much as possible. Because foundations of physics should be simple, the task of math is to describe quantum gravity on a t-shirt. Or, answer, why universe exists, should be short one. This would also confirm trend in physics until now. Smolin is also naturalist, as I am, but he think that elementary physics is not simple. What do you think about this?

My essay

Best regards,

Janko Kokosar

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You wrote very exhaustive presentation of mathematics in physics. At this search it is also importantly to find the most precise words, which describe our intuition. One good example of your precise words is: ''Without abstraction, our species with a limited brain is unable to reflect the world.'' Thus math is a process of abstraction. Thus, my conclusion is that the essence of math in pyhsics is to be abstract and simple as much as possible. Because foundations of physics should be simple, the task of math is to describe quantum gravity on a t-shirt. Or, answer, why universe exists, should be short one. This would also confirm trend in physics until now. Smolin is also naturalist, as I am, but he think that elementary physics is not simple. What do you think about this?

My essay

Best regards,

Janko Kokosar

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Dear Janko Kokosar,

thanks for the comment. I also had the chance to have a look into your essay.

Interesting mixture of topics. I remember on a discussion with bio-physicists. Now there is more and more evidence that Consciousness (as caused by thehuman brain) is strongly related to quantum mechanics. The quantum nature of some processes in the brain is maybe the root of Consciousness.

I think that at the end elementary particle physics can also explained simple. Currently we work on a topological model (based on the braid model of Bilson-Thompson). Maybe it is a way in this direction.

I rate your essay with seven.

Best

Torsten

thanks for the comment. I also had the chance to have a look into your essay.

Interesting mixture of topics. I remember on a discussion with bio-physicists. Now there is more and more evidence that Consciousness (as caused by thehuman brain) is strongly related to quantum mechanics. The quantum nature of some processes in the brain is maybe the root of Consciousness.

I think that at the end elementary particle physics can also explained simple. Currently we work on a topological model (based on the braid model of Bilson-Thompson). Maybe it is a way in this direction.

I rate your essay with seven.

Best

Torsten

Dear Torsten,

I think Newton was wrong about abstract gravity; Einstein was wrong about abstract space/time, and Hawking was wrong about the explosive capability of NOTHING.

All I ask is that you give my essay WHY THE REAL UNIVERSE IS NOT MATHEMATICAL a fair reading and that you allow me to answer any objections you may leave in my comment box about it.

Joe Fisher

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I think Newton was wrong about abstract gravity; Einstein was wrong about abstract space/time, and Hawking was wrong about the explosive capability of NOTHING.

All I ask is that you give my essay WHY THE REAL UNIVERSE IS NOT MATHEMATICAL a fair reading and that you allow me to answer any objections you may leave in my comment box about it.

Joe Fisher

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Joe,

the boundary of a 3D object is a surface. In this point I agree with you. Of course this is the reason why we see only surfaces at the first. But at the other there is a lot of experimental evidence for three (space) dimensions. I would expect that it is part of reality too.

I'm quite sure that at the fundamental level (around Planck length) the world is 2D. But I remember on former discussions...

Torsten

the boundary of a 3D object is a surface. In this point I agree with you. Of course this is the reason why we see only surfaces at the first. But at the other there is a lot of experimental evidence for three (space) dimensions. I would expect that it is part of reality too.

I'm quite sure that at the fundamental level (around Planck length) the world is 2D. But I remember on former discussions...

Torsten

Dear Torsten,

I just red your post to me about a wild embedding and a corresponding quantum state. Although my familiarity to your field is weak at the moment I am fully confident in your approach.

Cheers,

Michel

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I just red your post to me about a wild embedding and a corresponding quantum state. Although my familiarity to your field is weak at the moment I am fully confident in your approach.

Cheers,

Michel

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Dear Michel,

I think that we both have the same goal: to understand quantum mechanics from a geometrical point of view. At the end, our approaches will be converge.

BTW, there is a new Springer journal Quatum Studies

(they send me an email). Maybe interesting for you?

Best

Torsten

I think that we both have the same goal: to understand quantum mechanics from a geometrical point of view. At the end, our approaches will be converge.

BTW, there is a new Springer journal Quatum Studies

(they send me an email). Maybe interesting for you?

Best

Torsten

Dear Torsten,

Yes: Quantum studies: mathematics and foundations.

The editor in chief Yakir Aharonow writes in the preface:

"Finally, there is the approach championed by Dirac and repeated successfully by Feynman and later by Freeman Dyson, namely “playing with equations” as Dirac puts it. This approach sometimes causes equations to reveal their secrets as in the Dirac equation. Dirac took this approach and created results that mathematicians and physicists are still digesting. Feynman, first with the Lagrangian approach to quantum mechanics, the so-called path integral approach, and later with QED and most of the subsequent papers he wrote, operated in this manner. The same could be said of what Dyson did when he “cleaned up ”QED into a methodology usable for calculations. Playing with the problems of quantum mechanics often leads to the creation of new mathematics."

and “Think, reconsider, explore, create deep questions, use paradoxes as a tool for understanding, and finally: publish in this journal!”

A priori this is a good journal for us. My own essay has quotes from Dirac and Dyson, and implicitely to Feynman that anticipated quantum information theory: "There's Plenty of Room at the Bottom" (in 1959). May be I can submit my Monstrous Quantum Theory and you?

Best,

Michel

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Yes: Quantum studies: mathematics and foundations.

The editor in chief Yakir Aharonow writes in the preface:

"Finally, there is the approach championed by Dirac and repeated successfully by Feynman and later by Freeman Dyson, namely “playing with equations” as Dirac puts it. This approach sometimes causes equations to reveal their secrets as in the Dirac equation. Dirac took this approach and created results that mathematicians and physicists are still digesting. Feynman, first with the Lagrangian approach to quantum mechanics, the so-called path integral approach, and later with QED and most of the subsequent papers he wrote, operated in this manner. The same could be said of what Dyson did when he “cleaned up ”QED into a methodology usable for calculations. Playing with the problems of quantum mechanics often leads to the creation of new mathematics."

and “Think, reconsider, explore, create deep questions, use paradoxes as a tool for understanding, and finally: publish in this journal!”

A priori this is a good journal for us. My own essay has quotes from Dirac and Dyson, and implicitely to Feynman that anticipated quantum information theory: "There's Plenty of Room at the Bottom" (in 1959). May be I can submit my Monstrous Quantum Theory and you?

Best,

Michel

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Dear Torsten

Thank you for 7 points, but I gave you 10 points yesterday. I did not send message, thus this is in rules of FQXi. :) Thus, this that you give points to me today, is a coincidence.

The main reason is because you used right words, that I continue to describe relation between math and physics.

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Thank you for 7 points, but I gave you 10 points yesterday. I did not send message, thus this is in rules of FQXi. :) Thus, this that you give points to me today, is a coincidence.

The main reason is because you used right words, that I continue to describe relation between math and physics.

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Dear Torsten,

I very much enjoyed your thorough exposition and your conclusions. Math is indeed a creative process that evolved from a need to have abstract unifying representations of the world. Your speech seems full of passion for the topics you study and that is very admirable for me. Your encyclopedic knowledge is just as impressive. I wish you best of luck in your research and in the contest!

You are more than welcome to read my essay and leave a comment, should you have the necessary time.

Alma

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I very much enjoyed your thorough exposition and your conclusions. Math is indeed a creative process that evolved from a need to have abstract unifying representations of the world. Your speech seems full of passion for the topics you study and that is very admirable for me. Your encyclopedic knowledge is just as impressive. I wish you best of luck in your research and in the contest!

You are more than welcome to read my essay and leave a comment, should you have the necessary time.

Alma

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Dear Alma,

thanks for reading my essay and your words. I want to make the unification of science using math very clear.

But I gave the complement back. I also read your essay and it is really great. Much easier to read then my essay (and maybe also easier to uderstand for any reader). I'm glad that the conclusion of our essay are (in principle) the same.

I wish you also the best luck for the contest (for that I gave a high rate).

Torsten

thanks for reading my essay and your words. I want to make the unification of science using math very clear.

But I gave the complement back. I also read your essay and it is really great. Much easier to read then my essay (and maybe also easier to uderstand for any reader). I'm glad that the conclusion of our essay are (in principle) the same.

I wish you also the best luck for the contest (for that I gave a high rate).

Torsten

Dear Torsten,

Your essay is very good. I want to comment on your attitude:

„As described above, the relation between math and physics is not accidental. But the discussion above also implies that math is a general concept for whole science. But at the first view, only physics has this strong relation, why?

I see the reason in the different complexity in science. Physics describes the dynamics of simple objects.“

This is a clear position and the answer is correct. The basis for your answer, you can look at the Ruđer Bošković, who some consider the father of modern science and the creator of the first theory of everything. You research as a hobby; you do not have shown references which I think is good. I bring you my experience that in literature even more can be understood from the theories of giants of natural philosophy than current scientists. With this approach I have come to a result in my essay.

Regards,

Branko

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Your essay is very good. I want to comment on your attitude:

„As described above, the relation between math and physics is not accidental. But the discussion above also implies that math is a general concept for whole science. But at the first view, only physics has this strong relation, why?

I see the reason in the different complexity in science. Physics describes the dynamics of simple objects.“

This is a clear position and the answer is correct. The basis for your answer, you can look at the Ruđer Bošković, who some consider the father of modern science and the creator of the first theory of everything. You research as a hobby; you do not have shown references which I think is good. I bring you my experience that in literature even more can be understood from the theories of giants of natural philosophy than current scientists. With this approach I have come to a result in my essay.

Regards,

Branko

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Dear Branko,

thanks for reading my essay and for the comments. I thought long what I can cite. Of course I'm influenced by many other thinkers (Hegel, Russel, Gödel, Einstein, Planck, Helmholtz etc.) but I don't find direct places to cite them. I'm sure that my thoughts were also thought by others but I had no time to find the places in the literature.

BTW, I'm a researcher and it is not only a hobby....

With your essay I have some problems. You try to relate numbers to observables like mass relations or the fine structure constant. I see your conclusion but I have problems with these numbers: Maybe your right but what did we learn from this numbers? What is a charge? If you calculate the fine structure constant then I would expect that you know it but I don't found any explanation.

Best

Torsten

thanks for reading my essay and for the comments. I thought long what I can cite. Of course I'm influenced by many other thinkers (Hegel, Russel, Gödel, Einstein, Planck, Helmholtz etc.) but I don't find direct places to cite them. I'm sure that my thoughts were also thought by others but I had no time to find the places in the literature.

BTW, I'm a researcher and it is not only a hobby....

With your essay I have some problems. You try to relate numbers to observables like mass relations or the fine structure constant. I see your conclusion but I have problems with these numbers: Maybe your right but what did we learn from this numbers? What is a charge? If you calculate the fine structure constant then I would expect that you know it but I don't found any explanation.

Best

Torsten

Dear Torsten,

Great essay! You offered a nice historical review of math and physics, and gave strong arguments for why math is a creative process of the human mind. We seem to agree in many points as my essay reflects. For example, I don't believe in the Platonist view of mathematics, and I think that mathematics only provides models for describing nature not an exact correspondence. I would be glad to take your opinion in my essay.

Best regards,

Mohammed

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Great essay! You offered a nice historical review of math and physics, and gave strong arguments for why math is a creative process of the human mind. We seem to agree in many points as my essay reflects. For example, I don't believe in the Platonist view of mathematics, and I think that mathematics only provides models for describing nature not an exact correspondence. I would be glad to take your opinion in my essay.

Best regards,

Mohammed

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Dear Torsten,

This is: „in medias res“:

„Of course I'm influenced by many other thinkers (Hegel, Russel, Gödel, Einstein, Planck, Helmholtz etc.) but I don't find direct places to cite them. I'm sure that my thoughts were also thought by others but I had no time to find the places in the literature.“ Me too.

Sorry for researcher, I thought in the sense of the...

view entire post

This is: „in medias res“:

„Of course I'm influenced by many other thinkers (Hegel, Russel, Gödel, Einstein, Planck, Helmholtz etc.) but I don't find direct places to cite them. I'm sure that my thoughts were also thought by others but I had no time to find the places in the literature.“ Me too.

Sorry for researcher, I thought in the sense of the...

view entire post

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Dear Torsten,

Your essay provides a clear assertion and rationale against the Platonic view of mathematics. Your discussion of the need for mathematical abstraction and the relationship to physical explanation, especially from the historical perspective, is excellent. I found your attention to how math and physics each developed further abstraction at various critical points along their historical connection, to be an intriguing insight that reveals the deeper connection to humanity. The specific examples you gave were very pertinent; in particular, the example of how entropy and thermodynamics became more abstract I think is pivotal, if not even profound. My essay also discusses changing the paradigm used in explaining physical phenomena and the subsequent effects on mathematical abstraction, and how changing the mathematical representation affects physical explanation. Your concluding discussion of topology was very informative and provided a great example of mathematical structure helping to qualitatively understand science. Your essay is an excellent contribution to this forum topic, and I give it the highest rating.

Please take a moment to read and rate my essay, as we have several points in common. Thanks,

Steve Sax

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Your essay provides a clear assertion and rationale against the Platonic view of mathematics. Your discussion of the need for mathematical abstraction and the relationship to physical explanation, especially from the historical perspective, is excellent. I found your attention to how math and physics each developed further abstraction at various critical points along their historical connection, to be an intriguing insight that reveals the deeper connection to humanity. The specific examples you gave were very pertinent; in particular, the example of how entropy and thermodynamics became more abstract I think is pivotal, if not even profound. My essay also discusses changing the paradigm used in explaining physical phenomena and the subsequent effects on mathematical abstraction, and how changing the mathematical representation affects physical explanation. Your concluding discussion of topology was very informative and provided a great example of mathematical structure helping to qualitatively understand science. Your essay is an excellent contribution to this forum topic, and I give it the highest rating.

Please take a moment to read and rate my essay, as we have several points in common. Thanks,

Steve Sax

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Dear Steve,

thanks for the comments and for reading my essay. As you correctly saw, I'm not a fan of Platon and his idea about the world of ideas (independent of us).

I see math as part of humanity and of our brain. Aliens will also use math but (because of their other abilities) in another fashion.

I had other the chance to read your essay and rated it high.

You took agreat circle to explain your point of view: the problem of self-referentials, causality as main part of a computation and you dismissed the infinite universe of Tegmark.

Points about which we can agree.

Good luck for the contest

Torsten

thanks for the comments and for reading my essay. As you correctly saw, I'm not a fan of Platon and his idea about the world of ideas (independent of us).

I see math as part of humanity and of our brain. Aliens will also use math but (because of their other abilities) in another fashion.

I had other the chance to read your essay and rated it high.

You took agreat circle to explain your point of view: the problem of self-referentials, causality as main part of a computation and you dismissed the infinite universe of Tegmark.

Points about which we can agree.

Good luck for the contest

Torsten

Hello Torsten,

Well done on your essay in which you properly identified and discussed the relationship between math and physics in such an interesting way.

While, I may not be a professional mathematician or an expert like you on topology, I wish to ask, if there is a conflict between what math says and what physics says on a given subject, who are we to believe and why?

In my essay, I seem to identify a possible area where what math says could be different from what physics (or physical reality) tells us. You may want to give an opinion.

In math, an infinity value is recognized but in physics, nothing that is certainly infinite in value seems realizable. Or am I wrong?

I therefore suggest that when there is a conflict between math and physics on a question concerning physical reality, physics should be believed since you have yourself rightly pointed out that math is an abstraction and the outcome of a creative process. Or do you feel otherwise?

All the best in the competition.

Regards,

Akinbo

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Well done on your essay in which you properly identified and discussed the relationship between math and physics in such an interesting way.

While, I may not be a professional mathematician or an expert like you on topology, I wish to ask, if there is a conflict between what math says and what physics says on a given subject, who are we to believe and why?

In my essay, I seem to identify a possible area where what math says could be different from what physics (or physical reality) tells us. You may want to give an opinion.

In math, an infinity value is recognized but in physics, nothing that is certainly infinite in value seems realizable. Or am I wrong?

I therefore suggest that when there is a conflict between math and physics on a question concerning physical reality, physics should be believed since you have yourself rightly pointed out that math is an abstraction and the outcome of a creative process. Or do you feel otherwise?

All the best in the competition.

Regards,

Akinbo

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Dear Akinbo,

thanks for reading my essay and the comment. In principle, I agree with you that there is no real infinity. As you I see it as a concept to an value which can be arbitrarily large (but not fixed).

Certainly, if there is a conflict between physics and math I would prefer physics (if it is experimentally confirmed). But I think it is unlikely.

I also read your essay and rate them higher (8 points) but with no real effect on the number.

Good look for the contest

Torsten

thanks for reading my essay and the comment. In principle, I agree with you that there is no real infinity. As you I see it as a concept to an value which can be arbitrarily large (but not fixed).

Certainly, if there is a conflict between physics and math I would prefer physics (if it is experimentally confirmed). But I think it is unlikely.

I also read your essay and rate them higher (8 points) but with no real effect on the number.

Good look for the contest

Torsten

Dear Torsten,

I just read your essay liked how it interwove conceptual considerations into a historical narrative of the relationship between physics and philosophy. One advantage of such a bird's eye view is that it allows one to better appreciate that we are merely observing one "slice", as it were, of a continually evolving relationship between mathematics and physics which may take us into directions which we would not have dreamt of, just as it would have been the case with our ancestor's ability to predict contemporary mathematics.

I agree that mathematics is driven by intuition and creativity, but I would say that is also true for many other fields, and so by itself not the unique defining characteristic; rather it seems that the ability to use one's intuition and creativity in utter freedom except for constraints of consistency is what sets mathematics apart form other endeavors. Many other fields, like the arts, lack a consistency constraint, while the science lack the utter freedom to use one's imagination because the range of possibilities is limited by nature.

Finally, I agree with your view of mathematics as a unifying force for science, and I think this will become much more apparent in the future than it is now.

Best wishes,

Armin

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I just read your essay liked how it interwove conceptual considerations into a historical narrative of the relationship between physics and philosophy. One advantage of such a bird's eye view is that it allows one to better appreciate that we are merely observing one "slice", as it were, of a continually evolving relationship between mathematics and physics which may take us into directions which we would not have dreamt of, just as it would have been the case with our ancestor's ability to predict contemporary mathematics.

I agree that mathematics is driven by intuition and creativity, but I would say that is also true for many other fields, and so by itself not the unique defining characteristic; rather it seems that the ability to use one's intuition and creativity in utter freedom except for constraints of consistency is what sets mathematics apart form other endeavors. Many other fields, like the arts, lack a consistency constraint, while the science lack the utter freedom to use one's imagination because the range of possibilities is limited by nature.

Finally, I agree with your view of mathematics as a unifying force for science, and I think this will become much more apparent in the future than it is now.

Best wishes,

Armin

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Dear Armin,

thanks for reading my essay and for the comment.

You are right with your objection. My argument was to shortly presented. I had the idea to present a contradiction: math and creativity. Most people see math asa fully rational theory. But you are right also other areas share this property. Maybe one should add: math used creativity and intuition and also pure logic to realize these ideas. But I have to think about it more carefully.

I also read your thoughtful essay and rate them high (9 points). Great example to present the path integral.

Good luck for the contest

Torsten

thanks for reading my essay and for the comment.

You are right with your objection. My argument was to shortly presented. I had the idea to present a contradiction: math and creativity. Most people see math asa fully rational theory. But you are right also other areas share this property. Maybe one should add: math used creativity and intuition and also pure logic to realize these ideas. But I have to think about it more carefully.

I also read your thoughtful essay and rate them high (9 points). Great example to present the path integral.

Good luck for the contest

Torsten

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