How Accurately Can Mathematics Describe Nature? by Basem Galal and Mohammed M. Khalil

Dear Basem and Mohammed,

You have presented one of the best essays in the contest. Very clear, modest and not intrusive like many others. You deserve very high rating what you will observe in a minute. However I want to address some issues.

You present important objections to the view No. "2. Mathematics is discovered because it is part of nature just like physics." I agree with all objections if we define math as an abstract language of equations. Then the answer is No.3. The mathematics is invented as an abstract, platonic language used to describe reality and also for many other purposes. But pure geometry, in the meaning of shape and its dynamics and not equations or human language, is discovered in the sense that we perceive shapes and its dynamical changes. I think we need an universal, visual language, based on that geometry. It would be comprehensible to future supercomputers, aliens and maybe children as well. So far we have to use equations as our deficient language.

You claim: "...Mathematics is structured as theorems based on axioms. Axioms are the premise or starting point on which we build theorems" As you probably know, there were many attempts to formulate axioms also in physics (D. Hilbert, J. von Neumann, L. Nordheim, H. Weyl, E. Schrödinger, P. Dirac, E. P. Wigner and others). All these efforts failed. That is a pity, however a deductive system can consist not only of axioms but also other, already established theorems. So far theorems were reserved exclusively for mathematics. That means that we can use these established theorems only if we accept that the reality is isomorphic to mathematical structures. You argue that it is not the case and I agree. But we can use geometrical structures instead general notion of mathematical ones. Then we could try e.g. with the geometrization conjecture, proved by Perelman (so it is a theorem). And it generates testable predictions what you demand in conclusions. We have the set of 8 Thurston geometries. We can treat them as a space-like, totally geodesic submanifolds of a 3+1 dimensional spacetime. Then we use the correspondence rule to assign interactions and matter to the proper geometries. It seems to be oversimplified but you can find some technicalities in e.g. Torsten Asselmeyer-Maluga and Helge Rose's publications (arxiv.org/abs/1006.2230, arxiv.org/abs/1006.2230v6). In details it is really complicated.

If you are interested you can take a look at my essay.

I would appreciate your comments however I would understand if you were tired with the contest.

Jacek

Dear Mohammed,

The principle of minimum energy (really the second law of thermodynamics) states that for a closed system, with constant external parameters and entropy, the internal energy will decrease and approach a minimum value at equilibrium. It means that every object/structure shall deform to the shape that minimizes the total potential energy. That shape is an ideal shape. But the system cannot approach that ideal shape.

I am not a Platonist. Obviously, in nature we cannot find ideal shapes. As I have mentioned, the geometry is about shapes that we perceive in real world and not equations. However it is not practical or possible to make calculus on real shapes. To make predictions we need calculus. That is the reason we need idealizations (approximations) of real complexity. In my opinion, the lack of ideal shapes in nature is not an intrinsic feature of real objects (the second law) but the outcome of complexity , interactions and dynamics of interacting objects.

Best regards

Jacek

Dear Basem and Mohammed

Congratulations on an exceptionally thought-out and well-written essay. It helped me appreciate it that I agree in my own essay with many (but not all) of the points you made. For example you mention symmetry and universality as explanation of the effectiveness of mathematics. I go much further and speculate that at the deepest level mind, mathematics and nature share the same 'building blocks'. One idea I particularly liked is that you stress the power of abstraction in mathematics. As for the Kaluza-Klein theory the mathematics pointed to what I consider the solution to the problems of physics: adoption of a universal absolute ether matrix or lattice - the fifth dimension being the ether nodes. This corresponds to my own Beautiful Universe Theory so I am a bit biased to it! As you point out the K-K theory does not sit well with dynamic relativity, and I think relativity itself is just a mathematical re-formulation of an absolute universe with variable speed of light and Lorentz transformations.

Ùˆ الله اعلÙ...

Again congratulations for an excellent essay. Good luck with your studies.

Vladimir

Dear Mohammed,

thanks for reading my essay. As I see we are agreeing in many points. But more importantly, I also think that math is an invention. Thanks for bringing your essay to my attention.I rate your essay high.

Best

Torsten

Dear Basem and Mohammed,

Your point is very well argued and reasonable, very close to the one by the great mathematician, physicist and thinker Henri Poincaré in Science and Hypothesis

http://www.gutenberg.org/files/37157/37157-pdf.pdf

"Rôle of Hypothesis.--Every generalisation is a hypothesis. Hypothesis therefore plays a necessary rôle, which no one has ever contested. Only, it should always be as soon as possible submitted to verification. It goes without saying that, if it cannot stand this test, it must be abandoned without any hesitation."

I am impressed by Poincaré's insight. In our time, physics is much more mathematical. I think that it is the result of a collective cognitive effort, may be an adaptation of our specie to an ever changing environment. I like the view of Vincent Douzal in this respect.

I had a pleasant reading and give you now my best appreciation.

If you have time, I created a dialogue about a topic of interest for mathematical physicists. I am curious to see if you will like it.

Best regards,

Michel

Hi,

You wrote me that you looked at my paper and I am very appreciative. I read your paper and I like it.

I wish you would work out some more of your idea that simplicity and beauty in physical theories can be understood from the computational complexity point of view. Has anyone else talked about this?

Thanks again!

All the best,

Noson Yanofsky

Dear Noson,

Thank you for reading our essay

In our article we suggest that we can use computational complexity as a measure of simplicity because we use computers today for almost all physical computations and simulations .Hence , it's reasonable to choose measure of simplicity relative to computers, I am not sure if anyone else talked about that.

All the best,

Basem

Dear Mohammed, Basem,

It is a well established fact that people will mostly speak of things that impress them, either good or bad. I couldn't agree more with your point that Wigner's speech is not shedding light on theories that have not worked as hoped. For every successful theory like relativity or quantum mechanics, there is a bunch of other theories that we know are wrong, like N=4 super Yang Mills or your SU(5) example. I like your idea of quantifying the elegance of a theory through the number of dependent variables and the relations between variables. It sounds like the most compact network model possible. Your arguments do a very good job getting the point across as well as your clear and enjoyable writing style.

Warm regards,

Alma

Hi Mohammed and Basem--

An absolutely brilliant essay. I concur fully with your analysis. In fact, your essay is so good that I'm glad I didn't write on your precise topic. You would have put me to shame. And, needless-to-say, I certainly agree with your last bullet on page 3.

I can't believe you two are only undergraduates. Your professors are lucky to have you. I predict shining futures for both of you. Keep up the good work!

Best regards and best of luck in the contest,

Bill.

Dear Basem and Mohammed,

Though I disagree with your instrumentalist perspective on physics and your antiplatonist view of mathematics, I am genuinely impressed by your eloquent defense of your positions. It is all the more impressive considering that, if I infer correctly, you are not even native English Speakers.

Despite our philosophical differences, there is much in your essay that I do concur with, especially the notion that observation and experiment trumps any other consideration in science.

Your ideas about quantifying aesthetic notions like beauty are interesting and deserve more detailed treatment.Finally, I could not agree more with your last paragraph as my own research effort is geared precisely to developing new mathematics that helps us model and especially understand fundamental aspects of reality.

Overall, you did a great job

Best wishes,

Armin

Dear Mohammed,

Thank you for your kind remarks on my essay and I enjoyed reading yours as well. As you know, we agree on many points. It's very true that many theories which were considered beautiful or simple ended up failing to explain observations and experimental data. I liked your discussion on simplicity and computational complexity, and on how the same phenomenon can be described with different mathematical formulations. You present throughout a very interesting essay, and I rate it highly.

Kind regards,

Steve Sax