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

Hi Mohammed,

Thank you for reading my essay. You have written a very well essay although you probably know that our philosophies are different.

I have seen that you have co authored with Das and Faraj, that is very impressive. How did you do that? Is it possible to show my idea to Faraj and also get Basem to run the simple simulation(at the end of each section written "program link") to confirm the results.

Thanks and good luck.

Dear Basem Galal and Mohammed M. Khalil,

This is by far the best essay to argue that mathematics is invented which I have read, and, believe me, in researching for this contest, I read a lot of them!

Please take the time to check out and vote on my own essay:

http://fqxi.org/community/forum/topic/2391

Best of luck in the contest!

Rick Searle

Mohammed (& Basem),

Thanks for commenting in my forum. I am at a loss, however, to know why you think our ideas are opposed -- I found your excellent essay to reflect an entirely rationalist view of science, as does mine.

I want to point out something to you: You quote Einstein on mathematics as a human invention:

"How is it possible that mathematics, a product of human thought that is independent of experience, fits so excellently the objects of physical reality?" (from *Sidelights in Relativity* 1922)

In my youth, I studied Einstein's and Leopold Infeld's popular book, *The Evolution of Physics* (1938) the way some people study religious texts. They write:

"Physical concepts are free creations of the human mind, and are not, however it may seem, uniquely determined by the external world. In our endeavor to understand reality we are somewhat like a man trying to understand the mechanism of a closed watch. He sees the face and the moving hands, even hears its ticking, but he has no way of opening the case. If he is ingenious he may form some picture of a mechanism which could be responsible for all the things he observes, but he may never be quite sure his picture is the only one which could explain his observations. He will never be able to compare his picture with the real mechanism and he cannot even imagine the possibility or the meaning of such a comparison."

So regardless of whether mathematics is discovered or invented, it is only the rational correspondence of the mathematical language to experience and experiment, that gives us rational knowledge of the world.

Your concluding statement begins, "Mathematics and physics are different; mathematics is a useful human construct, and physics tries to describe the laws of nature. Yet, mathematics is very effective in physics. It enables us to make accurate predictions about the outcome of experiments and even predict undiscovered phenomena."

How could you think that this view differs from mine? I hope you return to my essay with new comments.

All best,

Tom

Mohammed, I'm afraid you still miss the point. I don't support the mathematical universe hypothesis a priori. The philosophical question of whether mathematics is invented or discovered has *nothing* to do with the correspondence of mathematics to physics, i.e., the corresponding truth content of their respective models.

You wouldn't say that natural language has truth content independent of physics, would you? In other words, the string of symbols C-A-T is true if, and only if, there is a physical counterpart to the symbols. That is what Einstein was saying -- e.g., he favored the introduction of extra dimensional models, even in his day, " ... if there exist good physical reasons to do so."

The MUH is based on physical probability, not mathematical philosophy.

Best,

Tom

Dear Galal & Khalil,

You have tried to give a logical picture how mathematics has become effective in physics, and suggest the need for new mathematical inventions to solve the hitherto unsolved problems. You say correctly, "A great mystery about nature is that we can describe the same phenomenon with different mathematical formulations". Have you thought of the reverse possibility? A unique mathematical equation can have different physical interpretations. Refer my essay: A physicalist interpretation of the relation between Physics and Mathematics.

Are there any mathematical laws? Or are there only mathematical structures? I would answer that both exist, and there should be a clear distinction between the two: mathematical laws are discovered; but mathematical structures are invented. The laws are fundamental and eternal, that even an omnipotent creator cannot defy the laws. The structures depend on axioms, which are nothing but 'assigned properties'. But the evolution of the structure follows the eternal mathematical laws.

The same thing is true for the physical world. The physical world has certain properties; but its evolution (the series of changes) depends solely on mathematical laws, and this leads to structures that are mathematically explainable. We try to explain the physical world based on axioms (assigned properties). To arrive at these properties, we depend on 'mathematical relations' based on the observable natural structures. However, this can be tricky. My essay deals with this. If the assigned properties are correct, we will be able to explain everything.

Dear Basem and Mohammed

I agree with your that mathematics is not enough to describe physics. Thus, that many mathematical theories predicted something in mathematics, but predictions were wrong. My opinion is that math is only an abstract language, which tell more simple what happening in physics. (Torsten Asselmeyer-Maluga used the best words: ''' abstraction is necessary concept for our species: we have a limited memory in our brain and a limited number of sensors to sense the world. Therefore, we have to simplify many relations in the world to understand them. But abstraction is also the root of mathematics: numbers as an abstract count of objects was the beginning. ') But fundamental physics should be simple, thus I hope that quantum gravity should be simple.

Thus, your approach is naturalistic (also Smolin) what is closer to me. Although you find good examples where only mathematics gave wrong predictions, I wrote one example where mathematics gave good predictions: Units kg, meter and second needs matematization and simplification, thus Planck found how to eliminate them, thus he showed how physics can become closer to mathematics. But my example with rectangular triangle shows how that euclidean geometry is a consequence of physics.

What do you think if I change one your sentence: '' U(1) gauge theory was an extension of general relativity that naturaly leads to electromagnetism. ''

But, I disagree that our theories are only models of nature. Math is a true goal of physics, but it is not everything.

My essay

Best regards,

Janko Kokosar

Dear Basem and Mohammed,

I liked reading your essay. It is very well written, and you explain very well the role of mathematical models, and how they can turn out to be inadequate to describe the physical world. I also like that you let open the possibility that a mathematical theory well suited to describe the universe may exist, although we will never be sure it is the true one.

Best wishes,

Cristi

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