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

Great essay.

I think math is discovered. I think you present a very good analysis of the comparison of discovered vs. invented. I'll spend more time thinking about your `invented' arguments.

The following addresses your objections to `discovered' (``Math is a part of nature...".)

A There is a need to discover the truth of how nature works to advance our survival. There is much more to learn about the universe, math and physics. We don't have different math formulas for the same physics phenomena. We have different assumptions for the same phenomena. OR, the formulas produce the same predictions with an easier formulation. Liebniz postulated if the different formulas produce the same results, they are the same.

B Even math starts with axioms. So, derivation of principles is not done in physics or math.

C We have many mysteries. Saying they have no solution is inaccurate and misleading. More accurate is we haven't discovered (or invented) the math yet. For example, Fourier analysis was unknown but the Greeks had `harmony of the spheres' and circles within circles (Ptolomey).

D Your 4th objection is really the crux of the whole issue. Infinity is not a number. It means unbounded or the increase without limit. Whether the universe is unbounded or not is currently a metaphysical issue. I consider it bounded and flat (see my essay on how). Thanks to the essay by Ojo and conversation with him, I have come to think that division is an unnatural operation. I think irrational numbers are invented and therefore are not natural and not valid math. Its purpose is to be an inverse of multiplication, but multiplication is repetitive addition. Therefore, the inverse of multiplication is repetitive subtraction. It solves Zeno's paradox. That is, the invented parts of math are invalid and not natural.

Your section on Patterns and Regularities argues for math being a part of nature. Thanks also for the recognition of the fractal nature of the universe.

Statistics may be an invention. I take the nature of physics to be cause and effect. Statistics addresses a situation where a pattern in the data has been noted and no cause-effect model has yet been developed. For example, QM may indicate the type of cause-effect model such as the Bohm Interpretation (I've researched this wit a model of photon interference). I suggest Group models are like the periodic table in the 19th century. A pattern was recognized; the table was constructed based on properties, holes in the table predicted new elements and their properties. Further, the early 20th century saw the development of a cause-effect (structure) model of why the periodic table worked. The table itself was invented, but the cause-effect was natural. The same is true of the group model where the holes predicted the properties of undiscovered particles.

I particularily like the `Beautiful but wrong' section. I suggest all our current models are `wrong' in the sense of limited.

Your essay is one of the very few that addresses the topic. Well, done!

Dear Mr. Galal and Mr. Khalil:

Your essay is very well written and insightful. Are you really both undergraduate students?

Given that your essay focuses on the role of mathematics in providing models for physical systems, you might be interested in my essay, "Remove the Blinders: How Mathematics Distorted the Development of Quantum Theory". I argue that contrary to universal belief, a simple realistic picture of the microworld is possible, completely avoiding the paradoxes that plague orthodox quantum mechanics. QM is not a universal theory of matter; it is rather a mechanism for distributed vector fields to self-organize into spin-quantized coherent domains similar to solitons. This requires nonlinear mathematics that is not present in the standard Hilbert-space formalism. This also makes directly testable experimental predictions, based on little more than Stern-Gerlach measurements. Remarkably, these simple experiments have never been done.

So while mathematics can provide important insights into physics, an incorrect mathematical model that becomes established may be seen as virtually religious dogma which is not to be questioned. That prevents further progress.

Alan Kadin

Basem and Mohammed,

I salute you for your cogent and straightforward argument that math is invented.

I also believe that it is an effective invention to represent and model the natural world, providing all the advantages you mention. My views are similar in my essay, "Connections: math, physics and the mind"

Regards,

Jim

Dear Basem Galal and Mohammed M. Khalil,

Very interesting and thought provoking essay. Your initial premise that mathematics is invented leading to its effectiveness, is followed up by a very solid argument "we argue that the usefulness of mathematics in discovering new theories is limited, and that it does not provide us with a real picture of the world, but with models useful as calculational tools for making predictions.".

Your statement "Our theories are models of nature. Some models are more useful than others." is the lead in to the geometric models in my essay here of the particles of the standard model. Hope you get a chance to read and comment on it.

Great essay, best of luck in the contest.

Regards, Ed Unverricht

I think that this is a remarkable essay for two undergraduates. It should signal a great future for both. Since my contribution takes a very similar position, but develops it historically, I really have no adverse comments. However, I find the claim that nature is symmetric a bit ambiguous. We use symmetry principles in developing theories especially the standard model of particle physics. Then we have to add qualifications because what we find are broken symmetries.

Ed MacKinnon

Dear Mssrs. Galal and Khalil,

You wrote: "A theory can have support from its theoretical foundation and solid mathematics,

but until it can produce predictions that can be compared with experiment, we must not evaluate it as the only truth. We should always consider alternative approaches even if they are less developed."

Please behold my alternative approach: This is my single unified theorem of how the real Universe is occurring: Newton was wrong about abstract gravity; Einstein was wrong about abstract space/time, and Hawking was wrong about the explosive capability of abstract NOTHING. Proof exists that every real astronomer looking through a real telescope has failed to notice that each of the real galaxies he has observed is unique as to its structure and its perceived distance from all other real galaxies. Each real star is unique as to its structure and its perceived distance apart from all other real stars. Every real scientist who has peered at real snowflakes through a real microscope has concluded that each real snowflake is unique as to its structure. Real structure is unique, once. Unique, once does not consist of abstract amounts of abstract quanta. Based on one's normal observation, one must conclude that all of the stars, all of the planets, all of the asteroids, all of the comets, all of the meteors, all of the specks of astral dust and all real objects have only one real thing in common. Each real object has a real material surface that seems to be attached to a material sub-surface. All surfaces, no matter the apparent degree of separation, must travel at the same constant speed. No matter in which direction one looks, one will only ever see a plethora of real surfaces and those surfaces must all be traveling at the same constant speed or else it would be physically impossible for one to observe them instantly and simultaneously. Real surfaces are easy to spot because they are well lighted. Real light does not travel far from its source as can be confirmed by looking at the real stars, or a real lightning bolt. Reflected light needs to adhere to a surface in order for it to be observed, which means that real light cannot have a surface of its own. Real light must be the only stationary substance in the real Universe. The stars remain in place due to astral radiation. The planets orbit because of atmospheric accumulation. There is no space.

Warm regards,

Joe Fisher

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