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....

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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!

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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...

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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

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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...

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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

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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...

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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

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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...

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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.

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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...

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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

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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...

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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

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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...

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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.

و الله اعلم

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

Vladimir

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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...

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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

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