Why Mathematics Works So Well by Noson S. Yanofsky

(I am copying my reply on your forum as a notice. I also made some observation above on the possibly touchy subject of the velocity of the observer in special relativity. Probably, you decided to ignore this for the moment, which is okay)

Dear Noson,

Thanks for finding the time to comment on my essay.

As regards, your first query why the real number system works so well in spite of all the discrepancies highlighted in my essay. My initial answer would be that most models would work well, if adhoc entities are invented to fill the loop holes in the modelling, even though paradoxes, counter-intuitive notions and inconsistencies may result in many cases. An example of this is the use of Calculus using the real number system to model motion. The adhoc entity in this instance is the infinitesimal, dx. For the real number system to work, dx must be capable of being both zero and not zero, i.e.

dx = 0 and dx тЙа 0

So if such contradictions are permissible, the real number system can work so well, but may be masking an aspect of reality, which if apprehended will do away with the adhoc improvisations used to cover the loopholes.

Regarding the second question, as I noted in my essay, physical space must exhibit a duality. It must be be capable of exhibiting discreteness and finite approximations being not infinitely divisible, BUT, physical space, the great separator of things into discreteness can itself not play this role which it plays for other entities on itself, hence it also exhibits a continuous nature. Hence my use of 'syrupy' to describe it. However, despite this parts of space are not eternally existing or so to speak, all parts of this syrup do not have the same expiry dates. It is the expiry dates that confers discreteness on the continuous syrup call space.

Finally, I love this quote from Roger Penrose, your fellow FQXi member. In his book, The Emperor's New Mind, p.113... "The system of real numbers has the property for example, that between any two of them, no matter how close, there lies a third. It is not at all clear that physical distances or times can realistically be said to have this property. If we continue to divide up the physical distance between two points, we should eventually reach scales so small that the very concept of distance, in the ordinary sense, could cease to have meaning. It is anticipated that at the 'quantum gravity' scale (...10-35m), this would indeed be the case", then further on,

"We should at least be a little suspicious that (despite the logical elegance, consistency, and mathematical power of the real number system) there might be a difficulty of fundamental principle on the tiniest scales", and "This confidence - perhaps misplaced-..."

It is the possibility that this confidence is misplaced that my essay tries to explore. I would have wanted your own opinion on how to divide a real number line, if there is always a third element between two elements and going by geometrical considerations these elements are uncuttable into parts, i.e. there is a point or number at each incidence of cutting and points cannot have parts or a part of it.

Many thanks for sharing your knowledge.

Regards,

Akinbo

Dear Noson,

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

Dear Noson,

You are suggesting an explanation why "any existing structure in our perceived physical universe is naturally expressed in the language of mathematics". Essentially your point is that since both physics and mathematics are about symmetries, a compatibility between the two sciences is reasonable.

However, Wigner's wonder about the relation of physics and mathematics is not just abut the fact that there are some mathematical forms describing laws of nature. He is fascinated by something more: that these forms are both elegant, while covering a wide range of parameters, and extremely precise. I do not see anything in your paper which relates to that amazing and highly important fact about the relation of physics and mathematics.

Regards,

Alexey Burov.

Noson,

Is it a paradox of symmetry that our presence is asymmetrical in a seemingly ordered universe? Curiously does our presence give it order through our observations?

Such questions are mind-boggling. In contrast,your essay is straightforward and logical.

The mathematician Hermann Weyl gave a succinct definition of symmetry:

"A thing is symmetrical if there is something you can do to it so that after you have finished doing it, it looks the same as before."

Some say we are an asymmetrical lump in that symmetry.

My essay shows my cowardice in avoiding such questions. I am straightforward in showing the connections of math, physics and the human mind.

Jim

Dear Noson,

Your Essay is one more argument as to why math is effective in the more philosophical conceptual sense, among many here and elsewhere. That is good, even those who are not real Platonist justify the effectiveness of math beautifully. Ironically the whole debate is an indication of the power of math looking at it from multitudes of angle. And that is precisely why people like Wolfram, Conway, Tegmark and others came to the natural conclusion that Mathematics has some very deep connection with reality.

Although I did not know about Wolfram and others at the time, just from basic interest in physics I took a similar guess as to math having a very intimate connection with reality. And so I took a guess as to the nature of such connection, but I was very lucky with my guess with everything turning out to be just right. And indeed my theory which shows the origin of the design of reality has a hint of your idea about the symmetry by semantics, and even at one point I so some connection with category theory, among many other connection(some in the essay). I will explain more once you get interested in it.

Essay

Thanks and good luck

Dear Noson,

Your denial of "that beauty plays a role in either physics or mathematics" strongly contradicts to the history of fundamental science, from Kepler and Newton to Einstein and Dirac. Your quotation of Einstein is an example of an extreme misinterpretation by taking out of context. The only reliable source where I know Einstein mentions this 'tailor' comparison is his preface to "Relativity: The Special and General Theory" (1920). His words follow:

"In the interest of clearness, it appeared to me inevitable that I should repeat myself frequently, without paying the slightest attention to the elegance of the presentation. I adhered scrupulously to the precept of that brilliant theoretical physicist, L. Boltzmann, according to whom matters of elegance ought to be left to the tailor and to the cobbler."

I suppose that it is perfectly clear that the specific elegance left here by Einstein "to the tailor and to the cobbler" has nothing to do with neglect of the mathematical elegance or beauty. It is hard to say anything further from truth than to claim aesthetic negligence of Einstein in general and in the matters of theoretical science in particular. There are many clear statements of Einstein about the role beauty played in his own thought and in the history of science. Take for instance the following, where Einstein defines mathematical beauty, or "inner perfection" of theory:

"Pure mathematics is, in its way, the poetry of logical ideas. One seeks the most general ideas of operation which will bring together in simple, logical and unified form the largest possible circle of formal relationships. In this effort toward logical beauty spiritual formulas are discovered necessary for the deeper penetration into the laws of nature." (Obituary for Emmy Noether (1935))

Thus, mathematical beauty/elegance is a unity of logical simplicity and richness of the content. In one or another way belief in the elegance of the laws of nature was expressed by many fathers of science, and this belief played a crucial role in the history of science. When Wigner wrote about 'unreasonable effectiveness of mathematics' he did not mean that laws of nature are somehow described by formulas; he meant that these formulas are both simple in form and rich in content.

The actual question, missed by many essays of this contest, including yours, is why 'the laws of nature are expressed by beautiful equations', as Wigner's brother-in-law put it. Symmetry is just a part of this logical simplicity. We may imagine a universe in which laws have nothing to do with any sort of symmetry and can be described by kilometer-long formulas only. Why the laws of our universe are so symmetric and so simple, which made them discoverable? This is the real question meant by Wigner, which your essay and so many essays here have completely lost.

Dear Prof. Yanofski,

Great essay! You offered a precise definitions of physics and mathematics and the relation between them. We seem to agree in many points, as my essay reflects, especially the importance of symmetry, and how regularities in nature allow mathematics to be so effective. I would be glad to take your opinion in my essay.

Best regards,

Mohammed

Dear Noson,

I enjoyed your crisply and clearly written essay very much. The fact that symmetries are lurking everywhere if we only look for them is a lesson that everyone who wants to understand the nature of reality should take deeply to hear. I had not previously thought of logical validity as a symmetry, or, for that matter, of the symmetry of applicability. While it is clear that the conserved "thing" in the former is truth, I am wondering what it is for the latter?

If I my give just one little piece of constructive criticism, I found ending the essay with an appeal to the anthropic principle a little bit of a downer. In its uncontroversial form it is a tautology, and if one wants to draw stronger conclusions from it, then it invites a host of questions the answers to which based on the strong versions stretch credulity.

However, you did mention at the very end that the anthropic principle does not answer all the deep questions.

On final note, I came across some reviews of your book "The outer limits of reason" which seems like a very interesting book and which I will add to my "to read" list.

Best wishes,

Armin

The idea of describing the mathematical character of physical laws in terms of the abundance of symmetries, was also present in the essay by Milen Velchev Velev. See my reply there. See also the references of essays with effective descriptions of "what is remarkable about the success of mathematics in physics" I collected in my review of this contest, which your essay does not account for.

Dear Noson,

I enjoyed very much your essay. One can't deny the success of mathematics, as you illustrated by predictions like those by Le Verrier, Maxwell, Dirac, Kepler, etc. You criticize well the positions of theologians and Plantonists, and especially the standard response, which presents mathematics as relative to the human experience. Indeed, the role played by symmetry must be fundamental, both in physics and mathematics, and may explain the connection between the two of them. I plan to read your references on the symmetry in math. I am particularly interested since some years ago I identified a symmetry in mathematics that connects many different structures under one umbrella, but I never find time to finish that paper (I think I thought that not many would consider it to be very useful to worth the effort). Of course, the idea is different than yours. I like your definition "a statement is mathematics if we can swap what it refers to and remain true" and "anything that satisfies symmetry of semantics, is mathematics". I agree with "Mathematics is invariant with respect to symmetry of semantics. We are claiming that this type of symmetry is as fundamental to mathematics as symmetry is to physics." Congrats for a so well written essay, which truly addresses the question of effectiveness of mathematics in physics.

Best wishes,

Cristi Stoica

Dear Noson,

I read your essay a while ago but I realized I forgot to comment. My sincerest apologies for this, because I enjoyed your writing very much! I found that there is much similarity of thought between us.

This is an extraordinarily precise and accurate analysis that you are making. You are saying that the problem only exists if "one considers both physics and mathematics to each be perfectly formed, objective and independent of human observers". This is absolutely true! Unless one starts with the assumption that they are unrelated, the problem disappears. You are going to the heart of the problem when you bring into discussion the symmetry and conservation laws of physics, because it is this reducibility to invariant quantities that is characteristic to both math and physics that makes them work together. Can the symmetry of syntax that makes math be a good support for physics be called a conservation law as well? I am thinking here on the lines of the internal logical consistency of math.

Wish you the best of luck in the contest!

Alma

Dear Professor Yanofsky,

Your example of James Bond leads to the obvious question whether you are saying that mathematical regularities, like well-known fictional characters, are in the end items of inter-subjective human agreement. Many people would say that mathematical entities, such as pi or the integer eight, appear to have a definiteness which is independent of human perception and agreement. In this way, the mathematical entities seem unlike fictional characters. I think a similar dissimilarity holds in the case of mathematical truths as contrasted with truths about fictional entities. But, as you say, the issues here deserve more thought. In any event, thanks for your stimulating essay here and for your book, "The Outer Limits of Reason."

Best wishes,

Laurence Hitterdale

Dear Noson,

first let me say that I enjoyed very much reading you book "The Outer Limits of Reason". It is one of the few books that I read more than 60% scratching my personal notes on it and highlighting the most interesting parts. The reason is that you penetrate the epistemology of physics and the structure of a scientific theory with focus and magnetizing style. In your nice essay I can recognize your style of the book.

I completely share the idea that the key to the solution of why mathematics works so well in physics is group theory and the notion of symmetry, as I also emphasized in my essay. I came to the conclusion that a lot of physics can be derived in terms of group representations. For example, we recently gave a definition of reference frame and boost that is purely group theoretical (without space, time, kinematics, and mechanics!) corresponding to a general invariance that only in a limiting situation (the so-called relativistic limit) leads to Lorentz invariance of QFT, but more generally is a nonlinear version of Lorentz group. This allowed us to extend the notion of boost to a discrete Planck scale.

My best wishes

Mauro