Why Mathematics Works So Well by Noson S. Yanofsky

Noson -

Thank you for the superb essay. I enjoyed your book "The Outer Limits of Reason" but was a bit disappointed that it did not probe more deeply into the metaphysical implications of all those fascinating limits. This essay takes us there --- to the features of symmetry common to math and physics and to the question of why the world is the way it is. Well done.

I've taken a more metaphorical approach and hope you get a chance to comment on my essay "The Hole at the Center of Creation." My thesis poses a challenge you have not addressed, and the question may be asked this way: What is the ultimate symmetry from which all others emerge? While I use a different vocabulary in my essay, I would ask whether you agree that zero and infinity share an interesting quality: symmetry with infinite degrees of freedom.

With immense respect - George Gantz

I postulate that the Hole at the Center

Dear Prof. Yanofski,

Your essay is a very enjoyable and thought provoking read. The concentration of symmetries for the definition of physics and then the extension of symmetries into the definition of mathematics provided a little different idea to the subject.

In your conclusion "In detail, for any physical law, symmetry of applicability states that the law can deal with swapping any appropriate object for any other appropriate object. If there is a mathematical statement that can describe this physical law, then we can substitute different values for the different objects that one is applying"

Your ideas of extending symmetry to help explain the relationship between math and physics is, in my opinion, very successful.

Good luck in the contest.

Regards,

Ed Unverricht

Dear Professor Yanofsky,

I agree with the appeal to the anthropic principle in the way that you present it. In order for human beings, or other similar entities, to ask and to answer questions about mathematics and physics, the beings asking the questions must exist. Furthermore, in order for these beings to exist, their environment must be sufficiently orderly and stable. As you say, "If the universe did not have some regularities, no life would be possible." Yes, but this fact would seem to lead to the question whether the regularities must be mathematical rather than some other kind. According to your argument, the type of regularity prominent both in physics and in mathematics is the type of regularity congenial to human ways of thinking. You explicitly reject the view that mathematical order has a Platonic transmundane reality. Are mathematical regularities nonetheless objectively real in the physical world? Are these regularities independent of human cognition? Whichever way we answer, we would seem to be left with a further question. If we say that mathematical order is objectively real in nature, then the further question is why it is this kind of order rather than some non-mathematical type of regularity. At least we would need to understand why the order of nature is so thoroughly mathematical, with apparently no allowance for other kinds of order. On the other hand, if mathematical order is not inherent in nature, but is based on a human affinity for thinking in terms of symmetries, then we would confront two further questions. We would wonder why human beings happen to think this way, and we would wonder why this human thinking works so well in application to nature. So, the role of symmetries both in mathematics and in physics is important, but perhaps other explanatory factors are also needed.

Laurence Hitterdale

Dear Noson,

A nice thought provoking essay. You are the expert, but if a non-expert may point out a few things, here are some;

"Galilean relativity demands that the laws of motion remain unchanged if a phenomenon is observed while stationary or moving at a uniform, constant velocity. Special relativity states that the laws of motion must remain the same even if the observers are moving close to the speed of light"

In founding special relativity following the historic 1887 MM experiment, none of the observers, receptors or instruments was moving at the speed of light. This statement may therefore need some tweaking.

Then, talking about the farmer and his apples and oranges, who eventually arrives at the mathematical expression 9 4 = 13. When you say, this pithy little statement encapsulates all the instances of this type of combination, would it apply at all scales? Would it apply to a farmer of quantum particles as well? An unstated caveat in that statement 9 4 = 13 is that things that are being added are eternally existing things. But supposing existing things do perish, will 9 4 always equal 13? Suppose, things not existing come to exist, will 9 4 still equal 13? Although, not the main theme of my essay I find this statement 9 4 = 13 as being under the Parmenidean spell that, 'what exists cannot perish'. But if the universe itself can perish, how much more an apple? If the universe that was non-existent comes to exist, how much more a quantum object coming to exist and distort the equation 9 4 = 13? I therefore agree with your suggestion that going forward, the only way to capture all of the bundled perceptions of physical phenomena of a particular law is to write it in mathematical language which has all its instances bundled with it.

Finally, instead of mathematics belonging to one universe and physical reality belonging to another, why cant both be in the same universe? That is, why can't a mathematical object be equivalent in all respects to a physical object? Why can't the objects of geometry, like points, lines, surfaces and bodies not be same with physically real objects? Are we humans not the cause of this dichotomy of universes? I suggest we are, and as a result Nature presents us with paradoxes to guide us. If you eventually get to read my essay, I mention a few. In particular, I will like to know your opinion on how a line can be physically or mathematically cut if it is constituted of an infinite number of points, which are indivisible?

I cannot but agree with your statement that, "The point we are making is that mathematics works so well at describing laws of physics because they were both formed in the same way". But I venture to say further, not only formed in the same way, but living in the same place.

Best regards,

Akinbo

Noson,

Our world views are so similar that I'm at a loss to think of what to say. I loved the essay of course, so I hope you can visit my forum and we can hopefully engage in a discourse of the rational idealism that motivates us.

Thanks, and please accept my best wishes and highest mark!

Tom

Dear Noson,

You introduce the idea of 'symmetry of applicability' for characterizing physics and that of 'symmetry of semantics' for maths. As for the first concept, do you consider it distinct from what we call 'universality'? As for the second concept, is not the 'universality' of mathematical concepts that allow us to apply them in different contexts? But that may just be a rewording of what you are writing.

I like your advanced examples about the changes of semantics of mathematical statements (the Hilbert's Nullstellentsatz) that capture well what is sometimes also called the tautology in maths. In your paper [YanZel2] you explain how the language of category theory formalize these facts.

And for the relation of maths to physics you write "Rather the regularities of phenomena and thoughts are seen and chosen by human beings in the same way" and later you explain that the adaptation of the human being to his environnement forces him to perceive and organize the regularities. Myself I am not a platonist and I tend to consider that phys and maths are just two different cognitive processes that are constrained by the world external to us.

My essay is of a different taste but does not contadict you. I hope you can find time to read it, as you are a mathematician you can understand many parts of it.

Michel

Dear Noson,

I know experts do not like amateurs who say things differently from what the authority have proclaimed so I hope this comment does not put you off before reading my essay.

When you say, "While the 1887 MM experiment might show that the motion of the observer "is not even in the equation"...

In explaining the null result using Special relativity, there is a v in the length contraction and time dilation equations of SR and that v is supposed to represent the velocity of the observer.

The length contraction equation of the Lorentz transformation is

L' = L в€љ(1 - v2/c2)

While the time dilation equation is

t' = t в€љ(1 - v2/c2)

Also in concluding the paper, Michelson remarked that the relative velocity between any possible stationary ether and the earth was certainly less than one sixth of the earth's (observer's) velocity.

Best regards,

Akinbo

Noson,

Your diaper story reminded me of one about Sierpinski. He and his wife were waiting at the train station, and he was upset because one of the bags seemed to be missing. His puzzled wife said, "What's bothering you? I see that all six bags are here." "No!" replied Sierpinski, "I've counted them several times -- zero, 1, 2, 3, 4, 5!"

Tom

(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