Why Mathematics Works So Well by Noson S. Yanofsky

Dear Prof. Yanofski,

Your essay is simply superb and sublime. You explain there is no mystery of the Siamese connection of math to physics because both derived from the symmetry of nature. However, I believe you have not answered why nature is symmetrical. Even more strange why do we even notice the symmetries, you simply pointed out of course we noticed them in fact because we are humans who are alive who want to to continue on living by noticing the vagaries of nature to stay alive. Why then we want to stay alive? You simply tautologically answer that if we are not we would not be alive to answer these questions, and so on. This is rather tautology, still superb answer but not satisfactory enough to answer Wigner's enlightens feeling of wonder why this connection exists? Something must be beyond our existential being. You have not dare to cross the taboo to go beyond your own church's dogma. I urge you to go beyond your church dogma and say that nature has its Creator. Who, what, how and why is it? This will answer our own question of wonder who, what, how and why I am?

However as a great mathematician with its tribe's culture and its religion and its church's dogma that as a member of the tribe you are obliged voluntarily obeyed its unspoken rule. If not you are sinned and you shall fall from your Eden by eating that forbidden fruit of knowledge "Apple". We are in the same conundrum when we started: we must follow the rule, following the rule we are only frogs in the well. We observe the sky through our own hole that we dig in ourselves.

I ask you to get out from your hole and cross the boundary and look for the Creator of this "Alice in Wonderland". Lewis Carroll wrote with his magic pen: 'But I don't want to go among mad people,' said Alice. 'Oh, you can't help that,' said the cat. 'We're all mad here.'

Sincerely,

Leo KoGuan

Dear Prof. Yanofsky,

My apology. I reread my comment, it sounds harsh. That is not what I want to convey. Like Moses before, I urge you to lead mathematician community to the promised land and search and define our Creator. Not the personal God, but Einstein's God of nature. I would add God as a Mathrmatician who infuses his creation with math. We are living in math world. Our Creator is living in our equations and breath fire to the equations.

I read yours with pleasure for your clarity of thought and logics. You have my vote.

Best wishes,

Leo KoGuan

Dear Prof. Yanofski,

Your argument that the similarity in the symmetries shown by laws of physics and mathematics is the reason for the effectiveness of mathematics in physics. Is the similarity a chance coincidence? Or is there anything more fundamental?

I argue that the similarity is not a chance coincidence. Changes happen in the physical world entirely by way of motion. No motion implies no changes, and hence no laws. Motion is a space- time relation that follows mathematical laws. So all changes follow mathematical laws. In short, the physical world has only properties but no laws, of its own. The laws applicable to the world are mathematical. In any system, whether real or imaginary, the laws are invariably mathematical. This causes the similarity.

I invite you to read my essay: The physicalist interpretation of the relation between physics and mathematics".

Dear Prof. Yanofsky

Thanks for such a beautiful essay considering symmetry as the primary connection between physics and mathematics.

We can imagine too such a primary symmetric rules in nature as if the symmetry in basic logical patterns that connect the hardware part (physics) and software part (mathematics) to unfold the same nature or universe including inseparable ourselves in it being the cognitive observers.

But such basic logical symmetries in nature also could be two types (instead only one) : deterministic or causal and probabilistic or broken causality. Therefore it seems that there would be a two types of basic logical symmetries as well - one like symmetries in causality and other symmetries in all broken-causality. Both are symmetries but could appear asymmetric from another.

Hence, there might be two connections, instead of one, in-between physics and mathematics e.g deterministic and probabilistic symmetries, because there are also two sets of respective physics and mathematics i.e. deterministic and probabilistic.

I also invite you to my submission "A tale of two logic".

Regards

Dipak Kumar Bhunia

Dear Prof. Yanofski,

I greatly enjoyed your essay tracing the efficacy of mathematics for physics to their shared features regarding symmetry.

Given your computer science background, I was curious as to what you thought of my understanding of Stephen Wolfram's view at computation could serve as an alternative mathematics for physics? My take on that view is that Wolfram sees our current mathematics as "just one among many possible ones" and that mathematics success in physics is historically contingent and occurring bases on the types of problems mathematics has sought to address. Current physical laws are formulated in mathematical laws that are time symmetric- you can reverse them- in the same way a mathematical formula is reversible. But a physics based on computation may lack this symmetry. If you ran them a second time you would get an at least somewhat different answer.

Also, if you have the time, please read and vote on my own essay. While not as rigorous as yours it addresses many of the same issues.

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

Best of luck in the contest!

Rick Searle

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