Is there something mysterious about mathematics? | Hacker News Reader

Is there something mysterious about mathematics?

ideas.aeon.co

72 pointsfbrusch14 comments

I think we can all relate to an unclear sense of awe upon first encountering something like e^(i * pi) = -1 (which might have occurred around the first time reading Contact). It takes exposure to a certain amount of math to switch paradigms: it would be weirder if that identity weren't the case.

Dazzling inexperienced students with magical identities and coincidences isn't teaching mathematics, it's teaching numerology. I really appreciated this article.

Yeah, once you work through the explanation of that kind of thing the magic kind of falls out of it when you realize how the definitions fit together to produce that result.

However, I think there are certainly some areas of Math that are still pretty mysterious to the human intellect, no matter how advanced. The distribution of prime numbers and chaos come to mind.

The Monster group is my favorite example. It may very well have the same nature of "just falling out of definitions in an unsurprising manner" but it's certainly too large for any human to ever feel that way. Monstrous Moonshine was this project and is mentioned in the article.

In an ideal world, anyone trying to dazzle students with e^(i*pi) should immediately be rebuffed with the demand: "define complex exponentiation". While I was teaching calculus, I found that students have a tendency to assume that any function sufficiently familiar to them, must be defined for whatever values you can think to stick in it; if you were teaching the mathematics of chairs and tables, they wouldn't bat an eye if you mentioned cos(chair). This phenomenon also explains why there's such a ridiculously big deal made by laymen about the undefinedness of division-by-zero.

On the one hand, I agree with you.

On the other hand, to answer your demand: I define complex exponentiation by the formula e^(it) = -1 for all real, nonzero t.

Clearly stupid. Why? There are a lot of properties that one might naturally ask e^(it) to satisfy (for example, that e^(it) * e^(iu) = e^(i(t + u)), or that its derivative is ie^(it)), and the usual complex exponential satisfies all the right ones. So writing e^(i * pi) with no context has, I submit, at least some content.

I really enjoy Saunders MacLane's Mathematics Form and Function for sight at the "answers" this article seeks. MacLane's argument is that mathematics arises from a relatively small number of meaningful forces and "good" mathematics arises when those forces all align in particular places. You could of course go further by asking what exactly it is that makes those forces all align—why in these fields?—but the point of his book isn't exactly to answer the philosophical question but more to provide a little bit of a humanistic POV on the development of mathematics with just enough historical happenstance stripped away to make things intelligible but not so much as to lose the line of sight on why things (probably) developed the way that they did.

Gregory Chaitin, the developer of algorithmic information theory, has repeatedly said that many or almost all mathematical facts are "true for no reason".

https://en.wikipedia.org/wiki/Gregory_Chaitin#Other_scholarl...

A simple way of trying to understanding Chaitin's view is that if you try to take mathematical facts and match them up with explanations or proofs that humans could understand or recognize as useful or elegant, most facts won't be able to be matched up with any such explanation, because the facts inherently outnumber the explanations, even in a set-theoretic sense.

But it might be better to take a look at Chaitin's explanation rather than my paraphrase.

It sounds like he's talking about the incompleteness theorem, that there are statements that are true but we can never find a proof. That's closely related to the halting problem and the idea of computability. There are some programs that will run forever without halting, but we have no method for identifying all of them. ("true and not provable" matches "runs forever without halting, but we can't prove it")

I am not really surprised that there are patterns and connections between seemingly unrelated concepts. Why? Because I think they are way less unrelated than it may seem. After all everything is build on top of a pretty small number of axioms. Every result in number theory - proven or yet unproven - is essentially a consequence of the Peano axioms and the definition of the operations on those numbers. The Riemann hypothesis is a statement about prime numbers, which numbers are prime is defined by the Peano axioms and the definition of multiplication. The real numbers are in some sense build on top of the natural numbers by going through the rational numbers. And you can build the natural numbers on top of set theory. So in essence I think we are just exploring the structure of one and the same object - or maybe a few objects - and every branch of mathematics does this by making a few more assumptions and therefore looking only on a part of the whole.

So, kind of a summary of/riff on last week's Nova? Then again, maybe just a coincidence.

http://www.pbs.org/wgbh/nova/physics/great-math-mystery.html

I wonder what opinion this author would have of constructivist mathematics

Why is that?

Physics is about the way things are, whereas math is about the way things have to be.

No.

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