

Is there something mysterious about mathematics? - fbrusch
http://ideas.aeon.co/viewpoints/1829

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ahelwer
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.

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xamuel
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.

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impendia
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^(i __t) to satisfy (for example, that e^(i __t) * e^(i __u) = e^(i __(t
+ u)), or that its derivative is i __e^(i __t)), 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.

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tel
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.

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schoen
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...](https://en.wikipedia.org/wiki/Gregory_Chaitin#Other_scholarly_contributions)

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.

~~~
fsk
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")

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danbruc
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.

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andrewstuart2
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](http://www.pbs.org/wgbh/nova/physics/great-math-mystery.html)

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platz
I wonder what opinion this author would have of constructivist mathematics

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wetmore
Why is that?

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calhoun137
Physics is about the way things are, whereas math is about the way things have
to be.

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ankurdhama
No.

