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 <i>weren't</i> the case.<p>Dazzling inexperienced students with magical identities and coincidences isn't teaching mathematics, it's teaching numerology. I really appreciated this article.
I really enjoy Saunders MacLane's <i>Mathematics Form and Function</i> 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 <i>these</i> 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".<p><a href="https://en.wikipedia.org/wiki/Gregory_Chaitin#Other_scholarly_contributions" rel="nofollow">https://en.wikipedia.org/wiki/Gregory_Chaitin#Other_scholarl...</a><p>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.<p>But it might be better to take a look at Chaitin's explanation rather than my paraphrase.
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.<p><a href="http://www.pbs.org/wgbh/nova/physics/great-math-mystery.html" rel="nofollow">http://www.pbs.org/wgbh/nova/physics/great-math-mystery.html</a>