I went to two of the talks at Mengerfest today, (at the urging of a professor). They were unenchanting. Ellis's talk was a research talk, and so came with the dryness to be expected, while the material was interesting. Trefethen's talk would have been exceptionally interesting, were I a numerical analyst. Alas, I am no such creature.
After the talk someone commented about Trefethen's talk, and quoted Menger (apparently from a conversation): "Functions are the numbers of the 21st century."
That statement says a lot about the process of abstraction, and mathematical thought in general. Mathematics, (and it seems, with it, all of human thought) is tending toward increased abstraction. Originally integers were the epitome of mathematical abstraction; then rational and real numbers; then the Cartesian plane, complex numbers, and then sets.... With the 20th century developments of general abstract nonsense, topos theory and type theory, mathematics has climbed higher up this mountain than ever before.
There's a certain similarity in rules underlying all of these developments and abstractions-- they all work more or less the same. This implies (at least) one of three things:
1)The way we do math requires structures to exhibit similarities.
2)There is an inherent structure behind math... That is, the universe is design in such a way that there is a tremendously simple, yet elegantly complex structure.
3)The structures we impose are mental constructs. The fact that we are doing math requires these structures-- our mind cannot process abstract data in any other way.
Being me, I tend towards 1 or 3. 2 is too deterministic. 1, I think, presents more interesting applications: Is there another way to approach math, so that other structures develop?
And when I step back and examine the broader implications of 1,2, and 3, I realize, once again that it just doesn't matter. Math will still be conducted the same way; humans will continue to act roughly the same and life will continue as normal-- no interrupt to the daily scheduled programming.
If the different systems (that is, math, or the universe, or anything else) end up acting the same way, we have an isomorphism, right? So the systems can be considered the same, no matter how different the rules.
And I realized that more and more, this is why it's hard for me to give a shit when someone spouts self-righteous or otherwise ignorant nonsense: they'll continue acting the same. The system in which I said nothing at all is more or less the same as the system in which I correct their error, in the name of "Truth."
And at this point I've lost my train of thought... something on everything being the same, and so on and such... and I need to count some orbits.
Arbtirary thoughts on nearly everything from a modernist poet, structural mathematician and functional programmer.
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