
Some musings on mathematics - ColinWright
http://www.penzba.co.uk/Writings/SomeMusingsOnMathematics.html
======
j2kun
I think many mathematicians would argue that there's an additional component
of aesthetics at play. It's not just that you want to know whether a theorem
is true (in fact, nobody cares whether the theorem is true). You want to know
why, and you want the proof to be so trivial that in hindsight it's obvious.

This is why people build up grand theories, so that when they get to the mean
value theorem they can just say "Tilt your head and apply Rolle's theorem!"
It's _proofs_ that mathematicians are after, not answers, and beautiful proofs
at that.

~~~
JadeNB
> You want to know why, and you want the proof to be so trivial that in
> hindsight it's obvious.

Certainly I agree with the first part of this sentence, but I'm not sure about
the second part. I think that, if this is accomplished, then it is viewed as a
victory; but I don't think any mathematician would describe it as his or her
mission. (Perhaps on a larger scale, the mission of mathematics is to find the
right definitions that make previously difficult things obvious.)

~~~
j2kun
The parenthetical is what I meant by my sentence. But also to a lesser extent
Paul Erdos was a famous proponent of always finding simpler and more obvious
proofs.

------
lmm
I never like this "want to solve equations" narrative. Maybe some equations
just don't have a solution; advanced mathematicians know why it's nice to work
with an algebraically closed field, but it's not at all obvious. The example
about convergent sequences is much more useful; intuitively it seems wrong
that we should be able to write a "smooth" function like sqrt(x - 2) and see
that it's above zero for some values of x, below zero for other values of x,
and never pass through zero.

The historical invention of complex numbers wasn't to solve equations that had
no solution; it was as an intermediate step, a tool, when applying the cubic
formula. Likewise I suspect that fractions and negative numbers were invented
not to solve equations that couldn't be solved, but so that the rules of
arithmetic worked for expressions like "5 + 3 - 6".

(I actually agree with the main idea, but I don't think these examples support
it)

------
discreteevent
Good article. Explains exactly what pure mathematics is. My first love was
physics so this is exactly the kind of mathematics I have no interest in. To
me mathematics is just a language we can use to describe and understand
nature. Its good that some of the brightest work on advancing that language
but I'm more interested in the territory than the map.

------
tokenadult
A previous discussion[1] 11 months ago (no comments):

[https://news.ycombinator.com/item?id=5008412](https://news.ycombinator.com/item?id=5008412)

There are some older submissions too, with a few comments each.

[https://news.ycombinator.com/item?id=1071734](https://news.ycombinator.com/item?id=1071734)

[https://news.ycombinator.com/item?id=2264243](https://news.ycombinator.com/item?id=2264243)
(this one later went dead)

I like the series this comes from, but I'm not competent to make an informed
comment on the substance of this interesting post.

[1] Colin, you know I just had to do that whenever I see you submit something
that has been submitted before.

------
mathattack
I like the idea that pure math is pure problem solving, for it's own sake.
This quote captures it nicely: _It can be of no practical use to know that Pi
is irrational, but if we can know, it surely would be intolerable not to
know._

But isn't this still somewhat practical? Proving things gives us a mental
workout, even if there is not apparent use. (Not the pure mathematicians would
say, "Yes, though it's still worth doing in the absence of this.")

~~~
saraid216
At that point, what _isn 't_ practical?

