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There’s more to mathematics than rigour and proofs (2007)

terrytao.wordpress.com

31–40 of 95 posts

Re: There’s more to mathematics than rigour and proofs (2007)

#31

I can't wait for theorem provers to be commonplace.

A proof that no one would understand in not a good proof. The ideal approach to proving theorems, at least according to how Grothendieck did it, is to build a beautiful theory in which the proof becomes elementary.

Re: There’s more to mathematics than rigour and proofs (2007)

#32
I feel like this is how all domain expertise works, no? Start with intuition which helps you solidify some of the foundation. Flush out the foundation and start building complicated structures. Now that you've built up the experience, go back and use your intuition to figure out new types of buildings to build.

Re: There’s more to mathematics than rigour and proofs (2007)

#33
post #31

I can't wait for theorem provers to be commonplace.

A proof that no one would understand in not a good proof. The ideal approach to proving theorems, at least according to how Grothendieck did it, is to build a beautiful theory in which the proof becomes elementary.

It's quite a big assumption to think truth can always be bent so as to satisfy our ridiculously limited cognition. And math has been used instrumentally from the very beginning, so results are often much more important than the process. Theoreticians may still value elegance because that gives them pleasure or whatever, but few other people care about that as long as they can use the results.

Re: There’s more to mathematics than rigour and proofs (2007)

#35
post #8
post #2

One day I'll retire and go back to school. The idea of learning Math - really learning & understanding Math - as a fun pastime is so appealing. What's stopping me now? That sweet overpaid SDE salary and the endless obligations that come from being an adult. I suspect I am not alone...

One of my biggest regrets is not having studied (more) math. But would I regret not studying CS had I studied math? You really can't win :/

Same. But we can keep looking for opportunities to learn math in our spare time. For me though, old CS books have a lure all their own :) Maybe an algorithms book (plus MathOverflow) or TLA+ as a gateway.

Re: There’s more to mathematics than rigour and proofs (2007)

#36

Earlier quoted context omitted.

keep the salary. if the goal it to change the world, to understand reality, to have an impactful life, have a good standard of living, etc. math is one of the hardest ways of achieving that. It's such a saturated field. Almost everything you can imagine has been done to the highest possible degree of abstraction. Every stone overturned except for things which may take a lifetime to even try to understand. Writing a b…

I'm not going to gainsay your experience, but it doesn't match mine. Much of what you do in graduate school is to discover where the accessible areas of research are--where do we have a foothold, and are making progress, and what are some achievable results? There are big and hairy problems that are bad investments for a young mathematician. I would steer students clear of the Collatz conjecture. But once you get up…

when I worked on math I found that no matter what problem I was working on, someone had already solved it completely or to high level of abstraction than I had. got discouraging after while.

Re: There’s more to mathematics than rigour and proofs (2007)

#37
post #2

One day I'll retire and go back to school. The idea of learning Math - really learning & understanding Math - as a fun pastime is so appealing. What's stopping me now? That sweet overpaid SDE salary and the endless obligations that come from being an adult. I suspect I am not alone...

keep the salary. if the goal it to change the world, to understand reality, to have an impactful life, have a good standard of living, etc. math is one of the hardest ways of achieving that. It's such a saturated field. Almost everything you can imagine has been done to the highest possible degree of abstraction. Every stone overturned except for things which may take a lifetime to even try to understand. Writing a b…

I totally agree with what you say, but there's a lot of low hanging fruit in mathematizing biology.

It's not easy, but it's certainly very impactful.

Re: There’s more to mathematics than rigour and proofs (2007)

#38
post #13
post #8

Earlier quoted context omitted.

One of my biggest regrets is not having studied (more) math. But would I regret not studying CS had I studied math? You really can't win :/

CS is math...

In a practical sense, not really.

Re: There’s more to mathematics than rigour and proofs (2007)

#39
post #2

One day I'll retire and go back to school. The idea of learning Math - really learning & understanding Math - as a fun pastime is so appealing. What's stopping me now? That sweet overpaid SDE salary and the endless obligations that come from being an adult. I suspect I am not alone...

Yes, you can't do something if you chose to do something else instead.

Clearly you find earning/spending money more appealing than math.

Re: There’s more to mathematics than rigour and proofs (2007)

#40
post #10

Earlier quoted context omitted.

keep the salary. if the goal it to change the world, to understand reality, to have an impactful life, have a good standard of living, etc. math is one of the hardest ways of achieving that. It's such a saturated field. Almost everything you can imagine has been done to the highest possible degree of abstraction. Every stone overturned except for things which may take a lifetime to even try to understand. Writing a b…

I think the whole point would be to retire and do math for fun, not desiring anything more than the joy of discovery, no footnotes, no recognition, just math. 99.99% of everyone won't be remembered for their "contributions" so why not do something you enjoy?

Even if no one remembers you, your work is a contribution. Learning for its own sake is a hobby for yourself, like watching TV or reading a book.
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