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> For centuries, mathematicians have strived towards a single goal: to understand how the universe works, and describe it. Um, no. Mathematics itself has absol

by brazzy 14y ago

> For centuries, mathematicians have strived towards a single goal: to understand how the universe works, and describe it. Um, no. Mathematics itself has absolutely nothing to do with "describing the universe". Mathematics is purely abstract. It has no inherent relationship whatsoever with reality. That certain mathematical constructs can be used to model certain aspects of the real world is basically a lucky coincidence, but not very interesting to mathematicians - that's what physicists do.

crntaylor 14y ago
I think there's a reasonable argument to be made that mathematical constructs are part of the universe. David Deutsch has a method for ascertaining whether something can can be said to exist or not - ask whether it "kicks back" when you interact with it, in the sense that simulating the response of the thing you're considering in a totally convincing way would involve an effort as large as building a new universe for that thing to exist in. Rocks "exist" because, if you wanted to build a totally convincing simulation of a rock, you'd need to include all of modern physics as we currently understand it. Other minds "exist" because simulating them convincingly would basically require you to construct a true articificial intelligence (strong AI). Mathematics "exists" because giving someone the genuine experience of doing mathematics when they really weren't would involve a simulation of almost unfathomable complexity. There is a very real truth in the statement that '1 + 1 = 2', or the statement that there are arbitrarily long arithmetic progressions of prime numbers, or any number of other mathematical results. The world of mathematics is a very real part of the universe, that "kicks back" by constantly surprising us when we think about it. So I don't think the article, which is very well written and researched, deserves the middlebrow "Um, no" scorn that you treated it to.
- brazzy 14y ago
  > I think there's a reasonable argument to be made that mathematical constructs are part of the universe. Well, one could say that we "create" them using our minds, which certainly are, but that's deep into philosophical territory. > Mathematics "exists" because giving someone the genuine experience of doing mathematics when they really weren't would involve a simulation of almost unfathomable complexity. Sounds to me like a misapplication of the method - the complexity arises from simulating the response of a person to math, not from simulating math. By that standard, all fiction is "a very real part of the universe". > So I don't think the article, which is very well written and researched, deserves the middlebrow "Um, no" scorn that you treated it to. That concerned on one statement, not the entire article (which I found fascinating as well). Yes, I have to admit that this is rather smartassy, but I actially feel that, quite independant of this article, it is an important and amazing realization that few people make.
  - mcguire 14y ago
    "By that standard, all fiction is 'a very real part of the universe'." Strangely, I believe that statement supports crntaylor's argument, being the point of fiction as far as I can see.
- pasquinelli 14y ago
  "I think there's a reasonable argument to be made that mathematical constructs are part of the universe." oh, sure, but everything is part of the universe. human thought, for instance, is part of the universe. and i'd say that's what mathematics is about.
- bbq 14y ago
  > I think there's a reasonable argument to be made that mathematical constructs are part of the universe. What if there are 0 universes? Does 0 exist without any universes?
pfortuny 14y ago
You are wrong. It is reality what drives mathematicians to work, not a pure abstraction. It is because of reality that maths is interesting. It is because of Geo-metry that algebra is so relevant. Not to speak of Calculus...
- freefrag 14y ago
  He's not wrong. To many mathematicians reality is largely irrelevant. For example G. H. Hardy, an extremely prominent British Mathematician, believed that true Mathematics is an art form and is not useful. He dismissed applications of mathematics as dull and boring. And I can relate to him. It's incredible how complex, beautiful structures arise from a couple of simple axioms. It doesn't matter if what you study will be relevant or not, what matters is that it's fun and stimulating to explore.
  - stephencanon 14y ago
    A (somewhat tongue-in-cheek) question to ponder: If Hardy were alive today, would he turn his back on Number Theory as too useful, and study something more theoretical like the existence of long-time solutions to Navier-Stokes?
- 30thElement 14y ago
  As someone with a degree in applied math, the pure abstract is more interesting than the applied. Applied math is like building really amazing and intricate sand castles on the beach. Pure math is like building the same sand castle, but in the sky and it's kept aloft purely by how beautiful it is, freed from constraints like "touches the ground" and "can support itself under gravity". A lot of my friends feel the same way, with some of them specifically avoiding having "real world" applications of their work, as if that makes it an even better sand castle. As to why I have an applied degree instead of doing pure math, numerical analysis makes a weird intuitive sense to me, and I figured building decent sand castles on the beach was better than making terrible sand castles in the sky that could barely hold themselves up. It also gets the grant money.
  - 14y ago
pseut 14y ago
The statement's a little hyperbolic, but I think you're misunderstanding the author's point. The point is that math is aimed at trying to build understanding in contrast to producing logically airtight proofs. The proofs are important to make sure that our understanding is right, but there's more to it than that. I agree that's not the literal sentence you quoted, but it's definitely the theme of the article: according to many of the people cited, Mochizuki's shirking his responsibility by not explaining how to understand the result, regardless of whether the proof is correct or incorrect. Caveat: I Am Not A Mathematician (IANAM)
- brazzy 14y ago
  > The point is that math is aimed at trying to build understanding in contrast to producing logically airtight proofs. The proofs are important to make sure that our understanding is right, but there's more to it than that. Understanding is necessary to be certain the proof is correct. After that is ensured, many mathematicians will happily use the proof and base their work on it without understanding it. > Mochizuki's shirking his responsibility by not explaining how to understand the result, regardless of whether the proof is correct or incorrect. The main problem caused by his "shirking" is that it leaves the correctness of the proof uncertain and trying to fully understand it is a lot of work that could turn out to be wasted. People want to confirm the proof, and Mochizuki could make it much easier and less costly for them, but for some reason declines to do so.
  - pseut 14y ago
    From TFA: "Even an incorrect proof is better than no proof, because if the ideas are novel, they may still be useful for other problems, or inspire another mathematician to figure out the right answer." edit: I can see taking issue with the word, "universe" in the original quote. I have trouble seeing how the word "understand" is debatable.
peterwoo 14y ago
Here's an amusing counterargument: http://pauli.uni-muenster.de/~munsteg/arnold.html http://pauli.uni-muenster.de/~munsteg/arnold.html
- brazzy 14y ago
  That's not an argument, that's an almost incoherent rant.
  - deleted 14y ago
    [deleted]
cscurmudgeon 14y ago
This is a philosophical position that almost no one engaged in serious math holds. This book has an extremely modern discussion of this http://www.phil.cam.ac.uk/teaching_staff/potter/staip.html http://www.phil.cam.ac.uk/teaching_staff/potter/staip.html Only lay people and arm chair philosophers passing through math land have this unfortunate view.

by brazzy 14y ago

crntaylor 14y ago

I think there's a reasonable argument to be made that mathematical constructs are part of the universe. David Deutsch has a method for ascertaining whether something can can be said to exist or not - ask whether it "kicks back" when you interact with it, in the sense that simulating the response of the thing you're considering in a totally convincing way would involve an effort as large as building a new universe for that thing to exist in. Rocks "exist" because, if you wanted to build a totally convincing simulation of a rock, you'd need to include all of modern physics as we currently understand it. Other minds "exist" because simulating them convincingly would basically require you to construct a true articificial intelligence (strong AI). Mathematics "exists" because giving someone the genuine experience of doing mathematics when they really weren't would involve a simulation of almost unfathomable complexity. There is a very real truth in the statement that '1 + 1 = 2', or the statement that there are arbitrarily long arithmetic progressions of prime numbers, or any number of other mathematical results. The world of mathematics is a very real part of the universe, that "kicks back" by constantly surprising us when we think about it. So I don't think the article, which is very well written and researched, deserves the middlebrow "Um, no" scorn that you treated it to.

brazzy 14y ago
> I think there's a reasonable argument to be made that mathematical constructs are part of the universe. Well, one could say that we "create" them using our minds, which certainly are, but that's deep into philosophical territory. > Mathematics "exists" because giving someone the genuine experience of doing mathematics when they really weren't would involve a simulation of almost unfathomable complexity. Sounds to me like a misapplication of the method - the complexity arises from simulating the response of a person to math, not from simulating math. By that standard, all fiction is "a very real part of the universe". > So I don't think the article, which is very well written and researched, deserves the middlebrow "Um, no" scorn that you treated it to. That concerned on one statement, not the entire article (which I found fascinating as well). Yes, I have to admit that this is rather smartassy, but I actially feel that, quite independant of this article, it is an important and amazing realization that few people make.
- mcguire 14y ago
  "By that standard, all fiction is 'a very real part of the universe'." Strangely, I believe that statement supports crntaylor's argument, being the point of fiction as far as I can see.
pasquinelli 14y ago
"I think there's a reasonable argument to be made that mathematical constructs are part of the universe." oh, sure, but everything is part of the universe. human thought, for instance, is part of the universe. and i'd say that's what mathematics is about.
bbq 14y ago
> I think there's a reasonable argument to be made that mathematical constructs are part of the universe. What if there are 0 universes? Does 0 exist without any universes?

pfortuny 14y ago

You are wrong. It is reality what drives mathematicians to work, not a pure abstraction. It is because of reality that maths is interesting. It is because of Geo-metry that algebra is so relevant. Not to speak of Calculus...

freefrag 14y ago
He's not wrong. To many mathematicians reality is largely irrelevant. For example G. H. Hardy, an extremely prominent British Mathematician, believed that true Mathematics is an art form and is not useful. He dismissed applications of mathematics as dull and boring. And I can relate to him. It's incredible how complex, beautiful structures arise from a couple of simple axioms. It doesn't matter if what you study will be relevant or not, what matters is that it's fun and stimulating to explore.
- stephencanon 14y ago
  A (somewhat tongue-in-cheek) question to ponder: If Hardy were alive today, would he turn his back on Number Theory as too useful, and study something more theoretical like the existence of long-time solutions to Navier-Stokes?
30thElement 14y ago
As someone with a degree in applied math, the pure abstract is more interesting than the applied. Applied math is like building really amazing and intricate sand castles on the beach. Pure math is like building the same sand castle, but in the sky and it's kept aloft purely by how beautiful it is, freed from constraints like "touches the ground" and "can support itself under gravity". A lot of my friends feel the same way, with some of them specifically avoiding having "real world" applications of their work, as if that makes it an even better sand castle. As to why I have an applied degree instead of doing pure math, numerical analysis makes a weird intuitive sense to me, and I figured building decent sand castles on the beach was better than making terrible sand castles in the sky that could barely hold themselves up. It also gets the grant money.
- 14y ago

pseut 14y ago

The statement's a little hyperbolic, but I think you're misunderstanding the author's point. The point is that math is aimed at trying to build understanding in contrast to producing logically airtight proofs. The proofs are important to make sure that our understanding is right, but there's more to it than that. I agree that's not the literal sentence you quoted, but it's definitely the theme of the article: according to many of the people cited, Mochizuki's shirking his responsibility by not explaining how to understand the result, regardless of whether the proof is correct or incorrect. Caveat: I Am Not A Mathematician (IANAM)

brazzy 14y ago
> The point is that math is aimed at trying to build understanding in contrast to producing logically airtight proofs. The proofs are important to make sure that our understanding is right, but there's more to it than that. Understanding is necessary to be certain the proof is correct. After that is ensured, many mathematicians will happily use the proof and base their work on it without understanding it. > Mochizuki's shirking his responsibility by not explaining how to understand the result, regardless of whether the proof is correct or incorrect. The main problem caused by his "shirking" is that it leaves the correctness of the proof uncertain and trying to fully understand it is a lot of work that could turn out to be wasted. People want to confirm the proof, and Mochizuki could make it much easier and less costly for them, but for some reason declines to do so.
- pseut 14y ago
  From TFA: "Even an incorrect proof is better than no proof, because if the ideas are novel, they may still be useful for other problems, or inspire another mathematician to figure out the right answer." edit: I can see taking issue with the word, "universe" in the original quote. I have trouble seeing how the word "understand" is debatable.

peterwoo 14y ago

Here's an amusing counterargument: http://pauli.uni-muenster.de/~munsteg/arnold.html http://pauli.uni-muenster.de/~munsteg/arnold.html

brazzy 14y ago
That's not an argument, that's an almost incoherent rant.
- deleted 14y ago
  [deleted]

cscurmudgeon 14y ago

This is a philosophical position that almost no one engaged in serious math holds. This book has an extremely modern discussion of this http://www.phil.cam.ac.uk/teaching_staff/potter/staip.html http://www.phil.cam.ac.uk/teaching_staff/potter/staip.html Only lay people and arm chair philosophers passing through math land have this unfortunate view.