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Roger Penrose – Is Mathematics Invented or Discovered? [video]

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Re: Roger Penrose – Is Mathematics Invented or Discovered? [video]

#11

Post by Y_Y »

jeffdavis wrote: post #9

The expression is invented but the underlying relationships are discovered. It's silly to think that the sqaure root of negative one is something real to be discovered. It's just a stand-in to express complex relationships more succinctly. The same for negative numbers, for that matter. It would be equally silly to think that the underlying relationships are invented. Nobody invented prime numbers, they were discover…

It's just as silly to think of the "real numbers" as being discovered. They don't exist in the real world, after all.

Re: Roger Penrose – Is Mathematics Invented or Discovered? [video]

#12

Post by dnautics »

tartoran wrote: post #5
DeathArrow wrote: post #2

Intuitionist mathematics claims that mathematics is purely the result of the constructive mental activity of humans rather than the discovery of fundamental principles existing in an objective reality. [0] In intuitionist mathematics there is only potential infinity, no actual infinity. Constructive set theory differs from Zermelo set theory. That has many consequences in practice. Applying intuitionist mathematics t…

From the wikipedia source: In the philosophy of mathematics, intuitionism, or neointuitionism (opposed to preintuitionism), is an approach where mathematics is considered to be purely the result of the constructive mental activity of humans rather than the discovery of fundamental principles claimed to exist in an objective reality. That is, logic and mathematics are not considered analytic activities wherein deep pr…

The mathematical definition of intuition is not the colloquial definition. For example, in intuitionistic mathematics, you can't proof by contradiction, which means that some things that are "intuitively true" in the colloquial sense, are inaccessible.

Re: Roger Penrose – Is Mathematics Invented or Discovered? [video]

#13

Post by mikorym »

jeffdavis wrote: post #9

The expression is invented but the underlying relationships are discovered. It's silly to think that the sqaure root of negative one is something real to be discovered. It's just a stand-in to express complex relationships more succinctly. The same for negative numbers, for that matter. It would be equally silly to think that the underlying relationships are invented. Nobody invented prime numbers, they were discover…

The personal expression can get very far too, further than perhaps an advanced alien race may know. Supposing an alien race never discovered category theory, their mathematics could still be more advanced than ours, but perhaps more verbose, longer to write down or otherwise bulky and inelegant. The question of whether elegant mathematics is more advanced I guess becomes tricky to answer. But pedagogically and for practical application, elegance is paramount.

I think the most celebrated mathematicians are such because of their personal expression. And that "style element" also inspires people to go further than before. Grothendieck, who featured here a few weeks ago, is a case in point.

Re: Roger Penrose – Is Mathematics Invented or Discovered? [video]

#14

Post by mikorym »

Y_Y wrote: post #11
jeffdavis wrote: post #9

The expression is invented but the underlying relationships are discovered. It's silly to think that the sqaure root of negative one is something real to be discovered. It's just a stand-in to express complex relationships more succinctly. The same for negative numbers, for that matter. It would be equally silly to think that the underlying relationships are invented. Nobody invented prime numbers, they were discover…

It's just as silly to think of the "real numbers" as being discovered. They don't exist in the real world, after all.

They are necessary for the continuum.

If you throw away real numbers, then you lose major things in physics. I think for example you lose the wave function in QM.

It is not a question of whether they exist in nature, but rather whether they are the more superior technique, or not, to explain nature.

Re: Roger Penrose – Is Mathematics Invented or Discovered? [video]

#15

Post by DeathArrow »

hyw wrote: post #4
DeathArrow wrote: post #2

Intuitionist mathematics claims that mathematics is purely the result of the constructive mental activity of humans rather than the discovery of fundamental principles existing in an objective reality. [0] In intuitionist mathematics there is only potential infinity, no actual infinity. Constructive set theory differs from Zermelo set theory. That has many consequences in practice. Applying intuitionist mathematics t…

> Applying intuitionist mathematics to physics we can come to the conclusion that time flows Can you expound on this a little more? I'm completely new to the idea of intuitionistic mathematics and much more so to its applications in physics; the constructive approach to thinking about objects and properties is very refreshing and I'd like to hear how you've related those principles with the paradox of time in the con…

>Can you expound on this a little more?

Unfortunately no. I got the info from Quanta Magazine article. I am not knowledgeable enough to respond on this. Even though I did some high level math courses in University, I wasn't interested enough.

I am sure other HN users understand this stuff better.

Re: Roger Penrose – Is Mathematics Invented or Discovered? [video]

#16

Post by mikorym »

DeathArrow wrote: post #2

Intuitionist mathematics claims that mathematics is purely the result of the constructive mental activity of humans rather than the discovery of fundamental principles existing in an objective reality. [0] In intuitionist mathematics there is only potential infinity, no actual infinity. Constructive set theory differs from Zermelo set theory. That has many consequences in practice. Applying intuitionist mathematics t…

The reason why intuitionist mathematics is useful in QM is because it can be viewed as resource based logic.

But remember that (as far as I know) you can do intuitionist mathematics in classical mathematics, but not the other way around. So you can think of intuitionist mathematics as being embedded in classical mathematics.

Re: Roger Penrose – Is Mathematics Invented or Discovered? [video]

#18

Post by hansbo »

Y_Y wrote: post #11
jeffdavis wrote: post #9

The expression is invented but the underlying relationships are discovered. It's silly to think that the sqaure root of negative one is something real to be discovered. It's just a stand-in to express complex relationships more succinctly. The same for negative numbers, for that matter. It would be equally silly to think that the underlying relationships are invented. Nobody invented prime numbers, they were discover…

It's just as silly to think of the "real numbers" as being discovered. They don't exist in the real world, after all.

Don't they, though? The path of an orbit in vacuum should follow pi, unless coordinates are discrete (which they might be, I suppose)

Re: Roger Penrose – Is Mathematics Invented or Discovered? [video]

#19

Post by h-cobordism »

mikorym wrote: post #16
DeathArrow wrote: post #2

Intuitionist mathematics claims that mathematics is purely the result of the constructive mental activity of humans rather than the discovery of fundamental principles existing in an objective reality. [0] In intuitionist mathematics there is only potential infinity, no actual infinity. Constructive set theory differs from Zermelo set theory. That has many consequences in practice. Applying intuitionist mathematics t…

The reason why intuitionist mathematics is useful in QM is because it can be viewed as resource based logic. But remember that (as far as I know) you can do intuitionist mathematics in classical mathematics, but not the other way around. So you can think of intuitionist mathematics as being embedded in classical mathematics.

I think in your first paragraph you're confusing intuitionistic logic and linear logic --- intuitionistic logic still lets you "clone" and "delete" propositions freely.

The second paragraph is somewhat true --- at its base, the modern use of the term "intuitionistic logic" refers merely to not accepting the law of the excluded middle. However, there are varieties of intuitionism that are very, very inconsistent with classical logic, like those adopting the idea that "every function between real numbers is continuous".

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