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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]

#111

Post by andreareina »

guerrilla wrote: post #39
goatlover wrote: post #28

Earlier quoted context omitted.

> Does it even matter if Math is invented or discovered? It matters for philosophical inquiry. Does philosophy matter? It matters for people who find it valuable, the same way art, music or literature matters.

You think logic, the art of making distinctions, the study of critical thinking and the foundation of mathematics only matter in the same way art does?

I think not the whole of logic, but this particular distinction -- what changes in the world if one is true versus the other? Is there even a difference that is manifest?

It's like the question of what underlies quantum mechanics -- is the Copenhagen interpretation correct? Everett? Some flavor of deBroglie-Bohm? If there's no way to tell the difference, does it matter?

It matters as far as the philosophy of it matters to you, but the concrete consequences of one way being true versus the other could very well be nil.

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

#112

Post by ken47 »

noodles_ftw wrote: post #65
guerrilla wrote: post #41

I'll never understand how someone otherwise so apparently intelligent can be so religious. Weird, the systems that we spent massive amounts of energy designing to precisely describe reality do that better than all the ones that we threw out along the way!

> I'll never understand how someone otherwise so apparently intelligent can be so religious. Why not? Religious people believe that God is the Most Wise. Mathematics in nature attest to that attribute of God (as well as to other attributes of Him). An intelligent and religious person would recognize that, knowing it is God who came up with all the rules that keep the universe in balance. I'll never understand why som…

And who/what created God -- e.g. who is God's God? And so on? It seems like humans trying to understand a hypothetical God are not unlike ants trying to understand quantum physics.

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

#113

Post by mikorym »

Y_Y wrote: post #71
mikorym wrote: post #14

Earlier quoted context omitted.

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.

Wavefunctions are in fact generally complex-valued. I don't know if that supports your argument or not.

Complex numbers have the same cardinality as the reals.

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

#114

Post by mikorym »

h-cobordism wrote: post #19
mikorym wrote: post #16

Earlier quoted context omitted.

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…

When you separate internal and external logic, for example in topos theory, the external logic is still classical.

For example, any topos has internal logic on subobjects of at least a Heyting algebra. In some cases you specialise to a Boolean algebra.

However, when you write things down such as your arguments around functors and constructions, you argue according to classical mathematics externally. When you enter into the category (which is chosen not to be Boolean) and argue on the subobject structure, then you are entering intuitionistic logic. This is what I mean when I say that classical logic can be argued to have intuitionistic logic embedded in it.

The cardinality of the reals being equal to the cardinality of the naturals is then something you have to construct inside the topos. So you need to construct along the spirit of a natural numbers object (perhaps a real numbers object) and then use the internal logic to argue about continuity.

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

#115

Post by mikorym »

h-cobordism wrote: post #19
mikorym wrote: post #16

Earlier quoted context omitted.

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…

I think you are right about the comment about linear logic. I must be confusing A U A with A U ^A.

I checked the paper that I was thinking about and the condition that I saw there is distributivity being dropped rather than double negation being dropped.

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

#116

Post by Y_Y »

mikorym wrote: post #113
Y_Y wrote: post #71

Earlier quoted context omitted.

Wavefunctions are in fact generally complex-valued. I don't know if that supports your argument or not.

Complex numbers have the same cardinality as the reals.

That's true, but irrelevant.

Complex numbers aren't "necessary for the continuum" as you put it, and some realists might argue that they don't hold the same "discoverability" as the reals.

I wouldn't. I think the reals and the complex numbers have the same "realness" and that neither represents any innate property of the physical world, despite how obviously useful they are in physical models.

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

#117

Post by gowld »

peterburkimsher wrote: post #94
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…

Tell me more about the difference between "potential infinity" and "actual infinity". Recursive functions in computer programming, or fractals in maths, are infinite but can be described. Each small part looks like the whole, but the whole can never be fully known within a finite universe. Is that "potential" or "actual" infinity? I've been thinking a lot about infinity this past week, mostly because of Conway's lect…

The video timestamp you linked does not say "free will exists". Is says "humans has free will implies element particles have free will".

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

#118

Post by magicalhippo »

lmm wrote: post #108
magicalhippo wrote: post #76

If B follows from A via logic, but A is invented, isn't B invented as well then? Just to take a recent example which was mentioned here, Geometric Algebra[1]. There you assume you have some objects which aren't numbers but which when squared equals a given number. By doing that a bunch of nice results have been discovered. However to me the basic premise, take some objects which aren't numbers but which square to a n…

> If B follows from A via logic, but A is invented, isn't B invented as well then? Not really, because you have no choice about B - it's true (or rather it follows from A) whether you want it to or not. You could have picked a different A, but once you picked A then B was fixed.

Yet if I pull the trigger we say that I killed someone, not that the bullet did.

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

#119

Post by DeathArrow »

hliyan wrote: post #67
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…

I've always thought math as we know it is a result of both deep underlying relationships between natural constructs and how we perceive them. For example, a species who has no sense of vision will not develop geometry the same way we would. A species that has a sense of vision that also includes a direct distance perception (as opposed to our stereoscopic vision) will probably come up with a very different form of ge…

Maybe vision is helpful with elementary geometry. I don't know how vision would help with differential geometry, geometry of affine spaces, topology, algebraic geometry.

Maybe visualization is more helpful, but you can visualize things without seeing.

A sense of depth and distance, or any kind of metric is helpful, and senses like vision and hearing might be helpful in forming that, but I'm not sure they are required.

From another point of view, math theories are congruent, and you can explain the same things without using geometry, but using algebra, or differential equations or number theory and so on. You certainly can explain any math concept using just modern set theory - maybe it is not the most convenient thing to do.

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

#120

Post by moron4hire »

goatlover wrote: post #28
DeathArrow wrote: post #24

I am not sure of the implications. Does it even matter if Math is invented or discovered? Maybe it's both, and it isn't a contradiction between invention and discovering? We describe things having a certain radiation wavelength as having the color yellow. In that sense, yellow it's an invention. That doesn't mean the radiation doesn't exist. But to complicate things a bit, some things don't exist unless we observe th…

> Does it even matter if Math is invented or discovered? It matters for philosophical inquiry. Does philosophy matter? It matters for people who find it valuable, the same way art, music or literature matters.

The construction of your statement implies that you do not think art, music, and literature matter in the same was as philosophy.
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