Roger Penrose – Is Mathematics Invented or Discovered? [video] • Hacker News Board

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

#191

Post by DeathArrow » Fri, Apr 17, 2020, 3:45 PM UTC

StandardFuture wrote: post #154

henrik_w wrote: post #136

Earlier quoted context omitted.

I really liked this perspective from Bartosz Milewski in this (very good) podcast episode: mathematics is not inherent to the world, it is inherent to the human brain (paraphrased, around minute 32 in the episode) https://corecursive.com/035-bartosz-milewski-category-theory...

And really one of the biggest pieces of proof of this is that some of the earliest mathematics developed was planar geometry. But, planar geometry (primarily involving triangles and squares) does not actually exist in nature. Which means that planar geometry is an approximation technique developed by the brain in order to begin to understand the actual much more complex geometries that appear in the real world. This…

>This also suggests that mathematics is 100% constructed by the human brain, even if it is highly-influenced by relationships found in the physical world.

As I see it, mathematics is both discovered and invented.

We can model every existing thing in every possible world using math. Even if both the set of all things that might exist and all possible mathematical constructions are infinite, the later is larger. That's because we can also construct mathematical models of things that doesn't exist.

So it looks that from the set of all possible mathematical constructions, we extracted a subset that maps to objects in reality. That looks like a discovery process.

But we also constructed mathematical models of things that don't have corespondents in reality, so that much be more of an invention process.

Let's pretend for a bit that we forget all what we know and tomorrow we will start inventing things again. Or that a species of aliens start fresh on a planet.

The mathematic theories and notions we and the aliens might discover, build or invent might be different than the theories and notions we know today but would probably be equivalent. That suggests that mathematics just exists somewhere in its own world waiting to be discovered.

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

#192

Post by maxiepoo » Fri, Apr 17, 2020, 3:47 PM UTC

naasking wrote: post #168

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…

> 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] The "mental activity" part isn't strictly necessary. A constructive mathematics will produce results that are largely the same. What matters most is that mathematical objects are proven by construction, and so pr…

I think you and OP are talking about different things because the term "intuitionist" has been used for both the philosophy that EJ Brouwer espoused (which matches what OP said) and the large field of constructive mathematics that has been greatly influenced by Brouwer's philosophy but has embraced approaches like formal logic which Brouwer himself disliked.

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

#193

Post by guerrilla » Fri, Apr 17, 2020, 3:48 PM UTC

iovrthoughtthis wrote: post #59

guerrilla wrote: post #46

Earlier quoted context omitted.

Good thing the founders of modern science, mathematics, ethics and political theory didn't share the same opinion.

I would like some stuff to read on this if possible.

SEP contains dense summaries of every major thinker with extremely high quality references. Random example https://plato.stanford.edu/entries/leibniz/

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

#194

Post by guerrilla » Fri, Apr 17, 2020, 3:50 PM UTC

voldacar wrote: post #109

guerrilla wrote: post #39

Earlier quoted context omitted.

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?

you think too poorly of art

I just think our quality of life would be much lower without the forementioned than without art. We'd be much worse off without any of them. My point is not to dismiss art but emphasize the utility of philosophy.

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

#195

Post by throwaway17_17 » Fri, Apr 17, 2020, 3:51 PM UTC

naasking wrote: post #171

TheOtherHobbes wrote: post #126

Earlier quoted context omitted.

Free will is a philosophical concept which is essentially dualist. It's an attempt to reconcile two subjective experiences - the feeling of making a decision, and the fact that sometimes we can't predict the actions of others - with universal determinism. There's no reason why universal determinism and free will need to be related. So trying to "prove" free will with math makes as much sense as "proving" free will wi…

> Free will is a philosophical concept which is essentially dualist. It's not, although many people think this. Consider that the majority of philosophers are actually Compatibilists, in which free will is compatible with determinism, and so they would disagree with your definition. If the majority of practicing experts disagrees with your view on their subject, it's probably time to revise your view.

First, I would take strong issue at your describing the majority of philosophers as Compatiblists. I can find no support for such a broad statement. Also, one of the primary criticisms of Compatiblism, dating back as early as Kant, is that what Compatibilists define as ‘free will’ is not free will as most people, including philosophers understand it. Pulling a quick quote, the Compatibilist position can typically expressed by the following:

Arthur Schopenhauer famously said, "Man can do what he wills but he cannot will what he wills." In other words, although an agent may often be free to act according to a motive, the nature of that motive is determined.

This position that the ‘free will’ people experience is the ability to ‘decide’ to take some and then take it, while then saying the the decision to do so was predetermined, is not what is meant (especially in the vernacular) by free will.

Therefore, while I will not support a metaphysical dualist basis for ‘free will’ it is clearly incorrect to dismiss an argument of such based on 1) an appeal to authority which is unsupportable (in that your claim about the position of a majority of ‘experts’ seems unsupported), and 2) that even if it was supported, that said appeal to authority is a valid refutation of an argument. Further, dismissal of an argument by choosing to substitute a contested definitional term (the meaning of free will) for one that is clearly not the same is not rhetorically valid.

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

#196

Post by deltron3030 » Fri, Apr 17, 2020, 3:59 PM UTC

roywiggins wrote: post #166

deltron3030 wrote: post #96

Mathematics is just a language and likely a human discovery method for rules that encapsulate the universe and cascade down. A method invented to discover discoverable rules.

You can build all sorts of bizarre structures in ZFC that seem to have no relationship with reality at all. You can prove that almost all real numbers can't be computed or even described. There is an entire tower of infinities bigger than the natural numbers that telescope up and up. The axiom of choice gives you all sorts of nonphysical results. The world of ZFC doesn't feel very constrained by the actual universe,…

Sure, math/ZFC can describe potential and impossible universes including ours. But the motiviation for its invention as a language was very likely to understand relationships in our reality, physics, trade etc.

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

#197

Post by yters » Fri, Apr 17, 2020, 4:02 PM UTC

lisper wrote: post #184

The question assumes that there is a difference between invention and discovery. There isn't. Invention is a kind of discovery. The apparent distinction arises from our separation of physical and mental processes. But this is a purely artificial separation, a human conceit, because (unless you're a dualist) mental processes are physical processes. Specifically, mental processes are computational processes, which are…

So, if there is a difference between invention and discovery, then dualism is true?

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

#198

Post by pdonis » Fri, Apr 17, 2020, 4:10 PM UTC

lisper wrote: post #184

The question assumes that there is a difference between invention and discovery. There isn't. Invention is a kind of discovery. The apparent distinction arises from our separation of physical and mental processes. But this is a purely artificial separation, a human conceit, because (unless you're a dualist) mental processes are physical processes. Specifically, mental processes are computational processes, which are…

> Invention is a kind of discovery.

Some inventions can be thought of this way, but I don't think all inventions can. For a relevant example, human mathematical notation is a human invention that I don't think can be usefully thought of as a kind of discovery. (Another poster upthread mentioned Tegmark's response drawing a distinction between the structure of mathematics, which we discover, and the language we use to describe it, which we invent.)

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

#199

Post by lisper » Fri, Apr 17, 2020, 4:11 PM UTC

yters wrote: post #197

lisper wrote: post #184

The question assumes that there is a difference between invention and discovery. There isn't. Invention is a kind of discovery. The apparent distinction arises from our separation of physical and mental processes. But this is a purely artificial separation, a human conceit, because (unless you're a dualist) mental processes are physical processes. Specifically, mental processes are computational processes, which are…

So, if there is a difference between invention and discovery, then dualism is true?

Yes, though that's a bit of an odd way of putting it. It's kind of like saying that if matter can't be broken down into smaller constituents beyond a certain point then atomism must be true. It's more of a definition of dualism/atomism than a logical entailment.

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

#200

Post by jeffdavis » Fri, Apr 17, 2020, 4:13 PM UTC

h-cobordism wrote: post #30

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 silly to think that the sqaure root of negative one is something real to be discovered. As you said, the complex numbers are the unique (up to isomorphism) algebraic closure of R. Given that they arise as the unique solution to (in my opinion) a pretty fundamental question about the real numbers, I think it's fair to say that they've been "discovered", not "invented".

Real numbers might be less real than you think. Once you get beyond the number of possible quantum states in the universe, what do higher numbers really mean?