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

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

#211

Post by lisper » Fri, Apr 17, 2020, 4:30 PM UTC

pdonis wrote: post #206

lisper wrote: post #201

Earlier quoted context omitted.

> human mathematical notation is a human invention that I don't think can be usefully thought of as a kind of discovery Why not? It's a discovery of visual features that the human brain and unaided eye can easily discern, and the discovery of a correspondence of those visual features to concepts that the human brain can easily remember. All of these are (on the assumption that dualism is false) physical processes.

> It's a discovery of visual features that the human brain and unaided eye can easily discern, and the discovery of a correspondence of those visual features to concepts that the human brain can easily remember. Hm. Yes, I suppose you could look at it this way. However, there is still an element of arbitrariness or choice involved that is not present in the discovery of mathematical structures themselves. All humans…

> there is still an element of arbitrariness or choice involved that is not present in the discovery of mathematical structures themselves

That's not true. There is this thing in math called the "axiom of choice" which you can, quite literally, choose to accept or not. And math is chock-full of this kind of thing. Even geometry comes in different flavors: Euclidean and non-Euclidean. And we don't even know which of them corresponds to physical reality on cosmological scales!

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

#212

Post by pdonis » Fri, Apr 17, 2020, 4:31 PM UTC

lisper wrote: post #207

pdonis wrote: post #204

Earlier quoted context omitted.

> The apparent distinction arises from our separation of physical and mental processes. I'm not sure I understand this either. I think our ordinary intuitions would say that both invention and discovery are mental processes. They're just different kinds of mental processes: invention has an element of arbitrariness or choice in it that discovery does not. I have no problem with saying that mental processes are physic…

> invention has an element of arbitrariness or choice in it that discovery does not Yes, I can understand how someone could reach this conclusion. I used to believe it myself. But if you think about it you will realize that this appearance of "arbitrariness or choice" is an illusion. Take the airplane for example. Was it invented or discovered? Is there a relevant distinction between the "invention" of the airplane a…

> if you think about it you will realize that this appearance of "arbitrariness or choice" is an illusion.

I don't think that's the illusion. I think the illusion is that something as complex, for example, as "the airplane" has to be pigeonholed as either an invention or a discovery, instead of being a complicated process in which both invention and discovery were involved.

> Any story of invention can be re-cast as a story of discovery.

I think this depends on how far you are willing to stretch the meaning of the word "discovery".

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

#213

Post by Tainnor » Fri, Apr 17, 2020, 4:34 PM UTC

guerrilla wrote: post #189

Tainnor wrote: post #150

Earlier quoted context omitted.

There is a distinction between logic, which is just the study of formal systems where you can prove theorems as in any other area of mathematics, and the philosophy of maths and particularly of mathematical logic. The same applies for set theory. The first one can assert that (in classical first order logic) e.g. there are uncountable models of the natural numbers; this is irrefutable. The second one asks things like…

Irrefutable and "unequivically" true if you take some classical FOL as a productive method of producing knowledge, a philosophical assumption. Philosophy came first, mathematical logic is just a formalization of that and the reason it was formalized at all and not somethimg else is epistemological. Your entire argument attempting dismiss philosophy is philosophy.

I didn't dismiss philosophy.

I said that as soon as you fix some axioms (such as those of classical FOL), the conclusions are irrefutable. This is where mathematics begins. The question where those axioms come from or whether they are "true" are philosophical. The two disciplines are related, but separate.

Lots of mathematicians have different "foundational" beliefs from each other; some have studied or deeply thought about the philosophy, others may just speak to their intuition. However, this doesn't change the fact that they all come to the same conclusions from the same premises. E.g. a constructivist wouldn't be able to claim that a classical proof is "wrong", only that it's non-constructive and therefore unacceptable for some (philosophical or practical) reason; in fact non-constructive proofs can be seen as constructive proofs of some meaningless strings (e.g. the constructivist will maybe dispute the fact that there are discontinuous functions, but they will certainly accept the existence of a first-order derivation of the string representing "not all functions are continuous" from the axioms of set theory), the constructivist would just dispute that there is any meaning to these strings...

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

#214

Post by red75prime » Fri, Apr 17, 2020, 4:35 PM UTC

kevin_thibedeau wrote: post #145

red75prime wrote: post #106

Earlier quoted context omitted.

Arithmetic existed long before its axiomatization. Arithmetic was useful and no one stumbled upon contradictions in it. So it was natural to suppose that it can be described by some axiomatic system. Peano found it.

It is a system for modeling concepts invented by man. Everything that falls out of such a system is a product of the invention. Numbers don't inherently exist. Everything derived from that concept can't be a "discovery".

> Numbers don't inherently exist

How do we know that it is true?

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

#215

Post by pdonis » Fri, Apr 17, 2020, 4:35 PM UTC

lisper wrote: post #211

pdonis wrote: post #206

Earlier quoted context omitted.

> It's a discovery of visual features that the human brain and unaided eye can easily discern, and the discovery of a correspondence of those visual features to concepts that the human brain can easily remember. Hm. Yes, I suppose you could look at it this way. However, there is still an element of arbitrariness or choice involved that is not present in the discovery of mathematical structures themselves. All humans…

> there is still an element of arbitrariness or choice involved that is not present in the discovery of mathematical structures themselves That's not true. There is this thing in math called the "axiom of choice" which you can, quite literally, choose to accept or not. And math is chock-full of this kind of thing. Even geometry comes in different flavors: Euclidean and non-Euclidean. And we don't even know which of t…

> There is this thing in math called the "axiom of choice" which you can, quite literally, choose to accept or not.

No, there are these different mathematical structures: set theory with the axiom of choice, set theory with the negation of the axiom of choice, and set theory with neither. (More precisely, "Zermelo-Frankel set theory", since there are other set theories.) All of those structures were discovered. And, as I said, different humans invented different notations to describe these different structures.

Humans choose which of these structures to use for particular applications, yes. I don't think that process is either invention or discovery. Not everything humans do has to be an invention or a discovery.

Similar remarks apply to geometry.

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

#216

Post by lisper » Fri, Apr 17, 2020, 4:38 PM UTC

dorgo wrote: post #210

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…

>The apparent distinction arises from our separation of physical and mental processes. You make the distinction based on (physical or mental) processes involved in the discovery/invention. I wouldn't care so much about how the discovery/invention came to be and look more into what the discovery/invention is about. Usually, you can discover things which are present in the real world. Mathematicians are rarely interest…

But mathematicians are present in the real world, so you can choose to see math as the discovery of the behavior of the brains of mathematicians.

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

#217

Post by kevin_thibedeau » Fri, Apr 17, 2020, 4:38 PM UTC

red75prime wrote: post #214

kevin_thibedeau wrote: post #145

Earlier quoted context omitted.

It is a system for modeling concepts invented by man. Everything that falls out of such a system is a product of the invention. Numbers don't inherently exist. Everything derived from that concept can't be a "discovery".

> Numbers don't inherently exist How do we know that it is true?

They are symbols that we assign arbitrary meaning to. They are useful because of the axiomatic framework constructed to support them.

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

#218

Post by lisper » Fri, Apr 17, 2020, 4:40 PM UTC

pdonis wrote: post #212

lisper wrote: post #207

Earlier quoted context omitted.

> invention has an element of arbitrariness or choice in it that discovery does not Yes, I can understand how someone could reach this conclusion. I used to believe it myself. But if you think about it you will realize that this appearance of "arbitrariness or choice" is an illusion. Take the airplane for example. Was it invented or discovered? Is there a relevant distinction between the "invention" of the airplane a…

> if you think about it you will realize that this appearance of "arbitrariness or choice" is an illusion. I don't think that's the illusion. I think the illusion is that something as complex, for example, as "the airplane" has to be pigeonholed as either an invention or a discovery, instead of being a complicated process in which both invention and discovery were involved. > Any story of invention can be re-cast as…

No, it's an observation that there is no sharp line between invention and discovery. If you want to dispute that, you'll have to tell me how I can distinguish between the two.

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

#219

Post by dorgo » Fri, Apr 17, 2020, 4:40 PM UTC

lisper wrote: post #207

pdonis wrote: post #204

Earlier quoted context omitted.

> The apparent distinction arises from our separation of physical and mental processes. I'm not sure I understand this either. I think our ordinary intuitions would say that both invention and discovery are mental processes. They're just different kinds of mental processes: invention has an element of arbitrariness or choice in it that discovery does not. I have no problem with saying that mental processes are physic…

> invention has an element of arbitrariness or choice in it that discovery does not Yes, I can understand how someone could reach this conclusion. I used to believe it myself. But if you think about it you will realize that this appearance of "arbitrariness or choice" is an illusion. Take the airplane for example. Was it invented or discovered? Is there a relevant distinction between the "invention" of the airplane a…

> "arbitrariness or choice" is an illusion

after reading a little about quantum mechanics I'm not so sure any more.

>Even the arts can be seen as a process of discovery, specifically, the discovery of what processes and artifacts produce favorable reactions in human brains.

That assumes that the artists test their work against real audience and adjust their style accordingly. Maybe they do, but I'm sure that there are artists out there who just create art.

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

#220

Post by ukj » Fri, Apr 17, 2020, 4:41 PM UTC

guerrilla wrote: post #190

ukj wrote: post #78

Earlier quoted context omitted.

You didn't make a point. You asked a loaded question. It backfired on you.

My question asking if you value products of philosophy that have become essential to our society neither contains an unjustified assumption and thus is not a loaded question and still stands and thus has not backfired.

It was loaded.

It contains an unjustified assumption that “logic, the art of making distinctions, the study of critical thinking and the foundation of mathematics" and [other?] art belong in different categories.

It also takes an incomplete view on "critical thinking". Drawing distinctions is complemented by abstracting similarities.

The creation of knowledge (in all its forms) is itself a form of artistic self-expression. It is essential to humans, and therefore essential to society.

As a programmer my medium of self-expression is software. I am an artist as much as I am a logician and a scientist.

What I do is create. It also happens to be useful to others, which is why it pays fucking well too.