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

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

#261

Post by amarte » Fri, Apr 17, 2020, 6:32 PM UTC

Imagine an interview with a great painter. The painter is asked about the nature of painting and he responds that when he sets out to work in his studio and his brush strokes a canvass, he is discovering the fundamental nature of reality. He doubles down and exclaims that, in fact, reality is actually JUST lines and curves and shades and colors, and his proof is, well, look at how accurate his paintings are! Let's pretend that he actually is a very skilled painter, and many critics have marveled at the extent to which his paintings are indistinguishable from his subject matter. Still though, the interviewer clears his throat after an awkward pause and continues on to the next question.

To me, math is a medium of description much like painting or writing. It's units are not colors or words but points. A point, or "that which has no part", is much finer and carries with it far less baggage than something like a word. Points can be assigned numerical values and played with in clever ways. You can even sprinkle a fine dust of them over anything observable and create a copy of it to arbitrary degrees of precision.

I know math is far more than just the study of points, but I'm not convinced that math is anything more than our capacity to distinguish and describe extended to its limit. I also don't mean to belittle the accomplishments of mathematicians and theoreticians, I just think it's more reasonable to say that math is JUST the limit of description than it is to say that reality is JUST math.

After reading through some of the comments, I think many of us are on the same page.

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

#262

Post by lisper » Fri, Apr 17, 2020, 6:33 PM UTC

LolWolf wrote: post #257

pfortuny wrote: post #254

Earlier quoted context omitted.

You say > mental processes are computational processes as if this were obvious but... If you accept the possibility of random events (something deeply related to Quantum Mechanics)... There can be lots of non-computable things out there in our minds...

Such as? "Computational processes" ≈ "arbitrarily approximable by a classical computer," and QM absolutely fits in there: I just solve Schrodinger's equation for the Hamiltonian you care about to whatever precision you need. It will take exponential time, but that seems perfectly reasonable.

Exactly. Furthermore, there is no evidence that quantum processes are at all salient to what the brain does. There is nothing in the I/O behavior of a human brain that provides any evidence of quantum processes going on. There are also good theoretical reasons to believe that quantum mechanic is not salient to the operation of the human brain. The brain is thoroughly thermalized, so it decoheres in time scales that are vastly faster than any of its interesting behaviors.

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

#263

Post by techbio » Fri, Apr 17, 2020, 6:34 PM UTC

alanbernstein wrote: post #242

lisper wrote: post #233

Earlier quoted context omitted.

OK let me invent something: gnoiwiof wefoin wf oinwfe oinfowin wfeoi fwenoifwe Do you count that as in invention? If not, what distinguishes it from your "invention"?

You seem to be protesting that the bird invention is nonsense, and no more meaningful than random letters. Regardless of the answer to your question, I think you might be overlooking a different point. Here is my invention: a new piece of jewelry, a double ring, which you wear on both a finger and a toe at the same time. This is a realizable physical object that (presumably) has never existed before. Is it a discover…

This double ring may be the instrument that discovers a market for itself, or a visual/tactile aid to remind oneself of something, but in itself this invention is not a discovery of anything that can be said to have existed.

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

#264

Post by symplee » Fri, Apr 17, 2020, 6:47 PM UTC

karmakaze wrote: post #247

symplee wrote: post #172

Earlier quoted context omitted.

Isn't it the other way around? Axioms are chosen because there are no observable counter examples in the real world.

Not at all, pure math is in part about exploring axiomatic systems that may or may not have a physical counterpart. The latter is immaterial.

Can you give some examples of axioms in pure math that run completely counter to our physical world? For example:

  It is NOT possible to draw a straight line from any point to any other point.
  It is NOT  possible to extend a line segment continuously in both directions.
  etc...

or

  Things which are equal to the same thing are NOT equal to one another.
  If equals are added to equals, the wholes are NOT equal.
  The whole is LESS than the part.

Note that the original forms of the above axioms "make sense" to us because everything in our physical experience agrees with them. So when you said that the "physical counterpart ... is immaterial", I was curious to see an example of a "physically impossible" axiom.

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

#265

Post by pron » Fri, Apr 17, 2020, 6:49 PM UTC

analbumcover wrote: post #236

pron wrote: post #180

Earlier quoted context omitted.

I'm not. I'm saying that devices whose function is derived from mathematics (algorithms) have the same status as devices that derive their function from physics or chemistry. In either case, "application" doesn't mean some intellectual use but commercial use. The purpose of patents is to protect commercial applications in exchange for sharing the knowledge that led to their creation. The knowledge itself is never pro…

But that's the whole point, patents are awarded to things deemed inventions not things deemed discoveries. If you can't delineate between the two you can't say what is and what is not patentable. Fourier analysis couldn't have been patented, despite numerous commercial applications. It's just the patent system is setup to only reward low-level innovation, so it arbitrarily excludes research level innovation by termin…

Yes, what is patented is some commercial application of some knowledge, and that application is the invention. It doesn't matter whether the knowledge itself is invented or discovered. When an algorithm is patented, we're no more patenting math than we're patenting Newton's laws when a car brake is invented. We're patenting a commercial application of either, and that must be the invention.

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

#266

Post by hyperpallium » Fri, Apr 17, 2020, 6:50 PM UTC

Operational theory of mathematics: you define (invent) some rules, of starting points and ways to change them. Like the rules of a game.

In a sense, all possible outcomes already exist (only to be discovered).

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

#267

Post by phaemon » Fri, Apr 17, 2020, 6:54 PM UTC

cossatot wrote: post #152

phaemon wrote: post #110

Earlier quoted context omitted.

And yet, most blind mathematicians are geometers, which suggests your theory is incorrect.

We could perhaps modify the theory, thinking of the mechanics of research. Progress is made by an individual in the niches where others have not managed yet. Therefore the advantage in geometry for a blind mathematician may be in a different mental model of spaces. It is not surprising at all that almost all blind mathematicians are geometers. The spatial intuition that sighted people have is based on the image of th…

Ah! That's where I read it! Thanks, I read that book a year ago and couldn't remember where I'd heard that factoid.

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

#268

Post by bloomberg2020 » Fri, Apr 17, 2020, 6:57 PM UTC

amarte wrote: post #261

Imagine an interview with a great painter. The painter is asked about the nature of painting and he responds that when he sets out to work in his studio and his brush strokes a canvass, he is discovering the fundamental nature of reality. He doubles down and exclaims that, in fact, reality is actually JUST lines and curves and shades and colors, and his proof is, well, look at how accurate his paintings are! Let's pr…

Agreed

The crux of it is obvious: math is true from the perspective of a human consciousness

It’s defined after cognitive measure.

I won’t say it’s whittled down.

I would say ... it’s sight, sound, touch... attempt at quantifying our qualified experience

All the best math nerds were often deeply gifted and practiced forms of human art: poetry, music, sculpture... and many artists good at math, Brian May, for one example

IMO to get math one needs to build all those abilities

It’s really the people I know that don’t do that see math as a boring toolbox of rules and points

Not saying you’re not that

Math is not a thing like a rock because our ability to behave like more than a rock gave us the ability to define math. It has to be more than a simple this or that

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

#269

Post by bregma » Fri, Apr 17, 2020, 6:57 PM UTC

jbotz wrote: post #93

hliyan wrote: post #67

Earlier quoted context omitted.

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…

"two of our own species [discovered calculus]"..? Which two species are those? I assume by "own own" you mean Terran species, and one of them is Homo Sapiens, which is the other one? Really curious...

Take your stinking paws off me you damn dirty ape!

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

#270

Post by Koshkin » Fri, Apr 17, 2020, 6:58 PM UTC

The linguistic conundrum here arises due to the recently adopted mindless use of the word 'invented' instead of 'created' or 'designed.' (Indeed, Linus created Linux rather than 'invented' it as many would say today.) Invention is a form of discovery. So, the question should be, "is mathematics created or discovered?"