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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]

#301

Post by jeffdavis »

thomasahle wrote: post #296
jeffdavis wrote: post #187

Earlier quoted context omitted.

"additive inverses (negative numbers) are a very real part of addition" Inverses are just an abstraction that helps you rearrange operations. You'd think something like accounting would need negative numbers, but nope, it gets along just fine without them.

They have the concept of "minus", which is just another way of adding with a negative. Also deficits and surpluses seem like normal accounting terms.

"They have the concept of "minus", which is just another way of adding with a negative."

You have chosen to generalize it into the anstract concept of negative numbers. That doesn't mean that negative numbers are a fundamental truth about nature to be discovered.

Think about it like functional programming. Is that discovering some natural law of computing? Or is it inventing a new way to express computation? Of course it must be the latter, because you can compute anything without the use of functions.

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

#302

Post by lmm »

magicalhippo wrote: post #118
lmm wrote: post #108

Earlier quoted context omitted.

> 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.

People invented the electric chair because people discovered that high-voltage electricity can kill people. You might invent a particular axiom because it has an interesting mathematical consequence (e.g. the axiom of choice was invented to allow Hilbert's basis theorem to be proven), but what you invented was the axiom, not the consequence.

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

#303

Post by ambulancechaser »

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.

Presumably you would agree that not all of discovery is also invention. So doesn't that restore the original question of asking whether mathematics is an invention or discovery which is not invention?

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

#304

Post by ShamelessC »

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…

While there is technically no proof, there is a lot of evidence that the universe is infinite. In particular it's curvature is flat within the margin of measurement error which implies an infinite universe outside the observable universe.

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

#305

Post by alanbernstein »

cmonnow wrote: post #295
alanbernstein wrote: post #242

Earlier quoted context omitted.

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…

Your comment, made by alanbernstein 3 hours ago on this website hackernews on 4/17/2020 has never existed before. Is it an invention ?

If I must choose between invention and discovery, then yes.

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

#306

Post by cmonnow »

alanbernstein wrote: post #305
cmonnow wrote: post #295

Earlier quoted context omitted.

Your comment, made by alanbernstein 3 hours ago on this website hackernews on 4/17/2020 has never existed before. Is it an invention ?

If I must choose between invention and discovery, then yes.

All you did was discover that if you typed those words on your computer, they would appear on that website on that day.

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

#307

Post by pm321 »

Phenomenit wrote: post #230
hliyan wrote: post #90

Earlier quoted context omitted.

Yes, but it's possible to perceive distict objects without vision, except the separation is in time, rather than space. Or, depends on the number of sense organs available (e.g. a blind person can sense two different objects simultaneously by holding one in each hand).

Agreed, I wonder if the same amount of abstract concept would be possible though, can you feel a graph? could you do trigonometry with just hearing and sense of touch?

"could you do trigonometry with just hearing and sense of touch?"

Can you get data on this from a Monte Carlo method of study on match play between sighted and blind players playing go and chess at distance and blind which requires a mental model of the board state and translations?

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

#308

Post by sorokod »

roywiggins wrote: post #162
sorokod wrote: post #151

Earlier quoted context omitted.

to see if Math is discovered or invented, the easy thing to do is to see if a set of axioms can be discovered or is invented A math theory arises from the axioms it is based on. You just rephrased the question and added the word "easy". Put it another way, starting from a set of axioms we get a simple ( by some semiobjective definition of simple ) pure math theory that predicts reality to a rediculous level of precis…

Math is only sometimes done that way. Often an interesting field gets axiomatized later. Calculus, arithmetic, geometry, algebra, all existed productively for centuries before axiomatization. Same way you can build a programming language without a formal spec. Yes, you might find an ambiguous piece of code later, or paint yourself into a corner. But you can do quite a lot without axioms.

Calculus, arithmetic, geometry, algebra, all existed productively for centuries before axiomatization.

and now, in the year 2020, have the axioms been invented or discovered?

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

#309

Post by alanbernstein »

cmonnow wrote: post #306
alanbernstein wrote: post #305

Earlier quoted context omitted.

If I must choose between invention and discovery, then yes.

All you did was discover that if you typed those words on your computer, they would appear on that website on that day.

So each distinct event can be discovered, regardless of how predictable it is based on previous events? Each sunrise is a discovery?

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

#310

Post by satori99 »

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...

Do Dogs Know Calculus?

http://www.math.pitt.edu/~bard/bardware/classes/0220/dkc.pdf

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