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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]

#231

Post by jhedwards »

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…

OK let me invent something:

A fignaft is a small pink bird from the island of slamp. It has a short triangular beak well adapted to plucking seeds out of the hard cones of the humnit bush. It was nearly hunted to extinction by the umaglots, until their chief shaman declared the bird to be a repository of the souls of men lost at sea. Since then it has made a remarkable recovery.

In contrast to the mental process required to invent an imaginary bird such as this, the folks who discovered DNA used tools to image cell internals such that the structure of the contents was revealed.

When DNA was discovered, something about the structure of the external world was made clear to us, and many other processes could be clarified as a result. The fignaft bird, on the other hand, only reveals the contents of my mind, and is constructed entirely of existing knowledge.

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

#232

Post by lisper »

dorgo wrote: post #223
lisper wrote: post #216

Earlier quoted context omitted.

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.

lol, yes, you discover how mathematicians think, but what they think is still an invention. In the end, I agree that if we expand the definition of the universe beyond physical world to contain "everything" then yes, every invention is also a discovery.

> what they think is still an invention

Is there anything that any human thinks (other than straight-out mimicry) that is not an invention?

If I type:

squirgle florb snozbar

would you say that I have invented something? Because if the answer to that is yes, then I'll concede the point (and observe that "invention" is not a particularly interesting concept). But if the answer is no, then I challenge you to draw a distinction between what I just did and what mathematicians do in a principled way.

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

#233

Post by lisper »

jhedwards wrote: post #231
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…

OK let me invent something: A fignaft is a small pink bird from the island of slamp. It has a short triangular beak well adapted to plucking seeds out of the hard cones of the humnit bush. It was nearly hunted to extinction by the umaglots, until their chief shaman declared the bird to be a repository of the souls of men lost at sea. Since then it has made a remarkable recovery. In contrast to the mental process requ…

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"?

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

#234

Post by jeffdavis »

Y_Y wrote: post #74
yiyus wrote: post #27

Earlier quoted context omitted.

Depending on how philosophical you want to get, it is not so clear what exists and what does not. Neither it is absolutely clear what means to be discovered. I think that ideas can be discovered. In my opinion, theorems, or more precisely the proof of these theorems, for instance, are discovered. In the sense that they are not an invention, they "emerge" from more fundamental definitions. You seem to be of the opinio…

On the contrary. I think that "real" and "imaginary" numbers are both discovered and neither concept has more reality than the other. I think that even chess rules have a meaningful "existence" (in a Platonic sense) even if they are in some sense arbitrary. We can't discover "the one true rule system" but we can discover various possible systems and also theorems about them.

Why aren't real and imaginary numbers both invented?

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

#235

Post by pdonis »

lisper wrote: post #218
pdonis wrote: post #212

Earlier quoted context omitted.

> 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.

> it's an observation that there is no sharp line between invention and discovery

This is true of pretty much any human categories, yes. That does not mean the categories aren't valid or that they should all be collapsed into one.

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

#236

Post by analbumcover »

pron wrote: post #180
analbumcover wrote: post #142

Earlier quoted context omitted.

I don't see how you are distinguishing between inventions and discoveries. The "discovery" of a mathematical proof is just an application of knowledge, why aren't they patentable? The "invention" of the light bulb, on the other hand, is just the discovery that putting various particles together in a certain configuration produces light in a predictable manner, why is this patentable? I don't see how a clear delineati…

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 terming it discovery.

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

#237

Post by nikofeyn »

Reedx wrote: post #176

Same question also answered by Max Tegmark: https://www.youtube.com/watch?v=ybIxWQKZss8 Stephen Wolfram: https://www.youtube.com/watch?v=nUCwtLTUPQ4 Steven Weinberg: https://www.youtube.com/watch?v=NpMk9G-ddiM I particularly liked Max's answer who neatly makes the distinction between the structure (we discover) and the language (we invent). We're free to invent the names, but not the structures.

> I particularly liked Max's answer who neatly makes the distinction between the structure (we discover) and the language (we invent). We're free to invent the names, but not the structures.

i came here to give my opinion, but this is pretty much it. i need to read max tegmark's books.

the idea that how we got to the structures and concepts we find is undoubtedly invented, because we've already seen that we've re-invented our methods multiple times. but the structures and concepts don't really change. we extend them, but there seems to be some set of structures and concepts that get settled to. if we found some alien species that has mathematics, we'd probably find that there's some mapping between our structures modulo small choices that were made.

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

#238

Post by lisper »

pdonis wrote: post #235
lisper wrote: post #218

Earlier quoted context omitted.

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.

> it's an observation that there is no sharp line between invention and discovery This is true of pretty much any human categories, yes. That does not mean the categories aren't valid or that they should all be collapsed into one.

Fair enough. But notwithstanding, I contend that my claim is not vacuous. I would say that this particular failure of categorization actually reveals a deep truth, namely, that computation is universal.

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

#239

Post by naasking »

Tainnor wrote: post #205
naasking wrote: post #178

Earlier quoted context omitted.

> Yep, the counterpoint to the unreasonable effectiveness of maths in the natural sciences is accompanied by the unreasonable ineffectiveness of maths in the social sciences, humanities, etc. Except it's not ineffective at all in these subjects. Social science experiments have so many variables that the small experiments that can be conducted given the financial resources are insufficient to infer a good model. This…

If it's ineffective in practice, I consider it to be ineffective. But even if you could design an experiment perfectly and come up with some strong statistical evidence and then had other means to tease out what is actually causal and what is only correlational, you'd only know what influences what and not necessarily why. But yes, I did say that statistics/probability is some rare exception. You can study group theo…

> If it's ineffective in practice, I consider it to be ineffective.

Except you have no way to conclude that it's ineffective. Analytical solutions require data to study. Without data, or with little data, what analysis are you going to perform? At best, broad statistical correlations, which is exactly what we find.

You're effectively claiming that spoons are ineffective at a restaurant that provides only forks. Well no, if a spoon were available, then it would probably work just fine.

> but I haven't yet heard of anyone who has studied topology or galois theory and found that incredibly useful for understanding social phenomena better.

After 5 minutes of Googling:

* Power laws: https://en.wikipedia.org/wiki/Power_law#General_science

* Network theory: https://en.wikipedia.org/wiki/Network_theory

* There's a journal specifically for mathematical social sciences: https://www.journals.elsevier.com/mathematical-social-scienc...

* Economics, game theory, and social choice theory are all examples employing heavy analytical problem solving to social problems

* Quantum mechanics applied to social sciences: https://www.cambridge.org/core/books/quantum-social-science/...

The main stumbling block is that mathematicians are interested in mathematical problems, and so they make a common but mistaken assumption that social sciences either don't have such problems, or they are too messy for elegant math. Take it from a mathematician, this is incorrect: https://www.mathtube.org/sites/default/files/lecture-notes/S...

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

#240

Post by lavp »

I find mathematics to be the most beautiful thing in the universe, because it is the only thing I can truly think of as being perfect in every way. Existing only as an abstract concept, it still manages to find its way into every single aspect of our lives.

Two completely remote civilizations will still have the same mathematics. Sure one may have a more developed understanding, but if both civs wondered about how to get the hypotenuse of a right-angled triangle, they would both end up with Pythagoras' theorem. The only thing invented in math is our language and representation of such concepts.

Many people believe that mathematics becomes invented the further up you go like pure mathematics and I can understand why this perspective would come through, but take the example of G.H. Hardy who famously said “The Theory of Numbers has always been regarded as one of the most obviously useless branches of Pure Mathematics”. It could be argued at the time that these branches were simply invented mathematics because there was no practical application of it, but 30 years following his death came breakthroughs in cryptography. All of a sudden, this was no longer an invention. Calculus is not seen as an invention but a discovery because of all of its insane number of applications in physics and other areas.

Many areas of math that have yet to find practical use will always come under the scrutiny of being 'invented' but that simply isn't the case.

I firmly believe that all areas of mathematics are practical to the universe and its wonders (and therefore discovered), whether or not we can achieve a level to use such mathematics (or experience it) is a different matter.

I believe that at its core, mathematics is the universe. To say that we have invented it is arrogant and completely strives it of its beauty.

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