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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]

#251

Earlier quoted context omitted.

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…

You may have a point. And I did forget about things like social choice theory or game theory (although I'd assume that partially this is also due to e.g. social choice procedures or "games" often being very limited and artificial settings where by their very nature the relevant space of options/outcomes can be explored in some systematic way, which is generally less the case in more organic, complex settings, such as e.g. gradual societal changes).

When it comes to economics, I know that a lot of people don't agree with the basis of many mathematical models that are used, but I'm not an economist, so I can't speak to that.

So maybe I was overzealous in discounting mathematics for the social sciences altogether. Still, I would contend (albeit with much less evidence):

- There's some measure of people trying to construe "nice models" of things in those subjects instead of trying to make sure they agree with reality. I have a degree in linguistics and I've seen this over and over. Most of these models haven't convinced me at all.

- The amount of maths that you either need or at least benefit from in order to do good research in such areas is still substantially lower than in, say, physics. I think it's still to be noted how we can describe much of physics just with a small number of economic and elegant models. I haven't seen anything comparable in any of the social sciences.

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

#253
post #244

Earlier quoted context omitted.

I would love to see a proof of it if you happen to have it!

https://fsw01.bcc.cuny.edu/luis.fernandez01/web/texts/dgcs.p... Ctrl-F for "2D" (2-dimensional creatures), although I can't find the "discovery", maybe I remember it wrong. Btw, this was one of the mindfucks I learned studying physics. Other ones are: - dimensional analysis: checking if formulas can be wrong, and more importantly, finding formulas (eg, period of a pendulum) just by analyzing units that can be involve…

This is brilliant, thank you!

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

#254
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…

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...

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

#255
post #49

If you only look at mathematics I think it's simply: - Axioms are invented - Conclusions are discovered The magic part for me is that some axioms have been chosen so well that their conclusions are confirmed in the real world.

Oh no: the axioms come much much later. The order is exactly the reverse one.

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

#256
post #67

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…

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…

Assuming a species is intelligent enough to discover math, if they have some significantly different way of experiencing the world I think they would take a different path but would eventually reach the same destination we are heading towards, likely merging with us by the point we are at now.

I could see the layman understanding of geometry being quite different than ours, limited by their own visual system. But their mathematicians and our mathematicians would discover the same math, through different paths and perhaps with different strengths and weaknesses and a different path of educating budding mathematicians from the grade school level to the post doc level.

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

#257
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…

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.

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

#258

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...

This. It is discovered in the same sense that humans "discover" facts about humans via science- we're not born with perfect self-knowledge. It is "created" because there is no guarantee that our efforts correspond to reality. In that sense we are just playing in the sandbox of what we can conceive. The fact that mathematics corresponds so "unreasonably" to objective reality is because what we call objective reality i…

>It is "created" because there is no guarantee that our efforts correspond to reality. In that sense we are just playing in the sandbox of what we can conceive.

But doesn't this dance kinda near the notion that I am a brain in a vat and none of you exist (read this question with me as the speaker or with yourself as the speaker)?

I would say our brains do have limits. I can't well envision a 4D object. But once we reduce things down to simple logical axioms and constructs, these exist as much as anything can be said to exist. Even if I was a simulation that didn't even have a brain, much less eyes, the concepts I come up with would exist more than the flesh I incorrectly thought I had.

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

#259
post #49

If you only look at mathematics I think it's simply: - Axioms are invented - Conclusions are discovered The magic part for me is that some axioms have been chosen so well that their conclusions are confirmed in the real world.

Oh no: the axioms come much much later. The order is exactly the reverse one.

Well, I have to agree. From a practical perspective.

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

#260
It is a dispute in ontology. Do mathematical objects (say, numbers, sets) exist in the world? Some say, yes; others say, no; some others say, they exist in another world--called Platonic world.

We see similar disputes in philosophy of (natural) sciences: for instance, instrumentalism doesn't subscribe to the ontology of realism.

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