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The ideas of sets being the universal foundation of math is arbitrary as any other bit of math. Which is to say all of it is arbitrary. Just as you ask, "where

The ideas of sets being the universal foundation of math is arbitrary as any other bit of math. Which is to say all of it is arbitrary. Just as you ask, "where in nature does one observe these weird things?" I would to ask where in nature does one find a one? or a pi? or any other number really. Where do you find a triangle? Or a coordinate? Or any of the mathematical constructs we use day to day. Sure you can point at something triangle shaped and go "there!" but is it _really_ a triangle? Or just an approximation? Sure you can count one of something, but that's not the same as the number one. Just like you can't have pi of something. All of math is just a model that is surprisingly applicable to the real world. All of math is a byproduct of our brains. It doesn't exist, out there, in some Platonic World. Anyone who thinks so, I'm looking at you Max Tegmark, is mistaking the map for the territory.

NotSuspicious 3y ago
> All of math is a byproduct of our brains. It doesn't exist, out there, in some Platonic World. Anyone who thinks so, I'm looking at you Max Tegmark, is mistaking the map for the territory. I could brashly say the exact opposite and would have proven just as much as you. Fictionalists often attempt to shunt the Platonic realm into an ill-conceived emanation of the mental realm (which just so happens to itself be an accident of matter). Somehow all of this works by virtue of following a kind of mathematical logic that just so happens to not exist or something. I suspect the fundamental problem here is some kind of neurosis that psychologically compels people to reduce the quality of their thought until hard problems disappear. I hope we one day are able to build a catalog of all the ways thought goes wrong so as to prevent such nonsense from proliferating in at least some section of the world. I think a reasonably compelling way to teach yourself how to actually see The Problem (tm) is to view it from the perspective of Roger Penrose's three worlds ( https://hrstraub.ch/en/the-theory-of-the-three-worlds-penrose/ https://hrstraub.ch/en/the-theory-of-the-three-worlds-penros... ) and actually think through the implications in a contemplative, meditative way over the course of several hours. Any analysis of this issue that doesn't involve a sustained look at both logic and phenomenology is a waste of time.
- smokel 3y ago
  Thank you for that reference! It is indeed not so simple. If I continue my thought experiments about sets not being universal or foundational, I run into a myriad of problems. For one, how can one reason about anything when rejecting concepts? How can one conclude anything when rejecting logic? How can one infer anything when rejecting time? These problems seem to point to some recursive or symmetric (or circular as the article suggests) dependency between the realist and non-realist perspectives. I don't yet fully understand why there would have to be three worlds -- I'd intuitively say that two (e.g. physical and mental) suffice. The platonic world might simply follow from the mental one, or vice versa. I'll put in several hours of thought and report back in the next post that touches upon this subject. I concluded for myself that logic, science, nor philosophy are going to be of much help with this. I therefore turned to contemporary art, where such thought still has some kind of validity. Let's see where that leads me :) Edit: It seems that the "three world" idea is originally an idea by Karl Popper. Wikipedia [1] explains this in some detail, from which it becomes clear why the thought experiment has three, not two worlds. [1] https://en.m.wikipedia.org/wiki/Popper%27s_three_worlds https://en.m.wikipedia.org/wiki/Popper%27s_three_worlds
  - zehaeva 3y ago
    Part of the reason that you're going to run into trouble with sets being universal, or foundational, rightly found in a theorem that was birthed from set theory, Gödel's Incompleteness Theorem[0]. I will confess that when I took discrete in college I was seduced by set theory. I literally had the, naive, thought to myself "you could prove all of math with just sets!". As my education continued I found out, much to my chagrin, and Hiblert's[1], that I was very, very, mistaken. This hasn't stopped everyone from trying to continue Hilbert's program, though with a bit more limited scope[2][3]. By and large, all math is undergirded by a set of axioms that have to be taken as true with no proof. Even as far back as Euclid's Elements basically starts with a set of axioms and then proceeds from there. Strange that something that is so real must have a bunch of rules given as true with no proof of their validity beyond "well, everyone can see that it's true". In a final, ish, dig, I'll just say leave it up to Penrose to take Popper's sensible cosmology and turn it into a quasi-religious one. [0] https://en.wikipedia.org/wiki/G%C3%B6del%27s_incompleteness_theorems https://en.wikipedia.org/wiki/G%C3%B6del%27s_incompleteness_... [1] https://en.wikipedia.org/wiki/Hilbert%27s_program https://en.wikipedia.org/wiki/Hilbert%27s_program [2] https://en.wikipedia.org/wiki/Zermelo%E2%80%93Fraenkel_set_theory https://en.wikipedia.org/wiki/Zermelo%E2%80%93Fraenkel_set_t... [3] https://en.wikipedia.org/wiki/Reverse_mathematics https://en.wikipedia.org/wiki/Reverse_mathematics

NotSuspicious 3y ago

> All of math is a byproduct of our brains. It doesn't exist, out there, in some Platonic World. Anyone who thinks so, I'm looking at you Max Tegmark, is mistaking the map for the territory. I could brashly say the exact opposite and would have proven just as much as you. Fictionalists often attempt to shunt the Platonic realm into an ill-conceived emanation of the mental realm (which just so happens to itself be an accident of matter). Somehow all of this works by virtue of following a kind of mathematical logic that just so happens to not exist or something. I suspect the fundamental problem here is some kind of neurosis that psychologically compels people to reduce the quality of their thought until hard problems disappear. I hope we one day are able to build a catalog of all the ways thought goes wrong so as to prevent such nonsense from proliferating in at least some section of the world. I think a reasonably compelling way to teach yourself how to actually see The Problem (tm) is to view it from the perspective of Roger Penrose's three worlds ( https://hrstraub.ch/en/the-theory-of-the-three-worlds-penrose/ https://hrstraub.ch/en/the-theory-of-the-three-worlds-penros... ) and actually think through the implications in a contemplative, meditative way over the course of several hours. Any analysis of this issue that doesn't involve a sustained look at both logic and phenomenology is a waste of time.

smokel 3y ago
Thank you for that reference! It is indeed not so simple. If I continue my thought experiments about sets not being universal or foundational, I run into a myriad of problems. For one, how can one reason about anything when rejecting concepts? How can one conclude anything when rejecting logic? How can one infer anything when rejecting time? These problems seem to point to some recursive or symmetric (or circular as the article suggests) dependency between the realist and non-realist perspectives. I don't yet fully understand why there would have to be three worlds -- I'd intuitively say that two (e.g. physical and mental) suffice. The platonic world might simply follow from the mental one, or vice versa. I'll put in several hours of thought and report back in the next post that touches upon this subject. I concluded for myself that logic, science, nor philosophy are going to be of much help with this. I therefore turned to contemporary art, where such thought still has some kind of validity. Let's see where that leads me :) Edit: It seems that the "three world" idea is originally an idea by Karl Popper. Wikipedia [1] explains this in some detail, from which it becomes clear why the thought experiment has three, not two worlds. [1] https://en.m.wikipedia.org/wiki/Popper%27s_three_worlds https://en.m.wikipedia.org/wiki/Popper%27s_three_worlds
- zehaeva 3y ago
  Part of the reason that you're going to run into trouble with sets being universal, or foundational, rightly found in a theorem that was birthed from set theory, Gödel's Incompleteness Theorem[0]. I will confess that when I took discrete in college I was seduced by set theory. I literally had the, naive, thought to myself "you could prove all of math with just sets!". As my education continued I found out, much to my chagrin, and Hiblert's[1], that I was very, very, mistaken. This hasn't stopped everyone from trying to continue Hilbert's program, though with a bit more limited scope[2][3]. By and large, all math is undergirded by a set of axioms that have to be taken as true with no proof. Even as far back as Euclid's Elements basically starts with a set of axioms and then proceeds from there. Strange that something that is so real must have a bunch of rules given as true with no proof of their validity beyond "well, everyone can see that it's true". In a final, ish, dig, I'll just say leave it up to Penrose to take Popper's sensible cosmology and turn it into a quasi-religious one. [0] https://en.wikipedia.org/wiki/G%C3%B6del%27s_incompleteness_theorems https://en.wikipedia.org/wiki/G%C3%B6del%27s_incompleteness_... [1] https://en.wikipedia.org/wiki/Hilbert%27s_program https://en.wikipedia.org/wiki/Hilbert%27s_program [2] https://en.wikipedia.org/wiki/Zermelo%E2%80%93Fraenkel_set_theory https://en.wikipedia.org/wiki/Zermelo%E2%80%93Fraenkel_set_t... [3] https://en.wikipedia.org/wiki/Reverse_mathematics https://en.wikipedia.org/wiki/Reverse_mathematics