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You could brashly say that the Platonic world does exist? I must be from Missouri, show it to me! Do that and you can bring me to your, and Max Tegmark's, side. Show me a Tree that has the pure essence of tree, show me a One, show me the pure ideal of a triangle.

I do take umbrage at your insinuation that I have not actually thought "through the implications in a contemplative, meditative way over the course of several hours" on this subject. I have spent many years, and not a few semesters of college, contemplating this very subject.

Consider this, where would your platonic world go if there were no sentient beings in the universe? Would it sit there, on some "higher plane" awaiting discovery by no one? Would the pure idea of a Tree get lonely? We're very into "does a tree fall in the forest with no one to hear it does it make a sound?" territory, however I think this is extremely important when trying to decide what is Real.

Plato, a man who lived ~2,400 years ago, decided that ideas were Real with zero proof and you're just going to accept his word on this, I'm supposed to just accept this? This does strike me as extremely life-centric, for lack of a better, all encompassing, term. If all life in the universe disappeared, the universe itself does not suddenly vanish in a puff of smoke. Stars will still fuse the elements, blackholes will still gobble up matter, the earth will continue to orbit the sun until it's consumed by the sun or is disrupted by some massive interstellar traveler. But the world of ideas wouldn't exist because there would be nothing to think of them. Mathematics wouldn't exist, because there would be nothing to conceive of them.

If you assume the platonic realm exists, sure, I would grant you all of Penrose's Three Worlds, I would grant Tegmark's belief that somewhere out there is a physical Platonic realm. I'd also probably believe in a lot of other things with out evidence as well.

But, maybe I'm just from Missouri. You're gonna have to show me.

Playing devil's advocate here. Mathematics and concepts can be shown in the regular way: in books, or in configurations of bits. Their meaning is generally lost though, unless there is someone to interpret it.

I think we should be careful about the contexts that we are discussing things in.

If we consider, for example, a string of text that contains the King James Bible, and we assume that the string "exists", that does not imply that the actual stories in the text exist, let alone that the characters featuring in it exist.

In the above example, existence has three different meanings.

It would be great if philosophy would be able to use strict type checking on their arguments :)

It would also be nice if I could simply state my philosophical dependencies in a plain text file.

I do agree that they can be "shown" or demonstrated, I don't think that's at argument here. And to be specific I chose the phrase Platonic World, or Platonic Realm specifically because it is the philosophical notion that ideas are real in the sense that the keyboard I am typing this message on is real, that those ideas exist independent of humans, and indeed independent of any living/sentient existence.

I called out Max Tegmark, partially because I find it humorous and, specifically because he has said that he believes that there exists a world where math is real, or maybe that the world is only math. It stems partly from his view of the multiverse. He makes an easy "punching bag" for this sort of thing.

So, yes, existence has many meaning in your example. I specifically called out one where ideas have a real existence independent of our reality or any subjects that operate within it.

I believe we are in agreement on this. Intuitively, and under the assumption that the physical world exists (although I have no clue why or how it exists), I feel that mathematics is formed inside of the physical world. The experience of being able to practice mathematics probably emerges from brain structures which have evolved in the physical world.

Still, there are some problems that I run into when taking these thoughts further. How, for instance can one apply deductive reasoning or apply Occam's razor in a context where these are not available?

I am also intrigued by your earlier remark that "math is just a model that is surprisingly applicable to the real world." (emphasis mine). This brings to mind "The Unreasonable Effectiveness of Mathematics in the Natural Sciences". Perhaps there is an easy way out for believers of anti-realism.

Would it be an interesting hypothesis to say that the real (physical) world that we observe is limited exactly by the way that our sensors and brains take shape in it? I'd like to think of this as the antithesis to "in the beginning there was nothing" -- I'd rather think that outside our physical world "there is all and everything"; we just seem to be able to reflect only on part of it. The unreasonable effectiveness of mathematics hints at a correlation between how brains work and what physical laws there are. Perhaps Emmy Noether's ideas on symmetry may lead to some clues here as well.

In this way it would not be surprising at all that mathematics is applicable to the real world, as it is so almost by definition. This is obviously not the same interpretation as Max Tegmark's, but it does hint at some kind of interplay between a mathematical world and a physical world.

Unfortunately, I can only make this theory work for myself intuitively. I have no grasp on what it means that the physical world is part of something bigger. In a way, it seems to be moving the goalposts, similar to how some people believe we are somewhere in a nested series of simulations. And I feel quite uncomfortable in using logic, concepts, abstractions and what have you, which are part of the human brain context, and possibly not of the context that I magically believe our physical world to reside in.

Something one should consider is that while math is surprisingly applicable, and as you correctly picked up I was making a reference to the "Unreasonable Effectiveness" quote, it is never exact.

Newton's laws are enough for us to fling rockets and robots to Mars, but they are not good enough for us to create our GPS system. And Relativity is amazingly good, but still not good enough to model black holes, dark matter, and dark energy. The breadth of equations in Quantum Mechanics are also supremely successful, and yet they don't work well in the realm of Relativity. The Standard Model doesn't know what dark matter or dark energy is.

So yes, all of this math we have is Unreasonably Effective. But it's still a model, and a model that is not 100% correct. We have gaps in our models and as we figure out better and better approximations for them we move to them.

In my first post I made a small comment about those who are Platonist "mistaking the map for the territory". This is a logical fallacy where one is confusing/conflating the semantics (in this case mathematics) with what it represents, reality.

Math, and by extension logic and any other model, or heuristic, that we use to make our way through this world is the map, it is amazingly effective. Just because a map is not the territory does not mean it's not useful.

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

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_... [1] https://en.wikipedia.org/wiki/Hilbert%27s_program [2] https://en.wikipedia.org/wiki/Zermelo%E2%80%93Fraenkel_set_t... [3] https://en.wikipedia.org/wiki/Reverse_mathematics

I wonder if you could prove it all with G. Spencer Brown's Laws of Form?

> I could brashly say the exact opposite and would have proven just as much as you.

go ahead