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Comment by smokel | Hacker News Reader

As much as I like philosophy of mathematics, I never feel quite at ease with sets being some universal foundation.

Lists, sets, graphs, all seem quite fundamental to us humans, but where in nature does one observe these weird things? A cave or a jug are highly complex things. Perhaps molecules resemble a graph, but if I understand physics correctly, atoms move like crazy and it's almost accidental that the graph structure is somewhat stable in most molecules.

This led me to believe that these fundamental containers are probably a byproduct of how our brains work, more than that they are fundamental outside of those.

Thanks to ChatGPT, I now know that this makes me a mathematical fictionalist, or mathematical anti-realist.

Anyone care to talk me out of this? :)

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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 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.

> 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-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.

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?

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.

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

go ahead

I prefer the HoTT/univalent foundations approach, which is about equivalences, which seem to me to capture a deep essence of what mathematics is about. Arithmetic starts with the idea that you can make some marks on the paper and manipulate them according to certain rules, and this somehow corresponds to how many sheep are in your field or what have you. Geometry starts with the idea that you can record some angles and then after your fields have flooded you can put the boundary markers back in places that are somehow the same as the places they originally were. Lists, sets and graphs are formal systems with arbitrary rules - but, crucially, they're equivalent to things we care about, and we can lift results about those formal systems (where they're easy to calculate) back along that equivalence to be facts about those real things (where they're useful).

My answer would be: when we distinguish something from something else. This is the root of all logic and cognition.

I like this answer. Been reading “Gödel’s Proof” and for a demonstration where the definition of “tautology” in the main text makes use of the concepts of True and False, there is an appendix explaining that you can arrive at the same result without those concepts, just by treating things as belonging to one class vs another (there is a one to one correspondence with True and False but the meaning is arbitrary)

"An engineer, a physicist and a mathematician have to build a fence around a flock of sheep, using as little material as possible. The engineer forms the flock into a tight circular shape and constructs a fence around it. The physicist builds a fence with an infinite diameter and pulls it together until it fits around the flock. The mathematician thinks for a while, then builds a fence around himself and defines himself as being outside.”

I think sets are fundamental to counting. An example: how many trees are in the forest? It depends on where you draw the boundary for tree. An Aspen Grove, for example have many stems of one large interconnected tree.

Try to count trees from the perspective of the sun, or from the perspective of an electron.

Trees are made up of billions of atoms, in extremely differing configurations, which can only be appreciated by machinery that is able to abstract it into the concept of a tree. To the rest of the universe it's just a bunch of atoms, with no clearly defined boundary.

So, counting is not necessarily a fundamental thing either. At least, according to how I like to interpret things :)

Excellent points. I think we are viewing "fundamental" differently. Instead of fundamental to the universe, I see counting as fundamental to the usefulness of math, which in itself is just an abstraction of the real world. (Whole Plato's cave thing)

> where in nature does one observe these weird things?

That's like asking "where in nature does one observe numbers"?

Literally everywhere.

Your jug partitions water into the set inside the jug and the set outside it.

Caves are a subset of rock formations.

All that partitioning and counting can only be performed by a system that is able to abstract and differentiate between different abstractions. Brains and computers appear to be good at this, but other than that, I don't see many other structures that are capable of doing this.

So, no, I don't observe numbers in the universe, if I take on the perspective of, say, a rock.

Um what?

Are you saying you want something that can be logically reasoned by a rock?

It's obviously not logical, and possibly not even reasonable, but it seems like a viable way to get out of the frame of reference from a human being.

I'm fine if this way of thinking is discarded as nonsense. I'd be happy to find more constructive ways to continue with this.

> I never feel quite at ease with sets being some universal foundation

That's not a big issue nowadays since there is univalent foundations, which is based on ∞-groupoids instead of sets.

Sets being the foundation is a historical accident. It is convenient and well-studied, and if you can encode your objects and axioms in set theory, you have a sure footing (at least as sure as it gets) with respect to consistency.

Otherwise, of course, lists, graphs, etc. are all objects on their own and exist independently of any encoding as sets.

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