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What is math? A teenager asked that age-old question on TikTok, creating a viral backlash, and then, a thoughtful scientific debate
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- What Is Math?
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I think it has something to do with the distinction between possible and actual existence. In “The Library of Babel” Borges described a library of all possible books. These books “exist” in a similar way that mathematical solutions “exist.” That is, they don’t really exist until you make them, but they’re possible. To say that a solution exists really means that a solution is possible to find, in some sense.
Some senses of possible existence are more practical than others. It gets a bit confusing when you have easy and automatic ways to construct things. For example, a computer program that calculates and prints a number can be said to represent that number without actually running the program. Many video games generate procedural worlds on demand. In some sense all possible Minecraft worlds might be said to exist, as possibilities anyway, but the region beyond your view won’t be generated until you get there.
The lines can get pretty blurry as the possibility of construction gets more tenuous. Do the contents of an encrypted file exist if you lost the key? Sort of, not in any practical sense, but it is possible in an astronomically unlikely sense of “possible” that you could decrypt it by guessing blindly, or that the algorithm isn’t as secure as we assume.
Does a mathematical solution really exist if you have no specific way to find it? Constructivist math is stricter about this than some other kinds of math, and it seems that it doesn’t matter what you pick, at least for any practical purpose. And even when constructivist math says a solution is possible, you might not be able to run the program in any practical way.
Even for something as seemingly simple to construct as the natural numbers, thinking too hard about existence results in some gray areas. You’re not going to print them all out, after all, and nearly all natural numbers are too large and random to define in any practical way, so their possibility of existing in any practical sense seems even more tenuous than the contents of an encrypted file without its key. But there’s no clean boundary between practical and impractical natural numbers, so it is more practical (mathematically) to assume their (possible) existence, without making any distinctions.
This is only the start of a slippery slope causing mathematicians to believe in all sorts of technically-possible (practically impossible) entities that we will never see. Our world is finite, so infinities are assumed to exist for mathematical convenience, not in any practical, real-world way.
That's all based on the premise that numbers need to be "practical", or have some kind of real world equivalency, though, which is not what modern (and by modern I mean like post 1800s) math ever tried to do. Modern math is more rigorous than that.
There are many numerical systems that stem from different axioms, some of which happen to be a good model for things in real life. Like the real numbers, for instance. But that is intentional; the real numbers were constructed by mathematicians because they had useful properties which were often reflected in the real world. These were chosen.
There are many, infinite rather, other systems. And they are useful; all encryption is done in either a galois field or an algebra based on elliptic curves, both of which will be very alien from the "real life". But they enable just about everything in computing today.
It's not a belief, though, because no mathematical system has any direct relation to reality. In the extended real number line, infinity exists (and ruins all the field axioms, but regardless)... and that's that. It exists, because in this system we enforce that it exists, and that 1/0 = infinity = -1/0. That choice propagates in all kinds of interesting ways.
I used natural numbers as an example that everyone is familiar with, but yes, there are a lot of entities that exist as mathematical terms that can thought of and written down. Symbols and rules can be instantiated just like anything else that can be written down and that’s quite concrete. It may be that we have a symbol for a set that contains entities that mostly can’t be written down, though, even in principle.
Even when we’re not talking about numbers, practicality will come in again in the sense that some proofs may conceptually exist that are far too long to ever be written down, but there is no mathematically clean way to distinguish between practical and impractical proofs, so we don’t, or at least not when describing proofs mathematically.
It also seems like a lot of mathematical terms are metaphorically related to practical terms? In particular, “exists”. Mathematical existence isn’t the same as real-world existence, but it may be useful metaphorically to ignore the distinction, and this might be why mathematicians sometimes sound like Platonists.
The way I think of “practical” existence is sort of like data in a non-lazy programming language. It’s natural for me to think of entities as actually existing when they take up memory, and lazy data structures as sort of a cheat that blurs the distinction between things that actually exist and things that could in principle be calculated on demand. Non-lazy data structures are finite but lazy data structures can be infinite, sort of, and mathematical entities seem pretty lazy, though the parts we manipulate have been instantiated.
(But on another level, the existence of bit patterns on a computer is not entirely concrete, what with virtual memory and all.)
To be frank, I have no idea what you're talking about. A proof, at least one that is accepted as canon, must be rigorous and have clear lines of logic, using the classical techniques. There aren't "impractical" proofs whose contents are a blur but everyone just believes anyway.
There is a difference between proofs which explain why, and proofs which just prove a theorem. For example, Nash's original proof of nash equilibria is beautiful, and is very much defined, and not "conceptually too long to be written down", but it also doesn't say about the specific mechanics. Which is annoying, because when applying it, then you need to dig deeper than Nash.
But, again, it's not a "lazy" list in programming language. You can download it; it's a few pages in a pdf.
Or do you mean like induction, where you can prove all elements of an infinite set have some property?
I'm also not sure what you're saying. Partially because "exists" is overloaded here; exists, as a math term, refers to belonging to a set.
In the general sense, mathematics is constructed; anything can exists in an arbitrary sense, whether or not you can construct something with all the properties you want is another question.
Notation is arbitrary, though? I'm really not sure what you're saying. There are no such entities that "can't be written down". That doesn't really make any sense. Consider X to be an entity that can't be written down. I just called it X, therefore it has been written down.
I feel like you're trying to say something along the lines of Godel incompleteness, but that's... really not what that is.
I want to push back a specific part of what you're saying because I think it gets to the meaning of mathematical existence in conversations like this.
I think you're being a bit too quick to simplify the definition of mathematical existence, and I think it is less well defined than you make it out to be. First, mathematical existence does not a priori mean being a member of set, because you can formulate mathematics without reference to set or elements (take type theory for example). But in such formulations we still talk about abstract mathematical objects, so the must be some question of whether the thing we are referring to exists. That is what we're talking about here.
(For what it's worth, I don't think it's true that anything can exist in an arbitrary sense. Let x be the first real number after √2. Does x exist? )
In any case I don't think whether or not you can write something down (or represent it in some way) has bearing on whether or not that thing exists, because the question is true even for things we can write down. When I write down √2, what am I referring to? Usually the claim implicit in these discussions is some sort of platonist one, where we agree that mathematical objects (numbers, groups, spaces) have some independent abstract existence. What you and @skybrian are talking about if I understand correctly is different. You're asking whether, if we take for granted these objects have those above qualities, can we represent them in some way which is fair and interesting but I think is different than the question at hand.
Yes, I was thinking of formal proofs about proofs, where a proof is modeled as a chain of formal statements of finite but unlimited length, as Godel did. Other mathematicians have gone on to prove other things about what’s provable. But yes, the proofs that mathematicians publish are concrete.
But you haven’t defined it sufficiently well to pick out any specific entity.
Let’s specify it slightly better by saying X is a natural number. But most natural numbers are “random” in that there is no way to specify them that’s significantly shorter than the number itself. Also, most natural numbers have more digits than there are atoms in the universe. Therefore, most natural numbers can’t be specified by us. We can only say things about them in bulk, using induction, not as individuals. It’s a tenuous, ghostly sort of existence.
If we write down N for the set of natural numbers, most of its members are like this. But we usually ignore the ghosts because we’re concerned with other things. What good are unusably large random numbers?
N isn’t concrete in the way that a finite set in a non-lazy programming language is concrete (at runtime). It contains entities that we have no practical way to generate.