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Stream: theory: philosophy

Topic: The generic triangle

John Baez (Oct 03 2024 at 15:55):

On Mathstodon, Boarders wrote:

Platonism vs nominalism: an old fashioned debate in philosophy is that of universals - talking of triangle as a property commits oneself to the existence of the abstract concept of triangle or perhaps a “generic triangle”.

In this case, we can, to some extent, resolve the question mathematically in a precise way - there is a mathematical object called the moduli stack of triangles, the generic triangle is the generic point of this moduli stack. We can answer various precise questions about this generic triangle such as what automorphism group it has etc. It also specializes to various more particular triangles with a precise notion of specialize due to algebraic geometers. It seems to me this is a better formal apparatus for thinking about mathematical universals than any kind of vague waffle available, but the idea has not managed to escape the purview of mathematicians.

David Corfield (Oct 03 2024 at 16:48):

There's a huge industry in philosophy in working out what's meant by when we say "Let $n$ be a natural number". A common position has it that $n$ refers to an "arbitrary natural number" which is a different thing from the individual natural numbers and has only generic properties.

Andrej Bauer at his curtest on the subject here:

What I am saying is that "a free variable is a projection" is a great deal better at explaining various perceived mysteries about free variables than fancy words from philosophy books. But your mileage may vary.

A two-pronged attack with Egbert Rijke writing:

The generalised elements of an object $A$ are morphism into $A$. Under this point of view, the generic element of $A$ is the identity morphism on $A$.

John Onstead (Oct 04 2024 at 02:43):

Category theory actually gives us the most powerful ever version of finding "the generic X". This process is known as "externalization" and it involves going from a specific instance of X to viewing it as a functor out of a particular category, known as the "generic X" or "walking X". In a sense, the "walking X" is the platonic ideal of X, the archetypal or prototypical X. Making statements about the walking X then you can specialize to all instances of X across mathematics. There's many examples ranging from the "walking morphism" A -> B which is the archetypal morphism, to the Lawvere theory of groups which is the archetypal group object. This is also related to the notion of classifying object, which is the formal dual of the walking object since every morphism into it represents an instance of X.

Julius Hamilton (Nov 16 2024 at 16:47):

That’s interesting. I feel like I’m often attracted to “maximum generality”, but I feel like quite a few people caution one that this is sort of a fool’s errand. I feel like I’ve heard quite a few people mention that there is no “single best generalization” of certain ideas; that the purpose of abstraction is to focus on what specifically is relevant to some task at hand; that mathematics has a kind of relativism where we can model structures in different languages but not claim one is more central than any other.

Mike Shulman (Nov 16 2024 at 17:01):

"Generic" is different from "maximum generality".

John Baez (Nov 17 2024 at 00:26):

Yes, the "generic X" is a perfectly well-defined thing, unique up to equivalence, after we've chosen X (the "theory") and the context in which to work with X (the "doctrine"). If you seek greater generality, you will tend to keep fiddling around with the doctrine and the theory.

(I'm not very attracted to maximum generality: I mainly want to understand things and also use math to help the world. Often these quests are aided by increasing generality - but also very often it's important to decrease generality and think very hard about a specific example! As Polya said: "if there's a question you can't solve, there's probably an easier question you can solve" - so switch to that and see if it helps. Often to find a good easier question you should decrease the level of generality. It's better to add extra assumptions than sit there stuck.)

Joe Moeller (Nov 19 2024 at 22:47):

I think there's a pretty straightforward analogy that's helpful here: abstraction is like elevation. When you realize two concrete facts are examples of one abstract fact, you've put two sticks together and you're standing at their apex. This abstract point could be stregthened by supporting it with further struts on the ground: examples. The higher you are, the more you could see (if you dare to look down). So a naively appealing idea is to just build straight upwards, so you could see everything. That's not likely to produce a very stable structure, and if other structures haven't been built in the surrounding areas, you'll likely stop being able to make out any details on the ground anyway. Obviously sometimes it pays to be daring and build upwards a bit.

Kevin Carlson (Nov 19 2024 at 22:49):

Great metaphor!

Julius Hamilton (Nov 25 2024 at 16:31):

David Corfield said:

What I am saying is that "a free variable is a projection"…

The generalised elements of an object $A$ are morphisms into $A$. Under this point of view, the generic element of $A$ is the identity morphism on $A$.

Why is a free variable a projection?

Julius Hamilton (Nov 25 2024 at 16:33):

John Baez said:

As Polya said: "if there's a question you can't solve, there's probably an easier question you can solve" - so switch to that and see if it helps. Often to find a good easier question you should decrease the level of generality. It's better to add extra assumptions than sit there stuck.)

That is an idea I will definitely explore.

Ryan Wisnesky (Nov 25 2024 at 16:55):

if you look at a term in context, say, x:Nat, y:String |- x:Nat, then the categorical semantics is a morphism Nat*String->Nat, that is in fact the first projection of the context (the variable x). Whereas if you had x:Nat, y:String |- 5:Nat, you'd have a constant morphism, not a projection.

David Corfield (Nov 26 2024 at 08:01):

Right, so when you say 'Let x be a natural number,...', you are working in a context $x: Nat \vdash ...$. Say you want to establish some property of a number in general, and you arrive at $x: Nat \vdash p: P(x)$. This corresponds to a morphism in the slice over $Nat$, $p: Nat \to P$, respecting the maps down to $Nat$. If $P$ is a proposition, this will only be possible if $P$ holds for all $x: Nat$.

Sidney Congard (Jan 06 2025 at 17:01):

It seems however that, when speaking about "the generic number" x, some philosophers do not want to be able to derive e.g. that the generic x is even or odd (as it's neither). May there be something related to a kind of parametric polymorphism, to obtain only "uniform" properties, not those obtained by case analysis? (I don't have further ideas to sharpen this distinction though)

David Corfield (Jan 06 2025 at 19:33):

Wouldn't we expect it to be a feature of generic entities that distributivity over 'or' doesn't hold? So we can say

The generic number is either odd or even,

but not

The generic number is odd or the generic number is even.

In the negative form,

The generic number is neither odd nor even,

is fine in the sense

The generic number is not odd nor is it even,

but not in the sense

The generic number is neither odd nor even.

Sidney Congard (Jan 06 2025 at 19:51):

In this setup, "the generic X" seems replaceable by "any X", but idk if everyone want to bite the bullet (but I kinda do so it's hard speaking in their stead)

John Baez (Jan 06 2025 at 20:41):

I'd hope the logical rules governing the modality 'generically' would match those for open dense sets, since in topology 'P is generically true' means P is true on an open dense set.

John Onstead (Jan 08 2025 at 01:18):

John Baez said:

In this case, we can, to some extent, resolve the question mathematically in a precise way - there is a mathematical object called the moduli stack of triangles, the generic triangle is the generic point of this moduli stack. We can answer various precise questions about this generic triangle such as what automorphism group it has etc. It also specializes to various more particular triangles with a precise notion of specialize due to algebraic geometers. It seems to me this is a better formal apparatus for thinking about mathematical universals than any kind of vague waffle available, but the idea has not managed to escape the purview of mathematicians.

I actually wanted to revisit this point now that I learned more about the Yoneda lemma, co-Yoneda lemma, and how it connects to representability. Let's say I wanted to find the "generic triangle" as an object of a category that represents triangles (IE, such that the sets of morphisms out of it represent the triangles found in other objects). My immediate first guess was to think of the simplex category, in which you can think of each object as a "generic simplex". You can then build a simplicial set by "gluing together" these generic simplices from each dimension. In this case, the 2-simplex is actually the "generic triangle", since by the Yoneda lemma, every morphism from its representable simplicial set to any other simplicial set "picks out" a triangle. Via the geometric realization, we can also pick out triangles in topological spaces, again with the 2-simplex as the representing object. (Of course, every morphism from the 2-simplex into something like the cartesian plane can be deformed by homeomorphism to a circle or really any other similar shape)

John Onstead (Jan 08 2025 at 01:21):

So for me, the 2-simplex object in the simplex category is the "de facto" generic triangle. But this really makes me wonder how this notion of "generic triangle" connects with the one Boarders was writing about. Some time ago I did try to make an effort to understand moduli spaces and stacks, but I think I must've given up. A stack is like a groupoid, right? So this would involve determining what properties a "generic point" of a moduli stack has as an object of this corresponding groupoid (if that's even a valid thing to ask). In any case, my question would be: how might Boarder's notion of "generic point" and the 2-simplex as a representing object for triangles relate as notions of a "generic triangle" (whether that be on a philosophical or mathematical level)?

John Baez (Jan 08 2025 at 05:27):

John Onstead said:

A stack is like a groupoid, right?

There's a confusing diversity of ways of thinking about stacks, but a stack is like an appropriately categorified kind of sheaf. So, to get the definition of a stack, you can take the definition of a sheaf of sets, replace the sets by categories, require the laws of a sheaf only up to natural isomorphism, and make those isomorphisms obey their own coherence laws.

If the categories are groupoids, you get a "stack in groupoids". And these are like generalized groupoids - in the same sort of way that sheaves are like generalized sets, and stacks are like generalized categories.

In algebraic geometry the most commonly considered stacks are stacks in groupoids, so there it is a very good idea to think of a stack as like a blend of a groupoid and a scheme (which is the algebraic geometer's favorite kind of space).

John Onstead (Jan 08 2025 at 06:30):

John Baez said:

So, to get the definition of a stack, you can take the definition of a sheaf of sets, replace the sets by categories, require the laws of a sheaf only up to natural isomorphism, and make those isomorphisms obey their own coherence laws.

Ok, so a stack in groupoids is a functor $C^{op} \to \mathrm{Grpd}$ that's like a sheaf. So the category of stacks on a site $C$ is a lex reflective localization of $[C^{op}, \mathrm{Grpd}]$.
In that case, for the moduli stack of triangles, what is the precise site $C$, what are the objects of $C$, and what groupoids does the stack functor send each object to?

John Baez (Jan 08 2025 at 06:37):

John Onstead said:

Ok, so a stack in groupoids is a functor $C^{op} \to \mathrm{Grpd}$ that's like a sheaf.

Basically yeah. A 2-functor, since $\mathrm{Grpd}$ is a 2-category. (Maybe that's what you meant by
"functor".)

So the category of stacks on $C$ is a lex reflective localization of $[C^{op}, \mathrm{Grpd}]$.

Maybe, but that's not how I think about them. I don't know enough about stacks to know if that's equivalent to the definitions I've seen. In fact I'm far from an expert on stacks, I just kinda get some of the basic ideas.

In that case, for the moduli stack of triangles, what is the precise site $C$, what are the objects of $C$, and what groupoids does the stack functor send each object to?

That depends on whether you want a topological stack of triangles, a differentiable stack, an algebraic stack, or some other sort of stack. Each has its own site, or sites.

So, before you get all stacky, you have to decide whether you want to think about a topological space of triangles, a manifold of triangles (or a smooth space), or an algebraic variety of them (or a scheme). Then you have to decide what kind of symmetries between triangles you want to consider.

But I am happy to try to make these choices if you want. In fact, that's probably the only way we'll get a version of the "stack of triangles" that I have a chance of actually understanding.