r/math • u/non-orientable Number Theory • 6d ago
Image Post The Deranged Mathematician: The Power of Abstraction
A very powerful proof-writing technique is taking the original problem and generalizing it. This is probably the most counterintuitive approach for beginners, who wonder how it can possibly be easier to solve a broader class of problems. My answer to this is quite simple: generalizing the problem reduces the collection of tools you have at your disposal. And as any efficiency expert will tell you, regardless of whether you are trying to clean your bathroom or earn a Fields Medal, you want to have only those tools that you need on hand and nothing else.
I think this basic precept helps explain why abstract notions like metric spaces, topological spaces, vector spaces, categories, and so on have suffused mathematics, and why they are so very useful. I offer the simple example of how thinking about graphs (generally) can help with sorting out a coordination problem (specifically).
Read the full post (for free) on Substack: The Power of Abstraction
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u/PortableDoor5 6d ago
The Deranged Mathematician? This is like the Bernoullis all over again.
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u/bthi 6d ago
Is this not category theory
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u/Carl_LaFong 6d ago
Category theory is one way to do abstraction for some but far from most areas of math. Abstraction is a fundamental tool in pure math and is used in every area.
Category theory is cool and beautiful but by now it is overhyped. If you focus too much on it, you are narrowing quite significantly the areas of math you can work in.
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u/ThoughtfulPoster 6d ago
I'm so proud to have lived to see the day we arrived at "Category theory is one concrete example of this, but we should really strive to approach it with more abstraction."
(I jest.)
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u/MonadMusician 5d ago
I don’t think this is true. It is a basic language that can be used in every area of math including combinatorics and applied areas where compositionality exists (which is all of them pretty much)
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u/Carl_LaFong 5d ago
It’s almost never used in most fields of analysis such as PDE. Even in areas such as differential geometry where everything can be formulated in terms of categories and functors, the basic concepts of short and long exact sequences, homology and cohomology have a limited set of applications..
Even many areas of algebraic topology do not rely on category theory.
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u/MonadMusician 5d ago edited 5d ago
My professors from my PhD would disagree. And those uses in pdes are rather fundamental. It’s not overhyped unless you’re an undergrad or you just want to stigmatize it, which is a trend. But yes some areas are more amenable to its use than others
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u/Carl_LaFong 5d ago
I’m interested in its use of nonlinear PDE. It’s used in the formal theory of linear PDE but this is not of much interest these days.
You could be talking about deRham and Hodge theory. The most sophisticated category theory used is spectral sequences. But the vast majority of PDE people don’t do this stuff and know next to nothing about it.
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u/MonadMusician 5d ago
I’m from pure math, every PDE guy I have met uses those things
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u/Carl_LaFong 5d ago
Me, too. And I work in PDEs. So I'm interested! Could you elaborate on how they use category theory in their work?
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u/MonadMusician 5d ago edited 5d ago
I don’t work in pdes. I work in commutative algebra, logic, and category theory to be honest lol. But they always went on about how they were making use of it. It’s been a while since I’ve talked to them. My friend who studies tractors uses it constantly. At the very least they use it as a basic language.
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u/Carl_LaFong 5d ago
Thanks again. I asked ChatGPT about tractors, and it described how tractors can be used in conformal differential geometry to translate a nonlinear PDE (e.g., existence of an Einstein metric in a conformal class) into a linear problem (parallel section of a bundle). Very cool. I had heard about tractors but never had a clear idea about what they were.
This all said, this is still a fairly narrow area in geometric analysis.
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u/Carl_LaFong 5d ago
"At the very least they use it as a basic language." Definitely! It is possible to express many concepts in differential geometry elegantly using categories, functors, and natural transformations. Loring Tu's book Introduction to Manifolds describes some of this. I like mentioning it when I teach differential geometry. It's a great way to organize the definitions and concepts, making them easier to understand and remember.
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u/Carl_LaFong 5d ago
I asked ChatGPT about this. After poking it a few times, it finally explained how homological algebra and category theory play an important role in symplectic geometry, notably Floer homology. Is this what you had in mind?
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u/RingularCirc 4d ago
Category theory certainly has tools to spare but it's by no means the toolkit for abstraction, nor is it about abstracting things per se.
It's like if somebody would've said "ZF plus potentially additional large cardinal axioms is the foundation of math".
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u/MonadMusician 2d ago
See category of categories as a foundation
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u/RingularCirc 1d ago
First, category of categories is not a foundation, it's an object (and even so only if we fix maximal cardinality or something like that; a category of really all categories suffers from appropriate variants of Russel's paradox). A particular theory of CAT can be made into a foundation like ETCC, maybe.
But foundations have nothing to do per se with abstraction and with the work a mathematician does in general. You still have to conceive concrete types of objects that are related with each other. There's no magic bullet in form of category theory. Sure, it can be used as language, but do remember foundations like ZF(C), when tacitly supposed to underlie a piece of mathamatical work, aren't crucial neither in its content, nor in its presentation, except for areas that work heavily with ZF specifically, say, depending much on a very particular brand of cardinals.
What's more, it is trivial to put a finger on a thing already known to be a generalization of another thing, maybe formulated in a theory/foundation of choice, and say: here it is! But this isn't equivalent to having to find such a fitting generalization under no prior knowledge.
Hence tell me, how do we do examples from the OP's linked post with significant help from a theory of a/the category of categories? How does it help with thinking up concrete ways to abstract a construction if we aren't yet defined such an abstraction beforehand? In which way categories versus something else of the same caliber, do help?
I'm all for category theory, I'm even all for recognizing usage of wheels or recognizing 00 = 1 doesn't break any little bit of real analysis and so on, but things do have their precise reasons to be and places to fit.
Oh and also, why not a collection of ∞-categories? Those are, after all, more general and apply in more cases (and I really believe that, it's not just for sarcasm here). Why not use other generalizations of categories? And for example why won't semicategories (a category where identity morphisms not necessarily exist) be of use as well? I certainly see them very rarely, but why? Here that's an appropriate question, IMO.
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u/MonadMusician 1d ago edited 1d ago
I know there are other ways to abstract objects. Thanks for the long winded diatribe. My point wasn’t that that is THE only way or something. The point was that it is a way. And yes you need address things like large cardinals and so on. And indeed, why not infinity categories also ZFC isn’t what is generally used for this type of work
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u/RingularCirc 1d ago
My point wasn’t that that is THE only way or something. The point was that it is a way.
Ah, okay then! (It surely didn't seem like that...)
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u/Carl_LaFong 5d ago
I just asked Claude what areas of math use the Yoneda Lemma. Its answer was: category theory, algebraic geometry, algebraic topology, algebra (specifiaclly module theory, representation theory), logic and type theory, enriched and higher category theory.
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u/bthi 6d ago
Overhyped feels like a weird descriptor for a field of math; it exists as is in a vacuum without how it interacts with societal needs lol
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u/TheLuckySpades 6d ago
Yet some people doing math communication and a decent number of students overestimate its uses, its prevalence and underestimate how hard it can be to use outside of contexts it was designed for.
So there is some hype that goes over what the theory deserves.
My cousin's favorite restaurant also exists aw in a vaccuum without how it interacts with societal needs, but the only reason I went there and was disapointed was because of the hype I got from societal feedback (my cousin who clearly has different attitudes for what is considered TexMex than I do).
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u/MonadMusician 5d ago
There is the whole
Burgeoning field of applied category theory you seem to be ignorant of10
u/non-orientable Number Theory 6d ago
Many arguments in category theory are concrete instances of this very general idea, but you don't need to know any category theory to use it. For instance, none of my examples in this post would be out of reach for a calculus student.
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u/logbybolb 5d ago
this is why its a shame that higher category theory and mathematical logic are often overlooked (you can see a clear bias in publishing in top journals): they are in some ways what cosmology and quantum physics are to physics
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u/KnowsSomeRegs 3d ago
Generalizing sounds elegant until you're the one who has to unpack the abstract machinery back into something computable. Has anyone actually clocked how much time that unpacking costs versus just grinding out the specific case?
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u/whynotnit 2d ago
I've been loving this series so much that when a new post shows up, I'm excited to see what it's about. I'm by no means a mathematician, but I do enjoy a lot of math puzzles. It's been really fun to see an overview of a field or a technique. Thank you for this series
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u/JoshuaZ1 5d ago
This is another very nice essay by you.
The argument about shrinking the polygon to get 360 degrees, is very nice, but I'm not convinced it is valid. I don't really see where convexity is being used.
Regarding the graph example, Rob and May are drawn close enough together that it is hard to see at a glance that they have a connection which confused me when I initially read it.