Category theory and set theory: just a different language, or different foundation of mathematics?

Question

This is a question to research mathematicians, as well as to those concerned with the history and philosophy of mathematics.

I am asking for a reference. In order to make the reference request as precise as possible, I am outlining the background and nature of my questions here:

I did my Ph.D. in probability & statistics in 1994, and my formal mathematics education was completely based on set theory. Recently, I got interested in algebraic topology, and have started to read introductory texts like Allen Hatcher, or Laures & Szymik, and others.

I am struck by the broad usage of category theory and started to wonder:

(1) Is category theory the new language of mathematics, or recently the more preferred language?

(2) Recognizing that set theory can be articulated or founded through category theory (the text from Rosebrugh and Lawvere), is category theory now seen as the foundation of mathematics?

(3) Is the choice between category theory language and set theory language maybe depending on the field of mathematics, i.e. some fields tend to prefer set theory, others category theory?

Edit: On (3), if such a preference actually exists, what is the underlying reason for that?

Would someone be able to give me a good reference for questions like this? I would be very grateful for that.

Later Edit: Just adding the link to a great, related discussion on MO: Could groups be used instead of sets as a foundation of mathematics? It discusses the question whether every mathematical statement could be encoded as a statement about groups, a fascinating thought. Could groups be used instead of sets as a foundation of mathematics?

Answer 1

Category theory and set theory are complementary to one another, not in competition. I think this 'debate' is a bit of academic controversialising rather than an actual difference. If you've done a bit of category theory, you will realize how important the category of sets is (for Yoneda's lemma, representability, existence of generators, etc).

Even if you completely buy into homotopy type theory as a foundation for ∞-categories and homotopy theory, the theory of sets reappears in other garb as the theory of 0-types. A theory of sets is too natural an idea to escape.

I just also want to note: If you write out the syntactic version of ETCS, you end up with something that is more or less equivalent to ZFC. The ETCC, on the other hand, is widely considered to be a dead-end.

From the nLab:

As pointed out by J. Isbell in 1967, one of Lawvere’s results (namely, the theorem on the ‘construction of categories by description’ on p.14) was mistaken, which left the axiomatics dangling with insufficient power to construct models for categories. Several ways to overcome these problems where suggested in the following but no system achieved univocal approval (cf. Blanc-Preller(1975), Blanc-Donnadieu(1976), Donnadieu(1975), McLarty(1991)).

As ETCC also lacked the simplicity of ETCS, it rarely played a role in the practice of category theory in the following and was soon eclipsed by topos theory in the attention of the research community that generally preferred to hedge their foundations with appeals to Gödel-Bernays set-theory or Grothendieck universes.

Edit: Just to clarify, I think most mathematicians working in category theory, homotopy theory, algebraic geometry, etc. are more or less agnostic about foundations, as long as they are equivalent in strength to ZFC (or stronger with universes). There have been arguments for ETCS(+Whatever) as a 'better' foundation, but when you get into hairy set-theoretic issues (for example, see the Appendix to lecture 2 of Scholze's notes on condensed mathematics), we are just as likely to work with ZFC because setting up ordinals in ETCS is an added annoyance. I added this edit just to clarify that I am not a partisan of either approach and appreciate both (and am not interested in bringing up this old argument about Tom's paper that I linked!!!)

Answer 2

I think that Penelope Maddy's article What Do We Want a Foundation to Do? is a good starting point if you want to read some literature. I don't agree with all of Maddy's conclusions but the terminology that she introduces in this article is exceedingly helpful, as well as the very simple but often overlooked point that the concept of a "foundation of mathematics" is a multifaceted one.

Proponents of foundations other than set theory often emphasize what Maddy calls "essential guidance." The argument is that category theory (or whatever) more accurately reflects how mathematicians actually think, or how they actually do mathematics, or what mathematical structures really are. They may be right (although set theory has more resources in this direction than its opponents sometimes acknowledge), but these alternative foundations don't always outdo set theory when it comes to other roles that we might want a foundation to perform. For example, there's "risk assessment"—what axioms do you really need to derive your theorems, and are those axioms "safe"? Or "generous arena"—maybe the proposed alternative foundations are good for homotopy theory but aren't so suitable for the numerical solution of PDEs or the computation of small Ramsey numbers.

Set theory did a remarkable job in the 19th and 20th centuries of unifying mathematics, putting it all on a common foundation, and providing a framework for analyzing questions of consistency and provability. Nowadays it's easy to take that achievement for granted, and assume that all mathematics is "safe" and that if we want to use methods from one branch of mathematics in another then we will always be able to find a way to do so. If one takes that attitude, then "risk assessment" becomes irrelevant and "generous arena" and "shared standard" drop in importance—I can just worry about finding foundations for the kind of mathematics that I care about, and if my foundations are cumbersome for my colleague's kind of mathematics, well, that's my colleague's problem and not mine. On the other hand, if one does still care about generous arena and shared standard and risk assessment, then set theory still has many advantages.

In short, whether to use set theory or category theory as a foundation depends largely on what you want to do. I agree with Harry Gindi that it's best to think of them as playing complementary roles. In particular, for many of the "traditional" roles that people expect from a foundation (e.g., "meta-mathematical corral" is another Maddy term), I don't think set theory has been superseded.

Answer 3

The best reference I can think of for this is MathOverflow.

Contrary to some of the comments made above, foundational issues are today often a concern in mathematics and computer science. Contrasting foundational schemes is an activity not just limited to researchers in metamathematics or in mathematical logic. It occurs in computer science repeatedly as workers there develop new programming languages, disciplines, and tools for analysis. Mechanical proof checking, program verification, prototyping languages, relation to resource utilization, rapid system development, and other activities benefit from the perspectives offered by one system or another, or by comparing them.

People who frequent this forum often want to understand things more deeply, look for connections or phenomena that may reveal a ubiquitous pattern, or sense of commonality, so that what works for a proof idea in one field can be adapted to other fields. However, the people are raised in different environments, so their perspectives and means of expression vary. It is this variety that is one of the lesser appreciated aspects of MathOverflow: exposure to a wealth of ways of thinking.

Although your questions have been considered before, they are broad enough that I imagine people have only been able to see pieces of the picture, and that the picture is still new enough that data gathering is still going on. If you search MathOverflow (and the Nlab, and perhaps repositories like ArXiv, or proceedings from relevant conferences in computer science as well as mathematics) you will find many of these pieces. For users whose knowledge on this is more extensive than mine, three names pop immediately to mind: Bauer, Blass, Jerabek. (After I get coffee, more names may occur to me.) Looking at some of their answers on this forum may lead you to specific references.

Elsewhere on this forum I have seen a query similar in intent to yours. I answered that really the theories should be considered more as perspectives than as foundational frameworks, because the entirety of mathematics is not captured by just one. These perspectives (or tools) have utility in their variance and possible interplay, not just in their ability to express part of mathematics. But I am unsure that this way of looking at looking helps you in your search.

Gerhard "Speaking As Observer, Not Researcher" Paseman, 2020.05.16.

Answer 4

(1) Is category theory the new language of mathematics, or recently the more preferred language?

Category theory has been proposed in 1940s and started taking over algebraic geometry and topology first in 1970s, and its application has only grown from there.

Whether it is the preferred language depends on which field of mathematics you are thinking about.

Generally, fields with an algebraic flavor prefer category theory. Examples include algebraic geometry, algebraic topology, category theory (duh), algebraic set theory, topological quantum field theory (new branch in physics), type theory.

Fields with an analysis/calculus flavor prefer set theory. Examples include basic calculus, differential equations, differential geometry, functional analysis, probability, set theory (duh).

Finally, fields with a geometric flavor seem to go via a third way, based on geometric intuition and "self-evident postulates". Examples include geometric topology (especially when low-dimensional), knot theory, general physics (I count it as the most important branch of mathematics), and some new fields that have "synthetic" in their names, like synthetic differential geometry.

(2) Recognizing that set theory can be articulated or founded through category theory (the text from Rosebrugh and Lawvere), is category theory now seen as the foundation of mathematics?

General mathematicians don't care much about the foundation. Currently there is a feeling that anything that has been worked on for a long time is sound. Mathematics is now seen as a massive graph of math-nuggets that connect to each other, floating in a vacuum with no special nuggets considered as "the true foundation".

However, this does not mean that all math-nuggets are equally significant, or foundation-worthy. Generally, significance is measured by these characteristics:

1. How densely a nugget is connected to other nuggets;
2. How "deep" the connections are;
3. How fashionable it is (synthetic geometry is not fashionable now, but was extremely fashionable 2000 years ago);
4. How close to physical reality it is (this makes basic calculus more significant than theory of prime numbers).

For a good discussion of what makes a nugget "significant", read Hardy's A Mathematician's Apology, starting at section 11.

As for foundation-worthiness, Maddy's What Do We Want a Foundation to Do? is a great place to learn about the details. I think that in short, some nuggets are better suited for foundation if they satisfy the following criteria:

1. Can encode the things that mathematicians want to work with. This is analogous to a programming language being "Turing complete". A foundation must be "mathematician complete" or get close to it.
2. Can encode elegantly. This is less objective but also very important. In the algebraic-flavored fields, category theory wins over set theory in this aspect.
3. Can be checked mechanically, that is, it is good for formal verification. This is not yet a very important consideration, but is something univalent foundations and other type-theory foundations are explicitly trying to do well. For more, this Quanta essay is good for a start. For another, A computer-generated proof that nobody understands

(3) Is the choice between category theory language and set theory language maybe just depending on the field of mathematics, i.e. some fields tend to prefer set theory, others category theory?

What I said.

Update:

I have changed my mind about the foundation-worthiness of set theory. My new view is that set theory is inappropriate as a foundation of mathematics done by non-set-theorists.

Saunders Mac Lane (co-discoverer of category theory) was very keen on philosophical issues. His view is that set theory is not a good foundation to "real mathematics", since it does not encode the language of math as done by non-set-theorists. Category theory does the job much better. He thought that ZFC is particularly "inappropriate" because it is too strong, and just like how we should have "appropriate technology", mathematicians should use "appropriate foundations".

From To the Greater Health of Mathematics:

Set theory is not the only viable foundation for mathematics. It is striking that Smorynski, for all his extensive knowledge, appears to ignore the Lawvere idea of replacing set-membership by composition of functions as a primitive notion in foundations; this is an idea now explicitly available in the theory of an elementary topos. This approach to foundations has the advantage that it is closer to the actual practice of mathematics. It emphasizes the idea that the study of foundations is not restricted to consistency and proof-theoretic strength, but also covers analysis of the conceptual structure of mathematics, poorly reflected in the usual set theoretic translation.

There is a category-theoretic construction of ZFC set theory, called ETCS (Elementary Theory of the Category of Sets), Todd Trimble wrote about ETCS extensively in a series of essays, and observed that ZFC is ridiculously "strong" because of the extensionality axiom. This strength is inappropriate, as real mathematics does not require it, and it creates complications.

The deep meaning of the extensionality axiom is that a “set” $S$ is uniquely specified by the abstract structure of the tree of possible backward evolutions or behaviors starting from the “root set” $S$. This gives some intuitive but honest idea of the world of sets according to the ZFC picture: sets are tree-like constructions. The ZFC axioms are very rich, having to do with incredibly powerful operations on trees, and the combinatorial results are extremely complicated.

...

ZFC is an axiomatic theory (in the language of first-order logic with equality), with one basic type V and one basic predicate ∈ of binary type V×V, satisfying a number of axioms. The key philosophic point is that there is no typed distinction between “elements” and “sets”: both are of type V, and there is a consequent very complicated dynamical “mixing” which results just on the basis of a short list of axioms: enough in principle to found all of present-day mathematics!

My own reaction is that ZFC is perhaps way too powerful! For example, the fact that ∈ is an endo-relation makes possible the kind of feedback which can result in things like Russell’s paradox, if one is not careful. Even if one is free from the paradoxes, though, the point remains that ZFC pumps out not only all of mathematics, but all sorts of dross and weird by-products that are of no conceivable interest or relevance to mathematics.

Update: John Baez gave a good alternative word for "foundation" in Foundations of Mathematics:

Personally I don’t think the metaphor of “foundations” is even appropriate for this approach. I prefer a word like “entrance”. A building has one foundation, which holds up everything else. But mathematics doesn’t need anything to hold it up: there is no “gravity” that pulls mathematics down and makes it collapse. What mathematics needs is “entrances”: ways to get in. And it would be very inconvenient to have just one entrance.

Answer 5

It is worth mentioning that category theory can be articulated through and founded on sets, in a relatively straightforward manner.

That said, category theory and set theory seem to be two sides of the same coin even at a research level. I asked about this comparison in this MO question and got some excellent discussion from category and set theorists (see the comments).

Category theory is a universal language for discussing the notion of structure on sets, and a universal setting in which connections can be seen between seemingly disparate areas of mathematics.

One connection is mentioned in the answers above, and actually links sets to categories -- the Yoneda embedding

$${\bf Hom}_\mathcal{C}(-,\ \ ):\mathcal{C}\to{\sf Set}^{\mathcal{C}^{op}}.$$

The Yoneda Lemma shows that this embedding is fully faithful, meaning that for any category $\mathcal{C}$ there is a 'picture' of $\mathcal{C}$ inside its category of presheaves ${\sf Set}^{\mathcal{C}^{op}}$, a presheaf being a functor from its opposite category into the category of sets and functions. This presheaf category has many desirable properties it inherits from ${\sf Set}$ like (co-)completeness; it actually inherits all the structure of a topos (a very specific and nice kind of category). We see immediately that sets and categories are not separate competing entities, but different pieces of the same whole.

Other well-known examples of categorical connections between seemingly disparate areas include the duality between the category of Stone spaces and the category of Boolean algebras, or locales and frames.

The notion of a category can actually be viewed as a sufficiently general way to structure a set so that it can emulate (and generalize) many standard structured sets. Groups, rings, vector fields, modules, quotient spaces, partially ordered sets, Boolean algebras and more can all be cast as certain types of categories, and then generalized to groupoids etc. -- random variables, observables, probability measures, and states can be understood as arrows in a certain category (see Frič, R., Papčo, M. A Categorical Approach to Probability Theory, Stud Logica 94 (2010) pp 215–230. for a reference on the probability claims).

So, for someone with a background in working with structured sets, a category can be viewed as a pair of sets together with some functions between them, structured in a very general way that allows them to emulate almost all other notions of 'structure on a set'. Further, this same notion of a category can then unify these structured sets into greater structures and explore connections between them on a larger scale.

Answer 6

As you are asking for references, the following book might be of interest to you:

Basic Category Theory from Tom Leinster (I think from 2014 or 2017).

I like the way he introduces category theory, and this might give a (partial) answer to your question. Here is a quote:

"Category theory takes a bird's eye view of mathematics. From high in the sky, details become invisible, but we can spot patterns that were impossible to detect from ground level. How is the lowest common multiple of two numbers like the direct sum of two vector spaces? What do discrete topological spaces, free groups, and fields of fractions have in common?

Answer 7

A small, belated answer: although I do agree with the several other good answers that literally address the question as asked, I do also think it is informative to "reframe" the question, etc.

That is, there's always the rhetorical question about what we might need "foundations" for math, anyway? Well, yes, we know some reasons, namely, that certain ideas, taken to their logical extremes, produce nonsense. We want to avoid this, obviously. So, can we make rules to avoid producing nonsense, but still producing good stuff?

In addition to that qualification for (to my mind) the point of "foundations", there is another version of "math" which is as a narrative of certain sorts of thinking, not only quantitative, which also allows a considerable degree of accurate "book-keeping/accounting". Such a language for narrative does not, at first approximation, need "foundations" any more than productive math, and celestial mechanics, electromagnetism, needed for a long time. But, yes, even as a "narrative language", it is conceivable that there is a hidden grammatical (?!?) fault that can produce nonsense. Not only if taken to logical extremes, but perhaps only if "turned on its head". This would be bad. (We do not our narrative/descriptive language to say that a thing is and is not... etc.)

But I am insufficiently expert to know whether foundationalists these days talk in such terms.

Answer 8

The underlying reason of the utility of category is structural. This arose in the abstract algebra approach of Noether where the notion of a structure preserving map was isolated - a morphism.

This idea is so basic to mathematics now, that it is baked into category theory whereas in set theory, the notion of a function is a derived concept, never mind that of structure preserving map.

Just as Marx-Engels is said to have turned Hegelian idealism upside-down (not quite true), Eilenberg-Maclane can be said to have turned set theory upside down to construct category theory.

I would also say neither set theory nor category theory is "the language of mathematics". The only language of mathematics is language itself. To see this merely remove all language from any paper and see how easy it is to understand. It isn't usually - it turns into symbol sludge.

One other aspect of category theory has an avatar in physics. This is the notion of covariance which Einstein used so effectively in thinking through what GR entails. One can say that category theory is also a general theory of covariance.

As for a reference, check out the SEP article on Category Theory.