foundationslo.logicsoft-question
What is some current research going on in the foundations of mathematics about?
Are the foundations of mathematics still a research area, or is everything solved? When I think about foundations I'm thinking of reducing everything to ZFC set theory + first order logic. Is this still contemporary?
On the subject of categorical versus set-theoretic foundations there is too much complicated discussion about structure that misses the essential point about whether "collections" are necessary.
It doesn't matter exactly what your personal list of mathematical requirements may be -- rings, the category of them, fibrations, 2-categories or whatever -- developing the appropriate foundational system for it is just a matter of "programming", once you understand the general setting.
The crucial issue is whether you are taken in by the Great Set-Theoretic Swindle that mathematics depends on collections (completed infinities). (I am sorry that it is necessary to use strong language here in order to flag the fact that I reject a widely held but mistaken opinion.)
Set theory as a purported foundation for mathematics does not and cannot turn collections into objects. It just axiomatises some of the intuitions about how we would like to handle collections, based on the relationship called "inhabits" (eg "Paul inhabits London", "3 inhabits N"). This binary relation, written $\epsilon$, is formalised using first order predicate calculus, usually with just one sort, the universe of sets. The familiar axioms of (whichever) set theory are formulae in first order predicate calculus together with $\epsilon$.
(There are better and more modern ways of capturing the intuitions about collections, based on the whole of the 20th century's experience of algebra and other subjects, for example using pretoposes and arithmetic universes, but they would be a technical distraction from the main foundational issue.)
Lawvere's "Elementary Theory of the Category of Sets" axiomatises some of the intuitions about the category of sets, using the same methodology. Now there are two sorts (the members of one are called "objects" or "sets" and of the other "morphisms" or "functions"). The axioms of a category or of an elementary topos are formulae in first order predicate calculus together with domain, codomain, identity and composition.
Set theorists claim that this use of category theory for foundations depends on prior use of set theory, on the grounds that you need to start with "the collection of objects" and "the collection of morphisms". Curiously, they think that their own approach is immune to the same criticism.
I would like to make it clear that I do NOT share this view of Lawvere's.
Prior to 1870 completed infinities were considered to be nonsense.
When you learned arithmetic at primary school, you learned some rules that said that, when you had certain symbols on the page in front of you, such as "5+7", you could add certain other symbols, in this case "=12". If you followed the rules correctly, the teacher gave you a gold star, but if you broke them you were told off.
Maybe you learned another set of rules about how you could add lines and circles to a geometrical figure ("Euclidean geometry"). Or another one involving "integration by parts". And so on. NEVER was there a "completed infinity".
Whilst the mainstream of pure mathematics allowed itself to be seduced by completed infinities in set theory, symbolic logic continued and continues to formulate systems of rules that permit certain additions to be made to arrays of characters written on a page. There are many different systems -- the point of my opening paragraph is that you can design your own system to meet your own mathematical requirements -- but a certain degree of uniformity has been achieved in the way that they are presented.
We need an inexhaustible supply of VARIABLES for which we may substitute.
There are FUNCTION SYMBOLS that form terms from variables and other terms.
There are BASE TYPES such as 0 and N, and CONSTRUCTORS for forming new types, such as $\times$, $+$, $/$, $\to$, ....
There are TRUTH VALUES ($\bot$ and $\top$), RELATION SYMBOLS ($=$) and CONNECTIVES and QUANTIFIERS for forming new predicates.
Each variable has a type, formation of terms and predicates must respect certain typing rules, and each formation, equality or assertion of a predicate is made in the CONTEXT of certain type-assignments and assumptions.
There are RULES for asserting equations, predicates, etc.
We can, for example, formulate ZERMELO TYPE THEORY in this style. It has type-constructors called powerset and {x:X|p(x)} and a relation-symbol called $\epsilon$. Obviously I am not going to write out all of the details here, but it is not difficult to make this agree with what ordinary mathematicians call "set theory" and is adequate for most of their requirements
Alternatively, one can formulate the theory of an elementary topos is this style, or any other categorical structure that you require. Then a "ring" is a type together with some morphisms for which certain equations are provable.
If you want to talk about "the category of sets" or "the category of rings" WITHIN your tpe theory then this can be done by adding types known as "universes", terms that give names to objects in the internal category of sets and a dependent type that provides a way of externalising the internal sets.
So, although the methodology is the one that is practised by type theorists, it can equally well be used for category theory and the traditional purposes of pure mathematics. (In fact, it is better to formalise a type theory such as my "Zermelo type theory" and then use a uniform construction to turn it into a category such as a topos. This is easier because the associativity of composition is awkward to handle in a recursive setting. However, this is a technical footnote.)
A lot of these ideas are covered in my book "Practical Foundations of Mathematics" (CUP 1999), http://www.PaulTaylor.EU/Practical-Foundations Since writing the book I have written things in a more type-theoretic than categorical style, but they are equivalent. My programme called "Abstract Stone Duality", http://www.PaulTaylor.EU/ASD is an example of the methodology above, but far more radical than the context of this question in its rejection of set theory, ie I see toposes as being just as bad.
Here's an example of something that I think Mumford might advocate in the foundations of mathematics: Solovay's model.
The axiom of choice is generally accepted by mathematicians, but it has always suffered from the nagging problem that it violates certain intuitions we have. Almost all these counterintuitive consequences of the axiom of choice are related in one way or another to the existence of non-measurable sets. (See this related MO question for more information, in particular Ron Maimon's answer.) Solovay's model shows that we can come close to having our cake and eating it too: We can simultaneously have the axioms "all Lebesgue sets are measurable" and the axiom of dependent choice. The former pretty much eliminates all the probabilistic paradoxes while the latter gives us almost all of the "desirable" consequences of the axiom of choice.
The reason that I think this is the sort of thing Mumford might advocate is that Mumford's discussion of Freiling's theorem shows that he really wants to preserve probabilistic intuition even at the expense of jettisoning a well-accepted axiom. In the paper he suggests getting rid of the power-set axiom, but my guess is he was probably not familiar with Solovay's model at the time, and if he were, he would have been favorably disposed towards it.
Best Answer
It is quite difficult to answer this question comprehensively. It's a bit like asking "so what's been going on in analysis lately?" It is probably best if logicians who work in various areas each answer what is going on in their area. I will speak about logic in computer science. I am very curious to see what logicians from other areas have to say about their branches of logic.
A hundred years ago logic was driven by philosophical and foundational questions about mathematics. The development of first-order logic and ZFC was one important branch of logic, but others were model theory and computability theory. In fact, computability theory led directly to the invention of modern computers. Ever since logic has had a very fruitful relationship with computer science.
One could argue that applications of logic in computer science are not about foundations of mathematics, but such an opinion can hardly be defended. Of course, many applications of logic in computer science are just that, applications, but not all of them. Some are directly related to foundations of mathematics (see below), while others are about understanding what computation is about (Turing got us on the right track, but there's a lot more that can be said about computation apart from "it's all equivalent to Turing machines"). I hope you can agree that computation is a foundational concept in mathematics, equal to such concepts as (mathematical) structure and deductive method.
We live in an era of logic engineering: we are learning how to apply methods of logic in situations that were not envisioned 100 years ago. Consequently, we need to devise new kinds of logic that are well-suited for new problems at hand. To give just one example: real-world computer systems often work concurrently with each other in an environment which acts in a stochastic fashion. Computer scientists have learned that Turing machines, first-order logic and set theory are not a very good way of describing or analyzing such systems (the computers are of course no more powerful than Turing machines and can be simulated by them, but that's hardly a helpful way to look at things). Instead they devise new kinds of logic which are well suited for their problems. Often we design logics that have good computational properties, so that machines can use them.
Another aspect of logic in computer science, which is closer to a mathematician's heart, is computer-supported formalization of mathematics. Humans are not able to produce complete formal proofs of mathematical theorems. Instead we adhere to what is best described as "informal rigour". Computers however, are very good at checking all the details of a mathematical proof. Modern tools, such as various proof assistants (Coq, HOL, Agda, Lean), are improving at a fast rate, and many believe they will soon become useful tools for the working mathematician. Be that as it may, the efforts to actually do mathematics formally are driving research in how this can be feasibly done. While some proof assistants do use first-order logic and set theory (e.g. Mizar), most use type theory. This is probably not just a fashion. It is a response to the fact that first-order logic and set theory suffice for formalization of mathematics in principle, but not so much in practice. Logicians have been telling us that in principle all mathematics can be formalized. And now that we're actually doing it, we're learning new things about logic. This to my mind is one of the most interesting contermporary developments in foundations of mathematics.