Charles Wells’ Blog
Posted by John Baez
Charles Wells is perhaps most famous for this book on topoi, monads and the category-theoretic formulation of universal algebra using things like ‘algebraic theories’ and ‘sketches’:
- Michael Barr and Charles Wells, Toposes, Triples and Theories.
It’s free online! Snag a copy and learn some cool stuff. But I’ll warn you — it’s a fairly demanding tome.
Luckily, Charles Wells now has a blog! And I’d like to draw your attention to two entries: one on sketches, and one on the evil influence of the widespread attitude that ‘the philosophy of math is the philosophy of logic’.
‘Sketches’ are a trick for describing a type of algebraic gadget (say a group, or a category) by drawing some commutative diagrams. People use this trick a lot in an informal way, but it has been made formal and is by now an important alternative to the traditional way logicians describe structures by listing axioms.
Toposes, Triples, and Theories explains sketches, and Michael Barr and Atish Bagchi are now writing another book on the subject: Graph-Based Logic and Sketches. You can read a user-friendly review of sketches here:
- Charles Wells, Sketches: outline with references.
but now you can start with an even gentler introduction on Wells’ blog:
- Charles Wells, Turning definitions into mathematical objects.
This means you can now ask questions! — something that papers and books don’t yet handle well.
On another note: you’ve probably heard David Corfield inveigh against the ‘foundationalist filter’ — the idea that philosophers of mathematics need only pay attention to the ‘foundations’ of mathematics, meaning mathematical logic and set theory. I agree with him whole-heartedly, so it’s nice to see that Charles Wells is on our side:
- Charles Wells, How “math is logic” ruined mathematics for a generation.
I can’t resist quoting a bit:
By the 1950’s many mathematicians adopted the attitude that all math is is theorem and proof. Images, metaphors and the like were regarded as misleading and resulting in incorrect proofs. (I am not going to get into how this attitude came about). Teachers and colloquium lecturers suppressed intuitive insights and motivations in their talks and just stated the theorem and went through the proof.
I believe both expository and research papers were affected by this as well, but I would not be able to defend that with citations.
I was a math student 1959 through 1965. My undergraduate calculus (and advanced calculus) teacher was a very good teacher but he was affected by this tendency. He knew he had to give us intuitive insights but he would say things like “close the door” and “don’t tell anyone I said this” before he did. His attitude seemed to be that that was not real math and was slightly shameful to talk about. Most of my other undergrad teachers simply did not give us insights.
In graduate school I had courses in Lie Algebra and Mathematical Logic from the same teacher. He was excellent at giving us theorem-proof lectures, much better than most teachers, but he never gave us any geometric insights into Lie Algebra (I never heard him say anything about differential equations!) or any idea of the significance of mathematical logic. We went through Killing’s classification theorem and Gödel’s incompleteness theorem in a very thorough way and I came out of his courses pleased with my understanding of the subject matter. But I had no idea what either one of them had to do with any other part of math.
I had another teacher for several courses in algebra and various levels of number theory. He was not much for insights, metaphors, etc, but he did do well in explaining how you come up with a proof. My teacher in point set topology was absolutely awful and turned me off the Moore Method forever. The Moore method seems to be based on: don’t give the student any insights whatever. I have to say that one of my fellow students thought the Moore method was the best thing since sliced bread and went on to get a degree from this teacher.
Re: Charles Wells’ Blog
No doubt there are philosophers of math who think they only need to pay attention to mathematical logic and set theory. No doubt many philosophers of mathematics as a matter of fact only pay attention in their work to mathematical logic and set theory. But most philosophers of mathematics in the second group don’t do this because they believe the former. They do this, among other reasons, because mathematical logic and set theory raise important philosophical issues about mathematics, or because they work on historical figures, mostly mathematicians, (Dedekind, Frege, Gödel, Hilbert, Poincaré, at al.), for whom logic and foundations were important. And many philosophers of mathematics don’t just look at logic and set theory. David Corfield is of course a prominent counterexample, but he’s by no means alone. Just look at the volume The Philosophy of Mathematical Practice, edited by Paolo Mancosu (OUP 2008), some of the essays in New Waves in Philosophy of Mathematics, edited by Otávio Bueno and Øystein Linnebo, or the work of Steve Awodey, Mary Leng, Chris Pincock, and Jamie Tappenden, among others.
I wish you had also quoted the bit of Wells’ post that comes before the bit you just posted: “My remark that the attitude that ”philosophy of math is merely a matter of logic and set theory” is ruinous to math was sloppy, it was not what I should have said. I was thinking of a related phenomenon which was ruinous to math communication and teaching.” If there’s a reason to think that the blame for the related phenomenon Wells bemoans can be laid at the feet of philosophers of mathematics, I’d like to see it.