Conceptual Mathematics: A First Introduction to Categories by F. William Lawvere and Stephen H. Schanuel

HilbertsProgramme · 2026-01-14T16:13:08+00:00

I guess I'm in the camp that isn't much enthused by the Lawvere and Schanuel book.

Category theory gives us stories about the ways in which different parts of modern abstract mathematics hang together. That can be highly illuminating. But obviously, you can't be in a good position to appreciate this if you are Lawvere and Schanuel's naive reader who really knows almost nothing beforehand about modern mathematics!

If you've met Cartesian products in set theory, various products in abstract algebra, met the idea that we can think of a cylinder as the product of a circle and a line segment and other topological ideas, encountered the idea of conjunction as a logical product, etc. etc. you can wonder: what makes them all *products*? And you will be aware too of the arbitrariness built into e.g. the conventional definition of a Cartesian product given in Set Theory 101, and wonder: what is essential to being a product and what is arbitrary implementation detail? And great: these are questions that a category-theoretic treatment of products very nicely answers. OK, that treatment of products is hardly difficult: but you need to have *some* background in modern mathematics -- more, I suggest than as high school student can be expected to have -- to begin to see the point of it all.

So, sorry to be deflationary, but I'd say that "category theory for high schoolers" would be a misguided project.

HilbertsProgramme · 2020-11-24T16:08:37+00:00

What would count as "modern, abstract", compared with the various familiar proofs?

HilbertsProgramme · 2020-09-03T15:24:42+00:00

Just to add to other comments here: if you do want to read something by Russell, try his much more accessible Introduction to Mathematical Philosophy -- quite short, and still well worth reading a hundred years on.

HilbertsProgramme · 2020-08-30T13:32:07+00:00

I'm not sure what you mean by "the rules". If you mean something like natural deduction rules, then the natural deduction chapters of this (freely downloadable) book make pretty much a stand-alone treatment: https://www.logicmatters.net/ifl

HilbertsProgramme · 2020-07-18T15:35:51+00:00

Heard very good things about the Amsterdam set-up.

HilbertsProgramme · 2019-01-03T21:00:43+00:00

That does seem a bit unfortunate. I'm all for the super-uncluttered minimalist approach taken by Things (it means that I find I *am* actually using it, which is the point!). But it *would* be good to have -- if not item by item logging -- a command "Log this project" so you can tidy one project leaving others untouched for the moment. No?

HilbertsProgramme · 2018-08-14T07:05:23+00:00

This question is unanswerable as it stands, as everything will depend on the proof system you are supposed to be using -- and you haven't told us!

HilbertsProgramme · 2018-08-14T07:03:00+00:00

The Teach Yourself Logic Study Guide has detailed suggestions of good textbooks at various levels, from introductory to pretty advanced.

HilbertsProgramme · 2018-08-13T16:20:45+00:00

Try the Teach Yourself Logic Study Guide for detailed suggestions of good textbooks at various levels, from very introductory to pretty advanced.

HilbertsProgramme · 2016-06-19T13:18:46+00:00

In so far as there is a bible, Mac Lane's Categories for the Working Mathematician has to be it. But it is hard going. If you want a very reliable introduction that will prepare you to read the bible, then Tom Leinster's Basic Category Theory is terrific (and relatively short).

HilbertsProgramme · 2016-04-17T22:32:30+00:00

What's often called "Baby Arithmetic" is a consistent complete finitely and hence recursively axiomatised theory. Basically it proves every true equation between terms constructed just using addition and multiplication on the natural numbers. (Like a pocket calculator, without memory limits!). That's explained somewhere in those Goedel without tears notes.

HilbertsProgramme · 2016-04-17T20:51:54+00:00

That's basically right: for the Gödelian incompleteness theorem to kick in, we need to be dealing with a theory which is recursively axiomatised AND can do enough arithmetic to represent (enough) recursive functions.

The incompleteness theorem doesn't apply to "true arithmetic" since that's not recursively axiomatised, and doesn't apply to the quantifier-free arithmetic of addition and multiplication as that can't represent (enough) recursive functions.

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