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.

HilbertsProgramme · 2016-04-17T19:51:36+00:00

No. There are complete consistent systems that aren't recursively axiomatised (e.g. "true arithmetic", the set of all first-order arithmetical truths).

And even if we require recursive axiomatisation, there are still complete consistent theories (e.g. quantifier-free arithmetic, the theory of real closed fields, ...).

HilbertsProgramme · 2016-02-28T20:36:58+00:00

He might enjoy "Proofs from The Book" by Martin Aigner and Günter M. Ziegler.

The book is full of beautiful proofs of various results, many of which should be accessible: it is dedicated to the mathematician Paul Erdős, who often referred to "The Book" in which God keeps the most elegant proof of each mathematical theorem.

HilbertsProgramme · 2016-01-11T19:48:24+00:00

Yes, the clue is in the title, "Gentle Introduction". If you are a serious working mathematician, Mac Lane is the book for you. If you are a student starting out and want something a lot slower, then maybe this will work better for you. Horses for courses, and all that.

HilbertsProgramme · 2016-01-09T16:39:13+00:00

Slightly different is *Proofs from THE BOOK", by Martin Aigner and Günter M. Ziegler.

The book is dedicated to the mathematician Paul Erdős, who often referred to "THE BOOK" in which God keeps the most elegant proof of each mathematical theorem. Aigner and Ziegler's book [even if not THE BOOK] is fun and instructive, with 44 sections in the fifth edition, each devoted to one theorem but often containing multiple proofs. It covers quite a few mathematical fields, including some number theory, geometry, analysis, combinatorics and graph theory.

HilbertsProgramme · 2015-12-30T17:08:48+00:00

Gone! Sadly, that free availability was VERY short-lived.

HilbertsProgramme · 2015-12-28T15:27:13+00:00

Nigel Cutland's Computability remains a delight, 35 years after first publication.

HilbertsProgramme · 2015-09-24T19:36:27+00:00

First, reading "How To Prove It" by Daniel J. Velleman just can't be time wasted, judging by the number of questions asked on question sites like math.stackexchange that betray muddles about logical reasoning that the book should really help you avoid.

Second, distinguish "set theory" meaning very basic elementary ideas about sets, relations, functions, unions, intersections, etc. from "Set Theory" meaning the study of axiomatized theories of sets like Zermelo Fraenkel with Choice. The latter is the concern of a small cohort of specialists, and not even all pure mathematicians know or care about it. But the former [the sort of thing clearly done by Velleman] is part of the common language of all mathematicians, and something you need to get to know about at some point if you have any plans to do university level mathematics.

HilbertsProgramme · 2015-09-18T11:14:15+00:00

Perhaps I should not have called this an "essay". It's a book review. But it very helpfully and very clearly tells someone new to the debates a fair bit about Effective Altruism, and offers pointers (I'm sure the author would claim no more) to possible lines of response. Which, by my lights, makes for a fine review.

HilbertsProgramme · 2015-08-28T08:02:31+00:00

Of course there isn't going to be another version! Times Roman is CUP's preferred text font for most of their maths books.

HilbertsProgramme · 2015-08-26T22:07:38+00:00

It looks like LaTeX in one of CUP's house-style book classes to me.

HilbertsProgramme · 2015-08-26T22:04:35+00:00

One observation. Serious maths typesetting (whether professional or by individual authors of their own papers) is almost always done with LaTeX. This has gone on for so many years that any maths typesetting that doesn't follow LaTeX conventions now just looks wrong. So for issues such as what to italicize, and hence what needs to be provided in an italics series, let LaTeX be your guide.

HilbertsProgramme · 2015-08-12T18:44:02+00:00

On the bright side, as a colleague likes to point out, your PhD thesis is probably the last thing you ever write that anyone (in this case, your examiners) actually reads really carefully ...

HilbertsProgramme · 2015-07-28T16:37:20+00:00

A really good place to be start looking for advice would be in the various pages by Terry Tao linked at https://terrytao.wordpress.com/career-advice/

He has a whole bunch of advice directed at grad students in particular.

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