Customer Reviews

Toposes and Local Set Theories: An Introduction (Dover Books on Mathematics)

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I have a maths degree, I've done some Set Theory, and I've had some exposure to Category Theory from a Computer Science perspective. But I really struggled trying to read this!

If you're looking for "Toposes for Dummies" (and I still am!), this isn't it.

John L Bell is one of my favorite maths authors. He's also co-authored "Logical Options", another very good book about alternative logics.

This book is what I've found to be the easiest route to understanding toposes and particularly how it relates to logic.

The first chapter describes category theory but it's so dense that it's practically useless unless you already know category theory. I think a background in category theory is required to understand toposes. For this, I recommend:
1. "Conceptual Mathematics" by Lawvere +
2. "Basic Category Theory for Computer Scientists" by Pierce
OR a few other similar books such as Walters, or Barr & Wells.

You also need some background in simple type theory, for which there are quite a few introductory books.

There are a few other books about Topoi:

1. Goldblatt claims to be introductory, but I actually find it VERY hard to follow.

2. Bart Jacobs' "Categorical Logic and Type Theory" is actually easier than Goldblatt.

3. This book is the best I found so far.

I'm still reading the 3rd chapter. Will review more when I have time. Hope this helps!

The Bourbaki structuralist approach is said by Amir C, Aczel to be:
"overly formal . too abstract. and much more rigorous than necessary. thus making it unnecessarily difficult to read and understand mathematics."
My immediate reaction / objection to this book is that this level of abstraction removes categories from the 'real world' reference of a new mathematics for ordinary people. For this approach to be effective one must already have had set theory, algebraic theory, symbolic logic and specifically for this book, the theory of topological neighborhoods.
Those qualifications put this approach to category theory above what is taught to graduates in the physical sciences in general and removes it to graduates( or very advanced undergraduates) in mathematics alone.
I don't think this could be what Grothendieck had in mind
when he wanted to put category theory in the place of set theory
as the base of mathematics in teaching.
I'll give an example of the kind this author fails vividly to give!
Category:
mathematics authors
sub categories:
1) humanist
2) antihumanist ( structuralist)
arrows:
1) theoretical examples
2) concrete examples
Now to make a Lewis Carroll ( Charles Dodgson) type sentence using
this:
A mathematics author is an antihumanist if the only examples he gives in his text are theoretical examples.
A mathematics author is an humanist if he gives both theoretical and concrete examples.
I'll let you figure out in which category I think J. L. Bell belongs.