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

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 PierceOR 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!

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.

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 mindwhen he wanted to put category theory in the place of set theoryas the base of mathematics in teaching.I'll give an example of the kind this author fails vividly to give!Category:mathematics authorssub categories:1) humanist2) antihumanist ( structuralist)arrows:1) theoretical examples2) concrete examplesNow to make a Lewis Carroll ( Charles Dodgson) type sentence usingthis: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.

Really great. The best book I've found on category theory accessible to a philosopher. As accessible as Goldblatt, but a lot better. The last chapter, especially, should be read by anyone doing analytic philosophy in the set-theoretic vein.

非数学専門家のものです本書は、だいたい読みましたといっても、ページ数だけで、内容は･･･？？？？という感じですが（やっぱ非専門家というか、サラリーマンには、ちと厳しい？）ただ、本書のいわんとするところはつかんでいるつもりです。以下、本書を買おうとしている人が、参考になる？かと思いコメントします。本書は、圏論から始まります。そして、その圏論も、結構しっかり書いてあります。その後に、トポスとその周辺に触れています。そして、トポスに触れた後で、現代の、述語論理の定理、公式が、次々と、圏や、トポス（どっちだったっけ？）の表現に変換できることを見ます。そして、「ある、論理構造を持つ述語体系はすべて、トポスに１対１に対応する」という定理を一応の、目標として、話を進めていきます。ここの部分が、圏論だけを述べている、マックレーンのcategories for the working mathmatitianとは、大きく違いますこの本の肝は、ある論理構造を持つ、述語体系はの部分の、論理構造について、どう、数学的に表現しているか、ですがごめんなさい、その部分は読み切れませんでしたただ、内容的には、竹内外史先生の「層・圏・トポス」にものすごく近いと感じました。