These are beginner level references for Category theory, ones that can be read by students of Computer Science. I haven't had a chance to look at all of this material myself. Ones that I have looked at have been marked with a star. Things that I think are better beginner material have more stars. At the time of this writing I am an absolute beginner in category theory and to most of abstract mathematics (with possible exception of Set theory), hence take my opinions here as just that - beginner level opinions. Finally, material that has been recommended to me by some of my professors has been marked with a +.

Categories
by T S Blyth (amazon) **

An Introduction to Categories in Computing
by Barry Jay (PS file)

Category Theory for Computing Science
by Michael Barr and Charles Wells (homepage of Charles Wells) (PS file of lecture notes) ++

Category Theory for Beginners
by Steve Easterbrook (PDF slides)

Elements of Basic Category Theory
by Alfio Martini (citeseer)

Computational Category Theory
by D.E. Rydeheard and R.M. Burstall (PDF)

A Categorical Manifesto
by Joseph A. Goguen (citeseer)

Basic Category Theory for Computer Scientists
by Benjamin Pierce (amazon) *+

Today evening, over pizza, Dr Hofstadter combined his Poetry class and his Group theory class and explained to the combined audience how Al-Khwarizmi visualized the solution to quadratic equations and how Tartaglia visualized the solutions to cubic equations. Using these simple visual models, he derived the solutions to the general form of the equations. He then went on to explain Tartaglia's 1534 AD poem that details his solution to cubic equations written in terza rima, the style of Dante Alighieri's La Divina Comedia. Needless to say, it was a fun class.

But what caused me to burst out laughing was this conversation I overheard where two students were discussing Group Theory and its application to Physics. Apparently there is a textbook by approximately that title and the university bookstore mislabeled it as "Group Therapy for Physics".

As I burst out laughing, something from my own ignorance struck me. Being somewhat culturally limited, I tend to think of some Chinese foods as "strange". Today morning I was trying to prepare some packaged Chinese food that I had purchased, when I noticed that the instructions on the cover said "Mix in the Soap Powder with constant stirring", and my instinctive reaction was "Oh, that's another strange thing that they'd eat". It was only several moments later that the awkwardness of of thing struck me. I went back and looked carefully at the instructions again. I had misread it, it really said, "Mix in the Soup Powder with constant stirring".

I am very moved by "Uncle Petros and the Goldbach Conjecture" by Apostolos Doxiadis. It's a work of fiction that makes several references to real people and real mathematics. It seems the author has done a fair bit of research to write this book. The book is about a mathematician "Uncle Petros" who spends his entire life trying to prove the Goldbach Conjecture.

In the course of the book, Doxiadis touches on the lives of G H Hardy, S. Ramanujan, Cantor, Kurt Godel and much of the pain, isolation, failure and achievement that is involved in doing research. The book is a fascinating read and is in many ways very true to life though the main characters are fictious.

The Goldbach Conjecture simply states that every even number greater than 2 can be expressed as the sum of two primes. Wikipedia has some more detail. The conjecture has been known for about 250 years now. There is still no proof, though the fact has been verified for very large numbers. It is a conjecture and not a theorem because it has no proof. Work still continues on this. Dare to try solve it? You may also enjoy looking at the Goldbach weave.

In the early 90's a long standing famous problem, Fermat's Last Theorem was proved by Andrew Wiles. It took him 7 years of exclusive work to prove the theorem. The theorem has been know since 1637! And has escaped all these centuries without proof despite many many mathematicians working on it. One of the things that made Fermat's last theorem famous is that Fermat has scribbled in the margin of his notebook a comment to the effect that he knew a proof but that the margin is not wide enough to note it. (Wikipedia) Wiles's proof is 150 pages and uses mathematics discovered in the 20th century - this is most likely not the proof that Fermat had in mind, if indeed he had a correct proof.

The interesting things about proofs like these are not just that they confirm the truth of statements that we always suspected to be true, but that they advance the state of mathematics. In the course of chasing hard problems, often new theories and new approaches to proof theory are discovered.

I have many many things to do before the end of the semester, which is a week away. I have papers to complete, assignments to finish, evaluations and grading to do and such. I don't have time to read books or write blog entries.

Pondering my predicament, I walk into the Swaine Hall library (where the CS, Physics and Math books are), looking slightly sheepish. I had ordered a book from another campus library and I was showing up too late. "I got this email about this book, its about Qualified Types by <ark P Jones. Well, I got the email over two weeks ago, do you think you still have the book for me?". The email said to pick it up in ten days. The person looks around and find the book. It was over 3 weeks ago... Anyway, he says that they usually keep the books around for a month or so if no one else has requested them. I say, "Let me look around a bit and I will pick it up on my way out."

I was looking for a book about universal algebras, a subject I know nearly nothing about. Having done my Computer Engg in India and spent some years in the "industry" before coming to college, I have little or no understanding of most of discrete math or most of abstract mathematics. It something about our system in India, and its almost always hard to explain to people at the univ as to why precisely this is the case. Most of the time the ask "You haven't heard of this?". No I haven't. So I am trying to make up for some of this.

I find a little book - an old one - the date says 1947. It was written in 1932. "Galois and The Theory of Groups", H. G. L. R Lieber. (Text by Lillian R Lieber, drawings by Hugh Gray Lieber)(for the unfamiliar, Galois is pronounced approximately Galwa, like in galvanize). It was such a strange little book. I picked it up and Mark Jones's dissertation. I had heard of Group Theory many times, finally I had a text that looked accessible.

Not only was it accessible, it was engaging, though it was written in a strange way. Most sentences were in fragments with strange usages of capitalization. It also had rather strange looking diagrams. I spent the afternoon reading the little book. I think I understand the essence of the idea. I need to look into some details for the proofs and such. Here is how the book starts by introducing the man - Evariste Galois.

What a fascinating subject - group theory. And what a fascinating little book. As I wrote this blog entry, I searched for the authors.

Lillian Rosanoff Lieber and Hugh Gray Lieber

This husband and wife team of a mathematician (Lillian) and illustrator (Hugh) influenced many generations of mathematically inclined readers, who stumbled on one of the Lieber books in their youth and were intrigued by their style of explaining complicated mathematics in simple language. Three of the most popular were The Einstein Theory of Relativity (1936 and later editions), Infinity (1953) and The Education of T C Mits (The Celebrated Man In The Street) (1942 and later editions).

I am glad I have my priorities right about work and other "distractions". :)

"Wikipedia resurrects the original concept of knowledge" — Anon.

What an intriguing thought.

What a wonderful little tale, meta-circular metaphors! By none other than Raymond Smullyan.

Planet Without Laughter
http://www-cs-faculty.stanford.edu/~knuth/smullyan.html

Once upon a time there was a universe. In this universe there was a planet. On this planet there was virtually no laughter. Nothing like ``humor'' was really known. People never laughed, nor jested, nor kidded, nor joked, nor anything like that. The inhabitants were extremely serious, conscientious, sincere, hard-working, studious, well wishing, and moral. But of humor they knew nothing. All except for a small minority who had some feeling for what humor was. These people occasionally laughed and joked. Their behavior was extremely alarming to everyone else and was regarded as an obviously pathological phenomenon. These few people were called ``laughers,'' and they were promptly hospitalized. What was so alarming about their behavior was not only the strange noises they made and the peculiar facial expressions they bore while ``laughing,'' but the utterly pathological things they said! They seemed to lose all sense of reality. They said things which were totally irrational, indeed sometimes logically self-contradictory. In short, they behaved exactly like anyone else who was deluded or hallucinated, hence they were put into hospitals.

(Hmm... I realized later that this is linked to Donald Knuth's webpage)

I am taking a Computational Complexity course this semester, taught by Prof Daniel Leivant. The course had not been going too well for me - I felt that though I understood many of the details, I didn't really learn anything since my intuitions drew a blank.

Towards the end of the semester (actually in the very last week), I got my hands on Dr Christos Papadimitriou's text on Computational Complexity. This was such a nicely done book. What also struck me was the subtle elegance of many things. In class, Dr Leivant had mentioned that Papadimitriou was a man of much good taste and style, and this does seem to reflect in his text.

For example the books cover is a the painting of Venus by Botticelli. The front cover looks complex and confusing. It is not immediately apparent what the picture is, much like the subject of computational complexity.

But if you were able to look behind complexity, if you turned the book over, you would see the beautiful face of Venus.