Interview with the authors of An Invitation to Applied Category Theory: Seven Sketches in Compositionality

I recently listened to this interview with the Brendan Fong and David Spivak, authors of  'An Invitation to Applied Category Theory: Seven Sketches in Compositionality', currently one of my favorite books. The interviewer, Cory Brunson, is excellent and I particularly appreciated the way he got the authors to explain the key ideas of the various chapters.  For anyone starting out on the book (or on the course) this interview would make a good introduction.

Indeed the whole New Books Network looks very interesting.

Category Theory without Objects

Our notion of category is that of [Eilenberg & Mac Lane, 1945]. We identify objects with their identity maps and we regard a diagram A--f-->B as a formula which asserts that A is the (identity map of the) domain of f and that B is the (identity map of the) codomain of f.

From the start of the PhD thesis of F. William Lawvere ( Functorial Semantics of Algebraic Theories and Some Algebraic Problems in the context of Functorial Semantics of Algebraic Theories, 1963).

Essence of Linear Algebra

I have just finished watching this excellent series of short videos by Grant Sanderson of 3Blue1Brown on the geometric intuitions behind linear algebra:

Essence of Linear Algebra

They remind me of some maths courses it took with the Open University 20 years ago but they included quite a few things that were new to me.  I particularly liked the matrix interpretation of the differentiation of polynomials in the last video.

The Cost of Electricity

I have been looking at setting up a web server at home.  As this will involve leaving a computer running all the time, I have been looking at low-power PCs.  While estimating what the impact on my electricity bill would be, I was pleased to discover that, at current UK prices, running a 1 watt appliance for 1 year will cost approximately £1.  This pleases me almost as much as when I found out that I walk at a speed of approximately 100 metres per minute.

Mistakes as Spurs to Research

Here are a couple of examples in which young researchers submitted papers for publication which contained potentially embarrassing mistakes.  But, rather than being put off by their mistakes and retiring into obscurity to to lick their wounds, both researchers were spurred into doing the work which made them famous.

From this biography of the mathematician Neils Henrik Abel:

While in his final year at school, however, Abel had begun working on the solution of quintic equations by radicals. He believed that he had solved the quintic in 1821 and submitted a paper to the Danish mathematician Ferdinand Degen, for publication by the Royal Society of Copenhagen.  Degen asked Abel to give a numerical example of his method and, while trying to provide an example, Abel discovered the mistake in his paper.

Abel went on to prove that there is no general solution in radicals to quintic equations (the Abel-Ruffini theorem). In his book Finding Moonshine, Marcus du Sautoy suggests that Degen actually recognised that Abel's solution was flawed and, by asking for a numerical example, he was gently prompting Abel to look at his work again, without discouraging him.

The second example is from here on the web-site of the computer scientist Leslie Lamport:

When I first learned about the mutual exclusion problem, it seemed easy and the published algorithms seemed needlessly complicated.  So, I dashed off a simple algorithm and submitted it to CACM. I soon received a referee's report pointing out the error.  This had two effects. First, it made me mad enough at myself to sit down and come up with a real solution.  The result was the bakery algorithm described in.  The second effect was to arouse my interest in verifying concurrent algorithms.

Lamport's bakery algorithm was just the first of many important contributions he made in the field of concurrent algorithms.