concept will be a new key word. A concept will basically specify what kinds of methods the class will need.

The interesting thing to me is they used the word concept. They were obviously directly influenced by the relatively new mathematical theory of “concept lattices” which is derived from lattice and set theory.

In that theory, a concept is defined by an attributes set and the elements it applies to. There is also a ‘context’ which essentially the biggest possible concept. A common example is the context being all the planets in the solar system and some list of properties about the planets. A concept from this context could be, ‘all the planets with moons’. Really to reference a concept you only need the attributes and it’s context, as the elements will be easily derived. I might be fudging this a bit, it’s been awhile.

In this case, the elements will be all the defined classes. The attributes will be members of all these classes. Mathematically a concept is a subset of a context. The interesting things about concepts defined this way is they form a lattice. They are inherently ordered. Most importantly though, they form a complete lattice. That is, all subsets (of the concept lattice) are closed under join and meet, or in this case all abitrary union and intersections of concepts yield a new concept (including infinite ones if that is a possibility in the context, not that this would happen in compsci).

This means that no matter how many weird concepts you have, you can require or exclude them and the resultant set of matching classes is easily determined. This may seem obvious I guess. What impressed me though is that the people coming up with this chose to implement it directly using concrete pure mathematics.

This probably the best example to me of why I like C++.

Complete lattices also have very useful properties. I’d be interested to know if any of them could be used for compiler optimisations when matching a class to a template.

[edit: Damn Wordpress thinks my C++ tag is a C# tag. so I have to use C plus plus]

Edwards suggests to “read the classics”. Meaning the most famous works of the most famous mathematicians. In particular he means Riemann’s original paper Ueber die Anzahl der Primzahlen unter einer gegebenen Grösse, obviously he means a translation (for mono lingual English speakers).

As insane as reading Riemann sounds, it’s not nearly as insane as reading Grassmann (which made no sense to mathematicians). I think I agree with him. There isn’t any other way to really get a taste for where the passion comes from. So many maths texts provide very minimal context for the development of the ideas. They might provide things in a chronological order of development and think that suffices, but it doesn’t.

To get a real taste for a subject you have to dig around and stumble down the same dead ends. It isn’t enough to know the topic if you don’t know why it’s interesting. This doesn’t mean you have to understand the original papers completely. I mean, Riemann? Good luck me.

I probably would have run to the hills at this idea two years ago, but last year I had to research Hilbert’s 10th Problem. I went on long tangents, reading all sort of ancient papers by people that have theorems named after them. I actually wrote a pretty crappy paper by the end of it. I got so side tracked by logic, I took too long get around to doing any number theory. It was a number theory course. At the time I regretted that, but in hindsight I’m glad. Who gives a fuck about marks anyway eh?

Also all the ancient logic got me confused when I came to write the paper. They had odd conventions that are very different to the way they’d be written now.

I started with Hilbert’s address and Hilbert himself. Then I read Gödel’s paper where he first proved his more famous incompleteness theorem. I read papers by Kleene, Church, Turing, Post etc, each one subtly changing the course of mathematical history. Then I read all the papers by Davis, Putnam, Robinson and Matiyasevich which led to the completion of the solution of Hilbert’s tenth problem (vast tracks of these got me lost). It was a true mathematical odyssey that far more fascinating by reading the source, rather then via popular science novels or even retrospective maths papers. All of the papers would be available semi publicly in almost any University library.

I wish I could find a similar hook for my thesis. Then I’d be set, I’m having so much trouble looking for a place to dig my fingers in. Grassmann seems too far removed from my thesis. I guess I should find out who really came up with the definition I’m looking at, since it’s supposedly been know for a while. Or at least I could find out which century it came into the mathematical consciousness.

So yeah I should read the classics, but time is always a factor. In our results orientated education system it also requires an assessment reason.
I’m not promising anything to myself.