Although my desire as wee lad to understand math was often clouded by alcohol fumes and wafts of smoke from irrationally prohibited substances, I really caught the bug and after a while it was like some kind of horrible but also wonderful nightmare where there were wheels within cogs within conical shapes that could only be described by men with unusual facial hair and wearing tank-tops (although the latter might be flashbacks from early Universidade Aberta's programs). Let this be a warning to any of you out there who casually wonder if that λ-Calculus stuff the old warhorse of a math teacher droned out to you in the [insert decade here] was actually of any use whatsoever. The same goes to those dreadful Schönfinkel’s S-and-Ks…These idle curiosities can be your undoing!

So for example try onomasti komodein. You can look it up on wiki, but doing so gives you very little real understanding of what it is and how it fits into the Athenian political system, or its cultural or literary context. Similarly I can look up 'fractals' and read the wiki entry, but I won’t really understand what 'fractals' means like someone with proper mathematical knowledge will. Such were the small beginnings of my terrible addiction. Then I started dreaming of understanding Fourier transforms and Taylor series even though I was still not sure why similar triangles weren’t also congruent triangles (or maybe they are; the internet is huge but sometimes conflicting). It's fairly unlikely that I'll win any prizes (although I still do the Euro Millions every month or two), but if I live long enough then one day I hope that I'll understand that bit of quantum physics and Schönfinkel’s S-and-Ks where something could come out of nothing…

Functional Programming and Haskell in particular, curried functions, higher-order functions, and λ-expressions, Schönfinkeled functions, anyone? Wolfram dedicates a few pages to some to these concepts but does not dwell much on them. I’d say that Schönfinkel's contribution is the most important one to λ-Calculus and functional programming in particular. Without Schönfinkel we’d still be in the Stone Age computer-science-wise. And what is a curried function you may wonder? I’m not very fond of the way Wolfram explains what a curried function is. In my book it’s a normal function with optional parameters that default to 0, nil, NULL, etc. or simply a normal function that passes to sub actions based on conditionals (if you think of an ATM a curried function would be a wrapper function that collects the input then passes onto the main action when all input is ready). Simple right?

The proofs of Alonso Church and Alan Turing came from earlier concepts like the concepts of Cantor's recursiveness, of the combinatorial logic of Schönfinkel and of the λ-Calculus with Haskell Curry. The methods introduced by Gödel and used by Kleene and Rosser show that Church's system was inconsistent, and it also prevailed in the negative solution of Church's decision problem. Church first demonstrated that a given expression of λ-Calculus with normal form is non-recursive. In the same paper, Church stated what is now known as the Church's Thesis, for example, that general recursive functions (and therefore λ-definable ones) are also exactly those that are effectively “computable”. The theorem and thesis combine to produce the result, that having a normal form is not a property effectively decidable. Kleene himself also emphasises the importance of Gödel in the work he and Rosser did in their contributions to recursion theory in the early 1930s. I remember writing a paper on this back in the day in college.

Only Wolfram to write a book about something that almost no one, outside of the Computer Science field, knows anything about in this day and of fast food thought. You got to love it!

NB: The nice people at Wolfram were kind enough to send me this book for me to have a go at it which I did. Most of it was read on a Cruise to Italy as well but I've just finished it today. Between Louise Glück's Averno and Wolfram's "Combinators", what a marvellous contrast! ( )