Not much, I fear. Chazelle is a very smart guy, but here he's trying waaaay too hard to be cute, and the information density comes out very low.

Here's a one-paragraph summary. Anything that doesn't immediately seem familiar is an indication that it might be worth skimming the corresponding bit of Chazelle's article, although frankly there isn't much more content in his article than in my paragraph.

Progress in computer hardware is slowing; progress in algorithms is where the fun will be in future. At a high enough level of abstraction, all computers are basically the same. You can think of a computer as a thing that applies programs to data. The correspondence between programs and meanings-of-programs is basically arbitrary, kinda like some early 20th-century French intellectuals said about natural language. You can treat programs as data, which e.g. allows you to write programs that make copies of themselves. This is a bit like how living things make copies of themselves. An important class of problems take the form "Find a thing satisfying such-and-such requirements" where it's clear how to check an alleged solution rapidly, but there's no obvious way to find a solution in the first place. For instance, finding mathematical proofs versus checking that they're valid. It turns out to be good to interpret "rapidly" as "in time that grows at most polynomially in the size of the problem specification". Making this all rigorous gets you the notion of an "NP-complete" problem; there are many such, and it turns out that in some sense solving one is the same as solving them all. No one knows whether such problems can after all be solved rapidly (in the lingo: if "P=NP" is true). If they could, then mathematics and protein-folding and many other such things would become easy. Also, e-commerce would collapse, because it depends on public-key cryptography, one version of which depends on the conjecture that large numbers can't reliably be factored rapidly. There is no such thing as a program that looks at another program and the data it operates on, and tells you whether it will ever terminate. Such undecidability seems kinda like intractability (i.e., insolubility in polynomial time) but it turns out that undecidability is useless but intractability is useful, because often if you replace "find something such that X" with "find something such that showing it isn't X is intractable" you get an easier problem but one whose solution is just as useful. For instance, generation of (pseudo-)random numbers. You can "prove" you know something without revealing what it is (e.g., demonstrate that you can factor a large number, without revealing the factors); the same ideas also allow, e.g., two people to determine which of them is wealthier without either revealing their actual wealth. Similarly, you can "prove" that you have a proof of a mathematical theorem without revealing any of the actual steps in the proof. All this depends on P!=NP, or some slightly stronger variant thereof. Algorithms are cool. Traditional mathematics isn't enough for modern science; in the future there'll be lots of simulations and computer models -- algorithms. But we know how to reason with mathematical formulae, and as yet we have no equally strong "calculus of algorithms". This needs more research.

(Some of the claims in that paragraph are pretty silly. I think that in every case that's because what Chazelle wrote is silly in the same way.)