Thanks for another great article. One point which puzzles me slightly (not related to your article in particular) is that of the distinction made between "pure" and "applied" science. If science is a PROCESS (and not the results which are a product of that process), then isn't the distinction rather tenuous? - the process is the same whether carried out in order to obtain "knowledge for its own sake" or a solution to a practical problem. My use of the terms "pure" and "applied" are only in connection with science funding and everyday practicalities, not to any fundamental property of science.

Pure science explores for the sake of exploration — for example, Bell Labs (dedicated to pure research). When Bell Labs scientists Penzias & Wilson crawled around in their microwave dish, scraping away bird droppings, they were doing pure research, not applied — they were exploring the subject of microwaves, not their applications. They accidentally discovered the Cosmological Background Radiation (CBR) — an inadvertent side effect. In the case of mathematics, the difference is more clearly defined. Applied maths takes propositions assumed to be true (for example, the laws of mechanics) and then demonstrates the proof of related propositions by a process of deduction. On the other hand, pure mathematics requires only that assumptions necessarily imply the theorems which are deduced from them. It's more accurate to say that pure mathematicians construct proofs on the basis of axioms, while applied mathematicians construct proofs on the basis of the relationships discovered by pure mathematicians. But in practice, applied mathematicians are more likely to apply proofs than create them.

In everyday usage, applied mathematics is more like engineering, and pure mathematics is more like pure scientific research. To that extent, you could suppose that pure maths is more of an art than a science, since there is no requirement that the assumptions correspond to reality. This is true only in a limited sense (the axioms must not conflict with each other), nevertheless it's one basis for the argument that mathematics isn't really a science. Whether mathematics is a science is still an open question — the absence of perpetual falsifiability in mathematics is another of several issues. Of course, it nearly always turns out that even the purest of mathematics is found to have practical application in some field, but I would argue that problem-driven research is more efficient than the pursuit of "knowledge for its own sake". The truth of that proposition ("more efficient") depends on (a) whether we have already discovered all of mathematics (no), and (b) whether pure mathematicians successfully avoid creating anything useful (no). The present field of cryptography depends on fundamental work with prime numbers, and General Relativity and string theory are built on a foundation of non-Euclidean geometry, both originally mathematical curiosities with no imaginable practical application.

So there is a parallel between pure mathematics and pure science — they are both more useful than they appear at first glance.