h2g2 The Hitchhiker's Guide to the Galaxy: Earth Edition

Is anyone else interested in quasi-empiricism as a philosophy of mathematics? Basically, it says that mathematics is, in some sense, analogous to science, in that the axioms are technically arbitrary, but in fact must be such as to generate a mathematical system that models our intuition for what mathematics ought to be like. So it's quasi-empirical. You might be interested to read Lakatos' "Proofs and Refutations: The Logic of Mathematical Discovery" or Tymoczko (ed) "New Directions in the Philosophy of Mathematics".

So, is this a theory that would account for which which mathematics actually gets done?

I get an idea, from your summary, of the set of possible lists of axioms, and for each set of axioms, the set of provable propositions following from them. (Those would all be countable sets, n'est pas?) The process by which we select which subset(s) to explore could be described as quasi-empirical, perhaps.

It seems this might become less true over time - for example, less true for Lobachevsky than it was for Euclid, who certainly chose axioms that matched empirical experience. Is modern mathematics less quasi-empirical than ancient mathematics?


GTB

I think your summary is quite accurate, and extraordinarily concise .

Regarding decreasing quasi-empiricality. I think what you say is certainly true of axioms. In fact, the current axioms of mathematics are probably sufficient for all mathematics for a good long time. We know you can use them to generate arithmetic, theory of the reals, complexes, geometry, algebra, etc. What more could we want?

In response to that, there are two points. Firstly, you never know what might happen. Future developments in mathematics might lead us back to the axioms, and then a quasi-empirical process might start again.

However, even if that doesn't happen there is still a quasi-empirical process going on in ordinary research (that is, research which isn't into foundations of maths), in the development of new concepts. The example Lakatos gives is not contemporary but gives the general idea. He talks about the simultaneous development of the concept 'polyhedron' and Euler's formula V-E+F=2. You start off with simplistic ideas about what a polyhedron is, and you realise that for all polyhedra Euler's formula holds. You then start finding counterexamples, like the 'picture frame' (a polyhedral torus basically), or the really weird polyhedron which breaks Euler's formula but can't be embedded into 3D space, only a 5D space, but that satisfies some of the early definitions of a polyhedron. There are various responses to this, and they eventually lead to development of the concept of homology and the generalised form of Euler's formula in terms of homology groups.

So, in this case you have an intuitive concept 'polyhedra' and an intuitive theorem 'Euler's theorem' which can be exactly defined, and the process of coming up with exact definitions is a quasi-empirical one. You want to make the theorem true without disturbing its meaning too radically, but you also want to make the definition not too far from what we think of as a polyhedron. Ultimately, you come up with concepts and theorems which are for more subtle and refined even than your initial intuition, but the process of getting there is guided by your intuitive notions of what things ought to be, and also your discoveries about what actually happens with different definitions.

So, I think modern mathematics can be at least as quasi-empirical if not considerably more so than ancient mathematics, but only as regards concepts, not axioms.

What do you reckon?