Mathematics is the combination of two traditions, the deductive greek science whose Idealtypus was geometry, and the manipulative algebraic techniques that Middle Age inherited from Arab science.

Freeman Dyson, in the Gifford lecture “Infinite in all Directions”(1985) talks about a somewhat related juncture: Manchester vs Athens .

Briefly, Athens – i.e. the tradition of high philosophical ideas and deep ideas necessarily embedded in the practice of Western science – is just one possible strategy, one locus, to approach mathematics.

Another one was historically embodied by Manchester, the cradle of Industrial Revolution, where a more engineering or applied style was given full citizenship.

As Frederick Engels said, “If society has a technical need, that helps science forward more than ten universities”. And that was indeed the case with Manchester.

This distinction operates across all mathematics. A big ideas and deep philosophical questions approach can be accompanied by a manipulative, algorithmic approach. That links to the very English taste for algorithms, see G. Boole characteristic polynomial of second order ODE, Cayley-Hamilton ideas about Characteristic polynomial of a matrix and indeed A. Turing, who gave full rank to the idea of algorithm (Turing worked in Manchester his last years).

Need to expand, talking about examples from within Mathematics. Is this same ideas as Gilbert Strang, of MIT mathematics fame, when he says that going forward Algebra is going to be more important than analysis?

That has ramifications in the problem of the gap Continuum vs Discrete. More later.

See Arnold Thackray “Natural Knowledge in Cultural Context” (American Historical Review, 1974(79:3))