![]() It's of course true that also the pure mathematicians need heuristic ideas before they build their abstract edifices. in the goal to investigate the status of the parallel axiom in Euclidean geometry, which is a completely pure-math academic question. E.g., for Einstein when he looked for the right description of gravity within relativistic physics, he could build on the notion of Riemannian geometry, which was developed by pure mathematicians like Gauß, Lobatchevsky, Riemann, Minkowski, Levi-Civita et al. For this you need to know the mathematical structures, found by the pure mathematicians, to find such a description. The goal of the theoretical sciences is to find the right mathematical structures to describe some particular aspect of nature, e.g., the motion of bodies and continua (fluids and solid) in Newtonian or relativistic classical mechanics. This is, of course, important also for the applications, because the proven theorems and the discovery of other theorems in proving provides the idea, how to apply mathematics to the description of the real world. ![]() Pure math aims at rigorous proofs of theorems from a given set of axioms (or even investigates the implications of this idea itself in terms of formal logics). ![]() As already discussed pure math has another aim than applied math.
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