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Indeed, real open ended creative mathematics - i.e. pure mathematics in the classical sense - is always messy and conceptual, while technical definitions through rigourous axiomatic formalization almost only always come after the actual discovery has already taken place.vanhees71 said:That's precisely what I meant before. The Bourbaki books an some of the textbooks of the members of Bourbaki are closer to scientific research work, and without doubt excellent research work, but they are lousy as textbooks. I'm sure that all these brillant mathematicians didn't come to the results presented in the waybof these books but in creative acts of thinking. Of course at the end the finding must be formalized in this way to be true pure math.
Formalism, a bastard of logicism, championed by Hilbert in the pure mathematics community started to drive away many of the greatest late 19th century pure mathematicians, from Poincaré - famously the last univeralist (generalist), because of his creative instead of rigorous mind - onwards towards physics and applied math. Both Poincaré and Hadamard wrote on this subject.
Formalism then, during the 20th century, came close to culmination in Bourbakianism, driving generalists almost fully into applied mathematics. This drive-away was in peak effect mid-century - during the time of Mandelbrot et al. - firmly making their contributions to pure mathematics to instead incorrectly be viewed as physics and applied mathematics.
Incidentally, the last great theoretical and mathematical physicists - Feynman, Wilson, Anderson, Dyson, Mandelbrot, 't Hooft and Penrose - all recognized and spoke out against these developments in mathematics, but very few listenend i.e. taking their warnings at face value as critiques of mathematics itself, when they were actually criticizing formalism and axiomatics.
In classical pure mathematics - and therefore in physics as well - formalism is useless in discovering novel concepts, because it already presupposes full completeness of theory; this is why formal pure mathematics is purely deductive opposed to classical pure mathematics. To paraphrase Atiyah and Weyl: Hilbert and his followers killed creative pure mathematics. Bourbaki however made things severely worse by imposing the formalist ideology on students as well through the rewriting of curricula and textbooks.
This caused a severe widening of the divorce between pure mathematics and physics, worsening extremely with the professionalization of academia and overspecialisation of the sciences. The love between physics and mathematics would only be rekindled somewhat late in the 20th century, for somewhat wrong reasons, i.e. in string theory. It is happening again though, but now between applied mathematics and physics - while the formalist scoffs at both.
In any case, it should be obvious why formalism does more harm in mathematics than good; it is a self-imposed censorship of the mind borne out of the idea that mathematics must be reducible to logic, axioms and deductive reasoning alone. This is also exactly why to the physicist - today seen as a non-mathematician by most mathematicians - axiomatics are at best an afterthought; its a shame that many physicists seem to have forgotten this.