“Galileo was no idiot” (or “Wir werden wissen”)

What is the exact meaning of Hilbert’s famous remark that:

“Galileo was no idiot. Only an idiot could believe that science requires martyrdom – that may be necessary in religion, but in time a scientific result will establish itself”?

Why Bólyai,Lobachevsky (and later Riemann) got same equations at the same time, as well as Wiener and Kolmogorov, or for that matters, De Giorgi and Nash  as well as countless other figures in the history of mathematics?
One explanation is of course that of the hidden agenda, or superior intelligence, nicely orchestrated in this piece by Shafarevich. This is the route of Pythagoras and via Plato, of some of Western canon’s deepest thinkers. One of its latest champions seems to have been irremovable (see this article by Stanford logician George Kreisel)
This is a route, but of course others exist. Consider this problem, from an introductory numerical analysis book:


Boris Hessen, in “The Social and Economic Roots of Newton’s Principia” articulated a materialistic epistemology which did not fare very well in the scientific community of his time, but that had its proponents in the Hegelo-marxian camp, especially coming from the Eastern Bloc. Lukács inspired Lakatos and even Popper’s falsificationism may be a restatement of certain ideas of Hegel. (The whole text of Hessen’s work to be found here).

Barrow’s “PI in the Sky” articulates just these ideas in a very cogent way. At the end one is left wondering whether Pythagoras had indeed stumbled upon some serious piece of truth, which humanity has not yet fully uncovered or his was just another optical illusion, because from the distance Kant’s apriori is just frozen path-dependency.

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