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Which philosophers considered mathematics an experimental science (as opposed to a theoretical/speculative science)?

It seems Kant thought that mathematics (or at least geometry) is purely a priori, and thus not experimental. Quine also, in "Two Dogmas of Empiricism," thought "mathematics does not seem to be known on the basis of experience".

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  • Obligatory xkcd: Purity
    – MichaelK
    Apr 3 '18 at 9:36
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    It was known as psychologism in the late 19-th century, Mill was the most prominent representative, Frege and Husserl raged against it. In the recent times Borwein et al. have a whole book on experimental mathematics with philosophical introduction which states "I no longer view proof as the royal road to secure mathematical knowledge".
    – Conifold
    Apr 3 '18 at 20:21
  • @Conifold Perhaps you could turn your comment into a full answer.
    – Geremia
    Apr 4 '18 at 4:57
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We can consider John Stuart Mill's anti a priorism and naturalism.

For Mill:

amongst the Laws of Nature learnt by way of inductive reasoning are the laws of geometry and arithmetic.

See John Skorupski (editor), The Cambridge Companion to Mill (1998) and John Skorupski, Mill (1999).

For some recent approach to the philosophy of mathematics that are neither Platonist nor formalist we can see :

All these books consider mathematics as a social-cultural-historical human activity, arising from the human intellect. They all - in various ways - reject the idea of a "transcendental" mathematics independent of human thought.

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Who is a philosopher? I think a mathematician who ponders about this topic is at least a philosopher of mathematics. Here are three:

Mathematics is a part of physics. Physics is an experimental science, a part of natural science. Mathematics is the part of physics where experiments are cheap. [V.I. Arnold: "On teaching mathematics" (1997)]

But there is hope for a grand-unification! In twenty years (perhaps sooner!), mathematics can once again become a unified religion. All we need is to worship the new God of Experimental Mathematics! [D. Zeilberger: "Opinion 113" (2011)]

Mathematics as monologue, dialogue, and discourse needs tools of describing and communicating ideas. MatheRealism denies the existence of entities which, in principle, can never be observed or communicated – in particular thoughts that no-one can think. [W. Mückenheim: "MatheRealism" (2018) p. 351]

The most important theorems of a mathematics oriented towards reality can be proved in good approximation by means of experiments, performed mainly on efficient computers. Computers are the telescopes and microscopes of mathematicians. They improve the perspective and allow to distinguish details that cannot be seen otherwise. [W. Mückenheim: "Mathematik für die ersten Semester", 4th ed., De Gruyter, Berlin (2015) preface]

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If you drop the word 'experimental' from your question, then Pythagoras is probably the closest you'll get. He believed everything was made of numbers. A close second would be Plato who believed mathematics to be the key to understanding the world. Both men might take issue with your term 'experimental science,' as they thought science was a purely mental activity.

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  • I'm asking about philosophers who consider mathematics an experimental science.
    – Geremia
    Apr 3 '18 at 14:39
  • @Geremia You are unlikely to find any. Science is the empirical branch of knowledge, and experimental science is the part of science concerned with carrying out the actual measurements. Mathematics, on the other hand is a subset of logic and not part of science at all. To find a philosopher who believes mathematics to be part of science requires finding a philosopher with a very special kind of ontology. To take this question further and add the word ‘experimental’ to the word science makes the question absurd. What would a mathematical experiment even look like?
    – njspeer
    Apr 3 '18 at 14:58
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    See what I added to my question. Quine is an example of one who does not think mathematics is experimental.
    – Geremia
    Apr 3 '18 at 17:50
  • @Geremia As I sated above, the concept of experimental mathematics makes no sense. Using the normal meanings of the words involved, it’s a contradiction—like asking what philosophers believe in four-sided triangles. Again: what would a mathematical experiment even look like? Providing an example of a philosopher who does not think mathematics is experimental hardly helps. All philosophers should think that mathematics is not experimental. To think otherwise would be to redefine the terms involved.
    – njspeer
    Apr 3 '18 at 17:58

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