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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".
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 :
Proofs and Refutations (1976) by the Hungarian philosopher Imre Lakatos;
The Mathematical Experience (1981), by Philip J. Davis and Reuben Hersh;
Mathematics, Form and Function (1986) by the American mathematician Saunders Mac Lane;
Where Mathematics Comes From (2000), by George Lakoff, a cognitive linguist, and Rafael Núñez, a psychologist.
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.
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]
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.
