Is mathematics independent of science? And, vice versa.

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Mathematics investigates number systems some of whose properties arecritical for measurement. And without measurement, we wouldn't be ableto provide our scientific theories with sharp reality checks.Furthermore, those theories are themselves shot through with extremelysophisticated mathematics; most central claims of the advanced sciencescannot even be stated without a generous helping of mathematics. Manyhave been amazed that mathematics, often developed independently ofempirical research, turns out to be so useful, indeed necessary, forscience. But whatever one's explanation, it's a fact that it has.

Mathematics,by contrast, appears independent of science in important respects. It's true thatmathematical inquiry was often initially prompted by scientific inquiryinto the natural world. But what inspires mathematics is one thing, andwhat it owes its justification to is something else entirely. Mostmathematicians will tell you that the only ground for accepting amathematical claim is that someone has offered a deductive proof of it,and that the business of proving a statement bears no relation to howclaims are justified in the natural sciences. Neither the methods ofjustification employed in natural science nor any contingent facts about how matters stand in theworld seems relevant to the business of justifying mathematical truths.In these respects, mathematics appearsquite independent of science.

I agree with Alex about the way mathematics is independent of science. Einstein proposed that space is curved and hence non-Euclidian, but this didn't undermine Euclidian geometry, because that geometry is about an abstract space defined by the axioms of the system, not about physical space. So Euclidian geometry turns out not to apply to physical space, but it has not been refuted by physics.

There is however another way in which science and mathematics are not independent. Mathematicians may choose which problems to work on with an eye to what kind of mathematics might be particularly useful in science, and even more frequently scientists choose which problems to tackle by reference to the mathematical tools that are available to them.

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