Tuesday, September 20, 2016

Philosophy is irrelevant to mathematics, and vice-versa

Eli Maor notes that when mathematicians finally began to accept the idea of infinite series, they began to toss the notion of "infinity" around very casually, with very little philosophical rigor.

Similarly, Leibniz certainly could not explain philosophically what he meant by an "infinitesimal." But so what? Employing them allowed him to develop calculus, one of the greatest inventions in the history of mathematics.

Berkeley mocked the mathematicians occasional self-impotrance, but he had no intention of showing that their mathematical results were wrong. When he wrote, in The Analyst:

"And what are these Fluxions? The Velocities of evanescent Increments? And what are these same evanescent Increments? They are neither finite Quantities nor Quantities infinitely small, nor yet nothing. May we not call them the ghosts of departed quantities?"

His goal was not to dispute the usefulness of calculus, but to point out that mathematicians were putting their faith in concepts that they could not philosophically justify. And thus they had no basis for knocking similar moves by others.

And even though, mathematically speaking, infinitesimals have been put on a more sound axiomatic basis, I don't think anyone in the 300 years since Berkeley wrote has offered a decent philosophical explanation of what their ontological status is supposed to be.

But again, so what? Mathematicians are doing mathematics, not philosophy, and they only need to justify their concepts mathematically, not philosophically. If a mathematical concept works to produce interesting and/or useful advances in mathematics, that is the only justification it needs!

Asking a mathematician to philosophically justify some piece of mathematics is a lot like telling your plumber, who has just fixed a bad leak, that he has to explain his work in terms of molecular biology before you will pay him. Or telling an NFL coach that his play calling is all wrong, because he hasn't taken the principles of metallurgy into account.

(And this is the complementary point to my previous post on this matter, noting that interesting mathematical advances, however useful they are mathematically, do not resolve philosophical problems.)

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