Philosophy is irrelevant to mathematics, and vice-versa

By -

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.)

Comments

Post a Comment

Popular posts from this blog

Libertarians, My Libertarians!

By -
Cruel to be kind means that I love you . Because, while I think you are mistaken, your hearts are in the right place -- yes, even you, Silas --  unlike some people . This Breitbart fellow (discussed in the link above), by all appearances, deliberately doctored a video of Shirley Sherrod to make her remarks appear virulently racist, when they had, in fact, the opposite import. I heard that at a recent Austrian conference, some folks were talking about "Callahan's conservative turn." While that description is not entirely inaccurate, I must say that a lot of these people who today call themselves conservative give me the heebie-jeebies.
Read more

"Pre-Galilean" Foolishness

By -
 I am currently reading The Master and His Emissary , which appears to be an excellent book. ("Appears" because I don't know the neuroscience literature well enough to say for sure, yet.) But then on page 186 I find: "Asking cognition, however, to give a perspective on the relationship between cognition and affect is like asking astronomer in the pre-Galilean geocentric world, whether, in his opinion, the sun moves round the earth of the earth around the sun. To ask a question alone would be enough to label one as mad." OK, this is garbage. First of all, it should be pre-Copernican, not pre-Galilean. But much worse is that people have seriously been considering heliocentrism for many centuries before Copernicus. Aristarchus had proposed a heliocentric model in the 4th-century BC. It had generally been considered wrong, but not "mad." (And wrong for scientific reasons: Why, for instance, did we not observe stellar parallax?) And when Copernicus propose...
Read more