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Berkeley on Calculus

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As is well known (at least in certain nerdy circles), the immaterialist philosopher George Berkeley sharply criticized Isaac Newton, and the branch of mathematics Newton had founded, in 1734. Berkeley's book of that year, THE ANALYST, said that calculus depends upon presuming that an "infinitesimal" is something at a certain point in one's reason, then assuming it is nothing at another point. That is internally incoherent.

Berkeley thus earned himself a place within the usual story about calculus. The story goes -- Newton proposed certain rough-and-ready ideas, not yet fully developed. He developed them just far enough, and just deep enough, to figure out orbital mathematics. But he left holes in his reasoning. Berkeley saw the holes and called Newton out on this. Subsequent theorists re-worked the foundations to render this branch of mathematics safe from Berkeleyan assaults.

That, as I say, is the usual story.

In 1987, a fellow named David Sherry wrote an article on the subject of Berkeley's critique of calculus, published in a journal called STUDIES IN THE HISTORY AND PHILOSOPHY OF SCIENCE. Sherry took a somewhat different view.  He puts this incident in the context of a pragmatic conception of what constitutes a "demonstration."

"In my view, many of Newton's demonstrations bring about understanding because they situate his results in a familiar context or rather an amalgamation of familiar contexts." Sherry objects to the notion that this was a bad thing, that it was a Newtonian defect that required Berkeleyan assistance.

Newton's demonstrations in particular, Sherry adds, "comprise representational techniques from kinematics, which would be familiar to anyone who grasped that subject. The purpose of Newton's demonstrations is to make plausible a particular manipulation of signs by showing that it is a natural consequence of representational techniques already employed."

The purpose, again, is understanding. Trying to make someone understand why a pitcher might in certain situations deliberately throw outside the strike zone in baseball. Only those who have some background in baseball and its vocabulary,  and likely who have played the game if only in a friend's back yard, could make sense of certain explanations, yet to people who meet that test, those demonstrations of the "intentional walk" would be perfectly cogent.

Berkeley simply didn't understand baseball well enough to understand the strategic value of an intentional walk -- he didn't understand kinematics well enough to grasp the idea of instantaneous velocity. "Baseball" in this analogy translates to such ideas as Oresme's diagrams, and Galileo's techniques in the discussion of momentum and speed.

Sherry's account seems a good deal less flattering to Berkeley than the standard view. His take on pragmatism in mathematics seems to suggest that 'foundations' are unnecessary so long as an existing practice exists. So the idea that Berkeley did something progressive by proving the inadequacy of the "foundations" of calculus falls by the wayside.


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