Berkeley’s Aristotelian Critique of Newton’s Physics

Abstract

Berkeley’s central critique of Newton’s calculus in his 1734 Analyst argues that Newton’s general method of calculating the derivative suffers from an inconsistency: It first assumes that the increment of the independent variable is finite, and later it assumes this increment to be zero. Yet, the consequence of the first assumption is retained even while applying the second, contradictory assumption and “nothing is plainer than that no just conclusion can be directly drawn from two inconsistent suppositions” (Analyst §15). The inconsistency is of this structure: By A₁ (first assumption, i.e., Δx ≠ 0) Newton gets consequence C₁ (Δy/Δx = [f(x + Δx)-f(x)]/Δx) , and then by adding A₂ (second assumption Δx = 0) he derives C₁ (dy/dx=lim_Δx=0 Δy/Δx). The explicit scheme is, therefore,

$$ \left[ {\left( {{A_1} \to {C_1}} \right) \wedge {A_2}} \right] \to {C_2} $$

which means that (A₁ ∧ A₂ → C₁) → C₂, but since A₁ ∧ A₂ is a contradiction, C₁ cannot be said to follow from it in a non-vacuous way, and consequently the same goes for C₂, which is the derivative of the function: Certainly when we suppose the increments to vanish [= A₂], we must suppose their proportions, their expressions, and everything else [= C₁] derived from the supposition of their existence [= A₁] to vanish with them. (Analyst, §13)