Peacock's Principle as a Conservative Strategy

TL;DR

The paper defends Peacock's principle of permanence as a conservative strategy in mathematics, arguing that non-commutative algebras do not invalidate it and illustrating Hamilton's quaternionic calculus as an example.

Contribution

It offers a reinterpretation of Peacock's principle as a conservative strategy, clarifying its resilience against objections involving non-commutative algebras.

Findings

Peacock's principle is a conservative strategy, not an absolute law.

Hamilton's quaternionic calculus exemplifies a conservative approach.

Non-commutative multiplication does not necessarily invalidate the principle.

Abstract

The view that Peacock's principle of permanence has been invalidated by Hamilton's introduction of non-commutative algebras has always seemed rather odd, in light of Peacock's favorable reception of quaternions and the endorsement of his principle by Hamilton. But the view is not just odd; it is incorrect. In order to show this, I critically analyze Peacock's attempts to reject possible exceptions to his principle, like the factorial function and an infinite series due to Euler. Then I argue that the principle of permanence is best understood as an expression of a conservative strategy, philosophically grounded in Hume's conception of the laws of reasoning, which advocates their preservation to the furthest extent possible, thus allowing exceptions, i.e., violations of these laws. On this reading, non-commutative multiplication does not invalidate Peacock's principle, if the reasons for violating commutativity outweigh the reasons for its preservation. Finally, I show that Hamilton followed a conservative strategy of precisely this sort when he developed his quaternionic calculus.