Peacock's Principle as a Conservative Strategy

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.

💡 Research Summary

The paper challenges the widely held belief that George Peacock’s “principle of permanence of equivalent forms” was falsified by the advent of non‑commutative algebras, particularly Hamilton’s quaternions. The author begins by tracing Peacock’s own formulations of the principle between 1830 and 1845, noting a shift from a two‑part statement (direct and converse propositions) to a single, more inclusive version. Peacock regarded arithmetical algebra as a subordinate, “suggestive” science that should inspire symbolic algebra but not restrict it; only the formal equivalence of expressions, not their semantic content, needed to be preserved when moving to the more general symbolic setting.

The paper then reviews contemporary criticisms, especially those of Philip Kelland, who argued that arithmetical meaning must also be preserved, and of Bertrand Russell, who dismissed the principle as a “mistake” because new number systems (e.g., complex numbers) do not obey all arithmetic laws. By revisiting De Morgan’s long review, the author shows that the principle does not demand universal preservation of every form; rather, it allows for forms that exist in symbolic algebra but not in arithmetical algebra (with respect to rules) and even for arithmetical forms that fail to survive symbolically (e.g., factorials, Euler’s infinite series). Peacock’s attempts to rescue the factorial via the Gamma function are presented as an illustration of the deliberative process required to decide whether an exception is admissible.

Crucially, the author reinterprets Peacock’s principle as a “conservative strategy” grounded in David Hume’s epistemology: the laws of reasoning should be preserved as far as possible because of their practical usefulness, yet exceptions are permissible when the costs of preservation outweigh the benefits. Under this reading, non‑commutative multiplication does not invalidate the principle; it simply represents a justified exception where the advantage of abandoning commutativity (e.g., representing three‑dimensional rotations) outweighs the desire to keep the law.

The paper then examines Hamilton’s development of quaternions. Hamilton initially tried to retain all arithmetic laws, but he eventually recognized that the gain in expressive power for spatial rotations justified discarding the commutative law. Hamilton’s own accounts of this decision match the conservative‑strategy model: he weighed the reasons for preserving versus abandoning a law and chose the latter when the latter’s reasons were stronger. This demonstrates that Hamilton, contrary to the claim that he betrayed Peacock’s principle, actually acted in accordance with it.

Finally, the author argues that Peacock’s principle remains philosophically robust and mathematically relevant. By viewing it as a conditional directive—preserve as much as possible, but allow well‑justified violations—it can accommodate modern structures such as non‑associative octonions, Lie algebras, and other abstract systems. The paper concludes that the principle was never truly invalidated by non‑commutative algebras; instead, it provides a meta‑mathematical guideline for when and how to relax longstanding algebraic laws in the service of new mathematical insight.