Calculus

Abstract

As noted in Chapter 1, there have been calls recently for inconsistent calculus, appealing to the history of calculus in which inconsistent claims abound, especially about infinitesimals (Newton, Leibniz, Bernoulli, l’Hospital, even Cauchy). However, inconsistent calculus has resisted development. There seem to be at least two reasons for this. First, as we have seen, the functional structure of fields interacts with inconsistency to produce triviality even in the purely equational part of theories, in a way which normal paraconsistentist contradiction-containment devices, such as weakening ex contradictione quodlibet, do not prevent. Stronger theories, including set membership, terms of infinite length, order, limits and integration, are then infected with the same triviality. Second, the functional structure of inconsistent set theory remains difficult to control, and seems to require sacrifice of logical principles in addition to, and more natural than, ECQ. (See Meyer et.al. [29], Slaney [53], but also below Chapter 14.) But unless there are distinctive inconsistent theories of the order of strength of classical analysis, then the claim that the history of the calculus supports paraconsistency is undermined. Inconsistency might well instead be a symptom of confusion.