Logicism - The Logistic Construction of The Natural Numbers

The Logistic Construction of The Natural Numbers

The attempt to construct the natural numbers is summarized succinctly by Bernays 1930-1931. But rather than use Bernays' précis, which is incomplete in the details, the construction is best given as a simple finite example together with the details to be found in Russell 1919.

In general the logicism of Dedekind-Frege is similar to that of Russell, but with significant (and critical) differences in the particulars (see Criticisms, below). Overall, though, the logicistic construction-process is far different than that of contemporary set theory. Whereas in set theory the notion of "number" begins from an axiom—the axiom of pairing that leads to the definition of "ordered pair"—no overt number-axiom exists in logicism. Rather, logicism begins its construction of the numbers from "primitive propositions" that include "class", "propositional function", and in particular, "relations" of "similarity" ("equinumerosity": placing the elements of collections in one-to-one correspondence) and "ordering" (using "the successor of" relation to order the collections of the equinumerous classes)". The logicistic derivation equates the cardinal numbers constructed this way to the natural numbers, and these numbers end up all of the same "type"—as equivalence classes of classes—whereas in set theory each number is of a higher class than its predecessor (thus each successor contains its predecessor as a subset). Kleene observes that:

"The viewpoint here is very different from that where we supposed an intuitive conception of the number sequence and elicited from it the principle that, whenever a particular property P of natural numbers is given such that (1) and (2), then any given natural number must have the property P" (Kleene 1952:44).

The importance to logicism of the construction of the natural numbers derives from Russell's contention that "That all traditional pure mathematics can be derived from the natural numbers is a fairly recent discovery, though it had long been suspected" (1919:4). The derivation of the real numbers (rationals, irrationals) derives from the theory of Dedekind cuts on the continuous "number line". While an example of how this is done is useful, it relies first on the derivation of the natural numbers. So, if philosophic problems appear in the logistic attempt to derive the natural numbers, these problems will be sufficient to stop the program until these are fixed (see Criticisms, below).