Editing Principia Mathematica (section)

=== Wittgenstein 1919, 1939 ===
By the second edition of ''PM'', Russell had removed his ''axiom of reducibility'' to a new axiom (although he does not state it as such). Gödel 1944:126 describes it this way:
{{Cquote|This change is connected with the new axiom that functions can occur in propositions only "through their values", i.e., extensionally (...) [this is] quite unobjectionable even from the constructive standpoint (...) provided that quantifiers are always restricted to definite orders". This change from a quasi-''intensional'' stance to a fully ''extensional'' stance also restricts [[predicate logic]] to the second order, i.e. functions of functions: "We can decide that mathematics is to confine itself to functions of functions which obey the above assumption". |source=''PM'' 2nd edition p.&nbsp;401, Appendix C}}

This new proposal resulted in a dire outcome. An "extensional stance" and restriction to a second-order predicate logic means that a propositional function extended to all individuals such as "All 'x' are blue" now has to list all of the 'x' that satisfy (are true in) the proposition, listing them in a possibly infinite conjunction: e.g. ''x<sub>1</sub>'' ∧ ''x<sub>2</sub>'' ∧ . . . ∧ ''x<sub>n</sub>'' ∧ . . .. Ironically, this change came about as the result of criticism from [[Ludwig Wittgenstein]] in his 1919 ''[[Tractatus Logico-Philosophicus]]''. As described by Russell in the Introduction to the Second Edition of ''PM'':
{{Cquote|There is another course, recommended by Wittgenstein† (†''Tractatus Logico-Philosophicus'', *5.54ff) for philosophical reasons. This is to assume that functions of propositions are always truth-functions, and that a function can only occur in a proposition through its values. (...) [Working through the consequences] it appears that everything in Vol. I remains true (though often new proofs are required); the theory of inductive cardinals and ordinals survives; but it seems that the theory of infinite Dedekindian and well-ordered series largely collapses, so that irrationals, and real numbers generally, can no longer be adequately dealt with. Also Cantor's proof that 2<sup>n</sup> > ''n'' breaks down unless ''n'' is finite." |source=''PM'' 2nd edition reprinted 1962:xiv, also cf. new Appendix C)}}
In other words, the fact that an infinite list cannot realistically be specified means that the concept of "number" in the infinite sense (i.e. the continuum) cannot be described by the new theory proposed in ''PM Second Edition''.

Wittgenstein in his ''Lectures on the Foundations of Mathematics, Cambridge 1939'' criticised ''Principia'' on various grounds, such as:
* It purports to reveal the fundamental basis for arithmetic. However, it is our everyday arithmetical practices such as counting which are fundamental; for if a persistent discrepancy arose between counting and ''Principia'', this would be treated as evidence of an error in ''Principia'' (e.g., that Principia did not characterise numbers or addition correctly), not as evidence of an error in everyday counting.
* The calculating methods in ''Principia'' can only be used in practice with very small numbers. To calculate using large numbers (e.g., billions), the formulae would become too long, and some short-cut method would have to be used, which would no doubt rely on everyday techniques such as counting (or else on non-fundamental and hence questionable methods such as induction). So again ''Principia'' depends on everyday techniques, not vice versa.

Wittgenstein did, however, concede that ''Principia'' may nonetheless make some aspects of everyday arithmetic clearer.