Editing Logicism (section)

== Overview ==
Dedekind's path to logicism had a turning point when he was able to construct a model satisfying the [[axiom]]s characterizing the [[real number]]s using certain sets of [[rational number]]s. This and related ideas convinced him that [[arithmetic]], [[algebra]] and [[analysis]] were reducible to the [[natural number]]s plus a "logic" of classes. Furthermore by 1872 he had concluded that the naturals themselves were reducible to [[set (mathematics)|set]]s and [[function (mathematics)|mappings]]. It is likely that other logicists, most importantly Frege, were also guided by the new theories of the real numbers published in the year 1872. 

The philosophical impetus behind Frege's logicist programme from the [[The Foundations of Arithmetic|''Grundgesetze der Arithmetik'']] onwards was in part his dissatisfaction with the [[epistemology|epistemological]] and [[Ontology|ontological]] commitments of then-extant accounts of the natural numbers, and his conviction that [[Kant]]'s use of truths about the natural numbers as examples of [[A_priori_and_a_posteriori#Relation to the analytic–synthetic distinction|synthetic a priori truth]] was incorrect.

This started a period of expansion for logicism, with Dedekind and Frege as its main exponents. However, this initial phase of the logicist programme was brought into crisis with the discovery of the classical paradoxes of [[set theory]] ([[Cantor's paradox|Cantor's]] 1896, Zermelo and [[Russell's paradox|Russell's]] 1900–1901). Frege gave up on the project after Russell recognized and communicated his paradox identifying an [[inconsistency]] in Frege's system set out in the ''Grundgesetze der Arithmetik''. Note that [[naive set theory]] also suffers from this difficulty. 

On the other hand, Russell wrote ''[[The Principles of Mathematics]]'' in 1903 using the paradox and developments of [[Giuseppe Peano]]'s school of [[geometry]]. Since he treated the subject of [[primitive notion]]s in geometry and set theory
as well as the [[calculus of relations]], this text is a watershed in the development of logicism. Evidence of the assertion of logicism was collected by Russell and Whitehead in their ''[[Principia Mathematica]]''.<ref>{{cite SEP |url-id=principia-mathematica |title=Principia Mathematica}}</ref>

Today, the bulk of extant mathematics is believed to be derivable logically from a small number of extralogical axioms, such as the axioms of [[Zermelo–Fraenkel set theory]] (or its extension [[ZFC]]), from which no inconsistencies have as yet been derived. Thus, elements of the logicist programmes have proved viable, but in the process theories of classes, sets and mappings, and higher-order logics other than with [[Second-order_logic#Semantics|Henkin semantics]] have come to be regarded as extralogical in nature, in part under the influence of [[Willard Van Orman Quine|Quine]]'s later thought.

[[Kurt Gödel]]'s [[Gödel's incompleteness theorem|incompleteness theorems]] show that no formal system from which the [[Peano axioms]] for the natural numbers may be derived – such as Russell's systems in ''[[Principia Mathematica|PM]]'' – can decide all the well-formed [[sentence (logic)|sentences]] of that system.<ref>[http://philpapers.org/rec/RAAOTP "On the philosophical relevance of Gödel's incompleteness theorems"]</ref>  This result damaged [[David Hilbert]]'s programme for [[foundations of mathematics]] whereby 'infinitary' theories – such as that of ''[[Principia Mathematica|PM]]'' – were to be proved consistent from finitary theories, with the aim that [[finitism|those uneasy about 'infinitary methods']] could be reassured that their use should provably not result in the derivation of a [[contradiction]].  Gödel's result suggests that in order to maintain a logicist position, while still retaining as much as possible of classical mathematics, one must accept some axiom of [[infinity]] as part of logic. On the face of it, this damages the logicist programme also, albeit only for those already doubtful concerning 'infinitary methods'. Nonetheless, positions deriving from both logicism and from Hilbertian finitism have continued to be propounded since the publication of Gödel's result.

One argument that programmes derived from logicism remain valid might be that the incompleteness theorems are '[[mathematical proof|proved with logic]] just like any other [[theorem]]s'.  However, that argument appears not to acknowledge the distinction between theorems of [[first-order logic]] and theorems of [[higher-order logic]]. The former can be proven using finistic methods, while the latter – in general – cannot.  [[Tarski's undefinability theorem]] shows that Gödel numbering can be used to prove [[syntax|syntactical]] constructs, but not [[semantics|semantic]] assertions. Therefore, the claim that logicism remains a valid programme may commit one to holding that a system of proof based on the existence and properties of the natural numbers is less convincing than one based on some particular formal system.<ref>{{cite book |last1=Gabbay |first1=Dov M. |title=Studies In Logic And The Foundations Of Mathematics |date=2009 |publisher=Elsevier, inc. |location=Amsterdam |isbn=978-0-444-52012-8 |pages=59–90 |edition=Volume 153 |url=https://www.sciencedirect.com/bookseries/studies-in-logic-and-the-foundations-of-mathematics/vol/153/suppl/C |access-date=1 September 2019}}</ref>

Logicism – especially through the influence of Frege on Russell and [[Wittgenstein]]<ref>{{Citation|last=Reck|first=Erich|year=1997|title=''Frege's Influence on Wittgenstein: Reversing Metaphysics via the Context Principle''|s2cid=31255155 |url=https://pdfs.semanticscholar.org/a5e1/f41223452caf0775fe03ed08417e3530a9b8.pdf|archive-url=https://web.archive.org/web/20180824183548/https://pdfs.semanticscholar.org/a5e1/f41223452caf0775fe03ed08417e3530a9b8.pdf|url-status=dead|archive-date=2018-08-24}}</ref> and later Dummett – was a significant contributor to the development of [[analytic philosophy]] during the twentieth century.