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Logicism
In philosophy of mathematics, logicism is a school of thought comprising one or more of the theses that – for some coherent meaning of 'logic' – mathematics is an extension of logic, some or all of mathematics is reducible to logic, or some or all of mathematics may be modelled in logic. Bertrand Russell and Alfred North Whitehead championed this programme, initiated by Gottlob Frege and subsequently developed by Richard Dedekind and Giuseppe Peano.
Dedekind's path to logicism had a turning point when he was able to construct a model satisfying the axioms characterizing the real numbers using certain sets of rational numbers. This and related ideas convinced him that arithmetic, algebra and analysis were reducible to the natural numbers plus a "logic" of classes. Furthermore by 1872 he had concluded that the naturals themselves were reducible to sets and 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 Grundgesetze der Arithmetik[clarification needed] onwards was in part his dissatisfaction with the epistemological and 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 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 1896, Zermelo and 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 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 notions 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.
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 Henkin semantics have come to be regarded as extralogical in nature, in part under the influence of Quine's later thought.
Kurt Gödel's 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 PM – can decide all the well-formed sentences of that system.[full citation needed] This result damaged David Hilbert's programme for foundations of mathematics whereby 'infinitary' theories – such as that of PM – were to be proved consistent from finitary theories, with the aim that 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 'proved with logic just like any other theorems'. 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 syntactical constructs, but not 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.
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Logicism
In philosophy of mathematics, logicism is a school of thought comprising one or more of the theses that – for some coherent meaning of 'logic' – mathematics is an extension of logic, some or all of mathematics is reducible to logic, or some or all of mathematics may be modelled in logic. Bertrand Russell and Alfred North Whitehead championed this programme, initiated by Gottlob Frege and subsequently developed by Richard Dedekind and Giuseppe Peano.
Dedekind's path to logicism had a turning point when he was able to construct a model satisfying the axioms characterizing the real numbers using certain sets of rational numbers. This and related ideas convinced him that arithmetic, algebra and analysis were reducible to the natural numbers plus a "logic" of classes. Furthermore by 1872 he had concluded that the naturals themselves were reducible to sets and 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 Grundgesetze der Arithmetik[clarification needed] onwards was in part his dissatisfaction with the epistemological and 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 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 1896, Zermelo and 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 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 notions 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.
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 Henkin semantics have come to be regarded as extralogical in nature, in part under the influence of Quine's later thought.
Kurt Gödel's 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 PM – can decide all the well-formed sentences of that system.[full citation needed] This result damaged David Hilbert's programme for foundations of mathematics whereby 'infinitary' theories – such as that of PM – were to be proved consistent from finitary theories, with the aim that 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 'proved with logic just like any other theorems'. 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 syntactical constructs, but not 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.