# Logic and Modern Philosophy of Mathematics - The Revival Of Logic From The 1820s, And Its Algebraic Flourishing, Set Theory And The Rise Of Mathematical Logic

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This article surveys many of the main positions that have been held in the logic and philosophy of mathematics from around 1800 up to recent times. Most attention
is given to symbolic logics of some kind. No position has been definitive; indeed, especially over the last seventy years the variety has continually increased. To compensate for this article's lack of exhaustiveness, the bibliography is wide-ranging.

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Some major philosophers gave logic a high status in the seventeenth and early eighteenth centuries; in particular, Gottfried Wilhelm von Leibniz advocated it as a lingua characteristica, with an attendant calculus ratiocinator, while John Locke took it as a case of semeiotiké, the theory of signs. The subject did not flourish, however; thus, while René Descartes stressed rules for correc…

Especially from the 1820s with the Frenchman Augustin Louis Cauchy, mathematicians had become more sensitive to the need for rigor in proofs, carefully stating assumptions and definitions and formulating theorems in conditional form. Cauchy's approach was refined from the 1860s by the lectures at Berlin University of Karl Weierstrass. The main context was the calculus and its extension into…

Meanwhile, contentment with Euclidian rigor was dissolving. From the 1860s non-Euclidean geometries had been accepted as legitimate theories, especially due to the insights of Bernhard Riemann. In addition, various mathematicians, including Peano, had noticed that Euclidian geometry itself needed several more axioms than Euclid had stated. These developments made mathematicians still more aware of…

More bad news for Hilbert arrived in 1931, when the young Austrian Kurt Gödel proved that his metamathematical program could not be achieved for arithmetic with quantification over integers, and thus a fortiori for almost all acclimatized mathematical theories T; for Gödel showed that, in order to prove the consistency of T, its metamathematics had to be richer in assumptions than T itse…

The consequences of Gödel's theorems were profound. For example, logicism was replaced, especially by the American W. V. Quine, by elaborate systems of set theory and logic such as those
developed in Quine's Mathematical Logic (1940); however, Russellian reductions of the former to the latter were not claimed. Quine adhered to classical logic, with its two truth-values. But non…

Especially since World War II, new philosophies have emerged separate from logicism, formalism, and intuitionism. One kind is called naturalism, in which mathematical objects are said to be accessible to ordinary sense perception. Forms of Platonism are advocated, partly inspired by the enthusiasm of Gödel. Disaffection from concerns with mathematical truth has stimulated conventionalism; Hen…

Whately would be astonished at the current range and variety of logics. However, there is often still a considerable professional distance between logicians and both mathematicians and philosophers; in the mid-1930s an Association of Symbolic Logic was created to provide a venue and a journal for those active in the field. Now perhaps too many logical and philosophical flowers are blooming, and ma…

Abramsky, S., Dov Gabbay, and T. S. E. Maibaum, eds. Handbook of Logic in Computer Science. 5 vols. Oxford and New York: Oxford University Press, 1992–2000. Aspray, William, and Philip Kitcher, eds. History and Philosophy of Modern Mathematics. Minneapolis: University of Minnesota Press, 1988. Barwise, Jon, ed. Handbook of Mathematical Logic. Amsterdam and New York: North-Holland, 1977. Ben…

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