By Erik Palmgren, Sten Lindstöm, Krister Segerberg, Viggo Stoltenberg-Hansen

- comprises essays via world-leading specialists within the philosophy and foundations of arithmetic, describing present advancements within the foundations of arithmetic in a ancient perspective

- Analyses the classical philosophical and foundational perspectives of Frege, Brouwer, Hilbert, Gödel and Tarski and examines their relevance for present developments

- presents an in-depth research of varied sorts of neologicist philosophies of mathematics

- encompasses a entire part on mathematical intuitionism and optimistic mathematics

- deals wide discussions, by way of a number of authors, of the proof-theoretic programme of Hilbert and Bernays

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Extra info for Logicism, Intuitionism, and Formalism: What Has Become of Them? (Synthese Library, Volume 341)

Sample text

Though suspect among certain philosophers, theoretical hypotheses, including existential hypotheses positing such unobservable entities as atoms and molecules and ions, have proved indispensable in the practice of science, and that is why scientific debate over the admissibility of such hypotheses has closed. A philosopher may nonetheless still ask whether what is infeasible in practice, the elimination of reference to unobservable posits, may yet be possible in principle. Could theoretical hypotheses be somehow eliminated?

Austin. Foundations of Arithmetic. Oxford: Blackwell, 1950. 20. , Die Grundgesetze der Arithmetik, begriffsschriftlich abgeleitet, 2 vols. Jena: Pohle, 1893/1903. Reprinted 1962. Hildesheim: Olms. 21. , Philosophical and Mathematical Correspondence, Oxford: Blackwell, 1980. 22. , ‘Die Widerspruchfreihet der Zahlentheorie’, Mathematische Annalen 112, 493– 565, 1936. 23. ), Future Pasts: The Analytic Tradition in Twentieth-Century Philosophy. Oxford: Oxford University Press, 25–41, 2001. ¨ 24. , ‘Uber formal unentscheidbare S¨atze der Principia Mathematica und verwandter Systeme I’, Monatshefte f¨ur Mathematik und Physik 38, 173–199, 1931.

As higher scientific theories do some observational work, making new predictions of the results of possible observations, so do higher mathematical theories do some computational work, making new predictions of the results of possible calculations. But is this still true if we restrict our attention to calculations that are not just possible in principle, but feasible in practice? That is to say, if we now restrict our attention to those ⌸0 1 sentences admitted by feasibilists, is it still true that higher theories give new ⌸0 1 sentences of this restricted kind?

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