Hostname: page-component-586b7cd67f-rdxmf Total loading time: 0 Render date: 2024-11-24T01:17:58.887Z Has data issue: false hasContentIssue false
  • English
  • Français

On Gödel's Way In: The Influence of Rudolf Carnap

Published online by Cambridge University Press:  15 January 2014

Warren Goldfarb
Warren Goldfarb*
Affiliation:
Department of Philosophy, Harvard University, Cambridge, MA 02138, USAE-mail: [email protected]
Get access Rights & Permissions [Opens in a new window]

Extract

The philosopher Rudolf Carnap (1891–1970), although not himself an originator of mathematical advances in logic, was much involved in the development of the subject. He was the most important and deepest philosopher of the Vienna Circle of logical positivists, or, to use the label Carnap later preferred, logical empiricists. It was Carnap who gave the most fully developed and sophisticated form to the linguistic doctrine of logical and mathematical truth: the view that the truths of mathematics and logic do not describe some Platonistic realm, but rather are artifacts of the way we establish a language in which to speak of the factual, empirical world, fallouts of the representational capacity of language. (This view has its roots in Wittgenstein's Tractatus, but Wittgenstein's remarks on mathematics beyond first-order logic are notoriously sparse and cryptic.) Carnap was also the thinker who, after Russell, most emphasized the importance of modern logic, and the distinctive advances it enables in the foundations of mathematics, to contemporary philosophy. It was through Carnap's urgings, abetted by Hans Hahn, once Carnap arrived in Vienna as Privatdozent in philosophy in 1926, that the Vienna Circle began to take logic seriously and that positivist philosophy began to grapple with the question of how an account of mathematics compatible with empiricism can be given (see Goldfarb 1996).

A particular facet of Carnap's influence is not widely appreciated: it was Carnap who introduced Kurt Gödel to logic, in the serious sense. Although Gödel seems to have attended a course of Schlick's on philosophy of mathematics in 1925–26, his second year at the University, he did not at that time pursue logic further, nor did the seminar leave much of a trace on him. In the early summer of 1928, however, Carnap gave two lectures to the Circle which Gödel attended, or so I surmise. At these occasions, Carnap presented material from his manuscript treatise, Untersuchungen zur allgemeinen Axiomatik, that is, “Investigations into general axiomatics”, which dealt with questions of consistency, completeness and categoricity. Carnap later circulated this material to various people including Gödel.

Type
Research Article
Information
Bulletin of Symbolic Logic , Volume 11 , Issue 2 , June 2005 , pp. 185 - 193
Copyright
Copyright © Association for Symbolic Logic 2005

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

[2001] Awodey, Steven and Carus, André W., Carnap, completeness and categoricity: the Gabelbarkeitssatz of 1928 , Erkenntnis, vol. 54 (2001), pp. 145–171.CrossRefGoogle Scholar
[1934] Carnap, Rudolf, Logische Syntax der Sprache, Springer, Vienna, 1934.CrossRefGoogle Scholar
[1963] Carnap, Rudolf, Intellectual autobiography, The philosophy of Rudolf Carnap (Schlipp, Paul A., editor), Library of Living Philosophers, vol. 11, Open Court, La Salle; Cambridge University Press, London, 1963, pp. 3–84.Google Scholar
[2000] Carnap, Rudolf, Untersuchungen zur allgemeinen Axiomatik, Wissenschaftliche Buchgesellschaft, Darmstadt, 2000, (Bonk, Thomas and Mosterin, Jesus, editors).Google Scholar
[1997] Dawson, John W. Jr., Logical dilemmas: The life and work of Kurt Gödel, A K Peters, Ltd., Wellesley, Mass, 1997.Google Scholar
[1929] Gödel, Kurt, Über die Vollständigkeit des Logikkalküls, doctoral dissertation, 1929, in Gödel 1986, pp. 60–101.Google Scholar
[*1930c] Gödel, Kurt, Vortrag über Vollständigkeit des Funktionenkalküls, *1930c, in Gödel 1995, pp. 16–29.Google Scholar
[1931] Gödel, Kurt, Über formal unentscheidbare Sätze der Principia mathematica und verwandter Systeme I , Monatshefte für Mathematik und Physik, vol. 38 (1931), pp. 173–198, in Gödel 1986, pp. 144–195.CrossRefGoogle Scholar
[*1933o] Gödel, Kurt, The present situation in the foundations of mathematics, 1933, in Gödel 1995, pp. 45–53.CrossRefGoogle Scholar
[1986] Gödel, Kurt, Collected works. Vol. I: Publications, 1929–1936, Oxford University Press, New York and Oxford, 1986, (Feferman, Solomon et al., editors).Google Scholar
[1995] Gödel, Kurt, Collected works. Vol. Ill: Unpublished essays and lectures, Oxford University Press, New York and Oxford, 1995, (Feferman, Solomon et al., editors).Google Scholar
[2003] Gödel, Kurt, Collected works. Vol. IV: Correspondence A—G, Oxford University Press, New York and Oxford, 2003, (Feferman, Solomon et al., editors).Google Scholar
[1996] Goldfarb, Warren, The philosophy of mathematics in early positivism, Origins of logical empiricism (Giere, R. N. and Richardson, A. W., editors), University of Minnesota Press, Minneapolis, 1996, pp. 213–230.Google Scholar
[1928] Hubert, David and Ackermann, Wilhelm, Grundzüge der theoretischen Logik, Springer, Berlin, 1928.Google Scholar
[1903] Russell, Bertrand, The principles of mathematics, Allen and Unwin, London, 1903.Google Scholar
[1920] Skolem, Thoralf, Logisch-kombinatorische Untersuchungen über die Erfüllbarkeit oder Beweisbarkeit mathematischer Sätze nebst einem Theoreme über dichte Mengen, Skrifter utgitav Videnskapsselskapet i Kristiania, I, Matematisknaturvidenskapelig klasse, no. 4, 1920, pp. 1–36.Google Scholar
[1967] van Heijenoort, Jean, Logic as calculus and logic as language, Boston studies in the philosophy of science, vol. 3 (1967), pp. 440–446.CrossRefGoogle Scholar