Introduction The following nine itineraries in the history of mathematical logic do not aim at a complete account of the history of mathematical logic during the period 1900–1935. For one thing, we had to limit our ambition to the technical developments without attempting a detailed discussion of issues such as what conceptions of logic were being held during the period. This also means that we have not engaged in detail with historiographical debates which are quite lively today, such as those on the universality of logic, conceptions of truth, the nature of logic itself etc. While of extreme interest these themes cannot be properly dealt with in a short space, as they often require extensive exegetical work. We therefore merely point out in the text or in appropriate notes how the reader can pursue the connection between the material we treat and the secondary literature on these debates. Second, we have not treated some important developments. While we have not aimed at completeness our hope has been that by focusing on a narrower range of topics our treatment will improve on the existing literature on the history of logic. There are excellent accounts of the history of mathematical logic available, such as, to name a few, Kneale and Kneale (1962), Dumitriu (1977), and Mangione and Bozzi (1993). We have kept the secondary literature quite present in that we also wanted to write an essay that would strike a balance between covering material that was adequately discussed in the secondary literature and presenting new lines of investigation. This explains, for instance, why the reader will find a long and precise exposition of Löwenheim’s (1915) theorem but only a short one on Gödel’s incompleteness theorem: Whereas there is hitherto no precise presentation of the first result, accounts of the second result abound. Finally, the treatment of the foundations of mathematics is quite restricted and it is ancillary to the exposition of the history of mathematical logic. Thus, it is not meant to be the main focus of our exposition.1 Page references in citations are to the English translations, if available; or to the reprint edition, if listed in the bibliography. All translations are the authors’, unless an English translation is listed in the references. We have received comments on an earlier draft of this paper from Mark van Atten, José Ferreiros, Johannes Hafner, Ignasi Jané, Bernard Linsky, Enrico Moriconi, Chris Pincock, and Bill Tait. Their help is gratefully acknowledged. 1
IntroductionThe following nine itineraries in the history of
mathematical logic do notaim at a complete account of the history
of mathematical logic duringthe period 19001935. For one thing, we
had to limit our ambition to thetechnical developments without
attempting a detailed discussion of issuessuch as what conceptions
of logic were being held during the period. Thisalso means that we
have not engaged in detail with historiographical debateswhich are
quite lively today, such as those on the universality oflogic,
conceptions of truth, the nature of logic itself etc. While of
extremeinterest these themes cannot be properly dealt with in a
short space, asthey often require extensive exegetical work. We
therefore merely pointout in the text or in appropriate notes how
the reader can pursue theconnection between the material we treat
and the secondary literature onthese debates. Second, we have not
treated some important developments.While we have not aimed at
completeness our hope has been that by focusingon a narrower range
of topics our treatment will improve on theexisting literature on
the history of logic. There are excellent accounts ofthe history of
mathematical logic available, such as, to name a few, Knealeand
Kneale (1962), Dumitriu (1977), and Mangione and Bozzi (1993).
Wehave kept the secondary literature quite present in that we also
wanted towrite an essay that would strike a balance between
covering material thatwas adequately discussed in the secondary
literature and presenting newlines of investigation. This explains,
for instance, why the reader will finda long and precise exposition
of Lwenheims (1915) theorem but only ashort one on Gdels
incompleteness theorem: Whereas there is hithertono precise
presentation of the first result, accounts of the second
resultabound. Finally, the treatment of the foundations of
mathematics is quiterestricted and it is ancillary to the
exposition of the history of mathematicallogic. Thus, it is not
meant to be the main focus of our exposition.1Page references in
citations are to the English translations, if available;or to the
reprint edition, if listed in the bibliography. All translations
arethe authors, unless an English translation is listed in the
references.We have received comments on an earlier draft of this
paper from Markvan Atten, Jos Ferreiros, Johannes Hafner, Ignasi
Jan, Bernard Linsky,Enrico Moriconi, Chris Pincock, and Bill Tait.
Their help is gratefully acknowledged.1
