Archive for History of Exact Sciences

, Volume 34, Issue 1–2, pp 141–192 | Cite as

“The True” in Gottlob Frege's “Über die Grundlagen der Geometrie”

  • William Boos


In this essay, I examine the metaphysical and metalogical ramifications of Gottlob Frege's controversy with David Hilbert and Alwin Korselt, over Hilbert'sGrundlagen der Geometrie. These ramifications include

(1) Korselt's original appeals to general metatheoretic Deutungen (interpretations);

(2) Hilbert's puzzling belief that whatever is consistent in some sense ‘exists;’ and

(3) Frege's ‘semantic monist’ conviction that theoretical sense and reference (mathematical and other) ‘must’ be “eindeutig lösbar” (“uniquely solvable”).

My principal conclusions are

(4) that Frege's position in (3) represented a pervasively dogmatic presumption that his newly discovered quantification theory ‘must’ have a propositional metatheory (‘the True’; ‘the False’); and

(5) that this needless assumption adversely affected not only his polemic against the moderate semantic relativism of Hilbert and Korselt, but also his reception of type-theoretic ideas, and greatly facilitated his vulnerability to the sort of self-referential inconsistency Russell discovered in Grundgesetz V.

These conclusions also seem to me to provide a conceptual framework for several of Frege's other arguments and reactions which might seem more particular and disparate. These include

(6) his arbitrary restrictions on the range of second-order quantification, which undercut his own tentative attempts to give accounts of independence and semantic consequence;

(7) his uncharacteristic hesitation, even dismay, at the prospect that such accounts might eventuate in a genuinely quantificational metamathematics, whose Gegenstände (objects) might themselves be Gedanken (thoughts); and, perhaps most revealingly

(8) his otherwise quite enigmatic, quasi-stoic doctrine that ‘genuine’ formal deduction ‘must’ be from premises that are ‘true.’

A deep reluctance to pluralize or iterate ‘the’ transition from theory to meta-theory would also be consonant, of course, with Frege's vigorous insistence that there can be only one level each of linguistic Begriffe (concepts) and Gegenstände (objects). With hindsight, such an assumption may seem more gratuitous in the philosophy of language (where it contributed, I would argue, to Wittgenstein's famous transition to ‘the mystical’ in 6.45 and 6.522 of the Tractatus); but its more implausible implications in this wider context seemed to emerge more slowly.

In the mathematical test-case discussed here, however, such strains were immediately and painfully apparent; the first models of hyperbolic geometry were described some thirty years before Frege drafted his polemic against Hilbert's pioneering exposition. It is my hope that a careful study of Frege's lines of argument in this relatively straightforward mathematical controversy may suggest other, parallel approaches to the richer and more ambiguous problems of his philosophy of language.


Conceptual Framework Quantification Theory Careful Study Undercut Semantic Relativism 
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Copyright information

© Springer-Verlag GmbH & Co. KG 1985

Authors and Affiliations

  • William Boos
    • 1
  1. 1.Department of PhilosophyUniversity of New MexicoAlbuquerque

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