Abstract
Michael Friedman’s project both historically and systematically testifies to the importance of the relativized a priori. The importance of implicit definitions clearly emerges from Schlick’s General Theory of Knowledge. The main aim of this paper is to show the relationship between both and the relativized a priori through a detailed discussion of Friedman’s work. Succeeding with this will amount to a contribution to recent scholarship showing the importance of Hilbert for Logical Empiricism.
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Notes
Friedman (1999) discussed issues related to implicit definitions or to Hilbert in virtually all chapters. However, the most pertinent one seems to be chapter 3, “Geometry, Convention, and the Relativized A Priori”, which was originally published as Friedman (1994), rather than the chapters dealing with Carnap’s views on logico-mathematical truth. Even more important for my purpose is Friedman (1992). Friedman (2002) contains extremely useful information, especially in the endnotes.
See Majer (2002, pp. 214, 219, 223). However, I fail to find any argument in Majer’s writings for his claim that implicit definition is merely a side issue.
Majer raises the question “Why didn’t the logical empiricists pay more attention to Hilbert and his axiomatic point of view?” (Majer 2002, p. 213).
There are also similar signs from other philosophers that indicate the urgency and timeliness of addressing this issue. For example, Yemima Ben-Menahem devoted a chapter on implicit definition in her recent book Conventionalism (Ben-Menahem 2006). William Demopoulos and Sathis Psillos write extensively on Carnap’s views on theoretical terms in science (Demopoulos 2003, 2007; Psillos 1999; Psillos and Christopoulou 2009).
Friedman (1992) is not included in Friedman’s masterpiece on logical positivism, i.e., Friedman (1999), probably because of the overlaps (especially with the final chapter on Carnap’s views on analytic truth). Further it is not found in the bibliography either. From my point of view, the exclusion of Friedman (1992) is rather unfortunate, as it seems the most extensive discussion of the problem of implicit definition in logical positivism.
“The former are constitutive a priori in that, first, they are not themselves subject to straightforward empirical confirmation or disconfirmation by measuring parameters and instantiating laws, and second, they first make possible the confirmation and disconfirmation of empirical laws properly so-called (viz., the axioms of connection)”. (Friedman 1994, p. 23; 1999, p. 61); Cf. Reichenbach (1920/1965).
Elsewhere Friedman explains that the abstract mathematical representations lying at the basis of Einstein’s general theory of relativity made both logical positivists and Einstein discern an intimate relationship between the theory of relativity and Hilbert’s axiomatic conception of geometry. He points out that “[o]n this new view of mathematics there is thus more need than ever for principles of coordination to mediate between abstract mathematical structures and concrete physical phenomena” (Friedman 2001, p. 71).
This is the italicized part in the paragraph I’ve just quoted from Friedman (1992, pp. 52–53). By “recent work in the foundations of geometry and relativity theory”, he seems to have Reichenbach in mind, but in order to have a larger perspective, one may also include the late nineteenth century luminaries such as Helmholtz, Poincaré, and Hilbert as well as Einstein, Reichenbach, and Schlick.
It is far beyond the scope of this paper to examine whether Friedman is correct here. Though it is highly controversial, to say the least, his suggestion deserves careful attention. For it seems to show the way to an in-depth understanding of the motivations of Carnap’s projects in both Aufbau and the Logical Syntax.
The former understanding can be found in Bernays (1961): “The general tendency of the Logical Syntax can be said to be an extension of the approach of Hilbert’s proof theory. For Hilbert the method of formalization is applied only to mathematics. However, in his lecture “Axiomatic Thought” Hilbert also said: “Everything at all that can be the object of scientific thinking falls under the axiomatic method, and thereby indirectly under mathematics, when it becomes mature enough to form into theory”. Carnap goes a step further in this direction in the Logical Syntax, by considering science as a whole as an axiomatic deductive system which becomes a mathematical object through formalization: the syntax of the language of science is metamathematics that is directed towards this object” (Bernays 1961, p. 186).
Reichenbach’s case is discussed in Stöltzner (2003), which deals with the problem of the principle of least action.
This question is raised because to Hahn and Frank Hilbert’s allusion to the (non-Leibnizian) pre-established harmony between our thought and the course of the world (or between mathematics and physics) was “thoroughly mysterious”. As what Stöltzner counts as “the main charge against Hilbert”, Hahn wrote: “Why should what is compelling to our thought also be compelling to the course of the world? Our only recourse would be to believe in a miraculous pre-established harmony between the course of our thought and the course of the world, an idea which is deeply mystical and ultimately theological” (Hahn 1987, p. 28; Stöltzner 2002, p. 251).
This way of setting up a dilemma is, of course, controversial. For example, Majer might object that there is no problem for Hilbertians grasping the second horn of the alleged dilemma. For they could understand Hilbert’s axiomatic as achieving a real progress over the old Euclidean axiomatic. However, in so far as there is an urgent need to distinguish sharply between the formalist position and the original Hilbertian axiomatic method, the threat of the dilemma seems genuine.
Majer might grasp this horn for an entirely different reason, because he believes that Carnap’s idea “to generalize the concept of an axiomatic system in such a way that geometry … turns out to be nothing but a logical theory” is “definitely not Hilbert’s point of view” (Majer 2002, p. 220).
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Acknowledgments
I am indebted to the editors and the two anonymous reviewers of Erkenntnis for their extremely helpful criticisms and suggestions. For the last 10 years, Young-Sam Chun, Junyong Park, Wonbae Choi, and Jinhee Lee shared with me enthusiasm for the history of the axiomatic method. I presented an earlier version of this paper at the international conference on “The Future of Philosophy of Science” held at Tilburg University in April 2010. I wish to thank the organizers and the referees of the conference as well as the participants including Michael Stöltzner, Jeongmin Lee, and especially Michael Friedman for their suggestions, advice, encouragement, and thought-provoking questions. Also, many thanks are due to the written comments of Nino B. Cocchiarella, which were truly instrumental at the final stage.
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Park, W. Friedman on Implicit Definition: In Search of the Hilbertian Heritage in Philosophy of Science. Erkenn 76, 427–442 (2012). https://doi.org/10.1007/s10670-011-9297-8
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DOI: https://doi.org/10.1007/s10670-011-9297-8