Abstract
Although profoundly influential for essentially the whole of philosophy’s twenty-five hundred year history, the model of a science that is outlined in Aristotle’s Posterior Analytics has recently been abandoned on grounds that developments in mathematics and logic over the last century or so have rendered it obsolete. Nor has anything emerged to take its place. As things stand we have not even the outlines of an adequate understanding of the rationality of mathematics as a scientific practice. It seems reasonable, in light of this lacuna, to return again to Frege—who was at once one of the last great defenders of the model and a key figure in the very developments that have been taken to spell its demise—in hopes of finding a way forward. What we find when we do is that although Frege remains true to the spirit of the model, he also modifies it in very fundamental ways. So modified, I will suggest, the model continues to provide a viable and compelling image of scientific rationality by showing, in broad outline, how we achieve, and maintain, cognitive control in our mathematical investigations.
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Notes
- 1.
This formulation largely follows that of de Jong and Betti (2010, 186).
- 2.
In “Logic in Mathematics” Frege (1914, 205) is less charitable, claiming that “Euclid had an inkling of this idea of a system; but he failed to realize it”.
- 3.
See Coffa (1991, 27–9) for discussion.
- 4.
See Detlefson (2008, especially p. 187).
- 5.
Quoted in de Jong (1996, 302).
- 6.
- 7.
See also Frege (1914, 221).
- 8.
In point of fact, Frege had had doubts but had managed to convince himself that the problems could be satisfactorily resolved. See my (2005, Chap. 5).
- 9.
The particular case Frege is referring to here concerns the notion of motion, but the point obviously generalizes.
- 10.
- 11.
Quoted in Shapiro (1997, 160).
- 12.
- 13.
See my (2005) for an extended argument for this claim. In my (2012a), I argue that it is Frege’s conception of mathematical language rather than that of the mathematical logician that we need if we are to understand the role of the practice of proving in coming to a better mathematical understanding. See also my (2014).
- 14.
See Chaps. 3 and 4 of my (2005).
- 15.
See, for example, Azzouni’s (2006, Chap. 6) discussion of what he takes to be the essentially differences between pre-twentieth century and twentieth (and twenty-first) century mathematics.
- 16.
This history is explored in detail in my (2014).
- 17.
- 18.
See my (2014, Chap. 5).
- 19.
- 20.
- 21.
- 22.
Frege did not introduce the Sinn/Bedeutung distinction until many years after the appearance of Begriffsschrift. Nevertheless, that distinction is already in play in the 1879 logic insofar as already in that logic Frege holds that we arrive at a concept word only by analyzing the thought expressed into function and argument. Even in Begriffsschrift sub-sentential expressions, whether simple or complex, designate something, have Bedeutung in addition to expressing a sense (Sinn), only in the context of a proposition and relative to a function/argument analysis.
- 23.
As Frege says in the long Boole essay, the proof of theorem 133 is “from the definitions of the concepts following in a series, and of many-oneness by means of my primitive laws” (1880/1, 38; emphasis added).
- 24.
- 25.
- 26.
- 27.
See Netz (1999, §2.2).
- 28.
- 29.
- 30.
An earlier, shorter version of this essay was presented at the first international meeting of the Association for the Philosophy of Mathematical Practice (APMP) in Brussels, Belgium, December 2010. I am grateful to the participants for very useful discussion.
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Macbeth, D. (2016). Frege and the Aristotelian Model of Science. In: Costreie, S. (eds) Early Analytic Philosophy - New Perspectives on the Tradition. The Western Ontario Series in Philosophy of Science, vol 80. Springer, Cham. https://doi.org/10.1007/978-3-319-24214-9_3
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