Abstract
Philip Kitcher has argued against the apriority of mathematical knowledge in a number of places. His arguments rely on a conception of mathematical knowledge as embedded in a historical tradition and the claim that this sort of embedding compromises apriority. In this paper, I argue that tradition dependence of mathematical knowledge does not compromise its apriority. I further identify the factors which appear to lead Kitcher to argue as he does.
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Notes
See especially Sect. 2.
Pust (2002) has also argued against Kitcher’s use of tradition independence to undermine a priori knowledge. While there are points of contact between his arguments and mine (see Sect. 3.2), our projects are quite different. His paper is focused on apriority in general and specifically on two theses concerning the tradition independence of knowledge and the tradition independence of warrant. He argues that Kitcher is wrong in what he maintains about these theses in various respects and that thereby his case against defeasible a priori warrants fails. I’m here focused specifically on Kitcher’s account of a priori knowledge as it applies to mathematics. This leads me to consider a number of issues Pust does not, including a diagnosis of how and why Kitcher rejects apriority in mathematics which is tied to his positive account of mathematical knowledge. I also give considerably more attention to the details of Kitcher’s account of a priori knowledge than does Pust.
Thanks to an anonymous referee for prompting me correct and clean up some of these clarifications.
By ‘sufficiently rich experiential circumstance’ I mean a sequence of experiences sufficiently rich for S for p. Thus, the expression is tacitly indexed to an agent and proposition. At present these are unproblematically contextually determined. I’m thinking of experiential circumstances on the model of possible worlds or situations. So, intuitively, a sufficiently rich experiential circumstance is any way S’s life experience might go such that S acquires the conceptual resources required to entertain p or to have the belief that p.
See Casullo (2003, Chapter 3). Note that the defeasibility at issue here is empirical defeasibility, defeasibility by experience, as opposed to defeasibility for mathematical or logical (i.e., prima facie a priori) reasons. To take a particularly unsubtle example, the logical inconsistency of naive set theory led us to revise our belief that unrestricted comprehension is true. Whatever warranted this belief, which was almost certainly a priori if anything is, was undermined by the discovery that unrestricted comprehension enables us to derive a contradiction.
It may be worth noting that Kitcher’s strong conception of a priori knowledge is modeled on knowledge by Kantian pure intuition (1983, §1.IV). Others have advocated intuition-based accounts of a priori knowledge in recent years, most notably Bealer (2000) and BonJour (1998). McEvoy (2007) outlines an intuition-based defense of a priori mathematical knowledge specifically in opposition to Kitcher’s anti-apriorism. (McEvoy’s defense turns on Kitcher’s conception of experience, which is just one way in which it differs from the project of the current paper.) Intuition-based accounts of a priori knowledge occupy a difficult position in the prevailing climate of philosophical naturalism. Kitcher’s rejection of the strong conception, however, would seem to improve the outlook for such accounts of a priori knowledge somewhat by freeing them from the obligation of satisfying conditions (b) and (c). But to pursue this here would take us too far afield.
I use `warrant\(_e\)’ in this discussion to make explicit that warrant is to be understood in a purely externalist sense.
The only candidates for the kind of possible lives Kitcher needs for the argument are ones that include experience sufficiently rich for the relevant agents and mathematical propositions.
For simplicity, I’ll just use \(\alpha \) going forward.
Note that this isn’t obviously the sort of case Kitcher intends. I include it in an effort to canvas all of the options that might be live.
Note that I’m not here claiming that S merely conceptualizes her experiences differently in \(T'\) than she would have in T (though this my be true). Rather, I’m claiming that being raised in \(T'\) she would have experiences different from those she would have had being raised in T. Crucially, in \(T'\) she would learn to value and treat as reliable (some) methods of inquiry which would be neither valued nor treated as reliable in T or she would learn to reject and treat as unreliable (some) methods of inquiry which would be valued and treated as reliable in T (or both). These experiences, learning which methods are to be trusted and which aren’t, are directly relevant to producing and sustaining S’s beliefs about the subject matter of her practice. Hence, they’re experiences with which Kitcher must be concerned.
The foregoing two paragraphs more or less take for granted that moving from one tradition to another (or modifying a tradition) involves new or different experiences. One might wonder whether this is so. While it might be notionally possible to change traditions independent of a change in experiences, it strikes me as borderline irrational. This is especially true in something like mathematics or the sciences. One simply doesn’t abandon one tradition for another without reasons for doing so, and these reasons will be grounded in experiences. Related to this and running through my arguments in the paragraphs: It’s not so easy as Kitcher makes out to separate experiences required to have the beliefs typical of a competent practitioner from the practice in which one works. So the very idea that one can have experiences \(e_1,\ldots , e_n\) appropriate to being a practitioner of T and then take just those same experiences and be a competent practitioner of a different practice \(T'\) is quite problematic. Thanks to an anonymous referee for prompting me to think more about this and rework these paragraphs.
See Roland (2008).
A similar conflation is identifiable in Field (2000). See Roland (unpublished).
That this is the case appears clear to me (as it did to Kripke). Whatever warrant or justification one has for the belief that the relevant number is prime depends on the reliability of the computing machine, which in turn depends on (at least) engineering and physics. I take these to be unproblematically empirical. Thus, the relevant warrant or justification is empirical.
Note I’m not here conceding that (\(\ddagger \)) is in fact false.
At this point one might object that I’m here holding Kitcher responsible for something he didn’t set out to do. Kitcher’s account of a priori knowledge (set out in Sect. 2) is an account of what it takes to actually know something a priori, not of what it takes for something to be a priori knowable (i.e., possibly known a priori). Set aside that these things are closely related. This objection simply misses the mark. The dialectic is important here. Kitcher argues that there are no a priori warrants in mathematics due to its tradition dependence. His argument most clearly emerges from his positive account of mathematical knowledge, rehearsed in Sect. 3.3. My critique in this section is aimed at this argument, not at Kitcher’s account of a priori knowledge. Conflating being known a priori with being knowable a priori allows him to take an argument based on his positive account of mathematical knowledge that is (let’s suppose) successful against (\(\ddagger \)) and apply it to (\(\dagger \)). But the conflation is illicit.
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Acknowledgements
Thanks to Otàvio Bueno, Jon Cogburn, Carrie Ichikawa Jenkins, Susan Vineberg, and two referees for this journal for comments and discussion. An ancestor of this paper was presented at the symposium on the a priori in mathematics at the 2015 Pacific Division meeting of the American Philosophical Association. Thanks to the participants of that symposium.
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Roland, J.W. Kitcher, Mathematics, and Apriority. Erkenn 84, 687–702 (2019). https://doi.org/10.1007/s10670-018-9977-8
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DOI: https://doi.org/10.1007/s10670-018-9977-8