Abstract
Set-theoretic pluralism is an increasingly influential position in the philosophy of set theory (Balaguer in Platonism and anti-platonism in mathematics, Oxford University Press, New York, 1998; Linksy and Zalta in J Philos 92:525–555, 1995; Hamkins in Rev Symb Log 5:416–449, 2012). There is considerable room for debate about how best to formulate set-theoretic pluralism, and even about whether the view is coherent. But there is widespread agreement as to what there is to recommend the view (given that it can be formulated coherently). Unlike set-theoretic universalism, set-theoretic pluralism affords an answer to Benacerraf’s epistemological challenge. The purpose of this paper is to determine what Benacerraf’s challenge could be such that this view is warranted. I argue that it could not be any of the challenges with which it has been traditionally identified by its advocates, like of Benacerraf and Field. Not only are none of the challenges easier for the pluralist to meet. None satisfies a key constraint that has been placed on Benacerraf’s challenge. However, I argue that Benacerraf’s challenge could be the challenge to show that our set-theoretic beliefs are safe—i.e., to show that we could not have easily had false ones. Whether the pluralist is, in fact, better positioned to show that our set-theoretic beliefs are safe turns on a broadly empirical conjecture which is outstanding. If this conjecture proves to be false, then it is unclear what the epistemological argument for set-theoretic pluralism is supposed to be.
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Notes
The “intended model” is not strictly speaking a model at all, since it is not a set.
For more on this, see Sect. 2.
Although this is the standard argument for set-theoretic pluralism, Hamkins (2012) suggests that the view also does better justice to set theorists’ experience working with different models of set theory. He writes,
This abundance of set-theoretic possibilities poses a serious difficulty for the universe view… one must explain or explain away as imaginary all of the alternative universes that set theorists seem to have constructed. This seems a difficult task, for we have a robust experience in those worlds… The multiverse view… explains this experience by embracing them as real (2012, 418).
But it is unclear what this argument comes to. Surely the real existence of the models in question does not help to causally explain set theorists’ psychological states. For more on this, see Sect. 2.
This formulation makes it sound as if the universalist accepts the Axiom of Foundation, which should not be built into the view. But the terminology is entrenched, so I stick to it here.
A similar problem plagues indispensability-based epistemologies inspired by Quine. Colyvan writes,
[L]et’s take a…charitable reading of the… challenge…to explain the reliability of our systems of beliefs….[W]e see that Quine has already answered it: we justify our system of beliefs by testing it against bodies of empirical evidence Colyvan (2007, 111).
Even if we can explain the justification of (or justify, in a dialectical sense) our belief in sets in in the way that we can explain the justification of (or justify) our belief in electrons, it does not follow that we can explain the reliability of our belief in sets in the way that we can explain the reliability of our belief in electrons. As Leng writes,
Mathematical objects are…acausal and non-spatiotemporal….These…features put them on a…different footing than electrons…[W]e should expect that the observed phenomena would be very different on the hypothesis that there are no such things [as electrons]…But if such counterfactual considerations have force against those sceptical about the unobservable physical objects posited by our theories, no analogous counterfactual is available against those sceptical about the mathematical objects our theories posit. A mathematical realist who starts a challenge, ‘If there were no numbers, then…’ will find it difficult to finish this supposed counterfactual in a way that could trouble those sceptical of mathematical objects (2010, 202, italics in original).
Assuming a “truth-value realist” interpretation, that is. For more on the distinction between “ontological” and truth-value realism, see Shapiro (2000), and the fourth point below. (I will speak of “our” set-theoretic beliefs in what follows to refer to those of the set-theoretic community—bracketing the familiar complication that it is unclear how to understand the notion of a scientific community’s beliefs. Field does not claim that we must explain the reliability of every individual’s set-theoretic beliefs, however outlandish, in order to answer his challenge).
It is sometimes suggested that Field’s formulation of Benacerraf’s challenge would not undermine our set-theoretic beliefs, but would merely undermine a particular construal of their contents (“truth-value realism”). But this is doubtful—at least assuming that truth-value realism was plausible to start with. In general, when faced with undermining—or, indeed, rebutting—evidence we do not simply reinterpret the contents of our beliefs. We give them up. For example, suppose we use a machine to test a liquid for a compound. We then learn that it only gave a positive reading because it was stuck on “positive”. Then our belief that the compound is in the liquid, not our belief that facts about it do not depend on our beliefs, seems undermined. Perhaps matters are different when a whole class of beliefs is on trial. But, if so, some explanation of the difference is needed. In any event, nothing non-semantic turns on whether we take the unanswerability of the reliability challenge to undermine our mathematical beliefs or merely to undermine our belief in mathematical realism.
Barton (2016) seems to conflate these challenges in his discussion of pluralism.
See Hellman (1989) or Chihara (1990) for ontologically-innocent interpretations of our mathematical theories. It is no wonder that Field takes the problem to depend on an ontologically committal interpretation of mathematics since he himself appeals to primitive modal ideology. See, again, his (1989, Introduction).
More exactly, this is all there is to know that is of any mathematical interest. One could still ask the semantic question of what the extension of “is a member of” is on an occasion of use. Alternatively, one might ask what is “packed into” our concept of set. But such questions are really just about us and lack any mathematical interest. They put no constraints on what sets (or set-like entities) there are.
This point does not seem to me to be sufficiently recognized in the literature. Balaguer (1998), for instance, seems to me not to be sufficiently sensitive to this.
See also Linksy and Zalta (1995, 25).
Of course, if one dismisses hyperintentional notions like “explanation”, in the present sense, as unintelligible, then Answer 2 does not get off the ground.
Moreover, we do seem to be able to show that the contents of at least some of our set-theoretic beliefs are implied by their explanation for the reason alluded to by Steiner in the quotation above. So, (a) is questionable as well.
For an argument that not even true arithmetic beliefs per se are hard-wired in us, see Clarke-Doane (2012).
This formulation assumes an ontologically-committal interpretation of set-theoretic claims. But, again, one can ask what beliefs we would have had had there been no non-vacuous set-theoretic truths, whether or not such truths turn on the existence of sets.
I will mention a more fundamental problem with this proposal and the next shortly.
See also Maddy (1988). In my (2016, 2.3), I suggest that whatever contingency there is in our set-theoretic beliefs is due to the contingency of our abductive practices. I argue that our elementary mathematical beliefs may be modally robust, and that our set-theoretic beliefs may “best systematize” the former. If the contingency of our abductive practices is undermining, then all manner of our beliefs—mathematical and otherwise—are undermined. However, our “abductive practices” may not stand or fall together. What distinguishes those which lead us to accept the Axiom of Choice from those which lead us to accept e = mc^2 may be precisely that the latter are more modally robust than the former. The latter, but not the former, are tested against a causally efficacious world.
This is far from settled, however. For relevant discussion, see Cosmides and Tooby (1991).
It might be thought that I have overlooked a way of understanding the reliability challenge, a formulation occasionally pressed by Field himself. In his (1989), he writes,
If the intelligibility of talk of “varying the facts” is challenged… it can easily be dropped without much loss to the problem: there is still the problem of explaining the actual correlation between our believing “p” and its being the case that p (238, italics in original).
I do not know what this means. It might be taken to involve showing that the correlation holds in nearby worlds, so the actual correlation is no “fluke”. But, in that case, we are just back to something like safety or sensitivity. Perhaps, then, there is a hyperintensional sense of “explanation” according to which one can intelligibly request an explanation of the “merely actual correlation” between our beliefs and the truths. But if that sense is not given by Answer 2 or 3, then I am not sure what it is. Even if there were such a sense, it is unclear how the apparent impossibility of offering such an explanation could undermine our set-theoretic beliefs, for reasons surveyed in Sect. 3. (Even if we cannot explain the “merely actual correlation” between our moral or mathematical beliefs and the truths, in some hyperintensional sense of that phrase, we might still be able to show that those beliefs are sensitive, safe (and objectively probable), realistically construed.) Finally, even if the above challenge can be made out, is distinct from those surveyed, and is worth taking seriously, whether the pluralist is better positioned to answer it would itself seem to depend on whether our non-logical set-theoretic beliefs are significantly more contingent than our deductive practices.
References
Balaguer, M. (1995). A platonist epistemology. Synthese, 103(3), 303–325.
Balaguer, M. (1998). Platonism and anti-platonism in mathematics. New York: Oxford University Press.
Balaguer, M. (1999). Review of Michael Resnick's mathematics as the science of patterns. Philosophia Mathematica, 7(1), 108–126.
Barton, N. (2016). Multiversism and concepts of set: How much relativism is acceptable? In F. Boccuni & A. Sereni (Eds.), Objectivity, realism, and proof (pp. 189–209). Berlin: Springer.
Beall, J. C. (1999). From full-blooded platonism to really full-blooded platonism. Philosophia Mathematica,7, 322–327.
Benacerraf, P. (1973). Mathematical truth. Journal of Philosophy,60, 661–679.
Boolos, G. (1999). Must we believe in set theory? In R. Jeffrey (Ed.), Logic, Logic, and Logic. Cambridge: Harvard University Press.
Chihara, C. (1990). Constructability and mathematical existence. Oxford: Oxford University Press.
Clarke-Doane, J. (2012). Morality and mathematics: The evolutionary challenge. Ethics,122, 313–340.
Clarke-Doane, J. (2016). What is the Benacerraf problem? In F. Pataut (Ed.), New perspectives on the philosophy of Paul Benacerraf: Truth, objects, infinity. Dordrecht: Springer.
Clarke-Doane, J. (2020). Morality and mathematics. Oxford: Oxford University Press.
Colyvan, M. (2007). Mathematical recreation versus mathematical knowledge. In M. Leng, A. Paseau, & M. Potter (Eds.), Mathematical knowledge (pp. 109–122). Oxford: Oxford University Press.
Cosmides, L., & Tooby, J. (1991). Reasoning and natural selection. In R. Dulbecco (Ed.), Encyclopedia of human biology (Vol. 6). San Diego: Academic Press.
Enoch, D. (2010). The epistemological challenge to metanormative realism: How best to understand it, and how to cope with it. Philosophical Studies,148, 413–438.
Field, H. (1989). Realism, mathematics, and modality. Oxford: Blackwell.
Field, H. (1998). Which mathematical undecidables have determinate truth-values? In H. G. Dales & G. Oliveri (Eds.), Truth in mathematics (pp. 291–310). Oxford: Oxford University Press.
Field, H. (2001). Which mathematical uncecidables have determinate truth-values? In Truth and the Absence of Fact. Oxford: Oxford University Press.
Field, H. (2005). Recent debates about the a priori. In T. S. Gendler & J. Hawthorne (Eds.), Oxford studies in epistemology (Vol. 1, pp. 69–88). Oxford: Oxford University Press.
Gödel, K. (1947). What is Cantor’s continuum problem? Revised and reprinted in P. Benacerraf & H. Putnam (Eds.), Philosophy of mathematics. Englewood Cliffs, NJ: Prentice-Hall, 1964.
Hamkins, J. D. (2011). The set-theoretic multiverse. arXiv https://arxiv.org/abs/1108.4223.
Hamkins, J. D. (2012). The set-theoretic multiverse. Review of symbolic logic 5:416–449
Hellman, G. (1989). Mathematics without numbers. Oxford: Oxford University Press.
Jensen, R. (1995). Inner models and large cardinals. Bulletin of Symbolic Logic,1, 393–407.
Koellner, P. (2013). Hamkins on the pluriverse. Manuscript.
Leng, M. (2009). Algebraic approaches to mathematics. In O. Bueno & O. Linnebo (Eds.), New waves in the philosophy of mathematics. New York: Palgrave Macmillan.
Leng, M. (2010). Mathematics and reality. Oxford: Oxford University Press.
Liggins, D. (2006). Is there a good epistemological argument against platonism? Analysis,66, 135–141.
Linnebo, Ø. (2006). Epistemological challenges to mathematical platonism. Philosophical Studies,129, 545–574.
Linsky, B., & Zalta, E. (1995). Naturalized platonism versus platonism naturalized. Journal of Philosophy,92, 525–555.
Maddy, P. (1988). Believing the axioms I and II. Journal of Symbolic Logic.,53, 481, 763–511, 764.
Mogensen, A. (2016). Contingency anxiety and the epistemology of disagreement. Pacific Philosophical Quarterly,97, 590–611.
Nozick, R. (1981). Philosophical explanations. Oxford: Oxford University Press.
Potter, M. (2004). Set theory and its philosophy. Cambridge: Cambridge University Press.
Pritchard, D. (2008). Safety-based epistemology: Whither now? Journal of Philosophical Research,34, 33–45.
Putnam, H. (1980). Models and reality. Journal of Symbolic Logic,45, 464–482.
Rieger, A. (2011). Paradox, ZF, and the axiom of foundation. In D. DeVidi, M. Hallet, & P. Clark (Eds.), Logic, mathematics, philosophy, vintage enthusiasms: Essays in Honour of John L. Bell (The Western Ontario series in philosophy of science). New York: Springer.
Sayre-Mccord, G. (1988). Moral theory and explanatory impotence. Midwest Studies in Philosophy,XII, 433–457.
Schechter, J. (2010). The reliability challenge and the epistemology of logic. Philosophical Perspectives,24, 437–464.
Schechter, J. (2013). Could evolution explain our reliability about logic? In S. G. Tamar & J. Hawthorne (Eds.), Oxford studies in epistemology (Vol. 4). Oxford: Oxford University Press.
Shapiro, S. (2000). Philosophy of mathematics: Structure and ontology. New York: Oxford University Press.
Steiner, M. (1973). Platonism and the causal theory of knowledge. Journal of Philosophy,70, 57–66.
Warren, J. (2015). Conventionalism, consistency, and consistency sentences. Synthese,192, 1351–1371.
White, R. (2010). You just believe that because. Philosophical Perspectives,24, 573–612.
Acknowledgements
Thanks to Joel David Hamkins, Achille Varzi, Jared Warren, and audience members of the Set-Theoretic Pluralism: Indeterminacy and Foundations conference at the University of Aberdeen for helpful discussion.
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Clarke-Doane, J. Set-theoretic pluralism and the Benacerraf problem. Philos Stud 177, 2013–2030 (2020). https://doi.org/10.1007/s11098-019-01296-y
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DOI: https://doi.org/10.1007/s11098-019-01296-y