Abstract
Hartry Field’s epistemological challenge to the mathematical platonist is often cast as an improvement on Paul Benacerraf’s original epistemological challenge. I disagree. While Field’s challenge is more difficult for the platonist to address than Benacerraf’s, I argue that this is because Field’s version is a special case of what I call the ‘sociological challenge’. The sociological challenge applies equally to platonists and fictionalists, and addressing it requires a serious examination of mathematical practice. I argue that the non-sociological part of Field’s challenge amounts to a minor reformulation of Benacerraf’s original challenge. So, I contend, Field’s challenge is not an improvement on Benacerraf’s. What is new to Field’s challenge is as much a problem for the fictionalist as it is for the platonist.
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Notes
Readers with concerns about the locution ‘knowledge of abstract mathematical entities’ are free to read that expression as shorthand for ‘knowledge of sentences that include terms that refer to mathematical entities’. Strictly speaking, Benacerraf himself speaks of knowledge of sentences. Thanks to an anonymous referee.
W.D. Hart does provide a potential argument for universal causal constraint in a discussion of Benacerraf’s epistemological challenge. It proceeds through three assumptions. First, “given that truth requires reference to objects, knowledge of truth is best understood in terms of some transaction, commerce or connection between the knower of the truth and the objects required for truth which justifies the knower’s belief about them” (1991: 94). Second, “the only generally agreed basic mode of commerce between people and objects that justifies true belief about them is perception” (p. 94). And third, “perception is by nature a causal process” (p. 96). But, on its most natural interpretation, the second claim of Hart’s argument is false. Perception of an object is not the only generally accepted way to acquire justified beliefs about it; inference is too. To use an example from Colin Cheyne (1998: p. 36), scientists knew, and hence justifiedly believed, that the chemical element germanium had certain properties before any instances of germanium had been discovered. Since instances of germanium had not been discovered, they had not been perceived. Scientists used their other knowledge of chemistry to justify their beliefs about germanium. Notice that, though Cheyne’s case is a clear counterexample to Hart’s argument, it is not clearly a counterexample to universal causal constraint itself. Perhaps scientists had causally interacted with germanium in some way, however indirectly. Thanks to an anonymous referee.
Potter used the case as a counterexample to the claim that knowledge requires causal interaction with the fact known.
I make a similar point in Nutting (2016: pp. 2145–2146).
By ‘perception-like knowledge’, I specifically include knowledge formed through intuition, where intuition is understood using a perceptual model.
Field actually seeks to avoid two concepts that appear in Benacerraf’s argument: ‘knowledge’ and ‘truth’ (1989: p. 230). Field’s challenge involves no discussion of knowledge, and I will not pretend it does. But Field primarily wants to avoid discussion of truth because he wants to avoid any presuppositions that truth involves correspondence. [Field (1986) endorses a deflationary account of truth.] Given Field’s motivation, I will use the terms ‘true’ and ‘truth’ freely while discussing his challenge, but I shall explicitly avoid presupposing a correspondence notion of truth, except where Field himself seems to presuppose it.
Sometimes Field uses the term ‘realist’, but I systematically use the term ‘platonist’ in this paper. Given Field’s characterization of realism, I take him to use the two terms interchangeably: “a realist view of mathematics involves the postulation of a large variety of aphysical entities” (1989: p. 230).
I preserve Field’s numbering.
Field suggests that, in well-developed branches of mathematics like elementary arithmetic, the platonist also should think the converse is almost always true, i.e., that most truths in those branches of mathematics are accepted by mathematicians. It’s not clear to me that the platonist should endorse this claim. I also do not think it plays an important role in Field’s epistemological challenge.
See Field (1989): “we need an explanation of how it can have come about that mathematicians’ belief states and utterances so well reflect the mathematical facts” (230).
Another approach, as in Armstrong (1973), is to model the kind of reliability relevant to justification on the reliability of an instrument like a thermometer. But this, too, is about the reliability of an individual, not the reliability of a group of people.
Clarke-Doane (2017) interprets Field’s reliability challenge as being about justification. (He is not exclusively focused on Field (1989); he also incorporates ideas from Field’s more recent work.) Some think that Clarke-Doane’s interpretation fits Field’s discussion, especially e.g. Field’s claim that “if it is impossible in principle to explain this [reliability], then that tends to undermine the belief in mathematical entities” (1989: p. 26). I disagree. I do not understand “belief in mathematical entities” to be ordinary mathematical beliefs, e.g., that 2 + 3 = 5. I understand it to be belief in mathematical platonism. [One can reject mathematical platonism while believing ordinary mathematical claims. I discuss general views in this category in Nutting (2017). For philosophical accounts in this vein, see Hodes (1984), Azzouni (2004) and Hofweber (2005).] See Liggins (2017) for a convincing case that Clarke-Doane misinterprets Field’s (1989) challenge, and that Field’s (1989) challenge is not about justification. Thanks to an anonymous referee for pressing me on this.
In claiming that mathematicians reliably agree about what mathematical claims to accept, I claim neither (a) that the relevant agreement is universal, nor (b) that mathematicians reliably agree that mathematical claims are true (especially not in any philosophically loaded sense).
With respect to (a), remarkable agreement among mathematicians is consistent with both (a1) disagreement about some mathematical claims, and (a2) fairly systematic disagreement among some subsets of mathematicians. For (a1), the agreement remains reliable even if there is significant disagreement about some claims in fairly isolated subfields—e.g., disagreement about some set-theoretic axioms. I assume only that there is agreement about the vast majority of mathematical claims that mathematicians have considered. For (a2), consider Clarke-Doane’s (2013: p. 474) example of Edward Nelson, who rejects the Successor axiom in arithmetic. Any small subset of mathematicians that includes Nelson will likely disagree about many mathematical claims. I do not take such cases to undermine the claim that there is remarkable agreement among mathematicians in general about mathematical claims in general. Rather, I take voices of dissent and contested claims to be outliers, and relatively rare; they are consistent with reliable overall agreement.
With respect to (b), a person can accept a claim without believing that it is true. Elgin 2004 gives several examples from science of claims that most scientists do (and should) accept, but that are not (and should not be taken to be) literally true; her examples involve, e.g., approximations and idealizations. In the case at hand, presumably fictionalists like Field accept some existential mathematical claims in the context of doing mathematics, though they believe those claims to be false. [They might accept them, e.g., as mathematically correct, or as “true according to standard mathematics” (Field 1989: p. 3). (The italics are Field’s.)]
Thanks to an anonymous referee.
Some philosophers, such as Clarke-Doane (2017), think that addressing Field’s challenge requires platonists to explain some kind of modal or counterfactual connection between mathematical truths and mathematicians’ beliefs (or, as I frame it, acceptance of claims). Such modal notions could be applied here. If one were so inclined, one could demand that a platonist’s explanation of mathematical agreement preserve the relevant kind of counterfactual connection between truths and claims accepted. The following discussion presupposes neither the inclusion nor exclusion of such modal or counterfactual assumptions. Thanks to an anonymous referee.
This suggestion is similar to what Linnebo calls ‘The Boring Explanation’ of the reliability of mathematicians: “Jones went to school, where he took courses in mathematics. Being a good student, Jones learnt a good deal of mathematics, and he learnt how to apply it to problems that are given to him. Moreover, the mathematical theory Jones was taught is true: its axioms are mathematical truths, and its rules preserve mathematical truth” (2006: p. 554). Linnebo correctly observes that the Boring Explanation is not an adequate solution to Field’s challenge.
There seems to be more disagreement among set theorists about axioms than there is among mathematicians in general about mathematical claims in general. Perhaps some set-theoretic axioms are outlier cases of mathematical claims that do not enjoy widespread agreement, even among experts (see note 14). I suspect that the unusual disagreement in such cases can be explained by the fact that axioms are often accepted on the basis of non-deductive inferences, which are less effective than testimony and proof at preserving agreement and truth.
Gödel seems to have such a view, with his talk of intuition or perception of the mathematical realm.
Suppose [as Linnebo (2012) suggests] that mathematical facts are grounded in facts about concrete objects. Then perhaps one’s interactions with concrete objects—the facts about which ground mathematical facts—lead one to accept mathematical claims.
Fictionalists might think that mathematical agreement results from some kind of social coordination or convention. Similarly, plenitudinous platonists might think that most (but not all) mathematicians have coordinated about which claims to accept, and hence, which mathematical objects to discuss.
Fictionalists would probably take social coordination or convention to explain mathematicians’ agreement.
It’s possible that Field tacitly assumes that platonists are committed to a perceptual model because he thinks that there are serious metaphysical problems with other attempts at platonistic epistemologies. In response to Lewis’s (1986) idea that we don’t need an explanation of mathematical knowledge because mathematical claims hold necessarily, Field objects to the idea that mathematical claims hold necessarily in any relevant sense (Field 1989: pp. 233–236). [Field (1984a) argues that mathematical knowledge is not merely logical knowledge, because mathematical claims have existential commitments.] In response to the possibility that mathematical objects can be known because they are somehow constructed by the mind, Field suggests that “it may be best to interpret such talk of ‘constructions’ as simply a picturesque way of saying that mathematical talk should be interpreted along fictionalist lines” (1989: p. 27). And in response to Wright’s (1983) idea that mathematical claims can be known through abstractionist principles, Field asks, “Why then don’t we need a Gödelian faculty of perceiving abstract objects to ascertain that there are abstract objects with the properties stated … ?” (1984b: p. 655). In short, Field seems to think that putative epistemologies of mathematics without perception-like cognition are liable to undermine platonistic ontologies.
I replace talk of ‘knowledge’ in the passage with talk of ‘acceptance’, in accord with Field’s stated preferences. Field explicitly sets out to raise a challenge “without use of the term of art ‘knows’” (Field 1989: p. 230).
When I say that a person ‘accepts a claim due to perception (or perception-like cognition) of an entity’, I mean more specifically that she accepts a claim to the effect that the object has certain properties because she perceives (or intuits) that the object has those properties.
Some contemporary platonists might disagree with this assumption. See e.g. notes 19 and 20.
(RF1) is also the clearest remnant of Field’s original challenge in this non-sociological remainder. The difference between mathematicians and non-mathematicians appears to be most significant for the sociological part of the challenge. Mathematicians and non-mathematicians have differing mathematical training, which presumably leads to differences in e.g. the mathematical claims considered, the kind and quantity of mathematical testimony received, and the number and quality of mathematical inferences performed.
Kasa actually says “accepting a true belief” (see Sect. 4), but I don’t assume accepting a claim requires believing it. See note 14.
References
Armstrong, D. M. (1973). Belief, truth, and knowledge. Cambridge: Cambridge University Press.
Azzouni, J. (2004). Deflating existential consequence: A case for nominalism. Oxford: Oxford University Press.
Benacerraf, P. (1973). Mathematical truth. The Journal of Philosophy,70(8), 661–680.
Burgess, J., & Rosen, G. (1997). A subject with no object. Oxford: Oxford University Press.
Casullo, A. (1992). Causality, reliabilism, and mathematical knowledge. Philosophy and Phenomenological Research,52(3), 557–584.
Cheyne, C. (1998). Existence claims and causality. Australasian Journal of Philosophy,76(1), 34–47.
Clarke-Doane, J. (2013). What is absolute undecidability? Noûs,47(3), 467–481.
Clarke-Doane, J. (2014). Moral epistemology: the mathematics analogy. Noûs,48(2), 238–255.
Clarke-Doane, J. (2017). What is the Benacerraf problem? In F. Pataut (Ed.), Truth, objects, infinity: New perspectives on the philosophy of Paul Benacerraf (pp. 17–44). Dordrecht: Springer.
Elgin, C. Z. (2004). True enough. Philosophical Issues,14(1), 113–131.
Field, H. (1984a). Is mathematical knowledge just logical knowledge? Philosophical Review,93(4), 509–552.
Field, H. (1984b). Critical notice of crispin Wright’s Frege’s conception of numbers as objects. Canadian Journal of Philosophy,14(4), 637–662.
Field, H. (1986). The deflationary conception of truth. In G. MacDonald & C. Wright (Eds.), Fact, science and morality (pp. 55–117). Oxford: Blackwell.
Field, H. (1988). Realism, mathematics, and modality. Philosophical Topics,16(1), 57–107.
Field, H. (1989). Realism, mathematics, and modality. Oxford: Basil Blackwell.
Goldman, A. (1979). What is justified belief? (Reprinted from Epistemology: An anthology, pp. 340–353, by E. Sosa, Ed., 2008, Oxford: Blackwell).
Hart, W. D. (1991). Benacerraf's Dilemma. Crítica: Revista Hispanoamericana de Filosophía, 23(6), 87–103.
Hodes, H. T. (1984). Logicism and the ontological commitments of arithmetic. The Journal of Philosophy,81(3), 123–149.
Hofweber, T. (2005). Number determiners, numbers, and arithmetic. Philosophical Review,114(2), 179–225.
Kasa, I. (2010). On field’s epistemological argument against platonism. Studia Logica,96(2), 141–147.
Lewis, D. (1986). On the plurality of worlds. Malden: Blackwell.
Liggins, D. (2006). Is there a good epistemological argument against platonism? Analysis,66(2), 135–141.
Liggins, D. (2010). Epistemological objections to platonism. Philosophy Compass,5(1), 67–77.
Liggins, D. (2017). The reality of field’s epistemological challenge to platonism. Erkenntnis. https://doi.org/10.1007/s10670-017-9925-z.
Linnebo, Ø. (2006). Epistemological challenges to mathematical platonism. Philosophical Studies,129(3), 545–574.
Linnebo, Ø. (2012). Reference by abstraction. Proceedings of the Aristotelian Society,112(1), 45–71.
Maddy, P. (1984). Mathematical epistemology: What is the question? The Monist,67(1), 46–55.
Maddy, P. (1988a). Believing the Axioms I. The Journal of Symbolic Logic,53(2), 481–511.
Maddy, P. (1988b). Believing the axioms II. The Journal of Symbolic Logic,53(3), 736–764.
Nutting, E. S. (2016). To bridge Gödel’s gap. Philosophical Studies,173(8), 2133–2150.
Nutting, E. S. (2017). Ontological realism and sentential form. Synthese. https://doi.org/10.1007/s11229-017-1446-4.
Potter, M. (2007). What is the problem of mathematical knowledge? In M. Leng, A. Paseau, & M. Potter (Eds.), Mathematical Knowledge (pp. 16–32). Oxford: Oxford University Press.
Wright, C. (1983). Frege’s conception of numbers as objects. Aberdeen: Aberdeen University Press.
Acknowledgements
Thanks to Corinne Bloch-Mullins, Ben Caplan, Sam Cowling, Lina Janssen, Kathryn Lindeman, May Mei, Reed Solomon, Audrey Yap, three anonymous referees, my colleagues at the University of Kansas, and an audience at Denison University. Thanks also to audiences that heard an ancestor of this paper at the University of Missouri, the 2014 Midwest Epistemology Workshop, and a workshop at the University of Umeå.
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Nutting, E.S. Benacerraf, Field, and the agreement of mathematicians. Synthese 197, 2095–2110 (2020). https://doi.org/10.1007/s11229-018-1785-9
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DOI: https://doi.org/10.1007/s11229-018-1785-9