Abstract
Mathematical apriorists usually defend their view by contending that axioms are knowable a priori, and that the rules of inference in mathematics preserve this apriority for derived statements—so that by following the proof of a statement, we can trace the apriority being inherited. The empiricist Philip Kitcher attacked this claim by arguing there is no satisfactory theory that explains how mathematical axioms could be known a priori. I propose that in analyzing Ernest Sosa’s model of intuition as an intellectual virtue, we can construct an “intuition–virtue” that could supply the missing explanation for the apriority of axioms. I first argue that this intuition–virtue qualifies as an a priori warrant according to Kitcher’s account, and then show that it could produce beliefs about mathematical axioms independent of experience. If my argument stands, this paper could provide insight on how virtue epistemology could help defend mathematical apriorism on a larger scale.
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Notes
In line with Kurt Gödel’s Incompleteness Theorem, some mathematical statements are not provable within an axiomatic system. The strongest possible form of this defense is that provable mathematical statements can be known a priori, since these statements have proofs.
Kitcher (1983), 36–40. “Apriority” means the property of being knowable a priori. I will use it in the same way throughout my paper.
Kitcher, 21–22. While experience is necessary to grasp the content of the proposition, but as long as that has been achieved, no further experience is relevant in the acquisition of the belief. See also George and Velleman (2002), 5–6.
Kitcher, Mathematical Knowledge, 22.
Here, β might be α itself. Kitcher himself defined a priori knowledge in terms of an “a priori warrant,” for which he provided a separate definition not dependent on a priori knowledge. Still, the definition might appear circular (although not viciously circular) so I combined the two definitions to eliminate the possible confusion. The original definition is found in Kitcher, Mathematical Knowledge, 24.
Kitcher, Mathematical Knowledge, 13–14.
He explained that he did this to maintain neutrality and generality. The point of this paper is not to provide that comprehensive account of warrants, but to adopt Kitcher’s psychologistic approach through virtue epistemology, particularly using a specific intellectual virtue which I discuss later. See Kitcher, Mathematical Knowledge, 17–18.
Battaly (2012), 4–5.
Ernest Sosa, “Lecture 2: A Virtue Epistemology” in A Virtue Epistemology: Apt Belief and Reflective Knowledge, Volume I (NewYork: Oxford University Press, 2009), 22–26.
Sosa, “Virtue Epistemology,” 25–26.
This is why the conditions conducive to the virtue’s performance (C in the above definition) is an important parameter in Sosa’s theory of knowledge.
He claimed that his definition I provided in (1) is a “simple normal form for a psychologistic account of knowledge.” See Kitcher, Mathematical Knowledge, 17.
Kitcher, Mathematical Knowledge, 18–21.
In this paper, I use “axiom” to refer to basic mathematical statements in general (including postulates and properties that need not be proven), and “theorem” to refer to derivative mathematical statements (including corollaries and lemmas). This qualification is important so as not to confuse my use of this words in the mathematical sense.
Modified version of Kitcher, Mathematical Knowledge, 39.
Kitcher, Mathematical Knowledge, 46–47.
Ibid.
The following is a summary of Ernest Sosa, “Lecture 3: Intuitions,” in A Virtue Epistemology: Apt Belief and Reflective Knowledge, Volume I (New York: Oxford University Press, 2009).
Sosa, Knowledge in Perspective, 275.
This is derived from Sosa’s definition of “rational intuition,” where I gave the name “intuition–virtue” to the “competence (an epistemic ability or virtue) on the part of S” that, for him, explains how intuition works. I did this to divert the attention from the attempt to explaining intuition to the competence itself. See the original definition in Sosa, “Intuitions,” 61.
Whether or not this intuition–virtue is an accurate model of intuition does not affect my argument.
Every proposition can only be either true or false (law of excluded middle), and cannot be both true and false (law of non-contradiction), thus the need to distinguish true propositions from false ones.
Note that I use “logical competence” rather than “logical virtue” so as not to confuse it with the intuition–virtue, although the logical competence could also be considered an epistemic virtue.
It is sufficient to have the minimum competence to apply the aforementioned principles of noncontradiction, excluded middle, and sufficient reason to consider one, for the purposes of this paper, “logically competent.” Henceforth, I use “logical competence” to refer to the capacity to use these three logical principles. Of course there are other, more complex logical principles that contribute to the formation of knowledge, but we are only concerned with those that contribute to the performance of the intuition–virtue.
This relationship between two epistemic virtues could be likened to the necessity of reliable faculties (e.g. vision, memory, induction, deduction) to attain high-grade knowledge (e.g. scientific, historical knowledge). See Battaly, “Virtue Epistemology,” 23. We learn here that some intellectual virtues, including the intuition–virtue, need other virtues for their proper functioning (in this case, logical competence).
I would have used “safe in any environment,” but this might be too strong, especially against skeptics.
I’m not saying here that the intuition–virtue is merely hypothetical or epistemologically construed. It’s just that as of now, it is difficult to determine physiologically where it is, or which mechanisms it involves. Perhaps, the intuition faculty may yet to be described precisely in future scientific developments.
Recall Kitcher’s definition that some process of the same type as the original must be available to produce the belief that p (1a). The original warrant in this case is the performance of the intuition–virtue. A similar process might be the performance of possibly some other epistemic virtue.
For the detailed argument, Norton-Smith (1991).
Norton-Smith, “Mathematical Apriorist,” 112. He does this by deriving from (1) sufficient conditions for someone to know a posteriori that p. Since the sufficient conditions derived are not satisfactory, it follows that the proposed necessary conditions for knowing a priori are too strong. See the entire paper for details.
I will be withholding discussion on conceptualism in this paper because of two important considerations. The more obvious one is that the intuition–virtue is more closely related to mathematical intuition (whether Kantian or Platonic), and comparing it with the conceptualist thesis might be veering away from the question. The other reason is that conceptualism would drive us to philosophy of language, which is beyond the scope of my paper.
Sosa, “Intuitions,” 45–47.
Ibid., 55.
Kitcher, Mathematical Knowledge, 48–57.
Ibid., 57–64. Although he argued only against Platonism because he claims that some non-Platonist forms of realism are defensible.
In the case of mathematical knowledge, intuitions are not based on the ontology of mathematical objects.
These were the common sources of knowledge enumerated by Sosa in his definition of intuition as an intellectual virtue (see definition 4). Since these might be argued to be the most feasible warrants, I believe it is enough to show that these sources are not essential in acquiring the belief. If ever there are other possible sources of knowledge, then I would just extend my strategy by showing that the intuition–virtue does not rely on these sources either.
We do not need for the belief to be acquired only if there is no reliance on another source of knowledge, apart from logical competence and sheer understanding of the proposition. As I have shown in testimony, a belief could be acquired by being taught, but it could also be acquired without being taught. Only the latter is relevant to my argument; I do not need to show that the former must be impossible.
Defending the claim that all mathematical axioms might be too tall an order, since one has to first enumerate all mathematical axioms upon which every theorem is based, and then evaluate whether the intuition–virtue (or perhaps some other epistemic virtue that is also an a priori warrant) could produce belief about each axiom. If the apriorist claims that most axioms are a priori, (and of course, defend premise 3b) then he/she can at least claim that most of mathematics is also a priori.
In this case, the field of propositions F is limited to mathematical propositions.
Cappelletti and Giardino (2007).
This might be of interest for a conceptualist, since language is essential in discussing concepts.
Cappelleti and Giardino, “Cognitive basis,” 77. Emphasis mine.
Bertrand Russell expressed the equation 1 + 1 = 2 in this way when he argued that every mathematical proposition are implications. See Russell (2010), 6.
With the limited knowledge of the subjects, I argue that r is equivalent to the belief that 1 + 1 = 2 is true. From their puzzled reaction to the presented proposition “1 + 1 = 1”, they must be expecting that one unit and another must make two units, not one. Note that we can see here the logical competence of the subjects, because they understand that the contradictory statements 1 + 1 = 1 and 1 + 1 = 2 cannot be both true.
This Cartesian introspection was presented briefly in the discussion of the Cartesian model of intuition. Sosa explains this more clearly in “Intuitions,” 56–57.
Russell defined “m + n” from Peano’s axioms in his Introduction to Mathematical Philosophy (1920; reprint, New York: Dover Publications, Inc., 1993), 5–6. I am applying his principles to 1 + 1 = 2: having defined “number” as the set of natural numbers and “successor” as the next number in the natural sequence, we have 1 + 1 being equal to the successor of 1, which is 2. Therefore, 1 + 1 = 2.
I must emphasize here that “intuitable” is defined in terms of the intuition–virtue, not in terms of intuition. Recall the very important remark that the intuition–virtue discussed here is not necessarily an accurate account of intuition.
See Kitcher, Mathematical Knowledge, 36–46.
References
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Clemente, N.L. A Virtue-Based Defense of Mathematical Apriorism. Axiomathes 26, 71–87 (2016). https://doi.org/10.1007/s10516-015-9274-y
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DOI: https://doi.org/10.1007/s10516-015-9274-y