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Published online by Cambridge University Press: 03 August 2015
In this paper, I argue that religious belief is epistemically equivalent to mathematical belief. Abstract beliefs don't fall under ‘naive’, evidence-based analyses of rationality. Rather, their epistemic permissibility depends, I suggest, on four criteria: predictability, applicability, consistency, and immediate acceptability of the fundamental axioms. The paper examines to what extent mathematics meets these criteria, juxtaposing the results with the case of religion. My argument is directed against a widespread view according to which belief in mathematics is clearly rationally acceptable whereas belief in religion is not. The paper also aims to make some of the implications of contemporary mathematics available to philosophers working in different fields.1
1 Many thanks to Ana Bajzeli, Claire Benn, Sharon Berry, Dani Rabinowitz, Stewart Shapiro, and Olla Solomyak for helpful comments on earlier drafts of this paper. I am also most grateful to John Hawthorne and all other participants of the ‘Etiology, Debunking, and Religious Epistemology’ workshop that took place as part of the Templeton project New Insights and Directions for Religious Epistemology at Oxford University in March 2015, for many helpful comments on my presentation of an earlier version of this paper.
2 Another obstacle for mathematical Platonism is the indeterminacy problem for the reduction of numbers to sets: even though we know that numbers are reducible to sets, the sets in question are indeterminate. The number three, for example, can be expressed as {{{ø}}} or as {ø, {ø}, {ø, {ø}}}. This indeterminacy has been taken as a point against the existence of determinate mathematical objects (cf. Benacerraf, Paul, ‘What Numbers Could not Be’, The Philosophical Review 74(1) (1965), 47−73CrossRefGoogle Scholar).
3 Neo-logicists like Crispin Wright and Bob Hale would disagree with this claim.
4 Benacerraf famously argues that a successful philosophical account of mathematics should provide both a semantics and an epistemology of mathematics which are coherent with an ‘overall’ semantics and epistemology, i.e. with a semantics and epistemology for ordinary assertions (cf. Benacerraf, Paul, ‘Mathematical Truth’, The Journal of Philosophy 70(19) (1973), 661−679CrossRefGoogle Scholar). My aim in this article is a somewhat more limited one: all I want to argue for is the conditional claim that, if we want to consider belief in mathematics rational, then we ought to consider religion rational as well.
5 Cf. for example, Friedrich Waismann, Ludwig Wittgenstein and the Vienna Circle (Oxford: Blackwell Publishers, 2005, 117ff and 129ff).
6 This is almost commonplace by now, but also commonplaces should be referenced: cf. Max Weber, ‘The Social Psychology of the World Religions' (1915) and ‘Religious Rejections of the World and their Directions' (1922), in H.H. Gerth and C. Wright Mills, From Max Weber: Essays in Sociology (New York: Oxford University Press, 1946) for some loci classici.
7 For a detailed discussion of what kinds of constraints evidence puts on our beliefs, see White, Roger, ‘Epistemic Permissiveness’, Philosophical Perspectives 19 (2005), 445−459CrossRefGoogle Scholar.
8 Cf. for example: Campbell, John, ‘Rationality, meaning, and the analysis of delusion’, Philosophy, Psychiatry, and Psychology 8 (2001), 89−100CrossRefGoogle Scholar; Davidson, Donald, ‘Rational animals’, Dialectica 36 (1982), 317−327CrossRefGoogle Scholar; Jacobson, Anne Jaap, ‘A Problem For Causal Theories Of Reasons And Rationalizations’, The Southern Journal Of Philosophy 31(3) (1993), 307–321CrossRefGoogle Scholar; Cohen, Stewart and Comesana, Juan, ‘Williamson on Gettier cases in epistemic logic and the knowledge norm for rational belief: a reply to a reply to a reply’, Inquiry 56 (2013), 400−415CrossRefGoogle Scholar; and Timothy Williamson, ‘Justifications, Excuses, and Sceptical Scenarios', in Julien Dutant and Daniel Dohrn (eds), The New Evil Demon (Oxford: Oxford University Press, forthcoming) for more recent discussions.
9 Stephen Hawking and Leonard Mlodinow, The Grand Design (New York: Bantam Books, 2010), 70.
10 Hartry Field, Science Without Numbers (Princeton: Princeton University Press, 1980).
11 Arguments of this kind are often discussed under the general heading of ‘Quine-Putnam indispensability arguments' (cf. for example, Hilary Putnam, ‘Philosophy of Logic’, reprinted in his: Mathematics, Matter and Method: Philosophical Papers, Volume 1, 2nd edition (Cambridge: Cambridge University Press, 1979), 323−357; W.V.O. Quine, ‘Success and Limits of Mathematization’, in his: Theories and Things (Cambridge, MA: Harvard University Press, 1981), 148−155. However, there are a number of slight variations of the argument out there, and not all of them would have been endorsed by Putnam and Quine.
12 Penelope Maddy, Defending the Axioms: On the Philosophical Foundations of Set Theory (New York: Oxford University Press, 2011), 9.
13 For John von Neumann, a contemporary mathematician of Gödel's, the theorem proved that ‘…there is no rigorous justification for classical mathematics' (von Neumann to Gödel in a letter dated November 29, 1930; see Miklos Redei (ed.), John Von Neumann: Selected Letters (Providence, RI: The American Mathematical Society, 2005), 8.
14 Edward Nelson, ‘Warning Signs of a Possible Collapse of Contemporary Mathematics', in: Michael Heller and Hugh Woodin, Infinity: New Research Frontiers (New York: Cambridge University Press, 2011), 81ff.
15 Thanks to Sharon Berry for helpful discussions on basic a priori knowledge in mathematics.
16 Op. cit. note 5, 140.
17 Bertrand Russell, Introduction to Mathematical Philosophy (New York: Digireads.com Publishing, 2010 [1919]), 8.
18 Cf. Joan Bagaria, ‘Set Theory’, in: Stanford Encyclopedia of Philosophy (Accessed on January 8, 2015, available at: http://plato.stanford.edu/ entries/set-theory/): ‘The notion of set is so simple that it is usually introduced informally, and regarded as self-evident. In set theory, however, as is usual in mathematics, sets are given axiomatically, so their existence and basic properties are postulated by the appropriate formal axioms. The axioms of set theory imply the existence of a set-theoretic universe so rich that all mathematical objects can be construed as sets. Also, the formal language of pure set theory allows to formalize all mathematical notions and arguments. Thus, set theory has become the standard foundation for mathematics, as every mathematical object can be viewed as a set, and every theorem of mathematics can be logically deduced in the Predicate Calculus from the axioms of set theory.’
19 Giuseppe Peano, ‘The principles of arithmetic, presented by a new method’, in Jean Van Heijenoort's From Frege to Gödel: A Source Book in Mathematical Logic, 1879–1931 (Cambridge, MA: Harvard University Press, 1967 [1889]), 83–97.
20 There are several alternative systems, conservative or proper extensions of ZFC, such as GBN (‘Gödel-Bernays-Neumann’), MK (‘Morse-Kelley’), NFU (‘New Foundations-Urelements'), KP (‘Kripke-Platek’), and TG (‘Tarski-Grothendieck’), but the Zermelo- Fraenkel axioms (plus the Axiom of Choice) are most commonly used and are widely considered to define the entire set-theoretic hierarchy. The concept ‘set’ is arguably even more fundamental than the concept ‘number’: there is wide agreement amongst mathematicians that practically all of mathematics can be expressed in terms of the axioms of set theory.
21 The axiom formulations are those of Thomas Jech, Set Theory (Berlin: Springer, 2006), 3−16.
22 The Banach-Tarski paradox states that we can discompose a three-dimensional sphere into a finite number of disjoint subsets and then put it back together in a different way so that we get two identical copies of the initial sphere.
23 In fact, it has been discussed lately whether mathematics needs new axioms and whether the current axioms of ZFC lack justification entirely. Cf. for example Feferman, Solomon, Friedman, Harvey, Maddy, Penelope, and Steel, John, ‘Does Mathematics Need New Axioms?’, The Bulletin of Symbolic Logic 6(4) (2000), 401−446CrossRefGoogle Scholar; Easwaran, Kenny, ‘The Role of Axioms in Mathematics’, Erkenntnis 68 (2008), 381–391CrossRefGoogle Scholar.
24 Buddhism, for example, is sometimes not counted as a religion because of its nontheistic nature.
25 In fact, it was always embarrassing that churches were more likely to be struck by lightning than pubs and brothels − because they have high steeples.
26 It is more accurate to say that the credence of religious beliefs is highly controversial and dependent on different societies. However, given that the goal of my argument is to align the credence of mathematical and religious beliefs, the fact that in some societies religious beliefs already have a very high credence only supports my conclusion. Hence, for the sake of argument, I shall assume a universally low credence for religious systems of thought, thus strengthening the conclusion of my argument to the max.
27 The foundational axioms of mathematics, i.e. Zermelo's set theoretic axioms, were formulated only quite recently, whereas mathematics has been practiced for at least three thousand years. Zermelo set out to formulate them when a specific need arose, viz. the need to prove Cantor's controversial well-ordering principle; cf. Maddy, Penelope, ‘Believing the Axioms. I’, The Journal of Symbolic Logic 53(2) (1988), 481–511CrossRefGoogle Scholar.
28 Classical controversies such as the apparent contradiction between Darwin's theory of evolution and the religious belief that God created the world as it is in six days are two fundamentally different paradigms, both of which try to explain the existence of the world in different ways. That does not mean that they are in principle irreconcilable with one another, and there are in fact several interpretations of the story of creation that render it compatible with science.
29 At this point I want to address an objection to my argument according to which my account of rationality sanctions not only belief in established religions but also in all kinds of bizarre things such as ghosts, Leprechauns, Voodoo, religious extremism, etc. My reply is that I argue that holding mathematical beliefs is equally (ir)rational as holding religious beliefs, given their epistemic and ontological neutrality. So long as a given abstract belief system fulfils the conditions of rationality set out in this article in the same way mathematics fulfils them, their epistemic status is indeed the same as mathematics. Consequently, lending higher credence to belief in Reform Judaism than, say, belief in radical interpretations of Islam, is a matter of cultural normativity, not a matter of epistemic objectivity. My account certainly doesn't help distinguishing ‘rational’ from ‘irrational’ religions (and political movements based on radical interpretations thereof), precisely because there is no fact of the matter as to which abstract belief systems are ‘more rational’ than others. The fact that different religions are externally inconsistent with one another is irrelevant for their rationality; what matters is that a given system ought not to stand in contradiction with science.
