Abstract
Having presented famous attempts at reestablishing mathematics on firm foundations in the previous chapter—the intense activity among mathematicians and philosophers of mathematics in the early part of the last century is often referred to as The Crisis in the Foundations of Mathematics—we now turn to a distinction among the views to which we have given relatively lesser attention, that is, whether logicists, formalists, and intuitionists are or are not realists. The question of mathematical realism vs. mathematical anti-realism, of course, has occupied us throughout—this book is about the mathematical realism of Thomas Aquinas, after all. Still, we have most recently focused upon the epistemological aspect of recent mathematical theories and not the ontology of mathematical objects, and it is with the latter topic with which we are directly concerned in this chapter. Here we introduce and consider some of the recent—and well-known—anti-realist arguments and distinctions which make the case for a mathematics which prescinds from object-claims, including the notion of mathematics as a mere heuristic, with avowedly fictitious objects.
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Notes
- 1.
Cf. the article of that name by José Ferreirós in The Princeton Companion to Mathematics, eds. Timothy Gowers, June Barrow-Green, and Imre Leader (Princeton: Princeton University Press, 2009), 142–56.
- 2.
Some argue that Aristotle himself is a mathematical anti-realist, for example, though, as we have seen, he is a realist, albeit not of a Platonic sort.
- 3.
Democritus famously denies the external reality of sensible characteristics like color and flavor, reducing them in re to the shapes, arrangements, and positions of atoms.
- 4.
As Stewart Shapiro notes, this article “…continues to dominate contemporary discussion in the philosophy of mathematics.” Stewart Shapiro, Thinking About Mathematics (Oxford: Oxford University Press, 2000), 31.
- 5.
Ibid., 25–33.
- 6.
Ibid., 31–2.
- 7.
Benacerraf, “Mathematical Truth,” 403.
- 8.
Cf. W. D. Hart, “Benacerraf’s Dilemma,” Crítica: Revista Hispanoamericana de Filosofía, 23, no. 68 (August 1991): 87–103.
- 9.
In Thinking About Mathematics, Shapiro supplies a helpful summary of prominent contemporary views, arranged along realist and anti-realist lines. “In recent literature on philosophy of mathematics, Gödel (1944, 1964), Penelope Maddy (1990), Michael Resnick (1997), and myself (Shapiro 1997) are thoroughgoing realists, holding both realism in ontology and realism in truth-value. … On the contemporary scene Hartry Field (1980), Michael Dummett (1973, 1977), and the traditional intuitionists L. E. J. Brouwer and Arend Heyting are thorough-going anti-realists, concerning both ontology and truth-value. Field holds that mathematical objects do not exist and that mathematical propositions have only vacuous truth values.” Shapiro, Thinking About Mathematics, 32. For our purposes in this chapter, we will focus upon Hartry Field’s anti-realism, as well as Quine-Putnam indispensability and Shapiro’s own structuralist realism.
- 10.
To be fair, Kant does not insist that space and time are not real, since he only concludes that we do not—cannot—know of their extra-mental reality, space and time being preconditions for sensory experience itself.
- 11.
Shapiro, Thinking About Mathematics, 29–30.
- 12.
One finds a parallel to this in Berkeley’s idealism. Berkeley seems genuinely puzzled as to why the inability to distinguish reality from fiction would befall a man who holds that the esse of unthinking things is their percipi. See his defense of the distinction in Treatise Concerning the Principles of Human Knowledge, §§28–33 and Three Dialogues Between Hylas and Philonous, Dialogue Three.
- 13.
A platonism, in fact. See Arthur W. Collins, “On the Question ‘Do Numbers Exist?’,” The Philosophical Quarterly 48 (Jan. 1998): 23–36, especially 25–28. We will turn to Quine-Putnam shortly.
- 14.
Real here means only that one’s ontological commitment to mathematical objects (like numbers) is at least as strong as one’s ontological commitment to the physical entities they measure. Such a commitment, notes Collins, is not incompatible with declaring the whole of mathematics—even the whole of physical science—a myth.
- 15.
Paul Benacerraf supplied a different response in “What Numbers Could Not Be,” a defense of his own structuralist account of mathematics. Some writers eventually raised the question whether perhaps even Aristotle is a mathematical fictionalist—though that is not a conclusion to which one would be naturally inclined. While fictionalism is decidedly anti-platonist, it suggests an exclusion of any sort of realism, Aristotle’s included. Mathematical fictionalists often point to Science Without Numbers as a point-of-origin for the contemporary debate. Field argues that mathematics is an unnecessary extension to our physical theories, that is, useful, but not required. Mathematical objects are therefore useful fictions. For more on Aristotle as realist or fictionalist, see Julia Annas, Aristotle’s Metaphysics Books Μ and Ν (Oxford: Clarendon Press, 1976), Penelope Maddy, Realism in Mathematics (Oxford: Oxford University Press 1990), and Edward Hussey, “Aristotle on Mathematical Objects,” in “Peri tôn Mathêmatôn,” ed. I. Meuller, Apeiron 24 (Dec. 1991): 105–34.
- 16.
Putnam’s case-in-point is Newton’s law of gravitation.
- 17.
Putnam, Philosophy of Logic (New York: Harper Torchbooks, 1971), 43. Emphasis mine.
- 18.
See Shapiro, Thinking About Mathematics, 229–37.
- 19.
Indeed, this applies to mathematical anti-realisms of whatever sort.
- 20.
Analytica Posteriora, I 1 71a11–16. διχῶς δ᾽ ἀναγκαῖον προγινῶσκειν· τὰ μὲν γάρ, ὅτι ἐστι, προϋπολαμβάνειν ἀναγκαῖον, τὰ δέ, τί τὸ λεγόμενόν ἐστι, ξυνιέναι δεῖ, τὰ δ᾽ ἄμφω, οἷον ὅτι μὲν ἅπαν ἢ φῆσαι ἢ ἀποφῆσαι ἀληθές, ὅτι ἐστι, τὸ δὲ τρίγωνον, ὅτι τοδὶ σημαίνει, τὴν δὲ μονάδα ἄμφω, καὶ τί σημαίνει καὶ ὅτι ἔστιν.
- 21.
This is unqualifiedly so for Putnam. For his part, Quine distinguishes between weak and strong indispensability.
- 22.
Putnam, Philosophy of Logic, 347. For more on Putnam, Quine, Maddy, and indispensability, see L. Decock, “Quine’s Weak and Strong Indispensability Argument,” Journal for General Philosophy of Science / Zeitschrift Für Allgemeine Wissenschaftstheorie, 33, no. 2: 231–50.
- 23.
Again, science is decidedly a posteriori, whereas logic and mathematics are, by contrast, necessary and a priori. In our view, that practicing scientists tend to mathematical realism is due to arguments like the indispensability argument.
- 24.
There are several dialogues we can point to in support of this claim. Cf. the Cratylus and its discussion of the changeability of natural things, the Timaeus’ division of forms into eternal and passing, and—perhaps most obviously—the analogies of the sun, divided line, and cave in the Republic.
- 25.
Summa Theologiae, Ia 84 1 corpus. “Videtur autem in hoc Plato deviasse a veritate, quia, cum aestimaret omnem cognitionem per modum alicuius similitudinis esse, credidit quod forma cogniti ex necessitate sit in cognoscente eo modo quo est in cognito. Consideravit autem quod forma rei intellectae est in intellectu universaliter et immaterialiter et immobiliter, quod ex ipsa operatione intellectus apparet, qui intelligit universaliter et per modum necessitatis cuiusdam; modus enim actionis est secundum modum formae agentis. Et ideo existimavit quod oporteret res intellectas hoc modo in seipsis subsistere, scilicet immaterialiter et immobiliter.
Hoc autem necessarium non est. Quia etiam in ipsis sensibilibus videmus quod forma alio modo est in uno sensibilium quam in altero, puta cum in uno est albedo intensior, in alio remissior, et in uno est albedo cum dulcedine, in alio sine dulcedine. Et per hunc etiam modum forma sensibilis alio modo est in re quae est extra animam, et alio modo in sensu, qui suscipit formas sensibilium absque materia, sicut colorem auri sine auro. Et similiter intellectus species, corporum, quae sunt materiales et mobiles, recipit immaterialiter et immobiliter, secundum modum suum, nam receptum est in recipiente per modum recipientis. Dicendum est ergo quod anima per intellectum cognoscit corpora cognitione immateriali, universali et necessaria.”
- 26.
Recall among Aristotle’s conditions for episteme that what we know cannot be otherwise.
- 27.
The question of access, first raised by Aristotle regarding Plato’s forms in general, remains one of the most vexing difficulties faced by contemporary mathematical platonists.
- 28.
Shapiro, Thinking About Mathematics, 32.
- 29.
Shapiro notes that other structuralists, Benacerraf and Hellman, “…do not presuppose the existence of mathematical objects.” Ibid., 257.
- 30.
Ibid., 258. Also 261: “…it is nonsense to contemplate numbers independent of the structure of which they are part.”
- 31.
We mentioned handedness from topology, earlier.
- 32.
Ibid., 262–263.
- 33.
Ibid., 259.
- 34.
Ibid., 261. Beyond these problems, Shapiro turns to the access question immediately following his consideration of ante- and in-rem structuralism.
- 35.
Ibid., 269. Shapiro distinguishes his view as being less relativistic than that of Michael Resnick—with whom he nevertheless has much in common. Resnick apparently sees the position-structure distinction along the lines of a Quinean web of belief. Cf. ibid., 267.
- 36.
Ibid., 269–70.
- 37.
For more on the uniqueness of structures, see Geoffrey Hellman, “Structures as Sui Generis Universals” in The Oxford Handbook of the Philosophy of Mathematics and Logic, ed., Stewart Shapiro (Oxford: Oxford University Press, 2005), chapter 17, §3, 541–6.
- 38.
Cf. Aquinas’ observation, above, regarding the platonists’ according to reality characteristics belonging solely to the intellect.
- 39.
Shapiro’s consideration of in rem structuralism is quite brief and framed mostly in terms of set theory. Given the requirement of what he calls its background ontology (the rem of the in rem structures), his chief difficulty seems to be its size. “…an eliminative structuralist account of arithmetic requires an infinite ontology. Similarly, an eliminative structuralist account of real analysis and Euclidean geometry requires a background ontology whose cardinality is at least that of the continuum. An eliminative account of set theory requires even more objects. Otherwise, the fields are vacuous.” Shapiro, Thinking About Mathematics, 272. Shapiro does consider—but ultimately rejects—a modal option.
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Rioux, J.W. (2023). Mathematical Realism and Anti-Realism. In: Thomas Aquinas’ Mathematical Realism. Palgrave Macmillan, Cham. https://doi.org/10.1007/978-3-031-33128-2_12
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