Abstract
In this paper I will review the developments in the foundations of mathematics in the last 150 years in such a way as to show that they have delivered something of a rather different kind: mathematical pluralism.
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Notes
- 1.
A good general reference for the standard material is Hatcher (1982).
- 2.
The Analyst, or a Discourse Addressed to an Infidel Mathematician (1734), §XXXV.
- 3.
For the material in this section, see Priest (1998).
- 4.
For the material in this section, see Zalta (2016).
- 5.
Earlier versions of type theory also required a somewhat problematic axiom called the Axiom of Reducibility. Subsequent simplifications of type theory showed how to avoid this.
- 6.
- 7.
About 20 years later, in the work of von Neumann and Zermelo, a model of sorts was found: the cumulative hierarchy. This did provide more coherence for the axioms, but it did nothing to save logicism. On the contrary, it appeared to give set theory a distinctive non-logical subject.
- 8.
For the material in this section, see Hallett (2013).
- 9.
Though one might also simply consider simply the constructive part of classical mathematics. See Bridges (2013).
- 10.
Further on all these things, see Iemhoff (2013).
- 11.
If the system is not a conservative extension it proves the negation of some true Δ0 sentence, and so the system is inconsistent. Conversely, if it is inconsistent, since it can prove everything, it is not a conservative extension.
- 12.
In these cases, the ideal elements are not statements, but objects. Hilbert discovered that quantifiers could be eliminated by the use of his ε-symbol. Thus, ∃xA(x) is equivalent to A(εxA(x)). One might—though I don’t think Hilbert ever suggested this—take ε-terms themselves to signify ideal objects. In this way non- Δ0 statements might be thought of as statements of Δ0 form, but which concern these ideal objects (as well as, possibly, real ones).
- 13.
For the material in this section, see Zach (2013).
- 14.
For the material in this section, see Marquis (2014).
- 15.
- 16.
On inconsistent mathematics in general, see Mortensen (2017).
- 17.
See Brady (1989).
- 18.
With naive comprehension we can define a set, c, such that x ∈ c ↔ (x ∈ x →⊥). Contraction and modus ponens then quickly deliver a proof of ⊥.
- 19.
- 20.
Perhaps surprisingly, Brady’s proof shows this strong form of comprehension to be non-trivial.
- 21.
See Priest (2006, ch. 17).
- 22.
Additionally, one would expect that the schema \(Pr(\left \langle A\right \rangle )\supset A\) would be provable in a theory of arithmetic in which Pr(x) really did represent provability. In a consistent theory, it is not, as Löb’s Theorem shows. However, the schema is provable in the above theories.
- 23.
A third foundational issue opened up by paraconsistency concerns category theory. Given that one can operate in a set theory with a universal set, it is possible to have a category of all sets, all groups, etc., where ‘all’ means all. The implications of this for the relationship between set theory and category theory are yet to be investigated.
- 24.
For a further account of some of these enterprises, see Dummett (2000, chs. 2, 3).
- 25.
On which, see Bell (2008).
- 26.
One answer to the conundrum is provided by non-standard analysis, an account of infinitesimals developed in the 1960s by Robinson. This deploys non-standard (classical) models of the theory of real numbers.
- 27.
Microcancellation follows. Take f(x) to be xa. Then, taking x to be 0, Microaffineness implies that there is a unique r such that, for all i, ai = ri. So if ai = bi for all i, a = r = b.
- 28.
See Mortensen (1995).
- 29.
Further on inconsistent boundaries, see Cotnoir and Weber (2015).
- 30.
For more on the following, see Mortensen (2010).
- 31.
And one can set things up in such a way that this does not imply that 90 = 0.
- 32.
It seems to me that given any formal logic there could, at least in principle, be interesting mathematical theories based on this. However, intuitionistic logic and paraconsistent logic (and perhaps fuzzy logic; see Mordeson and Nair 2001) are the only logics for which this has so far really been shown.
- 33.
In fact, the matter is arguably the case even in classical set theory. We can investigate set theory in which the Axiom of Choice holds, and set theory in which the Axiom of Determinacy, which contradicts it, holds. In that case, however, a monist might claim that we are simply doing model theory, and investigating what holds in models of certain set-theoretic axiom systems. One might consider a similar claim in the cases mentioned in the text: we are just doing model-theory using classical logic, albeit of models of non-classical logics. But this suggestion seems lame. First, intuitionist real number theory and paraconsistent set theory are not done in this way. So the suggestion gets the mathematical phenomenology all wrong. Secondly, the insistence that the model theory be classical seems dogmatic. Investigations could proceed with intuitionist model theory, which would give quite different results. (Note that intuitionist model theory is well established, but there is as yet no such thing as paraconsistent model theory.) Thirdly, it is entirely unclear how to pursue this strategy in the case of category theory, simply because the foundational problem with category theory was precisely that its ambit appears to outstrip the classical models.
- 34.
Some discussion of it can be found in Priest (2013) and Shapiro (2014). Shapiro takes mathematical pluralism to entail logical pluralism. I am not inclined to follow him down that path. Given a mathematical structure based on a logic, L, reasoning in accord with L preserves truth-in-that-structures. This may not be truth simpliciterpreservation.
- 35.
There are, as far as I can see, only two strategies for maintaining mathematical monism. One is the production of some kind of ur-mathematics to which all the kinds of mathematics we have met can be reduced. Maybe this could be some kind of foundational project for the twenty-first century; but nothing like this is even remotely on the horizon. The other strategy is simply to deny that the non-favoured kinds of mathematics really are mathematics. Now whether these theories are as deep, elegant, applicable, or whatever, as the favoured mathematics, might certainly be an issue. But it seems to me that denying that they are mathematics is the equivalent of the proverbial ostrich burying its head in the sand: these theories have clear mathematical interest.
- 36.
Many thanks go to Hartry Field, Arnie Koslow, and Zach Weber for very helpful comments on an earlier draft of this essay.
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Priest, G. (2019). From the Foundations of Mathematics to Mathematical Pluralism. In: Centrone, S., Kant, D., Sarikaya, D. (eds) Reflections on the Foundations of Mathematics. Synthese Library, vol 407. Springer, Cham. https://doi.org/10.1007/978-3-030-15655-8_16
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