Abstract
In this paper, I show that mathematicians can successfully engage in metaphysical debates by mathematical means. I present the contemporary work of Hugh Woodin and Peter Koellner. Woodin has argued for axiom-candidates which could, when added to our current set-theoretic axiom system, resolve the issue that some fundamental questions of set theory are formally unsolvable. The proposed method to choose between these axioms is to rely on future results in formal set theory. Koellner connects this to a contemporary metaphysical debate on the ontological nature of sets. I argue that Koellner connects mathematics to the philosophical debate in such a way that mathematicians can obtain a new philosophical argument by doing more mathematics. This story reveals an active connectedness between mathematics and philosophy.
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- 1.
Here are three examples of philosophers who have argued for the above point. Penelope Maddy analyses reasoning in set theory and discusses two heuristic principles in detail; (Maddy 1997). Dirk Schlimm tells us about the creative potential of axioms in Schlimm (2009). Kenneth Manders discusses diagram-based reasoning in Manders (2007).
- 2.
For example, Kollner holds that ‘the mathematical systems that arise in practice can be arranged in a well-founded hierarchy’ (Koellner 2014), which points towards some formalistic conception of mathematics.
- 3.
Notice that if ZFC were inconsistent, then every set-theoretical statement could be proven from ZFC and hence there would be no dilemma. Therefore Woodin seems to assume the consistency of ZFC here. I will do the same throughout this paper.
- 4.
Woodin calls this view the ‘set theorist’s view’; (Woodin 2009a, b). I prefer the term ‘non-pluralism’ because is better captures the idea behind the view. This term stems from Koellner, who defines it in terms of belief in the existence of theoretical solutions to the formally unsolvable questions of set theory; (Koellner 2014). He has an argument for ontological commitment attached to this view which I sidestep through my definition. The resulting philosophical differences are minor and play no role in the argument of this paper.
- 5.
‘Regular’ is a technical term, meaning that for every λ < κ there is no co-final function from λ to κ, i.e. there is no f: λ → κ such that ran(f) is unbounded in κ.
- 6.
For the real line, the measure problem is the question whether there is a measure on the reals, i.e. whether there is a function m from all bounded sets of reals to the non-negative reals such that m is translation invariant, countably additive and not identical to zero. Notice that in ZFC the Lebesgue measure does not solve the problem due to the Vitali set. It was eventually solved by Ulam; see Kanamori (2009, pp. 22–27) for a historical as well as technical discussion.
- 7.
Recall that Banach and Tarski showed that a three-dimensional sphere can be partitioned in such a way that, through rotations and translations (i.e. without changing the size of the pieces), two spheres can be obtained which are identical with the first. This paradoxical partition is a direct implication of the Axiom of Choice. By assuming certain large cardinals (namely infinitely many Woodin cardinals) one can ensure that the Axiom of Choice is used in a strong sense for these partitions (namely: the partition must be complicated, no projective sets suffice). See Woodin (2001) for a further discussion.
- 8.
For connoisseurs: the theory T is ZFC + SBH, whereby SBH denotes the stationary basis hypothesis.
- 9.
It should be noted that Woodin has a separate argument against a certain form of pluralism; (Woodin 2009b, pp. 16–20). Nonetheless, other forms of pluralism raise the above-mentioned issue.
- 10.
- 11.
From now on I will use ‘all large cardinals’ to mean ‘all large cardinal axioms consistent with ZFC’.
- 12.
For a presentation of one of the axioms for V = Ult − L and possibilities for its generalisation, see Woodin (2010a, p. 17).
- 13.
L(ℝ) is constructed just like L, but rather than starting the construction with ø one starts with ℝ instead. Note that A ∈ L[A] but, in general, A ∉ L[A], which vividly shows that, in general, L(A) ≠ L[A].
- 14.
The Axiom of Determinacy states that every subset of the reals is determined.
- 15.
Recall here the von Neuman hierarchy as given above.
- 16.
An elementary embedding between two models is a truth-preserving function between these models. For an elementary embedding j the critical point of j is the smallest ordinal α such that j(α) > α.
- 17.
See e.g. his Platonism and Anti-Platonism in Mathematics (1998).
- 18.
See e.g. his The Indispensability of Mathematics (2001).
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Rittberg, C.J. (2016). Mathematical Pull. In: Larvor, B. (eds) Mathematical Cultures. Trends in the History of Science. Birkhäuser, Cham. https://doi.org/10.1007/978-3-319-28582-5_17
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