Abstract
Philosophers of science since Nagel have been interested in the links between intertheoretic reduction and explanation, understanding and other forms of epistemic progress. Although intertheoretic reduction is widely agreed to occur in pure mathematics as well as empirical science, the relationship between reduction and explanation in the mathematical setting has rarely been investigated in a similarly serious way. This paper examines an important and well-known case: the reduction of arithmetic to set theory. I claim that the reduction is unexplanatory. In defense of this claim, I offer some evidence from mathematical practice, and I respond to contrary suggestions due to Steinhart, Maddy, Kitcher and Quine. I then show how, even if set-theoretic reductions are generally not explanatory, set theory can nevertheless serve as a legitimate and successful foundation for mathematics. Finally, some implications of my thesis for philosophy of mathematics and philosophy of science are discussed. In particular, I suggest that some reductions in mathematics are probably explanatory, and I propose that differing standards of theory acceptance might account for the apparent lack of unexplanatory reductions in the empirical sciences.
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Notes
Rantala (1992), p. 47.
See Nagel (1961).
Nagel (1961), p. 338.
See van Riel (2011) for a discussion of Nagel’s deductivism and its influence on his account of reduction.
Whether these cases count as bona fide intertheoretic reductions has been the subject of debate. I don’t want or need to take a stand on this issue here—letting go of some (or even many) individual examples is no problem for my purposes, as long as there are some genuine instances of reduction that look something like the ones philosophers have discussed.
For the sake of variety or readability, I’ll sometimes use expressions like “the reduction of numbers to sets” below. This is just loose talk. Unless the context indicates otherwise, such expressions always mean the same as “the reduction of arithmetic to set theory”—I don’t mean to invoke some alternative, ontological notion of reduction by speaking of objects instead of theories.
Potter (2004), p. 65.
Dipert (1982), pp. 366, 367. Emphasis in original.
The case of functions is more complicated and controversial than that of the ordered pair. Identifying a function with its set-theoretic graph, for instance, allows one to prove nontrivial theorems that weren’t available to earlier mathematicians operating with less clear or rigorous conceptions. (E.g., the theorem that there exist uncomputable functions.) The thesis that the reduction of functions is unexplanatory is probably most plausible when restricted to some class of “classical” functions, for instance differentiable or continuous functions of a real or complex variable.
Moschovakis (2006), p. 40. Italics in original.
This isn’t to suggest that the correct understanding of the concept in question need have been easy to determine. The modern notion of function, for instance, evolved slowly and rather tortuously from its roots in seventeenth century geometry and analysis to its current general form. For a brief overview of this history, see Kleiner (1989).
Anything except, perhaps, certain kinds of possible explanations that are too trivial, insubstantial, and pro forma to be worth caring about. For instance, if one subscribes to the view that facts of the form \(\exists xFx\) are explained by their instances, then one might think that the reduction of arithmetic to set theory explains why there exists a theory to which arithmetic is reducible. Maybe so. But my interest is in explanations that make a substantive contribution to mathematical understanding, and such cases clearly don’t fit the bill.
As philosophers have taken pains to show over the past couple decades, this type of explanation is encountered in pure mathematics no less than in the empirical sciences; working mathematicians often ask why theorems are true, and they often go to lengths in search of answers. (See the references in footnote 2 above.) To take a random recent example, Tao (2015) proposes a large-scale collaborative effort to find an explanation for certain surprising polynomial identities.
Some philosophers may want to deny that this second phenomenon is a genuine kind of explanation. If they were right, then my main thesis would be that much easier to defend, since there would then be fewer ways for the reduction of arithmetic to set theory to be prospectively explanatory.
Dauben (1979), pp. 58, 59.
Thanks to both Kenny Easwaran and an anonymous referee for Synthese for suggesting this example.
An anonymous referee suggests that results like Goodstein’s theorem—which are unprovable in first-order Peano arithmetic, but provable in the second-order setting using set-theoretic methods—might count as set-theoretic explanations of arithmetical facts, and hence might seem problematic for my view. I’m inclined to group this sort of case, together with the others mentioned in this paragraph, under the heading of “ways of applying set theory to arithmetic that may be explanatory, but which don’t depend on viewing numbers as sets”. So I don’t find such cases worrying.
To see why, it’s helpful to briefly describe the set-theoretic proof of Goodstein’s theorem. One starts by considering an arbitrary Goodstein sequence \(G\left( m\right) \), which is a certain sequence of natural numbers. One wants to show that \(G\left( m\right) \) eventually terminates, i.e. that it takes the value 0 at some point. (Goodstein’s theorem is the statement that all Goodstein sequences terminate.) To show this, one constructs a sequence of ordinal numbers \(O\left( m\right) \) with the properties that (1) \(G\left( m\right) \) can be shown to terminate if \(O\left( m\right) \) terminates, and (2) \(O\left( m\right) \) does in fact terminate. This is a sparse sketch of the proof, of course, but hopefully it’s clear from the sketch that the proof doesn’t exploit the set-theoretic representation of the natural numbers in any way. One could view the numbers as any sort of object whatsoever and the proof would still go through. In any case, I think it’s far from clear that the set-theoretic proof of Goodstein’s theorem is explanatory in the first place. The fact that a certain sequence of ordinals terminates, it seems to me, surely isn’t the reason why the associated Goodstein sequence terminates. The behavior of the two sequences is correlated, but what happens with the sequence \(O\left( m\right) \) can hardly be said to “ground” or “determine” what happens with the sequence \(G\left( m\right) \). So I’m doubtful that we’re even dealing with a case of mathematical explanation here.
The 1921 Kuratowski definition of the ordered pair was preceded by attempts in 1914 by Wiener and Hausdorff, who identified \(\left( a,b\right) \) with \(\left\{ \left\{ \left\{ a\right\} ,\emptyset \right\} ,\left\{ \left\{ b\right\} \right\} \right\} \) and \(\left\{ \left\{ a,1\right\} ,\left\{ b,2\right\} \right\} \), respectively. (The 1 and 2 in Hausdorff’s definition are arbitrarily chosen objects distinct from a, b, and each other.)
As I pointed out before, care should be taken not to confuse this point with the claim that set theory in general has nothing of explanatory value to contribute to number theory. I take the latter claim to be false, or at least highly dubious.
See Klein (2009) for an extended (and convincing) defense of this claim.
Balaguer (1998), p. 64.
Of course, Benacerraf’s argument has generated a great deal of discussion over the years, and some post-Benacerrafian philosophers have continued to hold views that ascribe some special metaphysical status to sets vis-à-vis the natural numbers. One might think that a proper examination of the reductionism issue would include some mention of these views. In fact, though, the views in question—or at least the ones I’m aware of—uniformly concede Benacerraf’s point that numbers can’t be uniquely identified with any particular sets in a principled way. Hence they’re not directly relevant to the line of thought I take up here. Some noteworthy examples of what I have in mind are the views of Penelope Maddy, W.V. Quine and Nicholas White.
Maddy’s early work (e.g. Maddy 1990) argues that numbers are (equinumerosity) properties of sets. This is indeed reductionism of a certain sort. But Maddy explicitly says that, for Benacerrafian reasons, there’s no hope of identifying numbers with sets themselves. (Whether numbers can or should be identified with properties of sets, and whether such an identification would have explanatory value, is an interesting question. But it’s not quite the question this paper is trying to answer.)
Quine (e.g. Quine 1960) held a view that can be described as claiming that “numbers are sets”, which sounds anti-Benacerrafian. But his version of reductionism basically amounts to the thesis that it’s convenient to identify numbers with sets, together with a pragmatic approach to ontology. On Quine’s view, we’re free to make any identification of numbers with sets that serves our purposes; if there are several equally handy choices, then each identification counts as equally correct, as long as we stick with the one we’ve chosen. So there’s no notion here of getting at a deep truth about what numbers “really were all along”. In particular, Quine agrees with Benacerraf that there’s nothing metaphysically special about either the Zermelo ordinals or the von Neumann ordinals. (Nevertheless, one might think that Quine’s view still imputes a type of explanatory value to set-theoretic reductions, even if it isn’t for the metaphysical reasons discussed in this section. See Sect. 2.6 below for more on this issue.)
A less well-known but notable reaction to Benacerraf is White (1974). White agrees with the anti-uniqueness part of Benacerraf’s argument, but from here he takes the unusual line that “the existence of multiple set-theoretic models of arithmetic should prompt us, not to say with Benacerraf that numbers cannot be sets, but rather to suggest that there are multiple full-blown series of natural numbers. Thus, for example, instead of there being only one three, there are after all many threes, and many thirty-sevens, and so on” (112). White’s view sounds at first like a version of reductionism, but it later becomes clear that this isn’t what he has in mind. His view is rather that objects of any kind count as numbers, insofar as they can be placed in an \(\mathbb {N}\)-like progression. So White doesn’t, after all, identify numbers with set-theoretic finite ordinals in particular (although these are among the objects that count as numbers for him). As he points out, the view is better described as a sort of Pythagoreanism than a type of set-theoretic reductionism.
Finally, it’s worth mentioning that another prominent claim from “What Numbers Could Not Be” is Benacerraf’s thesis that numbers aren’t objects of any sort at all. This is a claim with which many people have directly disagreed. I find Benacerraf’s argument for this view unconvincing myself, but its truth or falsity doesn’t directly bear on anything I say below, so there’s no need to canvass responses to it here.
Steinhart (2002), p. 355.
Steinhart (2002), p. 351.
Steinhart (2002), p. 351.
Steinhart (2002), p. 353.
Steinhart (2002), p. 352.
Maddy (1981), pp. 498, 499.
Moschovakis (2006), p. 33. The italics are Moschovakis’s, but the sentence in single quotes is displayed on a separate line in the original.
Kitcher (1978), p. 123.
An element of an integral domain is irreducible if it’s neither zero nor a unit, and if it isn’t expressible as a product of two non-units. (An integral domain is a commutative ring with no zero divisors, meaning that the product of nonzero elements is always nonzero. A unit is an element with a multiplicative inverse.)
Reck (2016).
Quine (1960), pp. 258–260.
Quine (1960), p. 259.
See Dipert (1982), fn. 23 for more on the early history of the ordered pair in modern logic.
Dipert (1982), pp. 367, 368. New Foundations is Quine’s system of set theory, which admits non-well-founded sets and has various other unusual features.
Hallett (1984), p. 300. Emphasis in original.
Quine (1960), p. 262.
Quine can’t be faulted for overlooking this example; Word and Object was published in 1960, while Robinson’s work on nonstandard analysis didn’t appear until later in the decade.
See Robinson (1974).
Quine (1960), p. 258.
Medvedev (1998), p. 664.
Bos (1974), p. 13.
Hallett (1984), pp. 300, 301.
Maddy’s early work, including the 1981 paper discussed above, espoused “set-theoretic realism”—a sort of reductionist view according to which numbers are properties of sets. Maddy has since abandoned set-theoretic realism and its metaphysical commitments, which explains why the recent papers mentioned here strike a quite different tone.
Maddy’s “elucidation” is apparently much the same sort of thing as Quine’s “explication”. For the record, I have some reservations about the extent to which Dedekind’s construction of the continuum is a good example of this phenomenon. One reason to be careful here is because, as Solomon Feferman has argued, there may in fact be several different, and equally worthwhile, conceptions of the continuum. So it’s not even obvious what would count as a successful explanation (or explication or elucidation) of continuity and its properties. Moreover, the device of Dedekind cuts does a poor job capturing at least one such conception, namely the one that’s operative in Euclidean geometry. As Feferman writes:
The main thing to be emphasized about the conception of the continuum as it appears in Euclidean geometry is that the general concept of set is not part of the basic picture, and that Dedekind style continuity considerations... are at odds with that picture. It does not make sense, for example, to think of deleting a point from a line, or to remove the end point of a line segment. Given two line segments L and \(L^{\prime }\), we can form a right triangle with legs \(L_{1}\) and \(L_{1}^{\prime }\) congruent to the given segments, respectively; but these share a vertex as a common point, each an end point. Thought of as a set, L is transformed into \(L^{\prime }\) by a rigid motion, and the same for \(L_{1}\) and \(L_{1}^{\prime }\). Thought of in that way, the vertex of the right triangle has displaced one of the end points, but which one? There are many similar thought experiments which dictate that lines, line segments and other figures in Euclidean geometry are not to be identified with their sets of points (Feferman 2009, pp. 174, 175).
Of course, “Dedekind’s continuum” was quite useful for the foundations of analysis and for other purposes. In any case, the issue is complex, and I can’t pretend to do it full justice here. (Thanks to the two anonymous referees from Synthese for prompting me to say more about this example.)
Maddy (2017), p. 305.
In light of the infinitesimal example from Sect. 2.6 and the other cases considered so far, one might wonder, as an anonymous referee did, whether the difference between explanatory and non-explanatory reductions is just the difference between cases where the relevant concepts weren’t or were originally “in good working order”. It seems to me that the example given below challenges this idea. We seem to have an explanatory reduction of algebraic varieties to schemes, but there was nothing unclear or contradictory about the notion of variety before Grothendieck came along. And I don’t think anyone would suggest that the notion of a scheme—which involves a lot of complex technical machinery with no immediately obvious geometric meaning—is simpler, clearer, more intuitive, or otherwise more epistemically or logically adequate than the notion of a variety. What’s going on in this case seems to be, rather, that schemes are richer in structure and that they carry more data than varieties, and this means that the scheme-theoretic viewpoint allows one to see further into the phenomena than the classical approach allows.
Vakil (2015), p. 12.
Perrin (2008), p. 213.
Eisenbud and Harris (2000), p. 1.
See Pincock (2015) for a defense of a dependence-style account.
Correia and Schnieder (2012) collects some recent work on the subject.
For an extended critique of this sort, see Woit (2006).
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Acknowledgements
Many thanks to Mahrad Almotahari, Kenny Easwaran, Colin Klein, Marc Lange, Daniel Sutherland, Lauren Woomer, and two anonymous referees for conversations and comments that helped improve the paper in countless ways. Thanks also to Jeremy Avigad, Alan Baker and the other participants in the 2015 Mathematical Aims Beyond Justification workshop in Brussels, where I presented an early version of some of this material.
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D’Alessandro, W. Arithmetic, set theory, reduction and explanation. Synthese 195, 5059–5089 (2018). https://doi.org/10.1007/s11229-017-1450-8
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DOI: https://doi.org/10.1007/s11229-017-1450-8