David Lewis's Philosophy of Maths (Preview of my SSHAP talk)

Introduction

Around 18 months ago my supervisor, Georg Schiemer, and I were catching up about the state of my work whilst taking a short walk through Vienna. Towards the end of this walk, Georg asked me if I was planning on submitting to a conference he was organising: the annual meeting of the Society for the Study of the History of Analytic Philosophy (SSHAP). I said about it but, now that he mentioned it, I did have a short little paper I was meaning to write on the history of Mathematical Formalism, why it fell out of fashion and why I thought David Lewis would have made a good Formalist. Georg liked the idea and so I wrote it. The paper was only ever supposed to be a bit of fun; more catharsis than serious scholarship as I think most of the serious points regarding Gödel's Incompleteness Theorems and Formalism were already well known by philosophers of maths. Alan Weir, for instance, has clarified most of this in several places. Nevertheless, the David Lewis connections were (to my knowledge) relatively novel, the paper only took a day or so to write and was rather fun to do so. The paper was simply entitled "why Formalism died too early and why Lewis should have brought it back". The conclusion is evident from the title.

To my surprise, I've encountered some degree of success, with my paper being accepted into the SSHAP conference (which was delayed by Covid but is taking place next week) and also into the Tilburg History of Analytic Philosophy conference. I suspect this was in no small part because the paper is quite light and fun! The Tilburg conference did something I rather like and had dedicated respondents. Mine was Frederique Janssen-Lauret, who was, at the time, a Post Doc on Helen Beebee's and Fraser MacBride's project "The Age of Metaphysical Revolution: David Lewis and his place in the history of analytic philosophy". I believe she has since got a position as a lecturer at Manchester. Alan Weir was also present for my talk.

Janssen-Lauret gave some helpful comments, and in the months since the discussion I've become convinced that (1) my initial argument was far too naive but that (2) my instinct was right that David Lewis's philosophy of maths is somewhat clunky, both as a view in its own right but also with respect to his wider views.

In preparation for the SSHAP conference, I've been reading a couple of papers suggested to me by Janssen-Lauret, by herself and MacBride on David Lewis and his philosophical views. The focus of their papers is more general and not specifically on the philosophy of maths, though do discuss the topic. What I'm going to try and do here is pull the parts of those papers relevant to the philosophy of maths and explain how Lewis's views on mathematics are, to be frank, somewhat of a mess.

I've avoided phrasing this post as a review of the Janssen-Lauret and MacBride papers simply because I have very little to add. I'm an uninspiring historian at best and have little to contribute to Lewis scholarship outside the philosophy of maths. Nevertheless, my one-sentence review is that the papers are good, they prompted me to change (or at least nuance) the way I think about Lewis and I think people should go and read them.

This post, then, is a sort of preview of parts of my talk next week at the SSHAP conference. Information about the conference can be found here: https://sshapvienna2021.univie.ac.at/ and my talk is at 10 to 4 on Wednesday in Room 5 (the Philosophy of Maths room).

What is David Lewis trying to do?

Mark Johnston described David Lewis as "the greatest systematic philosopher since Leibniz". Whilst I'm a little wary of superlatives like this so close in time to a philosopher's prominence (think of Wittgenstein, for instance, and how poorly proclamations of his greatness have stood the test of time), it's not hard to see what Johnston is getting it. Lewis was certainly an exceptionally good system builder. The Lewisian system, so says this analysis, is the combination of his Humean ontology, his unrestricted mereology and his possible worlds realism.

To jargon-bust: Lewis claims that what our world is made up of is a "mosaic" of properties (the "Humean mosaic") spread out over a 4D-spacetime. All objects are simply clusters of these properties. When one says "there is an electron at place X" this is true iff there is a particular distribution of charge and mass properties sitting at X in an electron-like manner. But, crucially, our world is not the only one. For every possible distribution of properties over the mosaic, there is some other world that really exists with that distribution. These worlds exist in exactly the same way as the real world. Our world is not special on Lewis's view. Being actual is just like being "here", it's merely an indexical. Most notably in his book "Plurality of Worlds" but also in his career more generally, Lewis showed that if you grant him this starting point he can provide answers to a great many (perhaps all) metaphysical questions.

As an undergraduate, I first encountered Lewis whilst thinking about trans-world identity. When I asked David Efird, one of my lecturers and eventual dissertation supervisor, why anyone would believe what Lewis believed about possible worlds he replied with an idea that I've come to call the Lewisian Gambit. David said something along the lines of "well, what Lewis says is if you give me this big concession at the start of the debate I'll be able to explain everything you want to explain and end up better in the long run". One can clearly see where this idea comes from if one reads the "Philosophers Paradise" introduction to Plurality. (Though several friends and colleagues of mine have, only half flippantly, suggested that Lewis is mocking metaphysicians when he does this: offering them everything they ever wanted whilst, true to the style of Greek tragedy, making it entirely unpalatable).

So in summary of the views of the systematic Lewis: there exists an infinity of Humean-mosaic worlds, at least corresponding to every arrangement of the mosaic and arbitrary construction over these mosaics is allowed. From this, Lewis will show how everything is reducible to the mosaic. Whilst the position seems initially troubling, given a difficulty justifying the existence of all these worlds, its eventual explanatory value and relative (type) parsimony will come to more than pay for this initial price. The Lewisian Gambit achieves sufficient compensation.

This is a nice story but a historically problematic one as Janssen-Lauret and MacBride point out in their paper David Lewis's Place in the history of late analytic philosophy: his conservative and liberal methodology. (I don't know why we're referring to the 80s-00s as "late" analytic philosophy. I don't see any reason to think that AP will finish soon!).

David Lewis didn't think of himself as a systematic philosopher, quite the opposite. At numerous points in his career in both letters and published works, Lewis disavowed this title. I will leave the detailed exposition of this to Janssen-Lauret and MacBride's paper but will pull out what seem to me to be the important points.

Lewis endorsed a claim by Priest that "It would be wrong to think of Lewis as a systematic philosopher... Lewis works like this: he gets interested in puzzles and problems; he likes to solve them; he does so by applying his technical expertise, his great ingenuity, his prowess in the thrust, parry, and counter-thrust of philosophical debate".

Lewis also talked of his desire to not have his entire collection of philosophical views stand or fall together. For instance, one should not believe his compatibilism about free will iff one believes his modal realism (or what-have-you). There might be some thematic similarities between his views. After all, he approached all these questions with similar methods, tools and preferences. However, there is not a unified system.

Some reconciliation can be done here between the systematic reading of Lewis and Lewis's own statements about the independence of his views. The answer would be to think of his views in a hierarchy. Up the hierarchy are his views on modality, language and the Humean mosaic (and perhaps others) and lower down are more separate views about exactly how certain reductions to the possible worlds mosaics are supposed to be done. On this reading, most of Lewis's views are independent of one another, but there is a common core that one must believe if one is to believe much by Lewis at all. The gambit is still in place, though Lewis leaves room for later scholars to disagree about exactly how it's best to recover compensation.

But there are deeper reasons why this reading of Lewis's system doesn't make sense. Lewis is even more sceptical of the value of philosophical systematisation than Quine was. In his famous "going down the hallway" passage (in Mathematics is Megethology) Lewis espouses a view of the role of philosophy where, ultimately, philosophy must bow very quickly to challenges by more established bodies of knowledge such as science or mathematics. Elsewhere Lewis said the following: "describe himself as a conservative in correspondence: ‘I am philosophically conservative: I think philosophy cannot credibly challenge either the positive convictions of common sense or the established theses of the natural sciences and mathematics". This is close to the sort of sentiment one might see in methodological naturalists such as Quine or Maddy. Where Lewis seriously differs from both of these people is that he extends his reverence not just to mathematics and natural sciences but also to common sense. Call this version of Lewis, Lewis the (methodological) naturalist.

This ultimately leads Lewis to a position where, in Parts of Classes, he accepts the existence of abstract classes. Though, he potentially attempts to reduce these classes to the mosaic in both the appendix of Parts and his later paper "Mathematics is Megethology". His argument for this is that it's common mathematical knowledge that classes exist and that philosophy is not able to challenge common mathematical knowledge.

But, if Lewis does accept abstract classes, the Lewisian Gambit has failed. Not enough compensation has been reached for the sacrifice as Lewis must also commit to additional demanding entities. There's clearly some kind of tension within Lewis's views here. It's somewhat confusing why, given Lewis's methodological naturalist views, Lewis would want to be a possible worlds realist or a Humean. The systematic Lewis of Plurality and the naturalist Lewis of Parts are in tension with one another.

Janssen-Lauret and MacBride pull this tension out in a lot more detail in their paper. I'll direct you there for further thoughts. What suffices for this paper is to understand the two kinds of projects one might want to consider when evaluating Lewis's work.

Lewis's Philosophy of Maths

Lewis didn't write a great deal on the philosophy of maths and, in my opinion, it remains the largest area of modern metaphysics upon which he has had little to no influence. What he has written can be found in Parts of Classes, especially the appendix co-authored by John Burgess (a titan of PoM) and A. Hazen, and in a later paper "Mathematics is Megethology" (MiM), which hits most of the main beats of the appendix to Parts. Janssen-Lauret and MacBride provide a nice summary of Lewis's position(s) in their paper "W.V. Quine and David Lewis: Structural (Epistemological) Humility" along with how they connect to wider themes in the work of both Quine and Lewis.

Lewis's initial view in Parts (not the Appendix) is that reality is divided into individuals and classes. There are also fusions of individuals and classes that are neither individuals nor classes, but everything is equal to the fusion of all its class-parts with all of its individual-parts. Just as with individuals, classes stand in mereological relations to one another. The fusion of two classes a and b is the class that contains all and only the members of a and b (the pair class). Moreover, Lewis provides an argument that the parts of classes are all and only those classes that are subclasses. From this, Lewis argues that all one really needs for set theory is a function that takes anything (individual or class) to its singleton and common mereological principles. To do "pure" set theory one also needs the empty set, which is an individual, not a class, on Lewis's view.

Most instructive is how this recovers the axiom of power. Start with some class x. Sufficiently strong mereological fission principles entail that every subclass of x exists and is a class in its own right. The singleton principle entails that there is a singleton of each of these subclasses. Sufficiently strong fusion principles entail that there is a class that is the sum of all of these singletons whose members are all and only the subclasses of x. Clearly, this is the powerset of x. I won't cover each axiom but defer to Lewis's own work.

As a quick technical note, the singleton function needs to be restricted appropriately so as to avoid Russel's paradox. Not all classes, for instance, have a singleton.

A proper class is a class that does not have a singleton. A set is either the empty set or an improper class.

This introduces two ontological commitments above and beyond the mosaic: classes themselves and singleton functions. In the appendix to Parts and also in MiM Lewis then finds ways of trying to reduce this to more "nominalistically" acceptable grounds (though one has to be careful when Lewis uses the word "nominalist" as he uses it highly idiosyncratically. See the first few passes of "Holes" by Lewis & Lewis). The reduction of functions and the reduction of classes are independent of one another.

The reduction of functions takes two parts. First of all, Lewis commits to an algebraic-structuralism about the singleton function. He abandons a desire to find the singleton function in favour of saying that any function that has certain sorts of properties is a singleton function. When mathematicians make claims about set theory they are (or should be) really saying "for all singleton functions f...". I've written about algebraicism more in my review of Mary Leng's "an i for an i, a Truth for a Truth".

Using tools provided by Burgess and Hazen, Lewis then shows how one can construct functions using mereology over individuals plus second-order logic. If one has full higher-order logic, then one can even make do without the mereology. But to do this, there still needs to be the elements of the range of the function. In other words, Lewis still needs to say something about classes.

The reduction of classes is more complex and I will skip some of the technical details because they are not necessary for present purposes. What matters, in short, is that if one allows possible worlds to be sufficiently large in terms of their numbers of atoms (inaccessibly large, to be precise), then there's a way of representing set theory in the mosaic. Effectively, one uses an order on these atoms to (relatively) arbitrarily define some of these atoms as singleton classes of the others. This would then allow a reduction of set theory to something like the mosaic.

There are two ways that one could read what's going on here.

The first takes the Lewisian gambit very seriously and says that Lewis always had some kind of reduction in mind, it just took him a little longer and the help of Burgess and Hazen to find it. In Parts, Lewis offers the first step of his reduction: the reduction of all classes to singletons. In MoM and the appendix, we get the promised other half of the reduction.

A second way of reading this takes Lewis's conservatism more seriously. One might think that what Lewis is trying to do here is provide an option for those more nominalistically inclined to accept. If one is comfortable with classes, one can enjoy the latter half of MoM as an interesting theoretical exercise (as Quine did). If one is not, then this is a serious endeavour to keep mathematics whilst not compromising one's nominalism.

The difference is if one thinks that Lewis's view in Parts that retains an ontology of classes (independent of the mosaic) is ever a view that Lewis seriously had or if this was only ever a stepping stone. I'll make it easy and consider both Lewi's "full" view (from Parts) and his "reductive" view (MoM). There is also a possible hybrid view that has the reduction of functions but not the reduction of classes. Discussion of this will be omitted.

In summary:

Both views:

• Reject that the empty set is a class.
• Hold that there are (in the first-order sense) proper classes.
• Are algebraic about the singleton function (and hence membership more broadly).

The full view:

• Commits to an independent ontology of classes.

The reductive view:

• Commits to second-order logic.
• Commits to inaccessibly large possible worlds.

Does Lewis's Philosophy of Maths achieve Lewisian aims?

Two Lewisian projects are being considered here (systematic and naturalist) and two views of mathematics (full and reductive). It would be useful to see what the former say about the latter. There are therefore four points of comparison. I'll go through them systematically.

The systematic Lewis, as discussed above, should not be impressed by the full view. The purpose of the Lewisian gambit was to give a substantial ontological sacrifice at the start of a discussion, but then to demonstrate the explanatory power of this sacrifice and to reap sufficient compensation. If Lewis is forced to adopt classes to explain mathematical facts, then it's clear that the gambit has failed to gain sufficient compensation and the wider project has been a failure.

It appears, at first sight, as if the systematic Lewis should be enthusiastic about the reductive view. It's certainly true that he should be more enthusiastic about it than the full view, as it keeps the gambit intact. But I don't think this is the case and there are certainly better views out there for him to adopt. This is for two reasons.

First, in many places, Lewis takes care to work using first-order logic, in line with his mentor Quine's views on higher-order logics. In the reductive view, Lewis abandons this and uses second-order logic. It's somewhat unclear to me how important the commitment to first-order logics is for the Lewisian gambit. If, like me, one is convinced by the kinds of readings of higher-order quantifiers that don't treat higher-order entities as sets (as Quine thought it must), then using higher-order logics is unproblematic for the gambit. If one agrees with Quine that higher-order logics are set theory in disguise then one might think that adopting second-order logic involves some kind of ontological commitment that breaks the Lewisian gambit. So, modulo the assumption that Quine is wrong about higher-order logics, this is not a problem for the systematic Lewis.

Second, the commitment to inaccessibly large possible worlds is an issue. In effect, it makes the sacrifice in the gambit greater. This is not a fatal flaw of the view, but a reasonable pro tanto reason against it. If there is a view in the philosophy of maths that can avoid such a commitment, it's better for the systematic Lewisian than Lewis's own view. There are several such views. Let's assume that fictionalism is ruled out because Lewis wants to show, via reduction, that one can genuinely get the truth (not fictional truth) of certain mathematical claims. Lewis could still be a modal structuralist or a formalist.

Modal Structuralism (Hellman) is the view that mathematical statements can be understood in the following way. Take any set of axioms A and any theorem T that is provable from A. Let A* and T* be the result of substituting the predicates of A and T with second-order variables and all the singular terms with first-order variables (i.e. the algebraic re-interpretations of A and T). The theorem T is understood as stating that it's necessarily the case that for all the variables in A* If A* then T*. Informally this says that a theorem, such as Pythagoras theorem, says that it's necessarily the case that for any structure that models Euclidean geometry, the right-angled triangles in that structure will obey Pythagoras theorem. Modal structuralism commits to modality (as does Lewis) and second-order logic (as does the reductive view). Does it do better in terms of the size of possible worlds?

An objection made by Parsons to Modal Structuralism is that it requires possible worlds to be big enough to model very large theories such as ZFC. If ZFC requires big models and all the possible worlds are too small, then there is no model of ZFC and every sentence will trivially be a theorem of ZFC on the Modal Structuralist view, by false antecedent. If the smallest model of ZFC is inaccessible, then Modal Structuralism is no better off than Lewis's reductive view. But it's potentially possible that there are smaller models of ZFC if one uses the downward Löwenheim-Skolem theorem. DLS proves that for any theory that has a model of some uncountable cardinality, it also has a countable model. If there are DLS models of big theories such as ZFC, then Modal Structuralists can make do with those without committing to big worlds.

It's complicated as to if there are DLS models of big theories such as ZFC. ZFC does not entail a countable DLS model of ZFC (this would amount to a proof of its own consistency). Generally, ZFC only "sees" cardinals up to a certain size and so can't see a cardinal big enough to see that it has a model of some uncountable cardinality and hence that it has a DLS model. More powerful theories that extend ZFC with some uncountable cardinal do see that ZFC has a model of some cardinality and hence a DLS model, but they can't see of themselves that they have a model of any uncountable cardinality.

In other words, DLS needs to be proven from some model theory and is a common theorem of any good (first-order) model theory. Model theory takes place inside some sort of set theory. Whichever axioms one chooses for that model theory, if it is consistent, it won't be able to prove of itself that it has a model of some uncountable cardinality, and hence a DLS model.

I will leave the details for another day and what the options for the Modal Structuralist are. It's sufficient for present purposes to say that this is an option for the systematic Lewis that is potentially better and certainly no worse than his reductive view.

But there's also an option that both avoids higher-order logics and avoids demandingly sized possible worlds: mathematical formalism. Formalism is the view that what mathematics amounts to is working out what results follow from which axioms, nothing more. Facts about proof can be made sense of in an entirely first-order manner, with no need for higher-order logics. Statements about proofs are simply statements about possible sequences of symbols with certain properties. Because proofs are finite, one only needs one's possible worlds to cover all possible finite symbol sequences, which is much less than Lewis's reductive view and either either as much as or less than Modal Structuralism.

I'll defend the claim that David Lewis should have been a Formalist in my SSHAP talk.

Lewis's naturalist must aim to stick as close to common sense and common mathematical knowledge as possible. Presumably, this also extends to how mathematicians generally think about mathematics, what one could call mathematical common sense. There's an empirical question as to exactly what mathematicians do think and what the content of mathematical knowledge is.

In MiM, Lewis gives his famous "going down the hallway" quote in which he encourages those sceptical of classes to go to their nearest mathematicians and explain their views. Lewis is confident that these views will be rejected by the mathematicians in question, at which point Lewis expresses a great degree of certainty in mathematical knowledge than in philosophical conclusions.

It's not at all clear that Lewis is correct about what mathematicians think. In their first paper mentioned Janssen-Lauret and MacBride perform Lewis's experiment going to their nearest maths department and speaking to Jeffrey Paris, a mathematician at their university of Manchester. Paris explained that he's a Formalist. The example I use is Kenneth Kunen's relatively standard 2011 textbook on set theory. In the opening section Kunen has an interesting, if slightly dated, discussion of the philosophy of maths. In this section Kunen expresses support for a kind of practical Formalism. Philosophical Formalism typically is, though needn't be, accompanied by a rejection of Platonism. If this is strictly part of the Formalist claim or if it's just a sociological fact that Formalists tend not to be Platonists is a matter for discussion. What's relevant is that the view Kunen outlines is agnostic on the existence of abstract mathematical objects. It just states that the Formalist facts are enough to be getting on with maths and if there are actual entities, abstract or otherwise, that model some axioms, well that's just a bonus!

So on a basic level it's not clear that the "going down the hallway" argument has the force that Lewis takes it to. But, even if it does, his own view is going to come foul of a similar kind of objection.

On the general points, I hesitate to speculate on exactly what mathematicians might think but hazard the following guess. First, whilst there's nothing obviously objectionable from the perspective of mathematical practice about Lewis's claims about mereological relations between sets, a practical mathematician might wonder if Lewis has somewhat missed the point. These mereological facts are not relevant to the proving of any important theorems, they are just not the kinds of facts, even if true, that are of that much mathematical interest. They might explain, in some philosophical sense, why some axioms are true, but why is this explanation to be of interest to mathematics? 

Second, when told that, in fact, the empty set is not a class but an individual, one might expect protest. Insofar as mathematicians accept the existence of classes (set theory is generally done without them, with class-talk being a shorthand), classes are those things whose identity is determined by its membership. The empty set is the same sort of thing as any other set, which, on Lewis's view, is an improper class. The distinction between things that have members and things that do not might be philosophically important for Lewis, but this seems perfectly open to a "hallway" argument as this differs from common mathematical understanding.

Lastly on the general points, as mentioned above, standard set theory does not actually contain (proper) classes. ZFC does not entail the existence of any proper classes and whilst class-talk is proper this is always shorthand for something else, typically a particular itteratively defined property that is not contained at any level of the cumulative hierarchy. Kunen is very clear on this point in his book. Lewis's very central view that there are proper classes is in contention with common mathematical understanding.

These criticisms apply to both the full and reductive versions of Lewis's position. 

Additional criticism can be levied against his reductive view, specifically against his reduction of classes to certain mereological atoms. In short, no set theorist thinks that this is the kind of structure that they are talking about when the do set theory. Whilst, if this structure did exist, is would be a model of ZFC, it's so conceptually distant from mathematical practice and mathematical understanding that it would undoubtedly fail the "hallway" test.

Thus, it's unclear why the naturalist Lewis should find his own proposal particularly appealing.

Summary

So, overall, Lewis's philosophy of maths is unappealing by the light of both of his "projects", the systematic one and the naturalist one. To me it remains a decidedly odd spot in Lewis's work. Lewis is a philosopher that I generally admire for the elegance and simplicity of his solutions. Some of his books and papers remain some of my favourites and he was certainly a brilliant philosopher. It just puzzles me, then, that his philosophy of mathematics was so odd. There are a wide range of other views that he could have taken that would have better fit different conceptions of his overall project.

I recommend reading a lot of Lewis, but I'd generally recommend steering clear of "Mathematics is Megethology" as I don't think it gives any really worthwhile insights.

As advertised a few places in this review, I'll be giving a talk on Lewis, the history of Formalism and why Lewis should have been a Formalist at the SSHAP conference this week. Information can be found here: https://sshapvienna2021.univie.ac.at/

The Lewis works mentioned are:

• Parts of Classes, Blackwell, 1990
• "Mathematics is Megethology" is in Philosophia Mathematica, 1993, and can be found here: https://philpapers.org/rec/LEWMIM

The papers mentioned by Janssen-Lauret and MacBride can be found here:

• https://philpapers.org/rec/JANDLP-2
• https://philpapers.org/rec/DENQSA (this is an edited volume by Janssen-Lauret , the Janssen-Lauret and MacBride paper is inside)

I welcome comments or friendly criticism below, or you're welcome to email me at gareth.pearce@univie.ac.at

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Picture Credit: schmooster, CC0, via Wikimedia Commons