Mary Leng's "an i for an i, a Truth for a Truth"

Introduction

Mary Leng's recent paper "an i for an i, a Truth for a Truth" is, in my opinion, a strong one. We read this paper as part of our weekly philosophy of maths reading group at the University of Vienna. My thanks, in alphabetical order, to Julie Lauvsland, Benjamin Marschall and Simon Weisgerber for an interesting discussion on the paper. The paper was my suggestion and I was excited to read it both because of its relevance to my work and because I was fortunate enough to be briefly taught by Mary during by BA at York.

The paper, I'm glad to say, didn't disappoint and I would certainly recommend it.

The paper responds to an objection raised by both Shapiro and Burgess to "algebraic" interpretations of mathematical statements. The idea behind algebraic interpretations is not to take mathematical statements at face value, but to ameliorate them in some way that will potentially make them nominalistically acceptable. A nominalist, for those unaware, is someone opposed, in some sense, to abstract objects. In the context of the philosophy of maths, this amounts to either rejecting the existence of or rejecting the possibility of knowledge of some abstract mathematical realm.

On the face of it, mathematical statements seem to commit one to the existence of (presumably abstract) mathematical entities. Consider the statement "7 is prime". This is a simple object-predicate sentence. Typically, these are true iff there is some object referred to by "7" that has the property picked out by "is prime". The nominalist has three options: (1) reject the claim that there are true mathematical sentences (2) accept that there are mathematical objects, but explain how they're not abstract or (3) avoid taking mathematical statements at face value.

In this paper, Leng opts for option (3).

There are several common re-interpretations of mathematical statements corresponding to different nominalist views. I, being a Formalist, understand mathematical truths as just being statements about what follows from which axioms. Other options include prefixing mathematical statements with fictional clauses ("within the context of mathematical fiction M...") or Hellman's suggestion of taking mathematical terms not to specifically refer, but to be variable terms for any old entity that could play the part of that term. Similarly, predicates are replaced with predicate variables. Call this an algebraic reading of the statement.

A well-known objection to this kind of approach is raised by both Shapiro and Burgess. These re-interpretations are either: (1) a descriptive theory about how mathematicians actually think about what they're doing or (2) a prescriptive theory about how they should.

(1), they claim, is wrong. (2), they claim, is laughably immodest. Call this the face-value objection.

Leng's claim in this paper is that Shapiro's own view must also read some mathematical statements algebraicly order to handle reference in structures (such as the complex plane) with non-trivial automorphisms. This would then even the scorecard on this point between Shapiro's view and Leng's.

Leng, herself, is a fictionalist but the objection applies to any nominalist taking option (3) and Leng's response doesn't rely on her fictionalism per se. Consequently, her response is available to any nominalist.

Overall, I do think there's a way out for Shapiro and he can resist this objection, but I think it does plausibly give some ground to the nominalist.

There are then two wider questions that I address towards the end of the review. First, does Leng's conclusion amount to a defence of her view from Shapiro's and Burgess's objection or a damning of Shapiro's view to the same fate? In other words, perhaps this is just an argument for some view other than both Shapiro's and Leng's. I argue that there are views that avoid this, though they have other problems. Second, how does Leng's strategy compare to other responses to Shapiro's and Burgess's objection? I argue that Leng's response does enough on the direct comparison between her and Shapiro but that other strategies are needed in the context of the wider debate.

Leng's response to the face-value objection

As mentioned above, nominalists have three options when thinking about mathematical truths. (1) reject the claim that there are true mathematical sentences (2) accept that there are mathematical objects, but explain how they're not abstract or (3) avoid taking mathematical statements at face value. Leng takes option (3), as to most mathematical nominalists (myself included).

But, argues Shapiro and Burgess, not taking mathematical statements at face value is either (1) a false descriptive claim or (2) a laughably immodest normative one.

In this paper Leng aims to even the score between her and Shapiro by showing that Shapiro is also committed to reinterpreting mathematical claims in cases of non-trivial automorphisms.

Briefly, automorphisms are bijective functions from a structure to itself that preserve the truth of all (first order) facts about that structure. So f is an automorphism iff phi(x) iff phi(f(x)) for all formulae phi.

Every structure has at least one automorphism: the trivial automorphism that takes f(x)=x. But there are many structures with interesting non-trivial automorphisms. Famously, the complex plane has many non-trivial automorphisms arising from certain symmetries between i and -i (in fact, the study of automorphisms of the complex plane is a substantive subfield in mathematics).

This yields a problem for realists when explaining how reference to complex numbers is possible. In the "standard" Kripke-Marcus theory of reference definite descriptions (sentences of the form "The X") are used to pick out particular objects and fix a name to that object. But if non-trivial automorphisms exist over a structure, then there is no way to uniquely pick out objects that are not invariant over these automorphisms. The number i is one such problematic object .

Shapiro's response to this issue is to reject the need for the term "i" to pick out a particular i. Instead the term can just be a stand-in for any object that is able to play that role.

But this, so Leng argues, also runs into the face-value objection. Just as the normalist has to re-interpret mathematical claims generally, Shapiro has to do so in the context of complex numbers (and other structures allowing for non-trivial automorphisms). This evens the score between her and Shapiro.

So Leng's argument runs as follows:

P1: Nominalists (of the right sort) globally commit to re-interpreting mathematical statements.
P2: Shapiro is committed to re-interpreting in local contexts where there are non-trivial automorphisms
P3: There are such contexts
P4: locally re-interpreting is just as bad as globally re-interpreting.
C: Nominalists and Shapiro are on a par with regards to the face-value objection

P4 is an obvious point for Shapiro to push back. I think there's also a way of Shapiro attacking P2 by rejecting his old view on reference in non-rigid structures.

Rejecting P4: How do you weight violations of the face-value objection?

Shapiro is committed to far fewer instances of re-interpretation than Leng. Leng has to re-interpret every mathematical statement, whereas Shapiro only has to re-interpret some.

There's clearly a prima facie case for both sides of this premise. On one hand, Leng might say that it's the principle not the degree that matters. There's no real sense in bean-counting the number of re-interpreted sentences. There's a virtue to never re-interpreting and anything less than that is the same. Either way, reinterpretation involves infringing on mathematics for philosophical reasons and this, so says the face-value objection, isn't acceptable. But equally Shapiro might push the other way and say that infringements do come in degree and that his position has fewer violations.

If this is the dialectic, I see no real way to break the impasse. From a practical point of view, that benefits Leng. The Platonist is the one advancing the objection and if they want to provide a reason for Leng to drop her view, the burden is on them to provide an all-things-considered convincing case that Shapiro's position is better.

But I think Shapiro has a more nuanced response available. This might result in giving some ground to Leng and other nominalists but it would leave the debate open in an interesting way, rather than locked at an impasse.

One option for Shapiro is to reject that mathematicians always interpret their own propositions "at face value". Automorphisms of the complex plane (or of the Riemannian Sphere) are the subject of mathematical investigation. This is plausibly evidence that mathematicians don't have a fixed notion of the number i. Rather there are different numbers that play that role under different automorphisms. Mathematics might, in this local context, actively endorse the kind of re-interpretation Shapiro wishes to endorse.

However, this strategy doesn't endorse global re-interpretation, only re-interpretation in contexts where there are non-trivial automorphisms. Statements about the natural numbers, which lacks such a feature, should still be read at face value.

I think the nominalist still has a way out here, though it takes a slightly different line from that adopted by Leng. I'll pick this up in a later section.

Rejecting P2: A stronger theory of reference

P2 follows from the Kripke-Marcus theory of reference. But there is no reason for Shapiro to commit to that view. It has a number of known problems and was plausibly never really intended to be a final theory, more a step in the right direction.

A natural candidate would be something along the lines of Lewis's an Sider's reference magnetism view. As I understand the view, though I am not an expert on Philosophy of Language so will likely miss certain nuances, the claim is that there are certain more or less natural ways for identity to work or for predicates to be assigned to properties. Our languages fix, like magnetism, onto these more natural setups.

The view obviously comes with a great deal of baggage but, even as stated, it's unclear that it would solve the problem. The issue is that i and -i (or other comparable points in the complex plane) have nothing to distinguish them. There is nothing to make one prospective candidate the more natural choice.

Nevertheless the strategy of rejecting K-M and adopting a stronger theory of reference is a good one. It's just that reference magnetism is a poor choice for an alternative. What is needed is a theory of reference that allows for something like the axiom of choice.

Choice is an axiom of Set Theory that allows one to arbitrarily pick members of sets. If one has a set of equivalence classes, choice allows mathematicians to work with arbitrary members of these equivalence classes. But if this could be used when determining reference, then Shapiro's worries would be over. Shapiro could take the set of all possible referents of "i" and use choice to pick an arbitrary one to be the real i.

This amounts to a baptism of the name that looks like this: "Let i be an object that is a square root of -1. Which one? Let the universe decide at random". Crucially, this is not saying that "i" is a stand-in name for any arbitrary object that could act as i. It's saying that there really is a singular definite object picked out by i but that we have left it up to this choice-like principle which it is.

If baptisms like these are linguistically acceptable, then Shapiro has an easy way out of the problems raised by Leng.

I'll leave it open if this is possible or not. I'm prima facie skeptical.

Wider thoughts: Is this a defense of Leng or an attack on Shapiro?

One of my first thoughts on finishing this paper is if it should be read as a defense of Leng's view (and the views of other nominalists) or as an attack on Shapiro's. A plausible response to Leng's paper is to see it as a reason both not to be a nominalist (of the appropriate sort) and and a reason not to be a Shapiro-style structuralist. I think the paper's interesting either way and I don't think it was necessary for Leng to address this point. Her claim is only that she is on a par with Shapiro and that holds true if her paper is a reason to drop both of their views, not a defense of both from the face-value objection.

But, nevertheless, I suspect Leng would prefer to use this paper as a basis for a defense of her view from the face-value objection. There would need to be a stronger argue for that than the one in her paper.

If there is such an argument, I suspect it looks something like this:

P1: Any Platonist view that can solve the Benacerraf reference problem must read some mathematical statements algebraically in cases of non-trivial automorphisms.
P2: There are cases of non-trivial automorphisms.
P3: Local algebraicism is just as bad as global algebraicism.
P4: global algebraicism is better than failing to solve the Benacerraf reference problem
P5: Leng's view is algebraicism algebraic
C: Leng's view is as good as or better than all Platonist views, with respect to the face-value objection.

Naturally, both of the above responses still apply.

P1 is clearly the contentious claim. One way out of it for realists might be via Set Theoretic realism. V, the universe of sets, does not have a non-trivial automorphism. It is also possible within V to reconstruct all of mathematics and to arbitrarily but definitely specify specific objects to count as, for instance, i. If there really is an abstract universe V then reference to it is made easier by it's lack of non-trivial automorphisms. It can plausibly escape the reference problem whilst also allowing for definite reference in other areas of mathematics, provided those areas are understood as occurring within V.

As it happens, I think there is a slightly different version of the reference objection that applies to Set Theoretic realism. In short, there are lots of epsilon-like relations definable over V. It's unclear how one can pick out the privileged membership relation. But this is an argument for another post.

My point here is to set up how I see the debate. I think there's a very live strategy for someone to extend Leng's argument but I think there are prima facie viable responses on the part of realists.

Wider thoughts: Is this the best way for a nominalist to respond to the face-value objection?

Leng's response is, in some sense, a very amicable one. She grants the face-value objection a great many of its premises. Specifically, she grants (at least for the sake of argument) the claim that not taking mathematical statements at face value is either (1) a false descriptive claim or (2) a laughably immodest normative one.

This is certainly a boon of her response. She meets her interlocutors largely on their own ground and argues that her view is at least as good as a rival from that position.

However, as I've argued above, I think Shapiro does have options for responding. A decisive response on one's opponent's ground is better than a decisive response on one's own, but, given the above responses, it's not clear Leng's reply is decisive. There are many ways Shapiro might respond.

As such, it's worth considering less amicable responses to supplement Leng's argument.

I don't think it's obviously true either that: (1) re-interpretation taken as a descriptive theory is wrong or (2) re-interpretation taken as a prescriptive theory is laughably immodest.

On the former, many mathematicians view what they are doing along the lines that Leng and other nominalists wish to "re-"interpret them. Kunen's 2011 book on Set Theory is a relatively mainstream textbook introducing Set Theory at a graduate level. Kunen opens with a short 3 page discussion of the philosophy of mathematics that is, in my opinion, worth a read. His understanding of the possible views is outdated but not unreasonable. Crucially, though, Kunen discusses the kinds of re-interpretation that Shapiro and Burgess oppose. Kunen endorses the nominalist (he calls this "formalist", in line with my use of the term) reading, though with a notable caveat that this view remain agnostic with respect to realism.

Now one ought not extrapolate too much from one author, but Kunen's discussion is at least enough to cast doubt on the claim that mathematicians obviously all take their statements at face value.

With respect to the prescriptive project, I think one must be careful. I enthusiastically endorse recent trends that aim to bring philosophy of mathematics closer to the study of mathematical practice, but I don't think this means that philosophers should avoid the kinds of immodesty that Shapiro and Burgess are worried about, provided that it's in the right context and in the right way.

Even if Kunen is an exception and mathematicians generally do take their claims at face value, I think there's a very reasonable case to be made that Kunen is in the right, arising from mathematical, not merely philosophical concerns.

I won't unpack this too much here but, roughly, mathematicians generally want to avoid philosophical debates and just get on with the maths. If re-interpreting mathematical claims algebraically allows them to get away from metaphysical debates about the real identity or reference of mathematical terms or predicates, then re-interpretation seems like a reasonable suggestion to make to the mathematical community.

Overall, though, what I wanted to state is that whilst Leng's paper is both interesting and useful, I think to win the wider debate nominalists need to challenge the claims of Shapiro and Burgess about face-value readings of mathematical claims.

Closing remarks

This is an interesting paper that does a lot of good work. It creates a genuine problem for Shapiro, though I do think there are strategies Shapiro can use to get out. In the context of the wider debate, more needs to be done. I've set out a blueprint for the kind of argument Leng could use to generalise her argument. I've also stated my own view that, as nominalists, we also need to be challenging the face-value objection more directly by rejecting Burgess and Shapiro's claims.

This was an enjoyable read and I look forward to reading more by Leng in the future.

For anyone wishing to read this article it can be found via these link:

Mary Leng is a professor at the University of York (UK). Leng, Mary - Philosophy, University of York

Shapiro's view is outlined in his 2005 work "Categories, structures, and the Frege-Hilbert controversy".

Burgess's view is outlined in his 2004 work "Mathematics and Bleak House"

Both are papers published in Philosophia Mathematica

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