Review: Pluralism in Mathematics by Michèle Friend

I am adding a few old(ish) papers to the blog section of the website for the sake of posterity. My first ever piece of published material was a little review of Michèle Friend’s Pluralism in Mathematics (you can find a pre-print here, or the journal version here). As I recall, I very much enjoyed the book. There are two issues that the book prompted me to think about and have stuck with me (I’m still trying to wrap my head around):

1. What might be the justification for holding a particular pluralism?

A first question that I think is very important is that one can have different `depths’ of pluralism. We might just be a methodological pluralists, in that we think that we should have multiple people working in multiple frameworks, as this is more effective for generating knowledge and results. (Roughly, this seems to be Chang’s position in Is Water H2O?, though he’s expressed something deeper in conversation.) However, we might have a deeper pluralism in that we might think that the structure of the world for (whatever reason) resists characterisation by a single perspective (this seems to be Kuhn’s position in some of his more `relativistic’ moments).

There are lots of challenges facing the `deep’ pluralist, but I’m especially interested in the possible differences of a deep pluralism in mathematics, versus that of science. There seems to be an important difference between the two, though we get different perspectives from both the empirical sciences and (foundations of ) mathematics, its only in the former that we have true incompatibility of our favourite theories (e.g. Beam mechanics, quantum vs. relativistic physics). In mathematics, we can instead (for many theories) simulate one within the other. For example, results discovered in the language of set theory can be translated into an algebraic theory of sets in category theory (e.g. ETCS), and conversely we can interpret any algebraic theory within the sets (possibly with the addition of large cardinals).

Despite this simulation, however, the two perspectives are very different when thought of as providing the “ontology” for mathematics. Set theory provides a system in which mathematical objects can be “built” and regards objects with very different internal structure as totally different in nature (e.g. \{\emptyset\} is a very different thing from \{ \omega \}, because their transitive closures look very different, but categorially in Set they are just both terminal objects, isomorphic, and hence categorially `the same’).

With this in mind, one question I’m still struggling with is the following:

Is there a motivation that could be supplied for a deep pluralism in the case of mathematics, and is this threatened by the disanalogy with the empirical sciences?

2. How to deal with contradiction between perspectives?

A second problem occurs once we have adopted a pluralism. Presumably then, the challenge is to figure out how to transfer useful information in a meaning-preserving (in some sense) way from one perspective to another, despite any conflicts. In short, we want to `adopt’ two contradictory perspectives at once, without dissolving into triviality.

The way Friend deals with this is to just take true contradictions on the chin, and adopt Priest’s Logic of Paradox (LP) to conduct her inferences. I have two problems with this:

A: True contradictions just aren’t appealing to me. I want to understand the world as given by multiple irreconcilable perspectives, not a single incoherent one. (I appreciate that lots of great stuff has come out of the paraconsistent and dialethist literature, but I just cannot get on board with a true contradiction.)

B: LP doesn’t validate modus ponens for the material conditional. I need modus ponens to reason properly (also, I feel it should be true on any perspective).

For this reason, I’m interested in taking a step back and rather looking at the process of “information transfer”. Friend does this a little by examining Priest’s method of “chunk-and-permeate” (roughly: hive off the relevant information and move it), but I wonder whether progress can be made by analysing the essential features of a foundational language. This, however, seems like a pretty difficult task.



Richness and Reflection

Just posting regarding an old paper that I think has some relevance: `Richness and Reflection’ contrasts motivations for reflection principles from both a multiversist and universist perspective.

I think this issue merits further examination. More generally, there are many positions between the `extremes’ of a one-universe perspective and an  almost-anything-goes first-order perspective. Justification  of new axioms in set theory is going to have a slightly different flavour for each.

Abstract: A pervasive thought in contemporary philosophy of mathematics is that in order to justify reflection principles, one must hold universism: the view that there is a single universe of pure sets. I challenge this kind of reasoning by contrasting universism with a Zermelian form of multiversism. I argue that if extant justifications of reflection principles using notions of richness are acceptable for the universist, then the Zermelian can use similar justifications. However, I note that for some forms of richness argument, the status of reflection principles as axioms is left open for the Zermelian.

You can find the paper here. If you don’t have access, you can find a pre-print on my here.