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Trends in Philosophy of Mathematics (day 3, talk 1)

On the third day the schedule was a bit more complicated, we had to choose between one of two parallel sessions. The choice was difficult, so I will be unable to comment on some of really interesting talks that I was unable to attend. If I don’t talk about a certain talk it’s because I was unable to make it to it because the alternative talk was more related to what I’m working on. For now, the first talk of the day.

Assadian: Crispin Wright and his Hero

Wright, defending the epistemic accessibility of prima facie impredicative Hume’s Principle tells a story of a fictional character (named Hero) who initially knows second-order logic and possesses a bunch of sortal concepts referring to concrete objects, but doesn’t understand the concept of number. Wright then argues that the Hero can process in stages in order to gain the understanding of the concept of natural numbers.

Stage 1 - The Hero introduces HP for the initial domain that he possesses a grasp of.

Stage 2 - The Hero now understands without circularity the truth conditions of Nx:Fx = Nx:Gx, where neither F nor G contain further occurrences of further numerical terms. The Hero also knows that Nx:Fx=s is false for any term s referring to an object present at Stage 1. In this sense, he seems able to solve the Caesar Problem for he comes to accept:

(NE) no object whose identity is grounded in anything else than HP can be identical to a number.

Stage 3 - The Hero moves on to understanding truth conditions of identity statements of terms containing embedded occurrences of numerical operators.

Assadian takes issue with Wright’s account of how the Hero learns NE, which is of key importance in solving CP. NE seems to hinge on the possession of a complete characterization of natural numbers (=identity conditions for them), and this is not something that the Hero can have at stage 2. For it seems that to understand the claim that the characterization of numbers doesn’t go beyond its identity conditions dictated by HP, one has to be able to grasp those identity conditions already.On the other hand, understanding of complex terms containing embedded numerical operators seems to require that NE be already known, if CP is to be solved. Moreover, if NE is not available at stage 2, the Hero is unable to solve CP at that stage.

A slightly different neologicist approach to the problem is to assume that different sorts of objects in general have different criteria of identity - once one postulates the existence of maximal categories of that sort and ads some fairly convincing assumptions about them, two theorems can be proven. The first says that for any category C for any two objects that fall under that category, they are identical iff they satisfy identity conditions corresponding to that category. The other one says that no object belongs to two different categories. It is needed if we are to exclude the possibility that both numbers and persons constitute a single larger category with its specific identity conditions.

Assadian argues that even though if those theorems are present, CP is solved, it’s quite implausible that those theorems are available to the Hero at stage 2, because theorem 2 already says something about the whole category of all numbers and their identity conditions. On the other hand, if those theorems are to be introduced only at stage 3, it is unclear why the Hero would be able to solve CP already in stange 2 to start with.

Although I generally agree that non-iterative approaches to abstraction principles have been so far unable to solve CP (among the iterative ones there’s Linnebo’s and mine, and I think mine can handle CP and it is quite unclear whether Linnebo’s does - but this is a whole different story), I really would have to see the proofs in detail - what would have to be checked is whether (i) the assumptions used to prove theorems 1 and 2 are convincing, (ii) theorems 1 and 2 follow from those assumptions, (iii) theorems 1 and 2 really allow to solve CP (i.e. to prove negations of mixed identity statements, or something to that effect), and (iv) whether no undesired consequences follow from the same assumptions. But the stuff seems interesting.

Qualms about the (non-)circularity of NE aside, what I’m rather worried about is the justification of the framework in which NE even makes sense. I mean, I have pretty hard time understanding the idea of objects such that there is nothing else to learn about them apart from their rather coarse-grained identity conditions. I am perfectly fine with coarse-grained or relative identity claims, or true fake identity between fake singular terms, but the idea that there really are objects such that the only way we can learn anything about them is through abstraction principles seems suspicious. Of course, it is an attempt to deal with epistemic challenges to Platonism about mathematics, but I don’t think a blunt answer of the sort "How can we know something about numbers? Well, we learn something about them through abstraction principles and there’s nothing else to learn" is satisfactory. I would need a more elaborate and convincing metaphysical story which would convince me to accept the existence of such things, and which would explain why those objects should enjoy this particular status.