Tuesday, 26 April 2011

Applicability, mixed & pure, and modality

Here are some thoughts on the applicability of mathematics. Important work on the applicability of mathematics by Quine, Putnam and Field clarified that mathematicized scientific laws, if we examine them closely, contain mixed predicates, whose interpretations are mixed relations between "concreta" and mathematical objects. A simple example of such a mixed predicate is the membership predicate $\in$. Furthermore, the mathematical objects that arise may be mixed or pure. Examples of pure mathematical objects are: natural numbers, integers, real numbers, complex numbers, infinite cardinals and ordinals; also various structures that turn up in algebra and geometry, so long as these are understood in an ante rem manner. Examples of mixed mathematical objects are sets, relations and functions whose transitive closure contains "concreta":
Pure: $7, -1, \pi, e^{i \pi}, \aleph_{57}, \omega^{\omega}, \mathbb{R}, \mathbb{Z}_3, cosine, SU(3), L^2[\mathbb{R}^3]$, etc. 
Mixed: the set of US presidents; the set of London underground stations; a 3-element graph whose nodes are {Frege, Hilbert, Noether}; the measurement scale $Mass_{kg}$; the electromagnetic field $F_{ab}$; the metric tensor $g_{ab}$, etc. 
What distinguishes the pure and mixed mathematical objects? It seems to be their modal & temporal status. The set of all people who have been US presidents now has 44 members, but (unless something weird happens) on Jan 20th, 2013, it will have either 44 or 45 members. And the set of all US presidents now in the actual world has 44 members, but it could have been different. So, there is some sense in which mixed mathematical objects, being anchored in concrete actualities or possibilities, can change, temporally and modally. But it seems right to say that pure mathematical objects don't change, temporally or modally. This is why we think that statements of pure mathematics are necessary.

There is a problem, however. It has to do with what are sometimes called "Cambridge changes". If x is my coffee cup, then x changes, by its temperature cooling from 50 Celsius to 20 Celsius between times t and t*. But can we not also say that the number 50 changed, from having the property "being the temperature-in-Celsius of cup x" at time t to not having this property at time t*? Similarly, 44 has the property of "being the number of all US presidents" on 26th April 2011, but will lose this property by Jan 20th, 2017. So, pure mathematical objects can change after all!

I'm not sure what the answer to this is, but I suspect it is connected to the "rigidity" of the pure relations between pure mathematical objects. The pure mathematical objects don't "change" relations amongst each other. But, when we consider their relations to concreta, they can "change" those relations, just as concreta can change their relations amongst each other. I may well be heavier than my colleague Dr X now, but after my diet, I will definitely be lighter. Admittedly, this is all rather unclear; and, in fact, there are some thinkers who have argued that mathematical objects (mixed and pure) contingently don't exist. Field has argued for this, and has also tried to explain the consequent apparent necessity of pure mathematics in terms of its conservativeness.


  1. "The set of all people who have been US presidents now has 44 members, but (unless something weird happens) on Jan 20th, 2013, it will have either 44 or 45 members. And the set of all US presidents now in the actual world has 44 members, but it could have been different"

    There is another option: it might be that the set of US presidents---that is, that particular object---has the members that it has of necessity, and instead, the term 'the set of US presidents' is non-rigid, just as 'the US president' is.

    This other option has the advantage that it doesn't go against some kind of modalised version of extensionality, which might be thought to be an essential feature of the set concept. The set which we will refer to by 'the set of US presidents' in 2017 will not be the same as the one we refer to now, in virtue of it having at least one additional element.

  2. Yes, that's definitely an option, and I should have included it - as it's the sort of line taken to show that languages have their syntactic and semantic properties essentially (which is why the T-sentences are necessarily true).
    I'm sympathetic to that view, and it keeps extensionality, as you say. That would mean "set-like" mixed mathematical objects don't really change after all; rather, descriptive terms denoting them are non-rigid.

    Consider other mixed mathematical objects, though, like the magnetic field B: a function from spacetime points to vectors. In different possible worlds, it has a different extensions. One could say that the term "the magnetic field" is non-rigid. I am not sure though.

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