This friday, I attended a thought-provoking talk by Katherine Dunlop about how Strawson and others (Ayer and Poincaré, for example) were interested in the relation between the following three questions:
- Reidentification: How is it possible to know that we are dealing with the same object, instead of two?
- Independece: How do we know that objects exist?
- Non-solipsism: How is it possible to make a difference between us and the (so-called external) world?
Unless I grossly misunderstood Ms Dunlop, we usually call them all objectivity, but they are clearly different things.
If we are empirists, we are specially interested in the empirical versions of these questions:
- Empirical reidentification: How is it possible to know that we are dealing with two appearences of the same object, instead of two different objects that are similar in appearence?
- Appearence independece: How do we know that unobserved objects exist (not just objects unobserved by us, but absolutely unobserved?
- Non-solipsistic Consciousness: How is it possible to make a difference between us and the (so-called external) world in the content of our (conscious) experience?
However, I think we could do the same thing for non-empirical realms, like mathematics. In mathematics, there is also a confusion between three different ways of cashing out mathematicsl objectivty:
- Mathematical reidentification: How is it possible to know that we are dealing with two instance of the same mathematical object?
- Formal independece: Does it even make sense to consider the possibility of there being mathematical objects whose existence is unprovable? Or, in the words of Douglas Bridges and Erik Palmgren, we should interpret the phrase “there exists” in mathematics strictly as “we can construct”?
- Non-solipsistic Consciousness: How is it possible to make a difference between us and the (so-called abstract) world of mathematics?
In mathematics, we also usually call them all objectivity, but they are clearly different things.
Furthermore, making this distinction might also help us understand why talk of "philosophy of mathematics" might be a misnomer, since different branches and practices of mathematics might require substantially different philosophical accounts. Consider mathematical reidentification. In arithmetics, it seems natural to think that there are no more than one for each number, i.e., that there are no more than two numbers three or two hundred and twelve. When we add two plus two, we do not think that we are combinign two different twos. However, as Marco Panza has recently stressed, the same cannot be said about Geometry, specially as it is developed diagrammatically (for example, by Euclid): when we draw a line or a triangle in a diagram, there is no way to tell whether such line or triangle is the same or different from any other line or triangle in any other diagram. Identiy is diagram-relative. We can tell figures within a single diagram apart, but not across diagrams. This is why Panza thinks Platonism cannot be the right metaphysics for Euclidean geometry.
A similar issue arises for formal independence. Constructivism seems prima facie more natural in certain branches of geometry and for certain periods of its historical development, while arithmetics and analysis seem to work on more realist assumptions.