During a conversation with Matthew Inglis, I said that I wanted to think of mathematical objects as social objects. This was in response to his saying that I had been a bit hard on mathematical Platonists during my seminar at the LSRI a few months ago. So the next step is going to have to be to explain what I mean exactly by that.
I know that I don’t mean the same as some philosophers do when they refer to Psychologism – the idea that mathematical objects are identical with particular biological occurrences in the human brain.
What I mean is most easily expressed with reference to the semiotic network that I’ve talked about before. Let’s start by assuming a post-structuralist account of meaning – that the meaning of a signifier consists in that signifier’s relationships with other signifiers. I like this as a start, because it not commit one to any particular research perspective (cognitive, socioculturalist, etc.).
So… mathematical objects have meaning by virtue of their relationships with other mathematical objects. On the surface, this isn’t difficult to accept – the meaning of ’3′ has a lot to do with it’s relationship with ’1′, ’2′ and ’4′, for a start. But these relationships are socially defined – and that’s the different bit.
So… the meaning of ’3′ consists in the signifiers that are activated within a community in response to the perception of ’3′. What exactly does this mean? I think this means 1 of 3 things: it’s either the union of all activated signifiers of all members of the community, or it’s the intersection of activated signifiers (activations common to all members), or it is somewhere between those two definitions. Whichever of those is the case, the social nature of the definition is clear. At the moment, I haven’t given a huge amount of thought yet to the question of where the balance is – exactly how the meaning is defined and what set of relationships constitutes the meaning of a given signifier. I have a bit of a sense that maybe it’s the wrong question to ask anyway – maybe unanswerable and maybe unaskable.
So that’s what I mean when I say mathematical objects are social objects. I’m thinking about what the implications of a definition like this are though. And one of things that I’m thinking is that in practice, a ‘social objects’ account of mathematics isn’t very different from a Platonic account.
