What a Number is...

Warren McCulloch asked, “What is a number that a man may know it, and a man that he may know a number?  The answer, he believed, lay in neurology, and that interface between mind and the world that Kant first talked about.  Our whole science, including the ideas we use to understand the brain, is built on mathematics.  But if mathematics is partly a function of our nervous systems, everything seems in danger of collapsing into a tangle of solipsistic loops.

With matters like these, philosophers press close to the edge of what it is [possible to know and to think about, It is easy to despair as Wittgenstein did, that science and all philosophies are ultimately closed systems of circular truths.  Since it is impossible to stop outside of these tautological webs, there will always be things we cannot know, stories that cannot be told.  The universe will always overwhelm the theory builders.  We are trapped inside our bodies and our nervous systems.  As Quine said, “We can’t get off the boat”. The best we can do, it seems, is to understand the nature of the filters and the mental structures, the mechanisms we use to convert experience into memories.

Assuming that there is an objective world, out there that works the same everywhere – at least – what would it be like for other creatures with other nervous systems?  Can we even assume that they would look at things as being embedded in space and time? Or is that being anthropomorphic?

“I think there are regularities of some kind in the universe,” Churchland said.  "And it might be that there are different ways of capturing those regularities or describing them.  But our brains, given the organization that they have, might be predisposed to a certain way of describing the regularities.  So that if we met a physicist on ARCTUTRUS 4 who has a very differently organized brain, his physics might look very different but it might be successful as ours.”

But what about mathematics? Here is one area where even the hard-core scientist tends to go Platonic, arguing that mathematics is indeed discovered, not invented, that there are regularities that seem to have some kind of independent existence.  The fact that there seem to be mathematical truths and perfect circles and triangles – things that you just don’t find lying around in the physical world – has long been used as a counterargument to empiricism.  Even the logical empiricists reserved a special place for mathematical truth, believing that it was separate from the truths induced by the senses.

But neural science continues to close in on philosophy; it offers a ways to explain mathematics without invoking spooky stuff.  A circle is a mental object, something that we never really experience in life.  But it doesn’t have to be pure mind stuff.  Rather, it could be a kind of compromise between the structure of the world and the structure of our brains.  We see a lot of things that come pretty close to being round.  The brain averages them according to its wired-in notions of order to come up with the concept circle. The fact that we use certain concepts – circle, triangle – and not others could be an accident of evolution.  People with migraines often report hallucinations of geometric objects; there seems t be something neurological about them.  There may be other ways to cut up the world that we can’t begin to imagine.  With different brains, these other shapes, or what ever they might be, would seem just as fundamental.

“There may be something we can tendentiously call mathematics that the people on ARCTURUS 4 do, and it doesn’t look like what we do,” Churchland said, “But it gets them around, and they manage to build spaceships, and they manage to use fire, and smelt metal.  So I’m not a Platonist with regard to mathematics at all.  I don’t want to be stamped by argument form ignorance -- namely, but what else would make the propositions of mathematics true? I don’t know what else, but maybe something in the brain. I think that about mathematics, and I also think that about logic.

“If you say that, of course, it usually sends people absolutely crawling up the walls.  But I don’t really see why. If the only alternative is that there are truths in Plato’s heaven, then it seems to me that the basic story has to be told in terms of the brain.  I just don’t quite see how it could be other wise”.

George Johnson. In the Palaces of Memory: How we build the Worlds inside our Heads. (PP 224-226) NY: Knopf (1991).