The mathematical language of the brain

Question

This question is similar, but not identical, to one I posted to the mathematics SE some time ago. I was originally unsure of where to post it. I believe this question is sufficiently different to warrant not migrating the other one.

There is a famous quote in von Neumann's The Computer and the Brain (published posthumously), that goes:

When we talk mathematics, we may be discussing a secondary language, built on the primary language truly used by the central nervous system.

Also

Thus logics and mathematics in the central nervous system, when viewed as languages, must structurally be essentially different from those languages to which our common experience refers.

I've read the relevant passages a few times, and far as I can see he is trying to convey a statement along the lines of: "The mathematical or logical langauge of the nervous system is not the same as that we use when we do/talk about mathematics."

From a modern perspective, we may say that the nervous system is a biological neural network, with (as von Neumann himself suggested in this book) low precision but high reliability. And then, presumably, this neural net is Turing-complete, and the actual way in which computations are performed is not important - they are still computations, and they could just as well have been run on some sort of a cellular automaton.

The people over at the mathematics SE seem to think there is no formal mathematical substance here, and that the issue is purely philosophical.

So my question is, are there philosophical positions that make this feeling of von Neumann more concrete? What, if anything, does this add/take way tofrom a theory of mind? I am thinking particularly here about the computational theory of mind.

Was von Neumann onto something fundamental here about the nature of mathematics as a language?

Answer 1

One aspect that enters in here is that energy is continuous, and logic never really is. The neural network that constitutes the mind cannot be identical to any logical language with a finite number of symbols, and all languages ultimately have a finite number of symbols.

Languages may point toward a countable set of values, but they cannot really constitute one, so we cannot even have a language that is 'dense' in the range of potential neural states (the way the rational numbers are dense in the reals). We cannot reasonably claim human language has a continuum of reference points, but the projected balance of ionic charges over time -- the stuff of which our memory is constituted, via Hebbian learning -- naturally does.

So there are basic aspects that biology or physics always involve but that cannot, ultimately be captured in language directly. We can only generalize about them indirectly. So whatever language we use to share or communicate the contents of the brain, it will have been simplified by the necessities of storytelling. This requires that it be rendered discrete.

Turing machines never compute anything continuous, unless we impose some definite limit on precision. So they are not a real model for biological activity that has not been reduced to language. The interal state of an analog machine can represent the full complexity of a fractional-dimensional solution to a differential equation, even if we can only read the output to a given precision.

This doesn't really help say anything useful about either of these two languages, but it does prove they have an essential difference. There is a maximum precision of language that can only approximate the precision of reality asymptotically.