I am increasingly fascinated by the mathematics of fundamental cognitive processes – like creatures finding their way to and from significant locations, or foraging for food, or foraging with the eyes, or comprehending the duration of an event. I’m excited by the fact that there are cognitive neuroscientists that have become focused on the architecture of these processes in particular. Their work seems to always suggest that our formal mathematical systems are growing out of these very same processes.
Gallistel is concerned with the abstractions of space, time, number, rate and probability that have been experimentally studied and found to be playing a fundamental role in the lives of nonverbal animals and preverbal humans. His premise is this:
the brain’s ability to represent these foundational abstractions depends on a still more basic ability, the ability to store, retrieve and arithmetically manipulate signed magnitudes.
He makes a point of distinguishing between magnitude and our symbolic numbers. Magnitudes are what he calls ‘computable numbers’ a quantity that “can be subjected to arithmetic manipulation in a physically realized system.”
Being a bit pressed for time, I’ll just reproduce some of his observations.
The representation of space, he says:
requires summing successive displacements in an allocentric (other-centered) framework, a framework in which the coordinates of locations other than that of the animal do not change as the animal moves. By summing successive small displacements (small changes in its location), the animal maintains a representation of its location in the allocentric framework. This representation makes it possible to record locations of places and objects of interest as it encounters them, thereby constructing a cognitive map of its experienced environment. Computational considerations make it likely that this representation is Cartesian and allocentric.
But in order to have a directive function, these representations of experienced locations must be vectors – ordered sets of magnitudes. And the organism accomplishes arithmetic with them.
A fundamental operation in navigation is computing courses to be run… Assuming that the vectors are Cartesian, the range and bearing are the modulus and angle of the difference between the destination vector and the current-location vector. This difference vector is the element-by-element differences between the two vectors. Thus, the representation of spatial location depends on the arithmetic processing of magnitudes.
Gallistel challenges the notion that time-interval experience is generated by an interval-timing mechanism, pointing out that
There is, however, a conceptual problem with this supposition: The ability to record the first occurrence of an interesting temporal interval would seem to require the starting of an infinite number of timers for each of the very large number of experienced events that might turn out to be “the start of something interesting”–or not
Instead, he proposes
that temporal intervals are derived from the representation of temporal locations, just as displacements (directed spatial intervals) are derived from differences in spatial locations. This, in turn leads to arithmetic operations on temporal vectors (see Gallistel, 1990, for details). Rats represent rates (numbers of events divided by the durations of the intervals over which they have been experienced) and combine them multiplicatively with reward magnitudes . Both mice and adult human subjects represent the uncertainty in their estimates of elapsing durations (a probability distribution defined over a continuous variable) and discrete probability (the proportion between the number of trials of one kind and the number of trials of a different kind) can combine these two representations multiplicatively to estimate an optimal target time .
I found one of the most interesting parts of this discussion to be the one on closure.
Closure is an important constraint on the mechanism that implement arithmetic processing in the brain. Closure means that there are no inputs that crash the machine. Closure under subtraction requires that magnitudes have sign (direction), because otherwise entering a subtrahend greater than the minuend would crash the machine; it would not be able to produce a valid output. Rats learn directed (signed) temporal differences; they distinguish between whether the reward comes before or after the signal and they can integrate one directed difference with another .
I find this particularly interesting because it took us some time to find signed differences in our symbolic system of subtraction or even to recognize the significance of closure.
I’ll end this with his brief conclusion. Some of the details of these studies can be found in the linked pdf.
It seems likely that magnitudes (computable numbers) are used to represent the foundational abstractions of space, time, number, rate, and probability. The growing evidence for the arithmetic processing of the magnitudes in these different domains, together with the “unreasonable” efficacy of representations founded on arithmetic, suggests that there must be neural mechanisms that implement the arithmetic operations. Because the magnitudes in the different domains are interrelated–in for example, the representation of rate (numerosity divided by duration) or spatial density (numerosity divided by area)–it seems plausible to assume that the same mechanism is used to process the magnitudes underlying the representation of space, time and number. It should be possible to identify these neural mechanisms by their distinctive combinatorial signal processing in combination with the analytic constraint that numerosity 1 be represented by the multiplicative identity symbol is the system of symbols for representing magnitude.