What sort of machine is the human brain, that it can give birth to mathematics ? Warren Mcculloch thought he knew part of the answer. Being a mathematician himself, he was eager to understand "how such a thing as mathematics could have seen the light ?" and he was convinced that the brain is a "logical machine".

In 1943, in an influential article , he stripped neurons of their complex biological reactions and reduced them to two functions: summing their inputs and comparing this sum to a fixed threshold. He then demonstrated that a network made up of many such interconnected units can perform calculations of an arbitrary complexity. In computer science jargon, such a network has the computational power of a turing machine -- a formal device which captures the essential operations at work in computers for reading, writing, and transforming digital data according to mechanical operations.

Mcculloch's work thus showed that any operation that can be programmed on a computer can also be performed by an adequately wired network of simplified neurons.

Mcculloch thus followed in the footsteps of George Boole who, in 1854, had set out as a research program for himself "to investigate the fundamental laws of those operations of the mind by which reasoning is performed, to give expression to them in the symbolic language of a calculus, and upon this foundation to establish the science of logic and construct its method"

Boole is the inventor of "boolean" logic, which describes how the binary values true and false, denoted by 1 and 0, should be combined in logical computations. Today, boolean algebra is seen as belonging to mathematical logic or to computer science. But Boole himself considered his research as a central contribution to psychology -- an "Investigation of the Laws of Thought", as his book was titled.

The metaphor of the brain as a computer had now acquired immense popularity, not only with the general public but even among specialists in cognitive science. It lies at the heart of the so-called "functionalist" approach to psychology, which advocates studying the algorithms of the mind without caring about the workings of the brain.

A classical functionalist argument stresses that any digital algorithm computes exactly the same result, regardless of whether it runs on a supercompuer or on a pocket electronic calculator. Does it matter, then, that the computer is made of silicon and the brain of nerve cells ?

For functionalists, the software of the mind is independent of the hardware of the brain -- and the mathematical results of Alonzo Church and Alan Turing guarantee that all functions that are computable by a human mind can also be computed by a turing machine or a computer.

In 1983, Philip Johnson-Laird went as far as to state that "the physical nature of the brain places no constraints on the pattern of thought" and that as a consequence, the brain computer metaphor "need never be supplanted".

Is the brain really nothing more than a computer or a "logical machine" ? Does its logical organization explain our mathematical abilities, and should it be studied independently of its neural substrate ?

On purely empirical grounds, the brain computer metaphor simply does not provide a good model of the available experimental data. There are many counterexamples that suggest that the human brain does not calculate like a "logical machine".

Rigorous calculations do not come easily to homo sapiens. Like so many other animals, humans are born with a fuzzy and approximate concept of number that has little in common with the digital representation of computers. The invention of a numerical language, and of exact calculation algorithms, belongs to the recent cultural history of humanity -- and, in several respects, it is an unnatural evolution.

Though our culture has invented logic and arithmetic, our brain has remained surprisingly refractory even to the simplest algorithms. By way of proof, one merely needs to consider the difficulty with which children assimilate arithmetic tables and calculation rules. Even a calculating prodigy, after years of training, takes tens of second to multiply two 6-digit numbers -- a thousand to a million times slower than the most sluggish personal computer. In domains in which the computer excels- the faultless execution of a long series of logical steps-- our brain turns out to be slow and fallible.

At the level of the neural circuits themselves, comparing the brain to a "logical machine" does not stand up to scrutiny. Each neuron implements a biological function considerably more complex than the simple logical addition of its inputs.

Above all, real networks of neurons depart from the rigorous of assembly of transistors in the electronic chips of modern computers. Logical gates are not primitive operations of the brain. If one had to look for a "primitive" function in the nervous system, it would perhaps be the ability of a nerve cell to recognize an elementary "shape" in its inputs by weighing the neuronal discharges it receives from thousands of the units.