Book Review Where Mathematics Comes From by Lakoff & Núñez (Basic Books)

So where does it come from according to these authors?   The answer seems to be :
“…[from] concepts in our minds that are shaped by our bodies and brains and realized physically in our neural systems” (p. 346).  One might consider this a rather obvious, not to say bland, conclusion but it is not a theory the mathematical establishment is going to accept any day soon. Why not?  Because it knocks out the ‘transcendental origin’ theory, otherwise known as Platonism, to which practically all professional mathematicians subscribe either openly or covertly. The authors point out that there is no way such a claim could be tested, so it cannot really be considered a scientific hypothesis.
The authors demonstrate fairly convincingly that many of the sophisticated mathematical
procedures we employ can be traced back to primitive schemas, such as the ‘Container Schema’ which underlies Set Theory and Boolean Logic, schemas which are themselves abstractions from physical sensations made by — wait for it — infants in arms. “Mathematics….is grounded in the human body and brain, in human cognitive capacities, and in common human activities and concerns” (p. 358).
All this is not news to me since I have, off and on, been advancing some such theory of the origin of mathematics for the last thirty years, but it is nice to see some of the details of these familiar cognitive ‘grounding metaphors’ fleshed out. The dreadful fact is that mathematicians, pure just as
much as applied, cannot get on without metaphors culled from sensory experience, and, far from ‘transcending’ these metaphors by abstraction, all too often mathematicians remain pathetically tied to these conceptions, the ‘metaphor’ of the Number Line being the most grotesque example. For, whatever numbers ‘really’ are, they certainly are not points on a line and they are not at a specific distance from a mathematical ‘origin’.
From my point of view, the book does not go far enough, since it (just) stops short of developing a truly empirical theory of mathematics, largely because of the excessive importance the authors give to what they call the ‘Basic Metaphor of Infinity’ (BMI) — for if there is one mathematical concept that is not grounded in our sensory experience, it is infinity.

Also, the book is too long — nearly 500 large pages —  though anything shorter would have been
dismissed by the establishment as superficial. For all that, this is a very welcome book and a brave one too, since the authors remark at one point,  with commendable understatement, that “it is not unusual for people to get angry when told that their unconscious conceptual systems contradict their fondly held conscious beliefs” (p. 339). Out of context, you might think Lakoff & Núñez were referring to hardline Creationists from the Bible belt in America — but no, ‘people’ in this quote simply means professional mathematicians.    Sebastian Hayes

Note:  This review appeared in M500, the magazine of the Mathematics department of the UK Open University.  S.H.