Where Mathematics Comes From - Critical Response

Critical Response

Many working mathematicians resist the approach and conclusions of Lakoff and Núñez. Reviews by mathematicians of WMCF in professional journals, while often respectful of its focus on conceptual strategies and metaphors as paths for understanding mathematics, have taken exception to some of the WMCF's philosophical arguments on the grounds that mathematical statements have lasting 'objective' meanings. For example, Fermat's last theorem means exactly what it meant when Fermat initially proposed it 1664. Other reviewers have pointed out that multiple conceptual strategies can be employed in connection with the same mathematically defined term, often by the same person (a point that is compatible with the view that we routinely understand the 'same' concept with different metaphors). The metaphor and the conceptual strategy are not the same as the formal definition which mathematicians employ. However, WMCF points out that formal definitions are built using words and symbols that have meaning only in terms of human experience.

Critiques of WMCF include the humorous:

"It's difficult for me to conceive of a metaphor for a real number raised to a complex power, but if there is one, I'd sure like to see it." - Joseph Auslander

and the physically informed:

"But their analysis leaves at least a couple of questions insufficiently answered. For one thing, the authors ignore the fact that brains not only observe nature, but also are part of nature. Perhaps the math that brains invent takes the form it does because math had a hand in forming the brains in the first place (through the operation of natural laws in constraining the evolution of life). Furthermore, it's one thing to fit equations to aspects of reality that are already known. It's something else for that math to tell of phenomena never previously suspected. When Paul Dirac's equations describing electrons produced more than one solution, he surmised that nature must possess other particles, now known as antimatter. But scientists did not discover such particles until after Dirac's math told him they must exist. If math is a human invention, nature seems to know what was going to be invented."

Mathematicians have also complained that Lakoff and Núñez have misunderstood some basic mathematical notions. The authors reply that the errors found in earlier printings of WMCF are now corrected.

Lakoff made his reputation by linking linguistics to cognitive science and the analysis of metaphor. Núñez, educated in Switzerland, is a product of Jean Piaget's school of cognitive psychology as a basis for logic and mathematics. Núñez has thought much about the foundations of real analysis, the real and complex numbers, and the Basic Metaphor of Infinity. These topics, however, worthy though they be, form part of the superstructure of mathematics. Cognitive science should take more interest in the foundations of mathematics. And indeed, the authors do pay a fair bit of attention early on to logic, Boolean algebra and the Zermelo–Fraenkel axioms, even lingering a bit over group theory. But neither author is well-trained in logic (there is no index entry for "quantifier" or "quantification"), the philosophy of set theory, the axiomatic method, metamathematics, and model theory. Nor does WMCF say enough about the derivation of number systems (the Peano axioms go unmentioned), abstract algebra, equivalence and order relations, mereology, topology, and geometry.

Lakoff and Núñez tend to dismiss the negative opinions mathematicians have expressed about WMCF, because their critics do not appreciate the insights of cognitive science. Lakoff and Núñez maintain that their argument can only be understood using the discoveries of recent decades about the way human brains process language and meaning. They argue that any arguments or criticisms that are not grounded in this understanding cannot address the content of the book.

It has been pointed out that it is not at all clear that WMCF establishes that the claim "intelligent alien life would have mathematical ability" is a myth. To do this, it would be required to show that intelligence and mathematical ability are separable, and this has not been done. On Earth, intelligence and mathematical ability seem to go hand in hand in all life-forms, as pointed out by Keith Devlin among others. The authors of WMCF have not explained how this situation would (or even could) be different anywhere else.

Lakoff and Núñez also appear not to appreciate the extent to which intuitionists and constructivists have anticipated their attack on the Romance of (Platonic) Mathematics. Brouwer, the founder of the intuitionist/constructivist point of view, wrote "Mathematics is a free construction of the human mind." Hence at least one person writing before Lakoff and Núñez were born concluded that mathematics emerged to serve human purposes and has no existence apart from this fact.