George Lakoff

I have been waiting anxiously for Dehaene's book to reach the local bookstores here. I am, however, familiar with his previous work and applaud it. I assume his current book is based on his earlier work and takes the case further. This research, and earlier research on subitizing in animals, has made it clear that our capacity for number has evolved and that the very notion of number is shaped by specific neural systems in our brains.

Dehaene is also right in comparing mathematics to color. Color categories and the internal structures of such categories arise from our bodies and brains. Just as color categories and color qualia are not just "out there" in the world, so mathematics is not a feature of the universe in itself. As Dehaene rightly points out, we understand the world through our cognitive models and those models are not mirrors of the world, but arise from the detailed peculiarities of our brains. This is a view that I argued extensively in Women, Fire, and Dangerous Things, back in 1987.

Rafael Nunez and I are now in the midst of writing a book on our research on the cognitive structure of mathematics. We have concluded, as has Dehaene, that mathematics arises out of our brains and bodies. But our work is complementary to Dehaene's. We are concerned not just about the small positive numbers that occur in subitizing and simple cases of arithmetic. We are interested in how people project from simple numbers to more complex and "abstract" aspects of mathematics.

Our answer, which we have discussed in previous work and will spell out in our book, is that other embodied aspects of mind are involved. These include two particular types of cognitive structures that appear in general in conceptual structure and language.

  1. Image-schemas, that is, universal primitives of spatial relations, such as containment, contact, center-periphery, paths, and so on. Terry Regier (in The Human Semantic Potential, MIT Press) models many of these in terms of structured connectionist neural networks using models of such visual cortex structures as topographic maps of the visual field, orientation-sensitive cell assemblies, and so on.
  2. Conceptual metaphors, which cognitively are cross-domain mappings preserving inferential structures. Srini Narayanan, in his dissertation, models these (also in a structured connectionist model) using neural connections from sensory-motor areas to other areas. Narayanan's startling result is that the same neural network structures that can carry out high-level motor programs can also carry out abstract inferences about event structure under metaphorical projections. Since metaphorical projections preserve inferential structure, they are a natural mechanism for expanding upon our inborn numericizing abilities.

Nunez and I have found that metaphorical projections are implicated in two types metaphorical conceptualization. First, there are grounding metaphors that allow us to expand on simple numeration using the structure of everyday experiences, such as forming collections, taking steps in a given direction, and so on. We find, not surprisingly, that basic arithmetic operations are metaphorically conceptualized in those terms: adding is putting things in a pile; subtracting is taking away. Second, there are linking metaphors, which allow us to link distinct conceptual domains in mathematics. For example, we metaphorically conceptualize numbers as points on a line. In set-theoretical treatments, numbers are metaphorized as sets. Sets are, in turn, metaphorically conceptualized as containers - except in non-well-founded set theory, where sets are metaphorized as nodes in graphs. Such a "set" metaphorized as a node in a graph can "contain itself" when the node in the graph points to itself.

We have looked in detail at the conceptual structure of cartesian coordinates, exponentials and logarithms, trigonometry, infinitesimals (the Robinson hyperreals), imaginary numbers, and fractals. We have worked out the conceptual structure of e to the power pi times i. It is NOT e multiplied by itself =BC times and the result multiplied by itself i times-whatever that could mean! Rather it is a complex composition of basic mathematical metaphors.

Our conclusion builds on Dehaene's, but extends it from simple numbers to very complex classical mathematics. Simple numeration is expanded to "abstract" mathematics by metaphorical projections from our sensory-motor experience. We do not just have mathematical brains; we have mathematical bodies! Mathematics is not "abstract", but rather metaphorical, based on projections from sensory-motor areas that make use of "inferences" performed in those areas. The metaphors are not arbitrary, but based on common experiences: putting things into piles, taking steps, turning around, coming close to objects so they appear larger, and so on.

Simple numeration appears, as Dehaene claims, to be located in a confined region of the brain. But mathematics - all of it, from set theory to analytic geometry to topology to fractals to probability theory to recursive function theory - goes well beyond simple numeration. Mathematics as a whole engages many parts of our brains and grows out of a wide variety of experiences in the world. What we have found is that mathematics uses conceptual mechanisms from our everyday conceptual systems and language, especially image-schemas and conceptual metaphorical mappings than span distinct conceptual domains. When you are thinking of points inside a circle or numbers in a group or members of set, you are using the same image-schema of containment that you use in thinking of the chairs in a room.

There appears to be a part of the brain that is relatively small and localized for numeration. Given the subitizing capacity of animals, this would appear to be genetically based. But the same cannot be said for mathematics as a whole. There are no genes for cartesian coordinates or imaginary numbers or fractional dimensions. These are imaginative constructions of human beings. And if Nunez and I are right in our analyses, they involve a complex composition of metaphors and conceptual blends (of the sort described in the recent work of Gilles Fauconnier and Mark Turner).

Dehaene is right that this requires a nonplatonic philosophy of mathematics that is also not socially constructivist. Indeed, what is required is a special case of experientialist philosophy (or "embodied realism"), as outlined by Mark Johnson and myself beginning in Metaphors We Live By (1980), continuing in my Women, Fire and Dangerous Things (1987) and Johnson's The Body In The Mind (1987), and described and justified in much greater detail our forthcoming Philosophy In The Flesh.

Such a philosophy of mathematics is not relativist or socially constructivist, since it is embodied, that is, based on the shared characteristics of human brains and bodies as well as the shared aspects of our physical and interpersonal environments. As Dehaene said, pi is not an arbitrary social construction that could have been constructed in some other way. Neither is e, despite the argument that Nunez and I give that our understanding of e requires quite a bit of metaphorical structure. The metaphors are not arbitrary; they too are based on the characteristics of human bodies and brains.

On the other hand, such a philosophy of mathematics is not platonic or objectivist. Take a simple well-known example. Are the points on a line real numbers? Well, Robinson's hyperreals can also be mapped onto the line. When they are, the real numbers take up hardly any room at all on the line compared to the hyperreals. There are two forms of mathematics here, both real mathematics. Moreover, as Leon Henkin proved, given any standard axiom system for the real numbers and a model for it, there exists another model of those axioms containing the hyperreals. The reals can be mapped onto the line. So can the hyperreals.

So given an arbitrarily chosen line L, does every point on L correspond to a real number? Or does every point on L correspond to a hyperreal number? (If the answer is yes to the latter question, it cannot be yes to the former question - not with respect to the same correspondence.) This is not a question that can be determined by looking at the universe. You have a choice of metaphor, a choice as to whether you want to conceptualize the line as being constituted by the reals or the hyperreals. There is valid mathematics corresponding to each choice. But it is not a matter of arbitrariness. The same choice is not open for the integers or the rationals.

Mathematics is not platonist or objectivist. As Dehaene says, it is not a feature of the universe. But this has drastic consequences outside the philosophy of mathematics itself. If Dehaene is right about this-and if Reuben Hersh and Rafael Nunez and I are right about it-then Anglo-American analytic philosophy is in big trouble. The reason is that the correspondence theory of truth does not work for mathematics. Mathematical truth is not a matter of matching up symbols with the external world. Mathematical truth comes out of us, out of the physical structures of our brains and bodies, out of our metaphorical capacity to link up domains of our minds (and brains) so as to preserve inference, and out of the nonarbitrary way we have adapted to the external world. If you seriously believe in the correspondence theory of truth, Dehaene's work should make you worry, and worry big time.

Dehaene's work is also very bad news for the theory of mind defended in Pinker's How The Mind Works (pp. 24-25), namely, functionalism, or the Computer Program Theory of Mind. Functionalism, first formulated by philosopher Hilary Putnam and since repudiated by him, is the theory that all aspects of mind can be characterized adequately without looking at the brain, as if the mind worked via the manipulation of abstract formal symbols as in a computer program designed independent of any particular hardware, but which happened to be capable of running on the brain's wetware. This computer program mind is not shaped by the details of the brain.

But if Dehaene is right, the brain shapes and defines the concept of number in the most fundamental way. This is the opposite of what the Computer Program Theory of Mind says, namely, that the concept of number is part of a computer program that is not shaped or determined by the peculiarities of the physical brain at all and they we can know everything about number without knowing anything about the brain.

Challenging the Computer Program Theory of Mind is not a small matter. Pinker calls it "one of the great ideas in intellectual history" and "indispensable" to an understanding of mind. Any time you hear someone talking about "the mind's software" that can be run on "the brain's hardware," you are in the presence of the Computer Program Theory.

Dehaene is by no means alone is his implicit rejection of the Computer Program Theory. Distinguished figures in neuroscience have rejected it (e.g., Antonio Damasio, Gerald Edelman, Patricia Churchland). In our lab at the International Computer Science Institute at Berkeley, Jerome Feldman, I, and our co-workers working on a neural theory of language, have also become convinced by results in our lab indicating that it is wrong. I refer to the results mentioned above by Regier and Narayanan indicating that conceptual structure for spatial relations concepts and event structure concepts are created and shaped by specific types of neural structures in the visual system and the motor system.

Dehaene's work is important. It lies at the center of some of the deepest and most important issues in philosophy and in our understanding of what the mind is and, hence, what a human being is. Consider one last implication. It has been taken for granted since the Greeks that what distinguishes human beings from animals is the capacity for reason. Since at least the Enlightenment, reason has been seen as a separate and unitary faculty, distinct from anything that animals have. Mathematics has been taken as the best example of reason. But if animals have even some significant part of our basic capacity for numeration, then they have part of our capacity for reason. That means that faculty psychology was wrong. It was wrong that the jump from primates to human beings was a jump from no-part-of-reason to total reason.

What is at stake in Dehaene's work?

  1. The objective existence of mathematics external to all beings and part of the structure not only of this universe but of any possible universe (Platonism).
  2. The correspondence theory of truth, and with it all of Anglo-American analytic philosophy. If the correspondence theory falls, the whole stack of cards falls.
  3. Functionalism, or The Computer Program Theory of Mind.
  4. The idea that human beings have all of reason and animals have none.

I can barely wait for his new book to get to my local bookstore.

The above is a recent reply by George Lakoff to comments on S. Dehaene's coming book "The number sense: How mathematical knowledge is embedded in our brains" (original French title: La bosse des maths). In this comment, posted for another group, Lakoff refers to the work he and I have been doing for the last 2 years on the cognitive science of mathematics (and which we believe has implications for math education).

Rafael E. Nunez
Institute of Cognitive Studies, University of California at Berkeley, Berkeley, CA 94720, USA

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Last Modified: 17th November 1997