Lakoff and Núñez, Where Mathematics Comes From (2000)

Preface: The idea of the book is to take cognitive perspective of mathematics, and to ask "Is the system of mathematical ideas also grounded in bodily experiences?" (p. xiv) They go on to say that "many of the most fundamental mathematical ideas are inherently metaphorical in nature" -- pushing the Metaphors We Live By stuff with examples like "the number line, where numbers are conceptualized metaphorically as points on a line" (xvi). The big challenge apparently, is that Núñez' wants to get all forms of "actual infinity" to work in the metaphor concept. Good luck, I say.

Introduction: Two questions frame the debate they're setting up on page 1.
Exactly what mechanisms of the human brain and mind allow human beings to formulate mathematical ideas and reason mathematically? "Is brain-and-mind-based mathematics all that mathematics is? Or is there, as Platonists have suggested, a disembodied mathematics transcending all bodies and minds and structuring the universe -- this universe and every possible universe?" (p.2)

So, let's question some of the assumptions in these two questions. Are the brain and mind are separate? Are they constructed of "mechanisms" (like a clock, I suppose)? Are they separate from human beings (in a power-position of "allowing")? The second question I make no bones about. I guess this is telling of my own thoughts/understandings (that there is no transcendental disembodied mathematics). They lay out the argument for the second question by considering that (essentially) *all* things carried out or investigated by humans is done with human language, human senses, human logic, and therefore biased. "It is sometimes assumed that the effectiveness of mathematics as a scientific tool shows that mathematics exists in the structure of the physical universe. This, of course, is not a scientific argument with any empirical scientific basis" (p.3).

[Ed note: It's clear to me that mathematics just isn't my bag. In Csikszentmihalyi's language of Flow, I'm apathetic to it. I suppose that if I knew it better I'd be more interested, but now I feel like I have too many other things that I need to focus on and no time to learn it, which is what I'd need to do in order to appreciate it more. It's like piano that way.]

They make a big deal about embodied realism (Lakoff & Johnson 1999) (reviewed nicely on this page this page:

Lakoff, a distinguished professor of linguistics at the University of California, Berkeley, and Johnson, head of the philosophy department at the University of Oregon, begin with "three major findings of cognitive science": first, that the mind "is inherently embodied"; second, that thought "is mostly unconscious"; and third, that abstract concepts "are largely metaphorical."

"More than two millennia of a priori philosophical speculation about these aspects of reason are over," the two declare in their introduction, for these findings "are inconsistent with central parts of Western philosophy. They require a thorough rethinking of the most popular current approaches, namely, Anglo-American analytic philosophy and postmodernist philosophy."

The best way to see why is to examine how these findings alter the concept of reason. "Reason," the authors state, is still viewed as "the defining characteristic of human beings." It includes "not only our capacity for logical inference, but also our ability to conduct inquiry, to solve problems, to evaluate, to criticize, to deliberate about how we should act, and to reach an understanding of ourselves, other people, and the world."

Reason, according to cognitive science, however,

is not disembodied, as the tradition has largely held, but arises from the nature of our brains, bodies, and bodily experience.... The very structure of reason itself comes from the details of our embodiment. The same neural and cognitive mechanisms that allow us to perceive and move around also create our conceptual systems and modes of reason. Thus, to understand reason we must understand the details of our visual system, our motor system, and the general mechanisms of neural binding.

Reason, in short, is not independent of perception and bodily movement, and neural associations take place between perceptual and inferential acts. Our bodies and brains determine the kinds of categories we will form for making sense of experience. So, for instance, our spatial notions of "in front of" and "in back of" derive from our being creatures with fronts and backs who project that distinction onto objects like cars and TVs. Reason is also "evolutionary, in that abstract reason builds on and makes use of forms of perceptual and motor inference present in 'lower' animals." That discovery "utterly changes our relation to other animals and changes our conception of human beings as uniquely rational."

According to Lakoff and Johnson, reason is therefore not universal in the sense of being transcendent--it is "not part of the structure of the universe." That led some early readers to tag the pair as "relativist" or even "multiculturalist," which they deny. They acknowledge that reason may be widely or universally (if contingently) shared by humans because of our similar bodies, a position they call "embodied realism," as distinct from the philosophical tradition's "disembodied realism." As such, reason is not "completely conscious, but mostly unconscious." It is "not purely literal, but largely metaphorical and imaginative," not dispassionate but rather "emotionally engaged."

If all this is true--and Philosophy in the Flesh attempts to demonstrate it by applying "embodied realism" to classic metaphysical puzzles (such as time and causation) and the history of philosophy itself--out goes much of our philosophical baggage from "major classical views of what a person is." Goodbye to the Cartesian subject, with a mind independent of the body. So long to Kant's radically autonomous person, because reason doesn't transcend the body. Adieu to the ideal utilitarian agent, since embodied humans don't control most of their reasoning, let alone the part that maximizes self-interest. Equally dispensable is the fashionable image of the mind as purely computational--fungible software working on fungible hardware--because real, embodied minds do not merely manipulate empty symbols.

So what does this all mean to me? It means that they're going to try to dispel the myth that mathematics is a Platonically disembodied Truth using what they know about cognitive science. It will be a tough task, because in order to dispel the myth of Science, and "Truth", you probably have to assert another "truth", which folks often take as only confirming the existence of Truth. In the end, you rarely convince any but those alread in the choir. Tough.