A brief survey of evolutionary explanations for why math works

Space has been given on this blog for Plantinga's argument about the difficulty natural selection has in explaining higher functions. My naturalist friends have dismissed such arguments, and this post is meant to explore that perspective. The perspective, to restate, is that natural selection is sufficient to explain the existence of higher-order cognition, such as mathematics. I will be summarizing from a paper given by Russel W. Howell, who was professor of Mathematics at Westmont College.

Recall that what I'm after are evolutionary explanations for rationality, and the first of these that Howell mentions is that natural selection selected for meaning. Evolutionary process selected for survival forms whose inner models of reality best matched reality itself. A variant of this, argued by Geoffrey Miller in his book The Mating Mind, says that any greater capacity can be sexualized. Therefore, rational operations such as logic or math may set one apart from the herd and increase one's chances for sex. Howell says of these, "These speculations, while certainly not disprovable, seem to have no good evidence in their support."

Stephen Mithin is next, with a module approach to rationality. Specialized psychological processes for this or that, once brought into contact with one another, create emergent psychological domains. Plantinga's criticism is leveled at this pillar.

And finally, the author mentions work by Pascal Boyer, whose theory he calls the byproduct hypothesis. The idea is that many higher functions of mind are not evolutionary adaptations in themselves, but are byproducts of the same. They piggyback on adaptedness. Howell writes, "If one if going to argue for something using an evolutionary framework, it behooves that person to supply a detailed model or story that will support it."

Perhaps, going forward, I can flesh out these criticisms and competing arguments with greater depth. But I hadn't had them in the blog yet, and this at least gives me a place to begin. Also, let me say this: natural selection has to have played a great role in human cognition and the existential fact of higher thinking in human beings. To deny this is to deny that matter matters and that natural processes and law have meaning and truly affect a true world. But there does not have to be an either/or. I deny sufficiency without cutting out the bone. And so I retain my Chalcedonian anthropology.

GEB 2: strange loops and the back end of materialism

GEB is a series of rooms, locks, and keys, and the first of these keys is the concept of a strange loop. Discussing the central musical idea of the work, the “Canon per Tonos” of J. S. Bach's Musical Offerings, Douglas Hofstadter writes,

What makes this canon different from any other [is that] it is so constructed that [its] ending ties smoothly onto the beginning. . . . These successive modulations lead the ear to increasingly remote provinces of tonality, so that after several of them, one would expect to be hopelessly far away from the starting key. And yet . . . the original key of C minor [is eventually] restored.

This mobius strip of a thing, traveling further and further from a point only to arrive there again, Hofstadter calls a strange loop. It is a phenomenon that “occurs whenever, by moving upwards (or downwards) through the levels of some hierarchical system [a tangled hierarchy], we unexpectedly find ourselves right back where we started.”

Strange loops occur in different ways. Hofstadter finds them in the work of M. C. Escher in, for example, “Ascending and Descending” or “Waterfall.” But implicit in all of them, he says, whether they encompass a dozen steps or only one, is the concept of infinity. “What else is a loop,” he writes, “but a way of representing an endless process in a finite way.” The representation of strange loops through modulation of musical keys or in the optical illusion of a waterfall existing on two and three dimensions at once is paradoxical—the conflict between finite and infinite. It smells like a higher mathematics and is, as he continues, a kind of translation of the Epimenides paradox or liar paradox, namely “All Cretans are liars,” when Epimenides himself was from Crete. Epimenides gives us a linguistic paradox. A self-referential statement of language which is neither true nor false. What is it?

Hofstadter goes on quite a while discussion twentieth century mathematician Kurt Gödel's discovery of a strange loop at the heart of mathematics. His description is hard to feel, being read today at the end of a continuum that begins with special relativity, quantum mechanics, and the literary term. Nevertheless “Gödel showed,” he says, “that provability is a weaker notion than truth, no matter what axiomatic system is involved.” If I understand him correctly, he is saying that Gödel discovered at the base and heart of mathematics itself that disorder is the beginning even if order is the end: that language shouldn't make sense, but does. That Escher shouldn't be able to draw his waterfall, but there it is. That Bach can write a canon without end, a canon that cannot be written by formula, a canon whose every note makes sense only as it is nested in a network of contexts and relationships, that “there is something deeper” than “mere fugality.” He scrawls a question mark across Euclid's planes and then asks how it can be done.

So what is Hofstadter after? Why paint a picture of the mystery of meaning arriving from an infinite variety of tangled disorder? Hofstadter is after one mystery: the mystery of intelligence. Where does intelligence come from? And can it arise from systems of logic when such systems exist only as they bracket acceptable possibilities—a bracketing unavailable to real intelligence. Life, the context of human intelligence, is not a bracketed thing. Life is messy and random—infinitely so. There are rules, and then there are rules about rules, and then there are rules for constructing new rules, and so on and so forth. “Sometimes the complexity of our minds seems so overwhelming that one feels that there can be no solution to the problem of understanding intelligence.” And, lest it be missed, Hofstadter is a materialist . . . a materialist face to face with the question of the meaning of life and the challenge of transcendence—which, for him, is not an option. He's painted himself into a metaphysical corner, how shall he escape?

J. S. Bach; intelligence; meaning; Kurt Gödel

GEB 1: stumble trip

Douglas R. Hofstadter, now a professor of cognitive and computer science at Indiana University, wrote Goedel, Escher, Bach: An Eternal Golden Braid in the late seventies. Today, it is a Pulitzer Prize-Winning book and something of a cult classic. Reading it gains one admission to a worldwide fraternal order. I hadn't known that. What I did know was that it asks questions about the self, about what “I” means, and about the origin of meaning and thought—and that it suggests answers. “In a word,” write Douglas R. Hofstadter, “GEB [Goedel, Escher, Bach] is a very personal attempt to say how it is that animate beings can come out of inanimate matter.”

epistemology; Douglas R. Hofstadter; materialism; subjectivity