Where Mathematics Comes From

The term cognitive science of mathematics originates with the book "Where Mathematics Comes From: How the Embodied Mind Brings Mathematics into Being" by linguist George Lakoff and psychologist Rafael E. Núñez.

They state that the 'cognitive science of mathematics' is "an embodied theory of mathematical ideas growing out of, and consistent with, contemporary cognitive science." It holds that "mathematics is rooted in everyday human cognitive activity instead of some transcendent Platonist netherworld."

Critics, such as Tom Seigfried claim that proponents of the cognitive science of mathematics "ignore the fact that brains not only observe nature, but also are part of nature.... [and fail to explain how math can] tell of phenomena never previously suspected." Like other proponents of the particle physics foundation ontology, Seigfried considers the powers of mathematics to predict what humans will perceive as proof of its objectivity:

"Many scientists suspect that math's success signifies something deep and true about the universe, disclosing an inherent mathematical structure that rules the cosmos, or at least makes it comprehensible."... to scientists. "If math is a human invention, nature seems to know what was going to be invented."

This argument is well known and was best summarized by physicist Eugene Wigner in "The Unreasonable Effectiveness of Mathematics in the Natural Sciences," 1960: "the enormous usefulness of mathematics in the natural sciences is something bordering on the mysterious and there is no rational explanation for it."

Some argue that the building of nuclear weapons and their accounting as "useful" is evidence that humans continue to extend models even minus such understanding. Postmodern critics including John Zerzan and feminists including Marilyn Waring argue that the radically autonomous view of Number is effective at manipulating the world only from a certain point of view. And that in recent times that view has been proven not so "useful".

This is, of course, a political argument, but science is not immune to politics.

Is it simply a matter of choice to see something as open to possibilities or determined by radically autonomous factors? What choice do we have about the mathematical structures we conceive? The cognitive science of mathematics removes the certainty, including perhaps certainty of conflict.

An important empirical research question is to what degree the cognitive phenomena on which mathematics is founded are shared with other Hominidae, the Great Apes, other apes, all primates, and broader membership in the animal kingdom.

Some of the arguments surrounding this theory resemble those advanced regarding nuclear fission when it became clear that the advancement of this science presented significant danger to life on the entire planet:

Ethically, we might also ask, if our mathematical models are really only "real" from a certain trained human point of view, what are the ethical and reasonable limits of testing them? Is it fair to create nuclear winter or black holes or ever-larger particle accelerators to determine whether or not a given theory predicts outcomes of dangerous acts, or huge outlays of limited resources? And if certain concepts are not tested widely and empirically, as a result of ethics or cost or other concerns, to what degree can a given mathematical model be valued over numerology?

Similar arguments gave rise to the Precautionary Principle - advances in a cognitive science of mathematics may be due to support for that principle.

Along with acceptance of ethical limits, an important feature of 20th century sciences was the discovery of limits to the human perceptive and cognitive capacity. If mathematics has such limits, as well, it would be unsurprising, even if they turned out to be ethical, as religion has consistently claimed. Faith and reason, as theology and philosophy, alternated as the ultimate arbiter of disputes in the natural sciences for millenia. If ethical choices shape our foundation beliefs about mathematics by shaping what experiments we undertake, and if this in turn shapes our mathematical notation and our acceptance of some ideas as "real", then the theories that we accept would in fact be our own, ethical, choices.

"The two possibilities, of union and of conflict, mentioned before, both of which are conceivable," Eugene Wigner's open door to the abandonment of a unified field theory, may also open the door to abandon prediction itself. At least, insofar as it applies to ourselves and other cognitive beings. Accepting and extending a cognitive science of mathematics has ethical implications that seem impossible to avoid: if we are inventing math and imposing it on others who invent it less, then science itself may be no more than a form of mental colonization. One which we may well abandon.

Numerical simulations, particle physics experiments, or a number of other human activities that rest on a relatively-objective concept of cognition, would appear to be in danger of being sacrificed to a political objective. Whether scientists accept this radical subordination to ethics and choices, and a re-classification of themselves as subsets of the cognitive sciences, is perhaps more of a political question than a scientific one.

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