Where Mathematics Comes From
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Where Mathematics Comes From: How the Embodied Mind Brings Mathematics into Being is a book by cognitive linguist George Lakoff and psychologist Rafael E. Núñez. The book seeks to establish a cognitive science of mathematics, or a theory of embodied mathematics. It is notable mostly for the dialogue it has begun between mathematicians, linguists and psychologists about the grounding of proofs.
The book calls for and attempts to begin a cognitive idea analysis of mathematics that analyzes mathematical ideas in terms of the human experiences, metaphors, generalizations and other cognitive mechanisms giving rise to them. Ultimately, the book argues, mathematics is a result of the human cognitive apparatus and must therefore be understood in cognitive terms. This idea analysis is distinct from mathematics itself and cannot be performed by mathematicians untrained in the cognitive sciences.
Platonism in the philosophy of mathematics is rejected: all we know and can ever know is human mathematics, the mathematics arising from our brains, and the question whether a "transcendent mathematics" objectively exists is thus unanswerable and almost meaningless.
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Math is reality. Why do I care about linguists or psychologists?
Among technically literate people, there is a consensus that mathematics is a neutral point of view, indeed that if logic itself is a valid mode of investigation, mathematics must equally be one. Mathematics is in some sense "useful", and insofar as it is equally useful to two humans, it is "neutral". However, throughout the early 20th century, the foundation ontology of algebra was in doubt: Alfred North Whitehead, Bertrand Russell, and Kurt Gödel established that logic and set theory were in some sense grounded on something else, something geometric and quite "real",
In the late 20th century, a literature of mathematics and its foundations began to grow in the field of cognitive science: Amos Tversky, Daniel Kahneman, and others challenged the strict Western/dualist view of subject/object relations that had dominated mathematics since Descartes, with a growing consensus that human cognition shared a great deal of bias.
In parallel, George Lakoff and Mark Johnson developed a critique of metaphors, and a more generalized subject/relation/object model of metaphor.
Meanwhile, the postmodernists, most notably Michel Foucault, developed a deep critique of Western ethics, theology and philosophy, which focused on the absence of any model of the living and acting human body. As if René Descartes' "cogito ergo sum" was a literal, God's-eye view, of the so-called "real world", and mathematics itself objective and unchanging: always discovered, never invented. This was contrary to a growing body of evidence in quantum physics that observers did in fact alter what they observed, and that the process of human cognition itself changed "reality".
An embodied theory?
The term "embodied" gradually came to reflect views that assumed an observing body, and which took into account limits imposed by its fragility and (in some analyses) its morality. Postmodern thought diverged from mathematical thinking sharply, and body philosophers such as Marilyn Waring and John Zerzan began to bluntly question the concept of Number itself as a guide to human choices.
A 'cognitive science of mathematics' would have to unify these diverse critiques, and bridge serious professional and cultural gaps — not only within the Western world, but also with the indigenous peoples and others whose world-view was not based on math. Some of these societies had persisted and thrived for millennia without any sophisticated notion of modern algebra, although they typically had simple geometry. This was one of many anomalies Western assumptions about culture failed to explain: for example, how sophisticated building projects had been conducted without most of the algebraic methods used by modern engineers.
Bodies and senses create math?
The 'cognitive science of mathematics' as defined by George Lakoff and Rafael E. Núñez 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." In other words, that the human body and senses are what create mathematics, and that it can be shared with other humans, only because they are so similar to us. "Mathematics may or may not be out there in the world, but there's no way that we scientifically could possibly tell," Dr. Lakoff claims. Math succeeds in science, Drs. Lakoff and Nunez argue in their book, only because scientists force it to. "All the 'fitting' between mathematics and the regularities of the physical world is done within the minds of physicists who comprehend both."
Some call this postmodern refusal to accept a radically autonomous universe that doesn't care about us at all.
Brains in nature
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."
What is the agenda?
The Santa Fe Institute credits George Lakoff and Rafael E. Núñez with "(1) the grounding of arithmetic, set theory and formal logic in the brain and body. (2) The cognitive structure of actual infinity (infinity as a "thing") for a wide variety of cases: the infinite set of natural numbers, points at infinity, mathematical induction, infinite decimals and the reals, limits and least upper bounds, infinitesimals and the hyperreals, and transfinite numbers. (3) The conceptual structure characterizing the meaning of <math>e^{ix}<math>, allowing us to characterize in cognitive terms what Euler's equation <math>e^{\pi i} + 1 = 0<math> actually means and why it is true on the basis of what it means."
This attempts to answer a question that has plagued philosophers of mathematics since Bertrand Russell and Alfred North Whitehead failed to ground arithmetic in set theory and formal logic in 1912. What is math based on?
The ambitious program of the cognitive scientists is to permit each and every proof in mathematics to be traced back to the "four distinct but related processes [that] metaphorically structure basic arithmetic: object collection, object construction, using a measuring stick and moving along a path." Or, for proofs that cannot be so traced, dismissal as numerology.
Mathematics and politics
This is not the first attempt to challenge mathematics and physics as the primary arbiters of shared human reality. Since at least the early 1960s, some have argued that secret military science, the building of nuclear weapons, and their accounting as "useful" is evidence that humans continue to extend models even long after they are shown to be counter-productive to human lives. 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 - that of a "dominator culture". 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. Nor regrets. After his breakthrough theory of special relativity was exploited to build the first nuclear weapon, Albert Einstein lamented:
- "If only I had known. I would have become a watchmaker."
Mathematics of doing, mathematics of feeling
Einstein's lament highlights the issue of ethical choices in experiments - a long-standing question in science that challenges falsifiability as a test of truth - if you don't dare test a theory, how can you know that it's true? If you test it by harming bodies that you or others care about, are you gaining assent to the theory by "proof" or by "fear", i.e. does agreement depend on terror of pushing the experimenter to a test? If so, how can the experimenter know what's real?
When the first observers of a manmade nuclear explosion in 1944 saw the mushroom cloud, and noticed that the atmosphere itself had not caught fire (which was impossible to predict until that moment), did they look at each other and happily nod agreement that the mathematics had been validated? Or did they, as body philosophers would prefer, calculate the likely impact of this action on bodies, empathize with it, then recoil in horror and quit?
"Mathematics may or may not be out there in the world," but we certainly feel its impact in our bodies, e.g. as bullets obeying F=MA or as cold due to a winter eviction we suffer for lack of some quantity of credit. The surest thing we can say about mathematics is that it describes something we do and that this is correlated with other things that we feel. But is it as real to plants as it is to animals? As real for ecologies as for infrastructures?
If we humans weren't writing it down, how much of mathematics would exist?
Does mathematics apply to other life forms?
Some challenge Lakoff's claim that there is "no way that we scientifically could possibly tell," and point out that the history of cognitive science is finding ways to test theses that previously relied only on self-reporting.
When researching "the role of embodiment and its biological and cognitive constraints", an important empirical research question is to what degree the specific cognitive phenomena on which mathematics is founded are shared with other Homininae, the great apes, other apes, all primates, and broader membership in the animal kingdom. Or, for that matter, robots and other entities we might accept as being radically autonomous actors.
Can this stuff ever be proven?
There seems to be some controversy over how rigorous these proofs can be. In a detailed and lengthy response to reviewer Bonnie Gold, Lakoff laid claim to "a different job than professional mathematicians have. We have to answer such questions as: How can a number express a concept? How can mathematical formulas and equations express general ideas that occur outside of mathematics, ideas like recurrence, change, proportions, self-regulating processes, and so on? How do ideas within mathematics differ from similar (but not identical) ideas outside mathematics (e.g., the idea of "space" or "continuity")? How can "abstract" mathematics be understood? What cognitive mechanisms are used in mathematical understanding?"
This clashes of course with mathematicians' desire to prove new math in terms of old math, rather than find metaphors common to old math and the human body.
René Descartes' "cogito ergo sum" seems to be under serious challenge - an embodied theory would necessarily start with breathing, seeing, hearing and basic decisions about what to pay attention to.
How would this change the science?
This line of research seems to stick mathematics with many category biases previously considered to be the domain of political science, theology and other fields that rely on complex ontology rather than clean axioms.
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 millennia. 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.
Does this imply that certain sciences are "over"?
- "Now I am become Death, destroyer of worlds" - Robert Oppenheimer
Considering the emotional choice among "embodied" reactions to the first nuclear explosion helps highlight the fact that humans define their own tests for what "works", and therefore that their shared belief in mathematics isn't necessarily a belief in more than their own cognition and culture. If there are sciences wherein falsifiability or case-based reasoning can't be understood or settled into some reasonable method that all humans would be said to experience similarly, that suggests we may abandon some inquiries.
Eugene Wigner's "two possibilities, of union and of conflict, mentioned before, both of which are conceivable," his 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.
If mathematics is subjective, is all science "cognitive"?
The Embodiment of all mathematical abstraction is a grand project, quite possibly one that will never be completed. Among other issues, there are no small number of constituencies that benefit from assuming that math can be safely built on math, with no attention paid to bodies or their messy ethics:
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 - a fully-embodied mathematics would impose an unreasonable burden of objectivity on any scientific process that routinely employed real or complex analysis.
Whether scientists accept this radical subordination to ethics and choices, and a re-classification of their work as a subset of cognitive science, is perhaps more of a political question than a scientific one.
Critiques
One reply to Lakoff's objection of Platonism is the view that any and all worlds containing cognitive beings capable of dreaming up mathematical concepts must operate according to the principles of sentential logic.
Also, if one accepts logicism in its only coherent form, one must reject Lakoff's outright denial of a transcendental mathematics, even if one accepts the findings of his research.
See also
References
- G. Lakoff, R. Núñez: Where Mathematics Comes From, Basic Books, 2000
External links
- Statement by Lakoff and Nunez (http://www.unifr.ch/perso/nunezr/warning.html)
- reviews of Lakoff, Nunez, 2000 (http://www.unifr.ch/perso/nunezr/reviews.html)
- Lakoff responds to Gold of MAA (http://www.maa.org/reviews/wheremath_reply.html)
- American Scientist review of " Where Mathematics Comes From: How the Embodied Mind Brings Mathematics into Being." (http://www.americanscientist.org/template/BookReviewTypeDetail/assetid/14366)
- "The Unreasonable Effectiveness of Mathematics in the Natural Sciences," Eugene Wigner, in Communications in Pure and Applied Mathematics, vol. 13, No. I (February 1960). New York: John Wiley & Sons, Inc. Copyright 1960 by John Wiley & Sons, Inc. (http://www.dartmouth.edu/~matc/MathDrama/reading/Wigner.html)