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LCSH

Mathematics / Social aspects

Subject

Mathematics / Social aspects

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Content

Then, if not in physical reality, could mathematical objects exist "in the mind"? Gottlob Frege famously derided this idea. If I add up a row of figures and get a wrong answer, it's wrong even if I think it's right. The theorems of Euclid remain after Euclid's mind is buried with Euclid. So where are the objects about which mathematics is objective? The answer was given by the French philosopher/ sociologist Emile Durkheim, and expounded by the U.S. anthropologist Leslie White [8]. But social scientists aren't cited by philosophers, nor by many mathematicians. (Ray Wilder was the exception.) The universe contains things other than mental objects and physical objects. There are also institutions, laws, common understandings, etc., etc., etc. -social-historical objects. (I say just "social" for short.) We cannot think of war or money or the Supreme Court or the U.S. Constitution or the doctrine of the virgin birth as either physical or mental objects. They have to be understood and dealt with on a different level-the social level. Social entities are real. If you doubt it, stop paying your bills-stop obeying the speed limit. And social entities have real properties. That's how we manage to negotiate daily life. Social scientists don't say "object." They say "process" and "artifact" and "institution." Social processes and artifacts and institutions are grounded in physical and mental objects-mainly the brains and the thoughts of people. But they must be understood on a different level from the mental or physical. In order to decide where mathematics belongs, I must consider all three-the physical, mental, and social. I need a word that can apply to all three-physical, mental, and social worlds. "Object" seems suitable. The common connotation of "object" as only a physical entity has to be set aside. Any definite entity-social, mental, or physical-whose existence is manifested by real-life experience can be called an object. Mental objects (thoughts, plans, intentions, emotions, etc.) are grounded on a physical basis - the nervous system, or the brain. But we cannot deal with our thoughts or the thoughts of each other as physical objects-electric currents in the brain. That is why there is a "mindbody problem." And social-historical objects are on still a different level from either the mental or the physical. Now, Martin, if you recognize the existence of social objects, you ought to ask, "Since mathematical objects are neither physical nor mental, are they social?" My answer is, "Yes, that is what they are." That's controversial. It's "maverick." That doesn't mean you can dispose of it by distorting or denouncing it. That mathematics is in the minds of people, including mathematicians, is not a novelty. Everyone knows that. It's in minds connected by frequent communication, in minds that follow the heritage of past mathematicians. My claim is this: to understand what mathematics is, we need not go beyond this recognized social existence. That's where it's at. Locating mathematics in the world of social entities DOESN'T make it unreal.
Or imaginary. Or fuzzy. Or subjective. Or relativistic. Or postmodern. Saying it's really "out there" is a reach for a superhuman certainty that is not attained by any human activity. A famous mathematician said to me, "I am willing to leave that question to the philosophers." Which philosophers? Professional philosophers who are not mathematicians?! To obtain answers meaningful to us, I'm afraid we'll have to get to work ourselves. Martin, 18 years ago you talked about "dinosaurs in a clearing," in order to prove that 2 + 3 = 5 is a mathematical truth independent of human consciousness. I answered that claim in my recent book. In your review of it in the L.A. Times, you ignored my answer. In your letter to The Intelligencer, you ignore it again. You just repeat your dinosaur anecdote. I will explain again. Words like '2' '3', and '5' have two usages. Most basically, as adjectives - "two eyes," "three blind mice," "five fingers." We call them "physical numbers," though they are also used for mental and social entities. It's a physical fact that two mama bears and three papa bears together make five great big bears. To put it in more academic terms, there are discrete structures in nature, and they can occur in sets that have definite numerosity. In mathematics, on the other hand, we deal with "abstract structures," not bears or fingers or dinosaurs. In mathematics, the words '2', '3', and '5' can be nouns, denoting certain abstract objects, elements of N. As I explain above, N and its main property are not found in physical nature. Counting dinosaurs uses physical numbers, adjectives, not the abstract numbers we study in mathematics. The physical numbers apply even if we don't know about them. They are part of physical reality, not human culture. Mathematical numbers, on the other hand, are a human creation, part of our social-historical heritage. They were created, we presume, from the physical adjective numbers, by abstraction and generalization. From time to time you call me a "cultural relativist." Cultural relativists say, "Western music (for instance) is not better or worse than New Guinea music. It's different, that's all." When I say mathematics is part of human culture, there's no relativism involved. More mysterious is your conclusion: "To imagine that these awesomely complicated and beautiful patterns are not 'out there' independent of you and me, but somehow cobbled by our minds in the way we write poetry and compose music, is surely the ultimate in hubris. 'Glory to Man in the highest,' sang Swinburne, 'for Man is the master of things.'
"This song of Swinburne seems to be "coming from left field." It suddenly denies your main contention. To understand it I look at your books, Order and 5urprise [2] and The Whys of a Philosophical Scrivener [3]. In Order and Surprise [2] you write, critiquing Ray Wilder, "One may, of course, adopt any way of talking one likes, but the fact is that mathematicians do not talk like Wilder except for a few who are motivated by an intense desire to make humanity the measure of all things ... to talk in a way so far removed from ordinary language, as well as the language of great scientists and mathematicians and even most philosophers, that in my layman's opinion adds nothing to mathematical discourse except confusion." The confusion here is your own. From the substantive issue, the nature of mathematical reality, you switch to mere convenience of language, without admission or apology. More significant, you are alert to any possible "desire to make humanity the measure of all things." You do not let that pass. You react by a gratuitous attribution of motives. Against Davis and me you raise the same non-issue of language, and make a similar gratuitous attribution of motives. "It is a language that also appeals to those historians, psychologists, and philosophers who cannot bring themselves to talk about anything that transcends human experience." We can talk about the transcendental Martin. We just don't think it explains mathematics. On page 72, you write, "The view that mathematics is grounded only in the cultural process slides easily into the 'collective solipsism' that George Orwell satirizes in his novel Nineteen EightyFour. For if mathematics is in the folkways, and the folkways can be molded by a political party, then it follows that the party can proclaim mathematical laws." This easy sliding is the notorious "slippery slope" pseudo-argument. Farfetched political insinuation degrades and cheapens this controversy. Later you write: "'Matter' has a way of vanishing at the microlevel, leaving only patterns. To say that these patterns have no reality outside minds is to take a giant step toward solipsism; for, if you refuse to put the patterns outside human experience, why must you put them outside your experience?" Apart from your dubious vanishing of matter, you again resort to "the slippery slope"-toward solipsism as well as Stalinism! (This time not just an easy slide, but a giant step!) (Some opponents of Social Security called it "the first step to socialism.")