Document (#28572)
 Author
 Hersh, R.
 Title
 Reply to Martin Gardner
 Source
 Mathematical intelligencer. 23(2001) no.2, S.35
 Year
 2001
 Content
 "Dear Martin Gardner, Thanks for your interest in my writings. As everyone knows, you're the most highly respected science journalist in the world. I just counted six of your books on my shelf. Yet for interesting, mysterious reasons, you seem unable or unwilling to understand my writing about mathematical existence. Your unhappiness with me is not new. You dissed The Mathematical Experience [1] by me and Phil Davis, in the New York Review of Books. In the most recent issue of The Intelligencer [4], you return to the task. You quote "myths 2, 3 and I from my Eureka article (reprinted in What is Mathematics, Really? [5], pp. 3739). Myth 3 is somewhat off the point; I will concentrate on 2 and 4. Myth 4 is objectivity. "Mathematical truth or knowledge is the same for everyone. It does not depend on who in particular discovers it; in fact, it is true whether or not anyone discovers it." Your reaction: "What a strange contention"  to call it a myth. Myth 2 is certainty. "Mathematics possesses a method called 'proof' ... by which one attains absolute certainty of the conclusions, given the truth of the premises." Your reaction: "Can Hersh be serious when he calls this a myth'?" In a way, I understand your difficulty. In common speech, when someone says, "That's just a myth!" he means something is false, untrue. But in scholarly writing, "myth" commonly has other meanings. I wrote, on the very next page, "Being a myth doesn't entail its truth or falsity. Myths validate and support institutions; their truth may not be determinable'' About certainty, I wrote: "We're certain 2 + 2 = 4, though we don't all mean the same thing by that equation. It's another matter to claim certainty for the theorems of contemporary mathematics. Many of these theorems have proofs that fill dozens of pages. They're usually built on top of other theorems, whose proofs weren't checked in detail by the mathematician who quotes them. The proofs of these theorems replace boring details with 'it is easily seen' and 'a calculation gives.' Many papers have several coauthors, no one of whom thoroughly checked the whole paper. They may use machine calculations that none of the authors completely understands. A mathematician's confidence in some theorem need not mean she knows every step from the axioms of set theory up to the theorem she's interested in. It may include confidence in fellow researchers, journals, and referees.
Certainty, like unity, can be claimed in principlenot in practice." Now, you're talking about certainty in principle. I do too. I recognize its importance as a positive and valuable guiding myth. I also talk about certainty in practice. The whole mathematical community recognizes its value and is engaged in seeking it. Immediately following in my book: "Myth 4 is objectivity. This myth is more plausible than the first three. Yes! There's amazing consensus in mathematics as to what's correct or accepted." On page 176 I elaborate: "Mathematical truths are objective, in the sense that they're accepted by all qualified persons, regardless of race, age, gender, political or religious belief. What's correct in Seoul is correct in Winnipeg. This 'invariance' of mathematics is its very essence." On page 181: "Our conviction when we work with mathematics that we're working with something real isn't a mass delusion. To each of us, mathematics is an external reality. Working with it demands we submit to its objective character. It's what it is, not what we want it to be." So we agree. Mathematical truth is objective! Then how can a sophisticated critic like you think I don't recognize the objectivity of mathematical truth? The question, of course, is what we mean by "objective." To you, "objective" means "out there." To me, "objective" means "agreed upon by all qualified people who check it out." But I'm unwilling to leave the matter at that. Like many other people, I think objectivity is to be understood by reference to objects, things that really have the properties we discover. Mathematical objects are simply the things mathematical statements are about. Numbers, functions, operators, spaces, transformations, mappings, etc. What sort of objects are they? They're not physical objects. Aristotle already explained that the triangles and circles of Greek geometry are not physical entities. The first few natural numbers are abstracted from physical sets. But the really big natural numbers are not found in nature. The set of all natural numbers, N is an infinite set, not found in nature. We made it up. The most important property of N is mathematical inductionan axiom said to be intuitively obvious. Intuition is "in there," not "out there." Certainly the use and interest of the abstract mathematical numbers come from their close connection with physical numbers. But meaning and existence can't be untangled without acknowledging the distinction between physical numbers and mathematical numbers. We also study infinitely differentiable infinitedimensional manifolds of infinite connectivity. These are not found in the physical world. Not to mention the "big" sets of contemporary settheorists.
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. socialhistorical 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 levelthe social level. Social entities are real. If you doubt it, stop paying your billsstop 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 objectsmainly 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 threethe physical, mental, and social. I need a word that can apply to all threephysical, mental, and social worlds. "Object" seems suitable. The common connotation of "object" as only a physical entity has to be set aside. Any definite entitysocial, mental, or physicalwhose existence is manifested by reallife 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 objectselectric currents in the brain. That is why there is a "mindbody problem." And socialhistorical 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 socialhistorical 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 nonissue 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" pseudoargument. 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.")
You go on: "I am an unabashed realist (for emotional reasons.) ... if all men vanished, there would still be a sense (exactly what sense is another and more difficult problem) in which spiral nebulae could be said to be spiral, and hexagonal ice crystals to be hexagonal, even though no human creatures were around to give these forms a name." "Exactly what sense" is exactly the issue! Leaving it at that is on a par with your "out there, never mind where." I turn to The Ways of a Philosophical Scrivener [3]. This book is a confession of faith. It is eloquent, touching, and immensely learned. I was impressed by the chapters "Faith: Why I am not an atheist" and "Immortality: Why I am not resigned." Starting on page 213, you write: "That the leap of faith springs from passionate hope and longing, or, to say the same thing, from passionate despair and fear, is readily admitted by most fideists, certainly by me and by the fideists I admire. ... Faith is the expression of feeling, of emotion, riot of reason ... How can a fideist admit that faith is a kind of madness, a dream fed by passionate desire, and yet maintain that one is not mad to make the leap? ... To believe what we do not know, what we hope for but cannot seethis is the very essence of faith. ... To believe in spite of anything! This is the essence of quixotic fideism ... With hope travels faith and with faith travels belief. But because it is belief of the heart backed by no evidence, it is never free of doubt. ..." After reading this, 1 finally appreciate your bitterly ironic quote from Swinburne. A selfnamed quixotic fideist has the hubris to tell me that saying man is the creator of mathematics is the ultimate in hubris! I'm sorry, Martin. 1 never wanted to disturb anyone's hope, faith, and belief. I'm sorry. P.S. 1 intended to answer your New York Review of Books article in my book, but my editor persuaded me not to. Thanks for this chance to respond in The Intelligencer."  Footnote
 Bezugnahme auf: Gardner, M.: Is mathematics "out there"? In: The Mathematical intelligencer 23(2000) no.1, S.78
 Field
 Mathematik
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