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Date

17. 7.1996 9:33:22

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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. 37-39). 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.