Letters to a Young Mathematician Read Online Free PDF Page A

is.
    Hersh’s answer is what he calls the humanist philosophy. Mathematics is “A human activity, a social phenomenon, part of human culture, historically evolved, and intelligible only in a social context.” This is a description, not a definition, since it does not specify the content of that activity. The description may sound a bit postmodern, but it is made more intelligent than postmodernism by Hersh’s awareness that the social conventions that govern the activities of those human minds are subject to stringent non social constraints, namely, that everything must fit together logically. Even if mathematicians got together and agreed thatπ equals 3, it wouldn’t. Nothing would make sense.
    A mathematical circle, then, is something more than a shared delusion. It is a concept endowed with extremely specific features; it “exists” in the sense that human minds can deduce other properties from those features, with the crucial caveat that if two minds investigate the same question, they cannot, by correct reasoning, come up with contradictory answers.
    That’s why it feels as if math is “out there.” Finding the answer to an open question feels like discovery, notinvention. Math is a product of human minds but not bendable to human will. Exploring it is like exploring a new tract of country; you may not know what is around the next bend in the river, but you don’t get to choose. You can only wait and find out. But the mathematical countryside does not come into existence until you explore it.
    When two members of the Arts Faculty argue, they may find it impossible to reach a resolution. When two mathematicians argue—and they do, often in a highly emotional and aggressive way—suddenly one will stop, and say, “I’m sorry, you’re quite right, now I see my mistake.” And they will go off and have lunch together, the best of friends.
    I agree with Hersh, pretty much. If you feel that the humanist description of math is a bit woolly, that this type of “shared social construct” is a rarity, Hersh offers some examples that might change your mind. One is money. The entire world runs on money, but what is it? It is not pieces of paper or disks of metal; those can be printed or minted anew, or handed into a bank and destroyed. It is not numbers in a computer: if the computer blew up, you would still be entitled to your money. Money is a shared social construct. It has value because we all agree it has value.
    Again, there are strong constraints. If you tell your bank manager that your account contains more than his computer says it does, he does not respond, “No problem,it’s just a social construct, here’s an extra ten million dollars. Have a nice day.”
    It is tempting to think that even if we consider math to be a shared social construct, it has a kind of logical inevitability, that any intelligent mind would come up with the same math. When the Pioneer and Voyager spacecraft were sent off into space, they carried coded messages from humanity to any alien race that might one day encounter them. Pioneer bore a plaque with a diagram of the hydrogen atom, a map of nearby pulsars to show where our sun is located, line drawings of a naked man and woman standing in front of a sketch of the spacecraft, for scale, and a schematic picture of the solar system to show which planet we inhabit. The two Voyager craft carried records with sounds, music, and scientific images.
    Would an alien recipient be able to decode those messages? Would a picture like o–o, two circles joined by a line, really look like the hydrogen atom to them? What if their version of atomic theory relied on quantum wave functions instead of primitive “particle” images, which even our own physicists tell us are wildly inaccurate? Would the aliens understand line drawings, given that humans from tribes that have never encountered such things fail to do so? Would they consider pulsars significant?
    In most discussions about such