Letters to a Young Mathematician Read Online Free PDF

questions, one eventually hears it argued that even if they grasped nothing else, any intelligent alien would be able to comprehendsimple mathematical patterns, and the rest can be built from there. The unstated assumption is that math is somehow universal . Aliens would count 1, 2, 3, . . . just as we do. They would surely see the implied pattern in diagrams like * ** *** **** .
    I’m not convinced. I’ve been reading Diamond Dogs , by Alistair Reynolds, a novella about an alien construct, a bizarre and terrifying tower, through whose rooms you progress by solving puzzles. If you get the answer wrong, you die, horribly. Reynolds’s story is powerful, but there is an underlying assumption that aliens would set mathematical puzzles akin to those that a human would set. Indeed, the alien math is too close to human; it includes topology and an area of mathematical physics known as Kaluza–Klein theory. You are as likely to arrive on the fifth planet of Proxima Centauri and find a Wal-Mart. I know that narrative constraints demand that the math should look like math to the reader, but even so, it doesn’t work for me.
    I think human math is more closely linked to our particular physiology, experiences, and psychological preferences than we imagine. It is parochial, not universal. Geometry’s points and lines may seem the natural basis for a theory of shape, but they are also the features into which our visual system happens to dissect the world. An alien visual system might find light and shade primary, or motion and stasis, or frequency of vibration. An alien brain might find smell, or embarrassment, butnot shape, to be fundamental to its perception of the world. And while discrete numbers like 1, 2, 3, seem universal to us, they trace back to our tendency to assemble similar things, such as sheep, and consider them property: has one of my sheep been stolen? Arithmetic seems to have originated through two things: the timing of the seasons and commerce. But what of the blimp creatures of distant Poseidon, a hypothetical gas giant like Jupiter, whose world is a constant flux of turbulent winds, and who have no sense of individual ownership? Before they could count up to three, whatever they were counting would have blown away on the ammonia breeze. They would, however, have a far better understanding than we do of the math of turbulent fluid flow.
    I think it is still credible that where blimp math and ours made contact, they would be logically consistent with each other. They could be distant regions of the same landscape. But even that might depend on which type of logic you use.
    The belief that there is one mathematics—ours—is a Platonist belief. It’s possible that “the” ideal forms are “out there,” but also that “out there” might comprise more than one abstract realm, and that ideal forms need not be unique. Hersh’s humanism becomes Poseidonian blimpism: their math would be a social construct shared by their society. If they had a society. If they didn’t—if different blimps did not communicate—could they have any conception of mathematics at all? Just as we can’timagine a mathematics not founded on the counting numbers, we can’t imagine an “intelligent” species whose members don’t communicate with each other. But the fact that we can’t imagine something is no proof that it doesn’t exist.
    But I am drifting off the topic. What is mathematics? In despair, some have proposed the definition “Mathematics is what mathematicians do.” And what are mathematicians? “People who do mathematics.” This argument is almost Platonic in its perfect circularity. But let me ask a similar question. What is a businessman? Someone who does business? Not quite. It is someone who sees opportunities for doing business when others might miss them.
    A mathematician is someone who sees opportunities for doing mathematics.
    I’m pretty sure that’s right, and it pins down an important difference