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[Mar. 9th, 2005|11:20 pm] |
TIME BANDITS
by JIM HOLT
What were Einstein and Gödel talking about?
Issue of 2005-02-28
Posted 2005-02-21
http://www.newyorker.com/critics/atlarge/?050228crat_atlarge
Isaac Newton believed that time was regulated by a sort of cosmic grandfather clock. “Absolute, true, mathematical time, of itself, and from its own nature, flows equably without relation to anything external,” he declared at the beginning of his “Principia.” Einstein, however, realized that our idea of time is something we abstract from our experience with rhythmic phenomena: heartbeats, planetary rotations and revolutions, the ticking of clocks. Time judgments always come down to judgments of simultaneity. “If, for instance, I say, ‘That train arrives here at 7 o’clock,’ I mean something like this: ‘The pointing of the small hand of my watch to 7 and the arrival of the train are simultaneous events,’” Einstein wrote in the June paper. If the events in question are at some distance from one another, judgments of simultaneity can be made only by sending light signals back and forth. Working from his two basic principles, Einstein proved that whether an observer deems two events to be happening “at the same time” depends on his state of motion. In other words, there is no universal now. With different observers slicing up the timescape into “past,” “present,” and “future” in different ways, it seems to follow that all moments coexist with equal reality.
how could one demonstrate that mathematics could not be reduced to the artifices of logic? Gödel’s strategy—one of “heart-stopping beauty,” as Goldstein justly observes—was to use logic against itself. Beginning with a logical system for mathematics, one presumed to be free of contradictions, he invented an ingenious scheme that allowed the formulas in it to engage in a sort of double speak. A formula that said something about numbers could also, in this scheme, be interpreted as saying something about other formulas and how they were logically related to one another. In fact, as Gödel showed, a numerical formula could even be made to say something about itself. (Goldstein compares this to a play in which the characters are also actors in a play within the play; if the playwright is sufficiently clever, the lines the actors speak in the play within the play can be interpreted as having a “real life” meaning in the play proper.) Having painstakingly built this apparatus of mathematical self-reference, Gödel came up with an astonishing twist: he produced a formula that, while ostensibly saying something about numbers, also says, “I am not provable.” At first, this looks like a paradox, recalling as it does the proverbial Cretan who announces, “All Cretans are liars.” But Gödel’s self-referential formula comments on its provability, not on its truthfulness. Could it be lying? No, because if it were, that would mean it could be proved, which would make it true. So, in asserting that it cannot be proved, it has to be telling the truth. But the truth of this proposition can be seen only from outside the logical system. Inside the system, it is neither provable nor disprovable. The system, then, is incomplete. The conclusion—that no logical system can capture all the truths of mathematics—is known as the first incompleteness theorem. Gödel also proved that no logical system for mathematics could, by its own devices, be shown to be free from inconsistency, a result known as the second incompleteness theorem. |
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