Word: maths
(lookup in dictionary)
(lookup stats)
Dates: during 1990-1999
Sort By: most recent first
(reverse)
Bizarre consequences, Godel showed, come from focusing the lens of mathematics on mathematics itself. One way to make this concrete is to imagine that on some far planet (Mars, let's say) all the symbols used to write math books happen--by some amazing coincidence--to look like our numerals 0 through 9. Thus when Martians discuss in their textbooks a certain famous discovery that we on Earth attribute to Euclid and that we would express as follows: "There are infinitely many prime numbers," what they write down turns out to look like this: "84453298445087 87863070005766619463864545067111." To us it looks...
...that different from such familiar notions as "prime number," "odd number" and so forth. Thus earthbound number theorists could, with their standard tools, tackle such questions as, "Which numbers are M.P. numbers, and which are not?" for example, or "Are there infinitely many non-M.P. numbers?" Advanced math textbooks--on Earth, and in principle on Mars as well--might have whole chapters about M.P. numbers...
...thus, in one of the keenest insights in the history of mathematics, Godel devised a remarkable statement that said simply, "X is not an M.P. number" where X is the exact number we read when the statement "X is not an M.P. number" is translated into Martian math notation. Think about this for a little while until you get it. Translated into Martian notation, the statement "X is not an M.P. number" will look to us like just some huge string of digits--a very big numeral. But that string of Martian writing is our numeral for the number...
...Godel's statement is true, it is not a theorem in their textbooks and will never, ever show up--because it says it won't! If it did show up in their textbooks, then what it says about itself would be wrong, and who--even on Mars--wants math textbooks that preach falsehoods as if they were true...
...interest--turn out to be incomplete because they are able to express statements that say of themselves that they are unprovable. And that, in a nutshell, is what is meant when it is said that Godel in 1931 demonstrated the "incompleteness of mathematics." It's not really math itself that is incomplete, but any formal system that attempts to capture all the truths of mathematics in its finite set of axioms and rules. To you that may not come as a shock, but to mathematicians in the 1930s, it upended their entire world view, and math has never been...