Mathematician KURT GODEL

He turned the lens of mathematics on itself and hit upon his famous incompleteness theorem--driving a stake through the heart of formalism

  • Share
  • Read Later

(2 of 3)

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 like one big 46-digit number. To Martians, however, it is not a number at all but a statement; indeed, to them it declares the infinitude of primes as transparently as that set of 34 letters constituting six words a few lines back does to you and me.

Now imagine that we wanted to talk about the general nature of all theorems of mathematics. If we look in the Martians' textbooks, all such theorems will look to our eyes like mere numbers. And so we might develop an elaborate theory about which numbers could turn up in Martian textbooks and which numbers would never turn up there. Of course we would not really be talking about numbers, but rather about strings of symbols that to us look like numbers. And yet, might it not be easier for us to forget about what these strings of symbols mean to the Martians and just to look at them as plain old numerals?

By such a simple shift of perspective, Godel wrought deep magic. The Godelian trick is to imagine studying what might be called "Martian-producible numbers" (those numbers that are in fact theorems in the Martian textbooks), and to ask questions such as, "Is or is not the number 8030974 Martian-producible (M.P., for short)?" This question means, Will the statement '8030974' ever turn up in a Martian textbook?

Godel, in thinking very carefully about this rather surreal scenario, soon realized that the property of being M.P. was not all 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.

And 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 X (about which the statement itself talks). Talk about twisty; this is really twisty! But twists were Godel's specialty--twists in the fabric of space-time, twists in reasoning, twists of all sorts.

  1. 1
  2. 2
  3. 3