The romantic's favorite mathematician.

A mathematician's guide to the news.
March 10 2005 7:27 AM

Does Gödel Matter?

The romantic's favorite mathematician didn't prove what you think he did.

Book cover

The reticent and relentlessly abstract logician Kurt Gödel might seem an unlikely candidate for popular appreciation. But that's what Rebecca Goldstein aims for in her new book Incompleteness, an account of Gödel's most famous theorem, which was announced 75 years ago this October. Goldstein calls Gödel's incompleteness theorem "the third leg, together with Heisenberg's uncertainty principle and Einstein's relativity, of that tripod of theoretical cataclysms that have been felt to force disturbances deep down in the foundations of the 'exact sciences.' "

What is this great theorem? And what difference does it really make?


Mathematicians, like other scientists, strive for simplicity; we want to boil messy phenomena down to some short list of first principles called axioms, akin to basic physical laws, from which everything we see can be derived. This tendency goes back as far as Euclid, who used just five postulates to deduce his geometrical theorems.

But plane geometry isn't all of mathematics, and other fields proved surprisingly resistant to axiomatization; irritating paradoxes kept springing up, to be knocked down again by more refined axiomatic systems. The so-called "formalist program" aimed to find a master list of axioms, from which all of mathematics could be derived by rigid logical deduction. Goldstein cleverly compares this objective to a "Communist takeover of mathematics" in which individuality and intuition would be subjugated, for the common good, to logical rules. By the early 20th century, this outcome was understood to be the condition toward which mathematics must strive.

Then Gödel kicked the whole thing over.

Gödel's incompleteness theorem says:

Given any system of axioms that produces no paradoxes, there exist statements about numbers which are true, but which cannot be proved using the given axioms.

In other words, there is no hope of reducing even mere arithmetic, the starting point of mathematics, to axioms; any such system will miss out on some truths. And Gödel not only shows that true-but-unprovable statements exist—he produces one! His method is a marvel of ingenuity; he encodes the notion of "provability" itself into arithmetic and thereby devises an arithmetic statement P that, when decoded, reads:

P is not provable using the given axioms.

So a proof of P would imply that P was false—in other words, the proof of P would itself constitute a disproof of P, and we have found a paradox. So we're forced to concede that P is not provable—which is precisely what P claims. So P is a true statement that cannot be proved with the given axioms. (The dizzy-making self-reference inherent in this argument is the subject of Douglas Hofstadter's Pulitzer Prize-winning Gödel, Escher, Bach, a mathematical exposition of clarity, liveliness, and scope unequalled since its publication in 1979.)



The Ebola Story

How our minds build narratives out of disaster.

The Budget Disaster That Completely Sabotaged the WHO’s Response to Ebola

PowerPoint Is the Worst, and Now It’s the Latest Way to Hack Into Your Computer

The Shooting Tragedies That Forged Canada’s Gun Politics

A Highly Unscientific Ranking of Crazy-Old German Beers


Welcome to 13th Grade!

Some high schools are offering a fifth year. That’s a great idea.


The Actual World

“Mount Thoreau” and the naming of things in the wilderness.

Want Kids to Delay Sex? Let Planned Parenthood Teach Them Sex Ed.

Would You Trust Walmart to Provide Your Health Care? (You Should.)

  News & Politics
The World
Oct. 22 2014 2:05 PM Paul Farmer Says Up to Ninety Percent of Those Infected Should Survive Ebola. Is He Right?
Business Insider
Oct. 22 2014 2:27 PM Facebook Made $595 Million in the U.K. Last Year. It Paid $0 in Taxes
The Eye
Oct. 22 2014 1:01 PM The Surprisingly Xenophobic Origins of Wonder Bread
  Double X
The XX Factor
Oct. 22 2014 10:00 AM On the Internet, Men Are Called Names. Women Are Stalked and Sexually Harassed.
  Slate Plus
Tv Club
Oct. 22 2014 5:27 PM The Slate Walking Dead Podcast A spoiler-filled discussion of Episodes 1 and 2.
Brow Beat
Oct. 22 2014 10:39 PM Avengers: Age of Ultron Looks Like a Fun, Sprawling, and Extremely Satisfying Sequel
Future Tense
Oct. 22 2014 2:59 PM Netizen Report: Twitter Users Under Fire in Mexico, Venezuela, Turkey
  Health & Science
Wild Things
Oct. 22 2014 2:42 PM Orcas, Via Drone, for the First Time Ever
Sports Nut
Oct. 20 2014 5:09 PM Keepaway, on Three. Ready—Break! On his record-breaking touchdown pass, Peyton Manning couldn’t even leave the celebration to chance.