Interactive theorem prover

He even taught me the Pythagorean theorem.

Switches off the definition of default blocks like theorem

According to Pythagoras's theorem, a2 b2 c2 B2 C2.

In fact, it turns out it's equivalent to proving Fermat's Last Theorem.

It turned out to be absolutely instrumental many decades later in proving Fermat's Last Theorem.

There is obviously Einstein, Niels Bohr, de Fermat's last theorem, and I'm not sure whether that's James Coburn or Richard Harris.

I'm a biologist, and the central theorem of our subject the theory of design, Darwin's theory of evolution by natural selection.

And you can verify that this right triangle the sides work out we know from the Pythagorean theorem, that 3 squared plus 4 squared, has got to be equal to the length of the longest side, the length of the hypotenuse squared is equal to 5 squared so you can verify that this works out that this satisfies the Pythagorean theorem.

So a kid who's 14 in high school gets this version of the Pythagorean theorem, which is a truly subtle and interesting proof, but in fact it's not a good way to start learning about mathematics.

In 1945, he wrote to T.S. Eliot T he full employment policy by means of investment is only one particular application of an intellectual theorem. You can produce the result just as well by consuming more or working less.

Social choice theory in its modern and systematic form owes its rigorous foundation to the work of Kenneth J. Arrow in his 1950 Columbia University PhD dissertation. Arrow s thesis contained his famous impossibility theorem, an analytical result of breathtaking elegance and reach.

And this kind of proof is the kind of proof that you need to learn when you're learning mathematics in order to get an idea of what it means before you look into the, literally, 1,200 or 1,500 proofs of Pythagoras' theorem that have been discovered.

Well, Shannon took us, in the '40s, from here to here from a telephone being a speaker wire that degraded with distance to the Internet. And he proved the first threshold theorem, that shows if you add information and remove it to a signal, you can compute perfectly with an imperfect device.

Scrutiny of the formal reasoning that establishes the theorem shows that relying only on the preference rankings of individuals makes it difficult to distinguish between very dissimilar social choice problems. The usability of available information is further reduced by the combined effects of innocuous seeming principles that are popular in informal discussions.

The reach and relevance of social choice theory is extensive. Rather than undermining the pursuit of social reasoning, Arrow s deeply challenging impossibility theorem, and the large volume of literature that it has inspired, has immensely strengthened our ability to think rationally about the collective decision making on which our survival and happiness depend.

Arrow s theorem shows that even very mild conditions of reasonableness in arriving at social decisions on the basis of simple preference rankings of a society s individuals could not be simultaneously satisfied by any procedure. When the book based on his dissertation, Social Choice and Individual Values, was published in 1951, it became an instant classic.

Nonetheless, Arrow s impossibility theorem ultimately played a hugely constructive role in investigating what democracy demands, which goes well beyond counting votes (important as that is). Enriching the informational base of democracy and making greater use of interactive public reasoning can contribute significantly to making democracy more workable, and also allow reasoned assessment of social welfare.

Cochrane s heaviest punching is reserved for Krugman s support for President Barack Obama s fiscal stimulus. He invokes the hoary old Ricardian equivalence theorem, revived by the Harvard economist Robert Barro, according to whom debt financed spending can t have any effect, because people, seeing the higher future taxes that must pay off the debt, will simply save more.

Research errors in economics are not uncommon, but they are usually caught at an early stage, as happened once to me in a prepublication draft. Sometimes errors are not discovered until later, when they are working papers, as with Reinhart and Rogoff, or after publication, as with Nobel laureate Ken Arrow, who had to correct a mistake in the proof of his famous impossibility theorem.