242 Are the Androids Dreaming Yet?

242 Are the Androids Dreaming Yet? “The hypercube of the hypotenuse is equal to the sum of the hypercubes of the other two sides” A picture of the hypercube might help you visualize things. It's quite difficult to get your head around this shape because it is hard to think in four dimensions. This seems strange because we have no problem seeing in three dimensions on flat, two-dimensional paper — it’s called a picture, but four dimensions on flat paper appears to stump us. Again there is no solution for a hypercube: no Pythagorean triple exists. Fermat’s Last Theorem asked whether this inequality for the cube and the hypercube is true for all higher dimensions — for the hyper- hypercube, the hyper-hyper-hypercube and so on. Tantalizingly, he claimed to have found a proof but wrote that it was too large to fit in the margin of his book. Its partly due to this arrogant annotation that it became the most famous puzzle in mathematics, frustrating mathematicians for nearly 400 years. Hilbert’s question back at the turn of the 20" century was whether a machine could find a proof of this conjecture by following a mechanical procedure, similar to our long multiplication example above. The puzzle was eventually solved in 1995 by Andrew Wiles, a mere 358 years after Fermat claimed to have solved it. Wiles’ proof runs to eighty pages of densely typed mathematical notation — considerably larger than the margin in which Fermat claimed his proof did not quite fit! There is an excellent book by Simon Singh — Fermat's Last Theorem - that tells the whole story. We now know for certain, thanks to Wiles, that the answer is ‘no. There are sixteen answers to the two-dimensional triangle puzzle but there is none for any higher dimension all the way up to infinity. How might a computer tackle this problem and find a proof? A computer could apply brute force and try many solutions; every combination up to 100 million has already been tried and no exception found. But, mathematicians are haunted by big mistakes of the past. There were theories they imagined to be true until someone discovered a counterexample. This sort of thing dogged prime number theorems. Mathematicians don’t like to look foolish and are suspicious of practical answers, “Well, I’ve tried it and I can’t seem to find an exception.” This sort of argument does not wash with them. That’s what engineers and physicists do. Mathematicians are better than that! Mathematicians want definitive answers; “It is certain no solution can exist’, and these sorts of answers require an understanding of the problem to see why no solution could exist. That’s a very high bar. What we need is a program that, rather than mechanically trying every possible HOUSE_OVERSIGHT_015932