Four color problem, odd Goldbach conjecture, and the curse of computing

GoldbachFor over twenty-three hundred years, at least since the publication of Euclid’s Elements, the conjecture and proof of new theorems has been the sine qua non of mathematics. The method of proof is at “the heart of mathematics, the royal road to creating analytical tools and catalyzing growth” (Rav, 1999; pg 6). Proofs are not mere justifications for theorems; they are the foundations and vessels of mathematical knowledge. Contrary to popular conception, proofs are used for more than supporting new results. Proofs are the results, they are the carriers of mathematical methods, technical tricks, and cross-disciplinary connections.

Of course, at its most basic level, a proof convinces us of the validity of a given theorem. The dramatic decisiveness of proofs with respect to theorems is one of the key characteristics that set mathematics apart from other disciplines. A mathematical proof is unique in its ability to reveal invalid conclusions as faulty even to the author of that conclusion. Contrast this with Max Planck’s conception of progress in science:

A new scientific truth does not triumph by convincing its opponents and making them see the light, but rather because its opponents eventually die, and a new generation grows up that is familiar with it.

Further, unlike in science, a mathematical conclusion is shown to be faulty by the proofs and derivations of other mathematicians, not by external observations. In fact, if we use Karl Popper’s falsification as the criterion for science then mathematics (as practiced) is clearly distinct. A typical statement, such as the recently proven odd (or, weak) Goldbach conjecture states:

Weak Goldbach conjecture

Every odd integer n > 5 is the sum of three primes.

Given any specific integer n, we can verify that it is the sum of three primes by looking at the finite list of primes less than n and seeing if any three of them (with repetitions allowed) add up to n. In other words, we could set a computer program running to check odd numbers one at a time, and if we ever came to one that can’t be written as the sum of three primes then we would have falsified (in the proper Popper sense) our conjecture.

“What makes this unscientific?”, you ask, “the weak Goldbach conjecture is precise and falsifiable, Popper would call it science.” The distinction is in how math is practiced, if the weak Goldbach conjecture only had to pass scientific standards of truth then it would have been an accepted result 271 years ago when Christian Goldbach first doodled it in the margins of his letter to Leonhard Euler. We wouldn’t need top mathematicians like H.A. Helfgott proving it.
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