Alternate Proofs

Is mathematics invented or discovered? The conviction that mathematical objects have an existence independent of any individual provides the last refuge for Platonism. The astonishing power and beauty of mathematics led the first mathematician to believe that it was discovered. Pythagoras declared "the universe is number." Mathematics discovered facts about the very substance of the universe. To Pythagoras, numbers were restricted to ratios of whole numbers. The first crisis in mathematics occurred when it was proven that the diagonal in a square could not be expressed as a ratio of whole numbers, which was a consequence of the Pythagorean theorem. The knowledge was scandalous, since it eroded Pythagorean ontology. Legend has it that when Hippasus revealed this discovery to the Greek world, Pythagorus killed him.

The numbers that could not be expressed as ratios of whole numbers came to be called irrational. Each time new numbers were needed to solve particular problems in mathematics, their existence was doubted, but they eventually became incorporated into mathematics. When mathematics required the use of the roots of negative numbers, their existence was called into question, but they eventually became incorporated as the imaginary numbers.

Each time mathematics experiences a crisis, the mathematical expands to include what was previously considered non-mathematical, and then the universe is again number. Any disturbances in mathematics get smoothed over, so that Platonists can claim the a posteriori coherence of mathematics was a priori all along. However, attention to the poetic ironies of mathematics betrays its ontological crises. Mathematical certainty is heralded as the paragon of logical rationality, but its crises leave a paradoxical linguistic residue.

The Queen of Science is irrational, imaginary, incomplete, and undecidable.

The same poetic irony is present in the symbol for equality, which first appeared in Robert Recorde's The Whetstone of Whitte in 1500: "And to avoide the tediouse repetition of these woordes: is equalle to: I will sette as I doe often in woorke use, a paire of parralles, or Gemowe lines of one lengthe, thus: =, bicause noe 2, thynges, can be moare equalle." The advent of non-Euclidean geometries demonstrated that those 2 thynges can be not equal, which negates the very authority of the equal sign. Hence, we can say unequivocally