“How is it possible that mathematics, a product of human thought that is independent of experience, fits so excellently the objects of physical reality?” Albert Einstein.
In 1939, at Cambridge University, Ludwig Wittgenstein was lecturing on the Philosophy of Mathematics. By this time, with messianic certainty, he was adamant that mathematics was just a lot of linguistic convention, a bunch of tautologies based on definitions and word play. He thought that seeking mathematical proofs, along with the quest to develop a mathematics without inconsistencies, was fruitless. Wittgenstein, teaching mathematics, was essentially against mathematics.
At the same time, Alan Turing, perhaps the most important mathematician of his time, was also at Cambridge, teaching a course in mathematical logic. He was also a student in Wittgenstein’s class. He had proven certain mathematical truths that would eventually be very important in code breaking during the war, and to the future of computer programming. He could not agree that being mathematically inconsistent didn’t matter.
“The real harm of a system that contains a contradiction, will not come in unless there is an application, in which case a bridge may fall down or something of the sort.”
Turing and Wittgenstein debated each and every class. The other students were bystanders. Wittgenstein would cancel class if Turing didn’t show up. Turing gradually realized that Wittgenstein saw debate, itself, as meaningless. He eventually stopped going to the class.
The Vienna circle philosopher, Moritz Schlick, told his friend Albert Einstein of his allegiance to Wittgenstein’s thinking, finding all philosophy ‘superfluous’ and all metaphysical thinking meaningless. Schlick was the dean of the Vienna school of ‘logical positivists’, philosophers who tried to believe that only in observation, verified by experiment was true. They just did not believe that thinking, itself, could lead to truth.
Einstein, like Turing, could not agree. He studied the philosophers Kant and Mach enormously helpful in finding truth. He continuously defended the role of both experiment and theory in scientific advancement. It was not one or the other. All living creatures used thinking in some way! Concepts, as well as facts, theory as well as data were necessary.
“Physics is an attempt to construct, conceptually, a model of the Real World, as well as its law governed structure. You will be surprised by Einstein the metaphysician, but in this sense every 4 and 2 legged animal is, de facto, a metaphysician.”
Turing’s legacy is computers, Einstein’s is space travel.
Computers that have logical inconsistencies of programming will crash.
Space ships, traveling over a billion miles to encircle and land on asteroids, with inaccurate calculations of fuel and trajectory will crash.
The SpaceX robot-guided Falcon 9 rockets ride into sun-synchronous orbit, deliver satellites to geo-synchronous orbit at the speed of a bullet, and then return, decelerating from a velocity of 120,000 feet per second to a velocity of zero, in a matter of minutes, landing, intact, on a platform that is 60 square yards in size, floating at sea.
Mathematics, a product independent of human experience, is the pilot.