I’m re-reading John Boyd’s The Last Starship from Earth, and an interesting scenario pops up in the middle. The character, a brilliant mathematics student named Haldane IV, is under investigation and trial. One of the jurors / investigators is a leader in the State’s Department of Sociology, and he is interested in Haldane’s research into building an electronic Shakespeare. Haldane believes there to be an underlying mathematics to aesthetics, which can be replicated and built into a machine. This machine would then replace the professional category of Poets. This Sociologist offers a “business proposition” to Haldane. In exchange for clemency in court, Haldane will help build a machine to replace the profession of mathematicians. The implications that a machine could be built to pursue mathematics sparked an interesting question in my mind. Could there be a finite number of mathematical axioms & proofs, or are they infinite?
There’s an amazing YouTube channel called Vsauce, and back in December 2015, host Michael Stevens released a video on Supertasks. In this video he explores the paradoxes of infinity and how it relates to the human mind and the physical universe.
Michael discusses Zeno’s Paradoxes. You may have encountered two of them, which tend to be muddled together: the Dichotomy Paradox, and Achilles and the Tortoise (I’m guilty of this too). These two consider running / walking a distance and how you must pass a halfway point, then half of the halfway, then half again and so on. In this way there’s an infinite number of points to pass and it creates the question, “can you actually finish the course?”.
In a smooth mathematical universe, this could go on forever. I can imagine this like being stuck in an infinite loop, or a program slowly crashing. But we don’t live in a smooth, mathematical universe. Races are finished, time flows. Part of the issue is that the fundamental unit of distance can’t be divided up forever. Our racers can’t take a partial footstep, and in the Universe, we theoretically can’t have a unit any smaller than the Plank distance, nor Plank time.
The human mind can still conceptualize these infinites. Yet our universe can’t manifest these into reality. In a way, our minds extend the capabilities of our universe. Mathematics is an extension of our minds and our ability to conceive beyond the physical realm. In the end, though, is Mathematics itself a finite set? And if so, is the human mind and by extension our imagination also finite and limited? I hope not.