Mathsquad is gaining form and impetus; in that it has gained visions. The last two meetings, with Maycock at the helm, covered proofs, last time ending powerfully on the inductive proof about the Towers of Hanoi game.
I imagine the Athenæum’s contribution to the queenbee of the sciences will be something unexpected, using our relatively scant knowledge base (in comparison to other math departments) to our advantage. Hopefully producing, like non-Euclidean geometry, like the complex field, like A. Robinson’s non-standard analysis, something interestingly tangential to the body of mathematics. Without a Gauss or Euler floating around, though, results will most likely be rather silly; or as they say in math journals “pure.”
My pragmatism extends into and over the philosophy of mathematics. I think mathematics— its language, rules, objects, and fantasies— is a human invention; a tool; shaped and hammered to form by utility and human modes of thinking. Mathematics is contigent, put another way. Contingent maybe less on historical than physical and physiological facts. Theoretically, I support the possibility of an alternate mathematics (or M* we’ll notate for fun). The degree of divergence possible between M and M* is kind of a hotbutton issue, especially because the history of mathematics has always been a dialectical reconciliation between M* and a pre-existent M.
Platonic-leaning mathematical realists readily agree that the symbolization of mathematical ideas can vary; but mathematics itself, mathematical “objects”— never. Formalists like David Hilbert, surrender more epistemically; saying that the axioms and rules of the game may vary. Once agreed upon, though, as in chess or checkers, the player is locked in (provided the rules and axioms are not too crappy to begin with). Near the orbit of the philosophy of mathematics, pragmatism seems closest in temperament to constructivists or intuitionists. They would probably hang out together, party together. But, the explications of constructivism and intuitionism have never really worded my hunches. Some constructivists have not gone far enough; constructivists like Leopold Kronecker who said that “the natural numbers come from God, everything else is man’s work.” I do not think the natural numbers come from God, for example. Before natural numbers would come concepts of identity and repetition, and these might just be ways the human brain grabs the world.
None of this— including the human invention of mathematics— dirties or discounts mathematics as a whole. The opposite is true: it gives one more reason for Man to openly brag before the Animal Kingdom.
Socrates gave up mathematics really early because, he said, he never understood how you could add one to another to get two things. Pundits and interpreters often take this as a quip; as Socrates being cute.
Nothing, however, better illustrates my point. The oldest warhorse of the mathematical realist is the undeniability of addition; that 1+1=2 is a truth that exists, embodied, in the world. But where is this embodied?
On your left is one porcelain teacup. On your right is another porcelain teacup. Now ask someone to add these teacups and what do they do? They put them closer together, with a satisfied look on their face: now they’re added. Are they sure the teacups are close enough?
Addition, in this case, seems dependent on the Gestalt psychological idea of groupings, here grouping by proximity. The teacups might not understand their own proximity. The universe might not understand why two teacups that are three feet apart are unadded; while two teacups an inch apart are perfectly added. Where is the threshhold of addition?
Other Gestalt psychological groupings might help clear shit up. We could put the teacups in a box; that way they share a common space.
This trail of ideas is not unfamiliar, not to anyone whose brushed passed the work of George Lakoff. This man is a cognitive scientist who does a brilliant job, in Where Mathematics Comes From and elsewhere, grounding the shapes and rules of mathematics into workings of the human brain, as an embodiment of mathematics.
Still though, Lakoff and Nunez are grounding mathematics, fixing it, demonstrating the necessity and inescapability of our schema and metaphors in some ways. So Lakoff is not radical enough for me; because to fully understand these schema and metaphors is, to my mind, to concieve of their negation, of alternative metaphors. Weirder metaphors that are more nonchalant about direct utility. Weirder schema that defy the Mathematical Oath of simplification. We would need to create a mathematics, M*, with axioms and rules and operations, that went against paramathematical values: elegance, simplicity, utility, feasibility. Or outside of deep cognitive concepts like identity, continuity, and repetition. That’s where genuine transgression can be made.
Sounds like a job for Mathsquad.
My private syllabus now, besides sunny detours through “A General Solution to the Game Mastermind” and “Soft Geometry,” will also begin at the beginning. Starting with arithmetic and related axioms and trying my hardest to swerve off the road, into ditches where numbers can be infinitely large, operations pointless; where new metaphors test-driven and even alternate, clunkier forms of notation hashed out. Perhaps we will make considerable progress on the mathematics governing doorknobs or accidentally trip over something John Conway would proudly call his own. We can pretend, after history has redeemed us, that our ignorance was really just Socratic irony all along.