It was my second year of graduate school. I had been in the US for a year. I had cracked my qualifiers. I knew the difference between pop tarts and Belgian waffles. I felt confident enough to start talking to non-mathematicians. After a few failed attempts, I met an economist (I will call her N) who liked me. You know in that way. Things that are important when you’re 23.
Things were going well until that fateful day when we were riding our bikes home from school. At that time I was a true believer; that the book of the universe was written in the language of mathematics. We were discussing N’s first year microeconomics class and all those “theorems” about choice and preferences. It was the perfect moment for me to put my foot into my mouth. I asserted that economics was second-rate mathematics, that no pure mathematician would ever consider economics to be the real thing.
N took my initial rubbishing of her dismal science in stride. She didn’t care much about standard economics anyway; her interests lay in chaos and complex systems. Both were hot topics those days. Complexity was the latest theory to attempt unifying everything from ecology to the economy.
I wasn’t taking any of that; I said to her that economics was about people, not software agents. Complexity is bunk, I said to her. That’s when N turned on me. She stopped her bike, looked daggers at me and hissed: “At least economics is undergoing a paradigm shift. Maths is such a conservative field; it’s been around for two thousand years and not a single paradigm shift ever. So boring!”
I was too much of a greenhorn to know what a paradigm was at that time, let alone paradigms that shifted. There was truth and then there were fads. I told her as much. Things went south between us soon after.
Why hasn’t mathematics had a paradigm shift?
N was right. Mathematics is a conservative field that doesn’t tolerate paradigm shifts. In mathematics, ontogeny recapitulates phylogeny. We begin our mathematical training with knowledge discovered (or invented, depending on what you think) a couple of thousand years ago. We continue onward from numbers to geometry to calculus to higher algebra and analysis and so on. There’s nothing a high schooler learns today that she couldn’t have learned two hundred years ago. In this age of disruption and paradigms of the week, the solidity of mathematics is refreshing in it’s retro appeal. The stability of mathematics is also it’s strength.
Still, it grates when someone asks you if there’s anything in mathematics beyond calculus. As every mathematician will tell you, new theorems are being proven everyday. I agree, but there’s something to N’s critique: while new theorems are discovered everyday, mathematicians are still proving theorems. They aren’t producing objects of some other kind. Consider this: quantum objects are rather different from classical objects. That’s why we call the shift from classical to quantum a paradigm shift. Setting aside metaphysical doubts about the existence of mathematical objects, mathematicians haven’t produced a new state of mathematical matter in a couple of hundred years. Can we think beyond the theorem?
You see this conservatism in the relative unimportance of foundations in the practice of mathematics. The foundations of mathematics had a brief period of importance in the late nineteenth and early twentieth century but now the dust has settled and mathematicians are back to proving theorems about algebraic groups and three dimensional manifolds. It’s almost as if practicing mathematicians are blind to the paradigm shifting possibilities of Cantor, Godel and Turing.
What a wasted opportunity for the intellectual revolutionary, right? Not quite. The foundational turmoil of the last century lead to the birth of computer science and informatics. Instead of a paradigm shift in an old domain, a new domain of knowledge was born.
The Theory Formerly Known as Mathematics
What we have today are two related but distinct sciences of form. The computer program is a genuine alternative to the theorem as a formal object. In some ways it’s better, for it interfaces with three-dimensional objects while mathematics is restricted to 2D interfaces such as paper and blackboards. Software can eat the world while theorems can’t, but software has a higher fad to substance ratio than mathematics. Can we combine the best aspects of the two fields?
Not right now. There isn’t an account of formal entities that takes the diversity of theorems and programs seriously while unifying that diversity into a coherent theory. What we need is a “theory formerly known as mathematics” that also happens to be the “theory formerly known as programming.” That unification will force a paradigm shift upon us, a new science of form that’s neither mathematics nor programming.
A disparate group of thinkers are already questioning the current mathematical and computational dispensation. Brian Cantwell Smith is talking about significance being more important to computation than algorithmic thinking. Brett Victor is talking about new interfaces for learning mathematics. Meanwhile, homotopy type theory is offering itself as an alternative foundation for mathematics from the highest end of mathematical prestige, . These streams of inquiry should coalesce into a larger assault on our understanding of form.
It’s possible that insights will come from the lowest rung of the mathematics ladder — from mathematics education. Sometimes, the lack of prestige can allow novel forms of experimentation. Keith Devlin has been talking about teaching mathematics with games instead of multiplication tables and place values. We know from Macluhan that the medium is the message. Once the medium of formal manipulation is a screen rather than paper, the message will also change. The new science of form might arise from the bottom-up, phylogeny reflecting ontogeny. Elementary school math’s teachers aren’t any going to win Fields medals, but perhaps there’s a greater revolution waiting to happen.