Infallibility: Newsletter 41
This newsletter is likely to a bit technical, but I hope the gist of the argument is available to everyone who persists through the technicalities. In my last newsletter, I said that mathematicians and physicists are the greatest fundamentalists in the world. That seems to have caused some heartburn and disagreement, and makes me think that the assertion needs more evidence in its favor. I will focus on mathematics, since it's the subject I know better of the two.
Modern mathematics - and fields such as economics that suffer from math envy - is presented axiomatically. It’s not uncommon at all for a book on group theory to start with the axioms of group theory. The claim is that the axioms of group theory characterize all groups (and nothing but groups) and therefore, reasoning about groups can proceed entirely at an abstract level without reference to any particular group. That model is taken to an entirely ethereal level with what’s called category theory, where we can reason about a whole range of mathematical structures entirely on the premiss that the structures at hand satisfy the axiomatic demands of a known category. There’s a reason why it’s called abstract nonsense. I happen to love that abstract nonsense, but as this essay argues, we should be alert to the theological underpinnings of the nonsense.
What’s an axiom, if not an article of faith? Russell, Hilbert and others were hoping that logic will satisfy the theological impulse while substituting rational criteria for faith. We know now through the work of Godel and others that the utopian dream of logic was unfounded, that there’s no theory that satisfies the criteria set out by its proponents. At some point, we have to take a leap of faith (borne out by experience - no bridge has fallen when the math was right) that mathematics provides the foundation that supports all the other sciences. The axiomatic method is terrible for learning and intuition, but it serves as an excellent marker of faith.
There’s a reason for the separation between the scientific and mathematical problems that motivate an axioms origins and determine its popularity on the one hand and its position at the high table on the other. One might motivate the axioms of set theory or group theory or analysis on aesthetic or applied grounds, but once it becomes an axiom it takes on a different name with a life of its own.
An axiom is like the pope, who is usually Cardinal XYZ, with the XYZ being the birth name of the individual, but once he becomes pope, he takes on a new name with a new ancestry. The current one was Cardinal Bergoglio of Buenos Aires and by taking on the name Pope Francis, he has highlighted his affinity for St. Francis, my favorite Christian saint and himself not a pope. The past is erased, deliberately, since the newly crowned axiom, like the pope, is no longer subject to critique. Just as the pope is infallible, axioms are infallible and the infallibility is tied to the institution, not the individual. The pope can’t be wrong and nor can Zermelo-Frankel. In other words, the power or prestige of an axiom comes from its being an axiom, not its origins in the streets of Beunos Aires. It’s that universal belief in the rigor, accuracy and certainty that comes with the axiomatic faith that I am calling fundamentalism.
Strangely, as the foundational project of reducing mathematics to logic has failed, the fundamentalist project of spreading axioms throughout mathematics has flourished, to the extent that it’s impossible to write mathematics at a higher level without starting with axioms. It’s a deep and powerful fundamentalism; we couldn’t have proven Fermat’s last theorem without it, but it also forecloses other routes for the subject, routes that are more likely to be riddled with mistakes and errors, routes that make mathematics more empirical. As far as I can tell, there’s only one practical alternative: mathematics that’s more closely tied to physics and other sciences, which is either dismissed by purists as second rate applied mathematics or reduced to magic when mathematicians wonder how physicists like Witten guess and prove unbelievable theorems.
I think both of those two approaches to the foundations of mathematics- pure and axiomatic or physical and applied - are entering the age of diminishing returns. I will continue to talk about alternate approaches to mathematics that are more cognitive, of mathematics as a particular way of bringing our mind to problems of the world.