The Structure of Mathematical Revolutions

Discussion of Kuhn’s account of scientific revolutions often focuses on the physical sciences, which is unsurprising given that Kuhn was a physicist by training.

I have also felt that this physics based focus also is a reflection of how physics has become the king of the sciences. The modern world view tends to be dominated by naturalism, but naturalism understood in decidedly physicalist terms. So when we see a fundamental reordering of our concepts of the physical, such as say space, time, matter, field and so on, or the introduction of new concepts, we get the most basic and most pure of scientific revolutions.

So we focus on examples, or counter examples, coming to us from physics as revolutions in physics go to the most basic of our categories regarding intelligibility of the natural world. For this reason, revolutions in physics also tend to open up the prospect of unification between different branches of inquiry.

For example, a revolution in physics might lead to a resolution of the mind-body problem. An interesting question, of course, is whether, in the absence of such a revolution, the mind-body problem is a “pseudo problem” to shamelessly borrow from the logical positivists.

Let us leave that problem, pseudo or no, aside.

I am interested here in exploring how it is that we might fit certain aspects of Kuhn’s model to mathematics, and was drawn to this question by thinking about the abc conjecture and epistemic coherence.

We have three positive integers, a,b,c, having no prime factors in common and that satisfy a+b=c. Denote d as the product of the distinct prime factors abc. The conjecture states that there are a finite amount triples (a, b, c) satisfying a+b=c with c>d.

That is a crude formulation born of my inability on this computer to reproduce notation. For more see Wolfram.

What is interesting about the abc conjecture is that it is considered an “important theorem” because its proof has implications for so many other theorems of number theory, that is those theorems are dependent upon the conjecture being true.

The abc conjecture therefore is critical for the coherence, or the stable structure of the web of knowledge as it were, of number theory. Critical to its coherence is its seeming implications for our understanding of the relationship between multiplication and addition. Furthermore, different formulations of it open the way to deeper understanding of the way that number theory and geometry are interrelated. That is to say, it promises to open the door to unification.

The conjecture we might say is “anomalous.” There it sits in the web of knowledge, critical to its tapestry, yet unproven. The anomaly needs to be addressed, hence its importance.

Now consider a purported proof by Mochizuki. The proof is impenetrable, as it is formulated in concepts and notation of the author’s invention. This does not mean it is wrong of course. But what is interesting is how the proof offers greater levels of abstraction and introduces new concepts that deepen understanding of arithmetic and geometry.

In other words, Mochizuki’s proof heralds a mathematical revolution as it deepens our understanding of fundamental mathematical concepts, offers deeper layers of abstraction, and introduces new concepts to boot.

The price of coherence, in other words, comes at the “cost” of an expansion in our conceptual armoury and/or a deepening of understanding of fundamental concepts of arithmetic and geometry, for instance.

The act of attaining coherence becomes “revolutionary.” This kind of makes sense. Theorems whose proof are crucial for coherence but remain anomalous suggest a limitation in the standard techniques and concepts of mathematical theory.

We could say that, in part, mathematical revolutions come about when an anomalous theorem, one whose proof is critical to epistemic coherence, is proved by altering our understanding of fundamental mathematical concepts and/or introduces new concepts not hitherto known.

The Riemann hypothesis, proof of which might tell us something of fundamental import regarding the distribution of prime numbers and much else besides, could also come under the category of “anomaly” in this Kuhnian sense.

Exercise: Of Hilbert’s problems presented in 1900 which were anomalous and of those resolved which led to revolution?