Not too much time ago, Caltech string postdoc Natalie Paquette posted an essay
That's a cute and simple story but she goes well beyond the treatment described in popular books. On the other hand, this picture of mathematics and physics as peaceful players who just care about the well-being of the other seems a bit one-sided and oversimplified. You might even argue that it's politically correct.
A week ago or two, Xi Yin, a young and brilliant Harvard professor (whom I know well after the years we were meeting) and an ultramarathon runner, wrote a more delicate response to Paquette's article:
So you may be reminded that lots of important mathematical and physical discoveries in the past were mocked by other people and groups – mocked as just mathematics, just theology, vague rubbish, and so on. Xi starts with mathematics in ancient Egypt and gets to the newest insights in quantum field theory and string theory that has incorporated quantum field theory.
To focus on the innovative big ideas, he argues that
the deepest and most far-reaching ideas of physics are not the most elegant or beautiful, but the ideas that are confusing, not rigorous, improperly formulated, or, in fact, utterly incomprehensible to mathematicians.Right, this is pretty much a definition of the "real disruption" within mathematics and physics. A genius has some good intuitive or rational reasons to push a really new idea. Exactly because it's revolutionary, it looks ugly and confusing to others. But it wins, changes the landscape of ideas, and is often polished so that it becomes a part of physics and/or rigorous mathematics.
The same basic theme is reiterated at the end of Xi's essay:
I believe that part of the job of a theoretical physicist is to make the lives of mathematicians miserable. There are, incidentally, few things I can think of that could make a mathematician more miserable than reading Leonard Susskind’s papers.Right. If you're a theoretical physicist, it's your job to make mathematicians sorry that they were born. To fulfill your obligations, you may dump tons of Susskind's papers on the mathematicians' heads – that could help. ;-) Susskind has produced lots of intuition, much of it has evolved into extremely important physical insights, and although I don't have the soul of a rigorous mathematician, even I often tend to choke when tons of Susskind's papers are dumped on my head – especially if you pick Susskind's papers that are just garbage.
These differences between mathematics and physics were recently touched on this blog when (in the discussion about Woit and quantum mechanics) I mentioned Feynman's monologue about the difference between the Babylonian and Greek mathematics. To simplify things a bit, he said that the Greeks – especially Euclid – found and elaborated upon the method to agree on axioms and fixed foundations which have to be taken very seriously and to build theorems upon them.
(Feynman's Greek-vs-Babylonian theme was taken from his Second Messenger Lecture out of seven at Cornell – I recommend you to watch the whole thing.)
In Babylonia, if we simplify a bit, they cared about getting as many truths about mathematics and Nature as they could and they didn't really care what was the axiom and what was the theorem. Feynman argued that the Greek approach has evolved into the modern rigorous mathematics – while the Babylonian approach was basically adopted by modern theoretical physics. In the modern mathematics of the Greek type (I hope that the Greeks won't sue late Feynman in the EU just like they harassed the Czech producers of the Greek yogurt and/or yogurt of the Greek type), you may start with a different set of axioms but in that case, you frame your activity as abandoning the old building and creating a new one from scratch.
In modern theoretical physics, too much rigor isn't too useful. Even more importantly, what's not important is what we call the axioms and what we call the derived facts. In physics and mathematics that is helpful to physics, the identification of the primordial postulates and the derived ones is largely arbitrary and the direction of the derivations may be reversed or permuted, often in very many ways.
You know, mathematicians care about the reduction of the number of axioms in their axiomatic systems. A smaller set of axioms makes their theory nicer and more concise. The theory looks more productive. But in physics, the ultimate goal is to produce as many correct predictions and explanations as you can. So although the "axioms of a physical theory" play a similar role as the mathematical axioms, they're just a subset of correct statements about Nature. The separation of the correct statements about Nature to axioms and derived propositions is largely arbitrary – a social convention – and we want the total number of correct statements that we can make to be as large as possible, anyway. So we shouldn't be too obsessed with the reduction of a subset, especially if the identity of the subset is largely arbitrary (and physically unmeasurable).
Powerful physical theories typically make it very hard or ambiguous to decide which propositions are axioms and which are derived. String theory is extreme in this sense. Important statements can be derived from others in many different directions. And one of the many dual descriptions of the same physics may be labeled as the primary while the other dual descriptions may be said to be derived. But in some sense, they are equally fundamental and equally true. When we learn about string theory, we need to start somewhere but the precise chronology of our learning is much less canonical and much less important than the total body of the truths.