Behold! A giant integral stretching off into the distance.
There’s not much worse than being a fledgling PhD student, approaching the daunting task of thesis-writing, finally beginning to stand on your own academic feet and have your own research ideas, and having a senior figure in your field shoot your work down in flames.
That’s not quite what happened to me recently, but it’s close. I’ve recently had the pleasure of being able to talk through some of my work to some visiting academics. Most met the discussion with the usual polite interest, but the most senior person I met with was quite critical of the methods I use, and it got me thinking a little more deeply about the limitations of my techniques.
As I understand it, his main objection was that I’m attempting to simultaneously use a multitude of complex techniques on a system that is itself highly complicated. It’s not that the professor thought I’d done anything wrong per se, more that he wasn’t confident that the results could be trusted. His advice was to be wary of building such a tall structure on such a shaky foundation.
I agree with his sentiment, of course, but I more strongly agree with another thing he said later. Ultimately, the only thing that justifies why we can do what we do is whether or not we get results. In the absence of the correct, formal mathematical tools, we do the best we can with the tools at hand. Time after time, the real world has revealed itself to not just be stranger than we suppose, but stranger than we can suppose, and often understanding the failure of a theory tells you more about the real world than its successes. If we don’t try, in other words, we don’t get anywhere at all.
That, of course, is the difference between a theoretical physics calculation and a pure mathematics proof. In maths, a proof is absolute and eternal. A mathematical calculation can’t be called a ‘proof’ until all the loopholes are closed and there’s no remaining possibility of it being wrong. In theoretical physics, though, there’s not really any such thing as a proof. Mostly because there’s so much of reality that we just don’t know about and don’t understand. At the start of a mathematical proof, you’ve got to choose your axioms, but in physics we don’t have a full set of axioms, just an incomplete knowledge of how our world works.
We take what we know of the world around us and marry it with what we can know from mathematics and together try to use both to understand what we see and predict new things. It’s very easy to do a piece of maths that is mathematically correct, but physically wrong. Like the famous Banach-Tarski paradox, for example, where you can take apart an object and put it back together only to find you suddenly have two objects, each identical in size to the one you started with. That one is an obvious case of a mathematical operation at odds with physics reality, but it’s surprisingly easy to accidentally do maths that contradicts reality in more subtle ways.
A work-in-progress draft of a current paper I’m working on. (Blurry on purpose – I’ve nearly been scooped a few times, I don’t want anyone seeing it clearly til it’s done!)
At the heart of all the research I’ve done so far lies a mathematical operation that, on first glance, looks like one of those. I’ll spare you the details, but it boils down to a mathematical trick that has unclear physical implications. It’s chief justification, though, is that it works. In cases where it can be verified by other theoretical methods, it gives the same results. In cases where the predictions can be experimentally checked, the experimental results seem consistent with the method. So as far as anyone can tell, this method works. But a mathematician would be induced to apoplexy by this rough and ready justification for why we’re allowed to apply this formula in the way that we do. (Wouldn’t be the first time a mathematician disagreed with a physicist over a mathematical formula.)
It comes down to a matter of style, and that I think is where the professor differed from me. Some prefer to solve comparatively simple problems exactly, using formally correct maths every step of the way, getting to a solution that is both mathematically correct and physically realistic. It’s a perfect solution, in other words, but one that’s only valid for an absurdly over-simplified problem. Others prefer to tackle more complex, realistic problems using whatever weapons they have in their arsenal, knowing that the problem can’t be solved exactly but hoping that our techniques will be good enough to gain some understanding, even if it is an incomplete one.
Naturally, it depends on what field you’re in and what problem you’re trying to solve, but given how naïve our starting points often are (such as the famous one I study), I think it’s somewhere between a miracle and a coincidence when a formal mathematical treatment of an overly simplified problem manages to reproduce exactly a feature we see in experiments.
Ultimately, all of our methods are approximations. Take an oversimplified starting point and make a precise mathematical treatment of it, and you’ll arrive at a theory that makes well-controlled predictions of something too simple to exist in reality. Or, take a more-realistic-but-still-simplified starting point and use techniques that are less well controlled to arrive at predictions that are less exact, but are at least predictions based on a realistic starting point. In either case, you just have to hope that somewhere in your calculation you’ve captured enough of the important physics to end up with a prediction that matches the real world in some way. Either way, we ultimately need experiments to check, and that’s the essence of science: verification.
That’s where theoretical physicists so importantly differ from mathematicians: we have the luxury of working in the real world which our experimental colleagues can experiment on, test, and verify the mathematics and the models that we use. A mathematician can’t directly experiment on a 196882-dimensional space, but we can actually plug wires into materials and measure them, confirming or negating our theories and telling us which bits of our mathematics work and which don’t.
For the moment, all of my predictions in both my published and unpublished work are consistent with the experiments out there, plus I’ve got a few more unpublished ones up my sleeve that experiments haven’t looked for yet. Those are the really interesting ones, the tests that will either kill my techniques stone dead or will turn them from pure speculation into established methods.
One day, maybe we’ll be able to solve all problems exactly, but until then, we’ll just have to compromise. For my money, until we treat the starting points of our theories with the same exactitude, I’m happy to leave stubborn insistence on mathematical rigour at the door when dealing with extremely complex problems. The cost of this is the loss of specific quantitative predictions, instead mostly only achieving qualitative ones, but in a field specialising in emergent phenomena, I think that’s an acceptable loss. It’s messy and aesthetically unsatisfactory, a far cry from the beauty of past methods, but it’s the best we’ve got right now for the complex problems we face in modern physics.
[A postscript: Let me be very clear that when I speak of relaxing mathematical rigour, I absolutely do not mean employing flagrantly invalid methods, only that using an unjustified mathematical trick can, in my view, be justified solely because it works. This is one of those rare cases in life where the ends can be said to truly justify the means. For specialists, the two specific techniques I’m referring to are uncontrolled 1-loop momentum-shell renormalisation group and the replica trick. These are both intensely mathematical techniques, but both lacking control parameters to ensure accuracy as you’d get in, for example, a four-minus-epsilon renormalisation group technique. But then, that technique has it’s own set of problems…]