Some Things I’ve Been Thinking About

When you think about it, math is pretty simple. I don’t mean to say that math is easy. “Simple,” rather, in a sort of ontological sense. Most of the basic formulas of mathematics are constructed from a remarkably small set of elementary operations – operations like addition, subtraction, multiplication, and division. What’s more, many of these operations are reducible to one another. When we talk about multiplication (of counting numbers, anyway), we’re really talking about iterated addition. Recall, 4 x 5 is just 4 + 4 + … + 4, five times. When we talk about exponentiation (e.g. 45), we’re really talking about iterated multiplication (4 x 4 x …  x 4, five times). Of course, there’s nothing particularly surprising about this underlying simplicity, since mathematics, in a certain sense, is invented. To a mathematical formalist, there are as many or few operations as mathematicians choose to define.

The simplicity of mathematics is an interesting observation only insofar as it relates to the structure of the physical world. The truly surprising observation is not that mathematics is generated by a small collection of interconnected operations, but that these operations appear to be sufficient for describing the natural universe. The fact is, even the most complex physical laws are expressible as equations, which are composed from a small handful of mathematical ingredients. Since our objects of study in physics are things like fields, vectors, and differential forms – often too complicated to be compressed into a single number or data point — these ingredients are more sophisticated than their arithmetic counterparts. Physicists deal with operations like gradient, curl, divergence, and surface integral. But as with arithmetic, these operations are closely intertwined. In fact the first three operations listed above are particular cases of the very general notion of a derivative, which is ubiquitous in physics. All of this leads to the rather natural question: if the laws of physics are constructed from a mere handful of mathematical operations, what do these privileged operations have in common, and what precludes other operations from making it into this list?

This is the driving question behind my research this year, and quite a bit is already known. What distinguishes the kinds of operations that play a role in physics from all the others one might wish to define is a special property called “naturality.” Although its technical definition is somewhat involved, the basic principle is simple. Mathematical objects often enjoy certain symmetries. A sphere, for example, is symmetric under arbitrary rotations in space. A mattress is symmetric under a group of symmetries generated by the three elementary “flips,” one for each axis of rotation (the technical name for this group of symmetries is the “Klein Group,” after the German mathematician Felix Klein). A natural operation is simply an operation which respects or preserves these underlying symmetries.

Thanks to the work of many mathematicians, we know a lot about natural operations. But there are a few salient gaps. My hope this year is to fill in some of these gaps, demonstrating that the operations used in physics are in fact the only natural operations, in the sense described above. Why would such a result matter? A result like this would rule out the possibility of some crucial operation that we’ve somehow missed or overlooked in physics (no one suspects this to be the case, but it would be nice to know for sure). A complete classification of natural operators would provide physicists of the future with a mathematical template for formulating physical laws. It would also, I think, raise interesting questions about the importance of naturality in physics, which I believe have not been adequately answered.

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