reflecting on a story I once heard

There is a story I once heard, for the first time many years ago. Time and again I’ve encountered this story, like commuters with a nodding familiarity with one another but never speaking, but I’ve been thinking about it a lot recently, in the light of current events.

Kurt Goedel was one of the great mathematicians of the twentieth century, a logician, an explorer of rules and the consequences of rules. For me, his incompleteness theorem is one of the great mathematical theorems of all time, an unexpected result and one that continues to ring in me, even though logic is not an area I’ve spent any time exploring.

The story I heard is known as Goedel’s loophole. The short version is that one his way to become sworn in as a US citizen in 1947, with Einstein as one of his witnesses, Goedel expressed the view that he had found a legal constitutional way for the US to become a dictatorship. Goedel seems to never have written down his loophole and so we don’t know precisely his proposed loophole was; I suspect, and this is a very personal view based on what little reading I’ve done, is that he may have found a formal route but not one that was necessarily a political feasible route.

But reading the news and watching the events of the day, I’ve been wondering about the security of constitutions written and unwritten. The US has a written constitution, both the original document and an ever expanding body of interpretation around that original document, going back to the Federalist papers. The UK has an unwritten constitution, and for both constitutions (and other constitutions underpinning national governments around the world), there is an inescapable question, just how secure are these constitutions.

For me, and again this is a very personal view, we as humans like to set formal rules for our behavior, back to the Ten Commandments and the Code of Hammurabi, and we then like to spend enormous amounts of effort thinking about the fractal hinterlands of these formal rules, and where precisely the boundaries are between acceptable and unacceptable behavior within that set of formal rules.

I’m not surprised that I find myself pondering these fractal hinterlands, if only because I’ve spent some time thinking about the nature of fractal sets in my day job as a mathematician.

The rules underpinning fractals often appear to be simple rules, but one common property these sets of rules have is that very small changes of inputs can result in widely and wildly differing outputs. So for instance two mathematical cases with similar but not identical sets of inputs might result in very different outcomes; this is why the Mandelbrot set is so delightfully complicated on smaller and smaller scales. Two legal cases might have similar but not identical sets of facts but end up with very different outcomes.

Seemingly simple sets of rules, so for instance John Conway’s game of life, can result, perhaps often result, in complicated behavior. And this same basic observation seems to apply not only to mathematical sets of rules but also to legal sets of rules, and that this includes sets of rules such as national constitutions.

Looking back, I’ll admit that I’m not entirely sure of the point I was trying to make here. Perhaps it’s that we need to acknowledge and respect and understand that any set of rules will result in difficult cases of similar inputs resulting in widely different outputs. Perhaps it’s that a naive faith in seemingly simple sets of rules to provide clear answers will never be satisfied. Perhaps it’s a plea, for sources of reading where legal scholars have taken Goedel’s loophole as a serious subject of inquiry, because I’m curious.

~ by Jim Anderson on 10 July 2022.

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