My back pain continued today. I did manage to get some things done, but was too tired to do any math besides reading more of Infinite Powers.
One thing Strogatz relates in that book is that Greek mathematicians emphasized geometry partly because they had no way of expressing irrational numbers, such as $\pi$, except in geometric terms: as lengths, areas, and their ratios. If you think about it, that has not changed much. We may have discovered more contexts in which $\pi$ or $\phi$ crop up, but we can still only express them exactly as solutions to particular problems. Any effort to write them down in isolation is only an approximation.
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