Pi: A Wrap-Up
A couple months ago, in a post on World Series odds (how are those working out, Michael?), a commenter by the moniker of Kurt Osis asked a random question:
Ok now to my random question for the day. Is all human knowledge based on Pi? This just occurred to me the other day, if knowledge is based on measurement and the only objective form of measurement we have is the ratio between a circle’s circumference and diameter then is all knowledge really based on Pi?
Naturally, this sounds like just the sort of woo that I’ve decried in The Tao of Physics and The Dancing Wu Li Masters. It also smacks of “mathing up” the fuzzy ideas to give them the veneer of rigor and respectability. I’ve seen politicians do it, we’ve all seen poststructuralists do it, and there’s a lot of others that do too. And one of the very few undeniably mathy words that almost everyone knows is that blasted Greek constant , so it gets called into service a lot.
Clearly, I had to nip this in the bud.
I pointed out that this idea of wrapping things up with “measurement” really gave away that this was nonsense. I cited that curvature of spacetime throws off exactly such measurements (a point I recently brought up with Todd, but he hadn’t thrown “measurement” out there himself). At that point Kurt backtracked and said that was an idealization, and the measured discrepancies were knowledge. Of course I had forgotten about how slippery arguments can be with someone who only cares for the veneer of rigor.
Still I pressed onwards. I pointed out that has nothing to do with any real, physical measurement. The Cabibbo angle, or the fine-structure constant — those are the real-world constants that are actually interesting because there is (as yet) no reason why they have to have the values that they do.
Then the discussion moved from an unrelated post on Michael’s weblog to an unrelated post on mine.
Again, Kurt advances the “epistemic ” hypothesis as if it’s remotely coherent. Now he asserts that he was “trying to think of something independent of the number system itself”, and finally I had something. Here I made my stand:
is far from independent of the number system. It is what it is exactly because of the way the real number system is structured.
Then and there I decided to stop what I was working on about linear algebra. Instead, I set off on power series and how power series expansions can be used to express analytic functions. Then I showed how power series can be used to solve certain differential equations, which led us to defining the functions sine and cosine. Then I showed that the sine function must have a least positive zero, which we define to be .
The assumptions that have led to the definition of are just those of the real number system: we are working within the unique (up to isomorphism) largest archimedean field. There is no measurement, no knowledge, no science, and no epistemology to it. Kurt’s real question — the one he hops onto other mathematical weblogs’ unrelated comment threads to ask — is really about philosophy. He’s asking for a final answer to the entire field of epistemic research. It’s not forthcoming; not on a math weblog, not on a philosophy weblog, not anywhere. It’s been around in its current form for hundreds of years, and I don’t see a resolution on the horizon. But it certainly doesn’t lie in an accidental quirk of the real number system that society has for some reason decided to exalt far beyond its true value.