[Avodah] Rambam and Pi

[Micha Berger]

On Thu, Apr 19, 2018 at 03:09:29AM -0400, Zev Sero via Avodah wrote:
: To put it in plain language, the very concept that one must have a
: rigorous mathematical proof for a proposition before one can state
: it as a fact didn't exist in the Rambam's day...

I fully agree. But there is an irony here.

When it comes to theolgy, the Rambam holds the chiyuv is to know, ie to
believe because they had proof -- not tradition, not pur faith, etc...

I think the Rambam believed he had a proof that pi was irrational.
However, he had a differnt definition of the word proof.

Continuing on this tangent... I think the definition of "proof" and
therefore of "Rationalism" changed so much since the Rambam's day,
the Rambam really wouldn't qualify as a "Rationalist" in our sense of
the word.

For example, science wasn't invented yet. The things the Rambam believed
about Natural Philosophy were not backed by anything comparable to the
rigor of scientific process.

(Which itself is only rigorous at narrowing down the search space.
Actual theories are constructed inductively, patterns found from a
number of examples, and only have Bayesian levels of certainty. There can
always be a black swan out there that, once found, requires replacing the
theory with a new one. But *disproving* theories? That black swan does
with certainty. Jumping back to before this parenthetic digression...)

Similarly in math. The Rambam didn't have a modern mathemetician's
definition of proof in mind. Even though he knew Euclid. But he
did and always expected knowledge to be backed by some kind of proof.

Archimedes spent a lot of time trying to "square the circle", i.e.
come up with a geometric way of constructing a square with the same
area as a circle. Numerous people tried since. This is the same
thing as failing to find the rational number that is pi.

Say the square they were looking for had sides of length s. So, the are
of the square would be s^2. To "square the circle" would mean to find a
square whose sides, s, are such that the ratio between s^2 and r^2 is pi,
so that the areas s^2 (the square) and pi * r^2 (the circle) are equal.

Repeatedly failing to find s by geometric construction eventually led
people to conclude it was a fool's errand. By the Rambam's day, it was
taken as a given that geometric construction could not find an s,
and therefore that the ratio between them, pi, was irrational.

(Weirdly, there still could have been a "black swan", the geometric
construction that hadn't yet been found. The level of confidence the
Rambam had would parellel that of a scientist believing the results of
repeated experiment, but not that of a modern methematician.)

Tir'u baTov!
-Micha

-- 
Micha Berger             Today is the 19th day, which is
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