[Avodah] Subject: Rambam and Pi - was Need For Secular Knowledge

[hankman]

R’ Jay F, Shachter wrote:
“We need to have a secular education in order to come to a correct
judgement about the credibility of the sages who preceded us.

Two examples will suffice.  Rambam in his commentary to `Eruvin 1:5
says that pi is irrational (I am not able to read his commentary in
the original Arabic, I am saying this based on my reading of a Hebrew
translation).  This is the Mishna that says that pi is 3.  Rambam
defends the Mishna by saying that pi is of course, not 3, but any
value we give would have to be an approximation, because pi is
irrational (he does not use that word, or more precisely the Hebrew
translator does not, but from his circumlocutions that is clearly what
he means), so the only question is how accurate an approximation we
need, and 3 is good enough for the halakha, since the exact value
cannot be calculated anyway.

A reader without a secular education would think that Rambam knew what
he was talking about.  He did not; he was guessing (as it happens,
correctly).  Rambam did not know that pi was irrational.  The
irrationality of pi was not proved until 1761, and the proof (and all
subsequent proofs) required mathematics that Rambam did not have.”

To this I respond:
First, I certainly would not wish to set myself up as one who is capable “to come to a correct judgement about the credibility of the sages who preceded us,” despite having some modern math and scientific knowledge. I think such a broad statement goes much too far. 

Not having any significant knowledge of Greek, I will refrain from commenting on your second example. However the first example you cite regarding Pi and the Rambam is one that I am willing to challenge. You are of course correct about the dating of the formal proof[s] (there are more than one https://en.wikipedia.org/wiki/Proof_that_%CF%80_is_irrational), date to modern, post Rambam times. However, it is very possible and probably likely that the Rambam intuited this on his own, even if without the benefit of a formal proof, that Pi had to be or was most likely to be irrational. I know I did so and probably most reasonably good students of math do so as well, long before they are aware of any formal proof for that fact. One reason for this is the well known method to approximate the value of Pi to any precision desired (even if very slow to converge) that requires little more than the knowledge of how to analyze a triangle. Simply inscribe a polygon of N equal isosceles triangles in a circle and calculate its perimeter and then divide by twice the length of the long side (= diameter of the circle). then do the same for N+1, and N+2 etc. a process in theory you could do infinitely many times for greater precision. So to intuit irrationality is very plausible even if not proven in our modern sense. Practically, note, that no matter how far you carry this exercise you will not get a repeating decimal that is the mark of the rational number.
So I think you are far from “judging the credibility” of the Rambam based on this example you cited.

Kol tuv
Chaim Manaster






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