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<DIV>R’ Jay F, Shachter wrote:</DIV>
<DIV>“<FONT face="Times New Roman">We need to have a secular education in order
to come to a correct<BR>judgement about the credibility of the sages who
preceded us.<BR><BR>Two examples will suffice. Rambam in his commentary to
`Eruvin 1:5<BR>says that pi is irrational (I am not able to read his commentary
in<BR>the original Arabic, I am saying this based on my reading of a
Hebrew<BR>translation). This is the Mishna that says that pi is 3.
Rambam<BR>defends the Mishna by saying that pi is of course, not 3, but
any<BR>value we give would have to be an approximation, because pi
is<BR>irrational (he does not use that word, or more precisely the
Hebrew<BR>translator does not, but from his circumlocutions that is clearly
what<BR>he means), so the only question is how accurate an approximation
we<BR>need, and 3 is good enough for the halakha, since the exact
value<BR>cannot be calculated anyway.<BR><BR>A reader without a secular
education would think that Rambam knew what<BR>he was talking about. He
did not; he was guessing (as it happens,<BR>correctly). Rambam did not
know that pi was irrational. The<BR>irrationality of pi was not proved
until 1761, and the proof (and all<BR>subsequent proofs) required mathematics
that Rambam did not have.”</FONT></DIV>
<DIV><FONT face="Times New Roman"></FONT> </DIV>
<DIV><FONT face="Times New Roman">To this I respond:</FONT></DIV>
<DIV><FONT face="Times New Roman">First, I certainly would not wish to set
myself up as one who is capable “</FONT><FONT face="Times New Roman">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. </FONT></DIV>
<DIV><FONT face="Times New Roman"></FONT> </DIV>
<DIV><FONT face="Times New Roman">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 <A title=https://en.wikipedia.org/wiki/Proof_that_%CF%80_is_irrational
href="https://en.wikipedia.org/wiki/Proof_that_%CF%80_is_irrational">https://en.wikipedia.org/wiki/Proof_that_%CF%80_is_irrational</A>),
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.</FONT></DIV>
<DIV><FONT face="Times New Roman">So I think you are far from “judging the
credibility” of the Rambam based on this example you cited.</FONT></DIV>
<DIV><FONT face="Times New Roman"></FONT> </DIV>
<DIV><FONT face="Times New Roman">Kol tuv</FONT></DIV>
<DIV><FONT face="Times New Roman">Chaim Manaster</FONT></DIV>
<DIV><FONT face="Times New Roman"></FONT> </DIV>
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