Some of Rabbi Menken’s eager adulations of Rabbi Meiselman’s book are hilarious. Rabbi Menken notes that an example of Chazal’s advanced knowledge of the natural world is that they presented Pi as being three, because this must have been because they knew it was an irrational number and cannot be expressed exactly!Rabbi Menken agrees that it would indeed be ludicrous to project current mathematical knowledge back into the distant past in order to excuse a coarse estimate, and then use that very projection to tout Chazal’s prescience. However, he claims, neither he nor Rabbi Meiselman said any such thing:
Rather, it was the Rambam who said so, 600 years before modern mathematicians reached this same conclusion. In the Rambam’s time, this statement was hardly projecting “current knowledge” back onto Chazal, because even then the nature of Pi remained unknown. On the contrary, the Rambam’s statement itself is evidence that “Chazal possessed knowledge of the physical world beyond what was known to other cultures.”He quotes Rabbi Meiselman as explaining how Rambam demonstrated that Chazal knew Pi to be an irrational number:
The Rambam gives no source for his information (that Pi is irrational). Scholars have presumed that he deduced it from Talmudic passages in which it is implied. In fact, the Rambam seems to say so almost explicitly. He writes that Chazal use an approximation for Pi rather than a fraction because it is irrational. This seems to imply that if Pi were rational there would be no justification for instituting a legal approximation rather than the appropriate fraction. The very fact that Chazal did so indicated to him that they knew it to be irrational.Accordingly, concludes Rabbi Menken, the Rambam’s statement itself is evidence that “Chazal possessed knowledge of the physical world beyond what was known to other cultures.” He is shifting responsibility away from himself and Rabbi Meiselman and instead onto Rambam. And he thus accuses me of mocking the words of Rambam.
Now, let's see what's actually going on here.
First of all, Rabbi Menken's claim that Rambam described Pi as an irrational number 600 years before anyone else is not true. What happened 600 years later was that Pi was proved to be irrational. But it was known to be irrational long before that. It seems that the early Greeks also hinted to Pi being an irrational number. Besides, where does anyone think that Rambam got the idea from that Pi is irrational? R. Meiselman quotes "scholars" as saying that he got it from the Gemara, but he suspiciously does not reference these scholars. Rambam surely didn't get it from the Gemara, or he would have said so. So presumably he got it from the mathematicians of his era, or worked it out himself. (In yet another ludicrous argument, Rabbi Menken takes this as further proof that Chazal knew Pi was irrational, because there is nowhere else that Rambam could have gotten it from!)
But what about the Gemara? Rambam says that Chazal knew that Pi was irrational, and therefore used an approximation. This is a reasonable position. Yet Rambam does NOT say, however, that the fact of Chazal using three proves that they knew it to be irrational. Rabbi Meiselman presents a highly speculative argument to this end: "This seems to imply that if Pi were rational there would be no justification for instituting a legal approximation rather than the appropriate fraction. The very fact that Chazal did so indicated to him that they knew it to be irrational." But there is simply no such implication in Rambam.
And there is a reason why Rambam would not make such an argument. That is because even if Chazal thought that Pi was rational, there would indeed be a justification for using an appropriate fraction. Perhaps Chazal thought that it was a complicated number, and therefore simply rounded it off (as many Rishonim indeed hold). One cannot prove from Chazal using the value of three that they knew it to be irrational.
And there is another possibility, too: that Chazal thought Pi was actually equal to three. Tosafos (Eruvin 14a) points out that, based on the context, the Gemara does not seem to be giving an approximation. Of course, there are various apologetics which argue otherwise, but Tosafos apparently didn't find them convincing. Thus, if someone wants to believe that the Gemara did not mean this, they can do so, but one cannot use the topic of Pi to prove that Chazal had superior knowledge of the natural world, unless one is willing to categorically dismiss Tosafos and Tosafos' arguments.
Furthermore, the Mishnah (Ohalos 12:6) says that "A square is greater than a circle by one-fourth," referring to the perimeter of each when the circle is drawn to the height of the square. This is true if Pi is assumed to be 3, but given a more accurate value of Pi, the perimeter of the square is actually closer to one-fifth longer than that of the circle.
(Some readers will doubtless find it hard to accept that Chazal believed Pi to be 3. The question is whether there is basis for their disbelief, and an analysis of the Gemara and Rishonim reveals that there were much more basic mathematical errors committed by some (but not all) of Chazal. Tosafos (Eruvin 76a) says that Rabbi Yochanan and the Gemara in Sukkah misunderstood a statement by the judges of Caesarea to mean that the diagonal of a square is equal to twice the length of its side. Tosafos states that Rabbi Yochanan subscribed to this understanding of the judges of Caesarea, and that the Gemara in Sukkah rejected it precisely because it is mathematically inaccurate. Rashba expresses surprise at Tosafos attributing a simple mathematical error to Chazal, and he gives an alternate explanation, but he does not deny that Tosafos does indeed say this! Ran likewise expresses surprise that the judges of Caesarea erred in a simple mathematical matter, and cites an alternate explanation of Rabbi Yochanan’s misunderstanding of what the judges of Caesarea were saying, which somewhat lessens the error, but still leaves Rabbi Yochanan making genuine errors of both interpretation and mathematics. Tosafos HaRosh states similarly.)
So, to conclude: We have a Rambam that does not say or imply that the fact of Chazal using Pi=3 proves that they knew it to be irrational. We further have other Rishonim who understand this Gemara as saying that Pi is a rational but complicated number, or that Pi does indeed equal three. So, yes, for Rabbi Menken/Meiselman to present this Gemara as proof of Chazal’s advanced knowledge of the natural world is indeed ridiculous. And, since Rambam did not say this, I am not mocking Rambam (though Rabbi Menken is indeed dismissing Tosafos and other Rishonim).
(Please note that I am currently in the middle of the jungle in Zimbabwe, and tomorrow I am heading to Botswana, so my internet access is sporadic!)
(The full series of critiques of Rabbi Meiselman's book is at this link)