Saturday, August 29, 2009

Mathematical Errors

Rewriting Jewish Intellectual History: A Review of Sefer Chaim Be’Emunasom

Part 9: Mathematical Errors

In the introduction to Chapter 5, R. Schmeltzer stresses that “attributing ‘error’, Heaven forbid, in any way, to Chazal’s words in the halachah and its reasons and its details that are explained in the Gemara, is heresy, Heaven forbid, in the concept of Torah min HaShamayim.” This may sound fairly normative, but R. Schmeltzer is not merely referring to accepted halachic opinions in the Gemara. He reiterates that “every single word and letter of Chazal was received from Sinai” (emphasis added). R. Schmeltzer states that this even applies to statements that the Gemara rejects as being refuted or that the Gemara determines were said badusa (“in error”).

Now, this is not only bizarre, it is also clearly not the approach of most Rishonim. In a footnote (any positions of the Rishonim that explicitly refute R. Schmeltzer’s approach and which I have managed to raise to public attention are only ever dealt with in a footnote), R. Schmeltzer discusses the statement of Tosafos (Eruvin 76b) that Rabbi Yochanan and the Gemara in Sukkah erred in interpreting a position stated by the judges of Caesarea. R. Schmeltzer places the word “erred” in quotes, and proceeds to explain that one should not, Heaven forbid, think that Tosafos means that it is an error in the ordinary sense of term. Instead, it was certainly a legitimate alternative viewpoint and was certainly something that was received at Sinai. In a circular argument, R. Schmeltzer claims that if Rabbi Yochanan’s statement was truly an error, it would be bittul Torah to study it (a view that numerous people have also heard from Rav Moshe Shapiro).

But this is clearly not the meaning of Tosafos. The kind of error being discussed by Tosafos is not one of sevara or methodology, where different viewpoints are possible, and where one can say that eilu v’eilu divrei Elokim chaim. Rather, it is a mathematical error. Tosafos 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. This is a simple mathematical statement, and it is one that is in error. 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.

Further confirmation of this understanding of Tosafos (as if any were needed) can be found in the other Rishonim. 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. Tosafos Rid expresses surprise that Rabbi Yochanan and the judges of Caesarea erred in such a simple mathematical matter, and leaves it as an unresolved difficulty, but does not say that it is inconceivable for any error to have been made. Yet R. Schmeltzer does not cite any of these Rishonim.

R. Schmeltzer does cite the Vilna Gaon, who states that one should not, Heaven forbid, state that there was an error here. But this is exactly why the Vilna Gaon gives an alternate explanation of the entire passage in the Gemara. The Vilna Gaon does not deny that Tosafos was indeed attributing a genuine error!

Thus, R. Schmeltzer has totally ignored the views of the Rishonim, and has misrepresented the view of the Vilna Gaon. But he was forced to do so; since they refute his insistence that everyone is obligated to believe that “every single word and letter of Chazal was received from Sinai.”


  1. The truth about these kind of mathematical sugyos is that the whole way that the Gemara and Rishonim discuss the subjects demonstrates without a doubt that they had no concepts of higher mathematics. Today, anyone with a grade-school education would just say, "square-root of 2" and know that that means about 1.414. Who would waste time arguing about such things and giving clever geometric proofs?

  2. What is your view of the most rational way to understand the development of the Torah she'baal peh? How much can logically be understood to have been given at Sinai and passed down through the generations, and how much can be assumed to have developed later?

  3. Daniel,
    It is sufficient in a critique to address the argument as it is presented by it's author and supporters. One need not have any counter-theory to the argument from authority that all of Torah is from Har Sinai. All one needs is a single piece of evidence to the contrary. Any evidence to the contrary invalidates the authority of the person making the argument.
    Assuming the author will not change his views, he needs to explain why HaShem and Moshe are transmitting errors from Sinai. You can't have it both ways.

    Gary Goldwater

  4. I know a counter-theory is not needed; I'm just interested how Natan himself understands Torah she'baal peh.

  5. Ephraim, the modern decimal system is very recent. The ancients had a lot of trouble doing arithmetic that we take for granted. Calculating square roots to arbitrary accuracy is really difficult without some form of positional notation or something similar. Nevertheless, this is an extreme example of a basic mathematical error. The Greeks and Romans had pretty decent estimate for the square root of 2. Indeed, even before that, the Babylonians estimated it as close to 1 + 24/60 + 50/60^2 using their base sixty system. This is accurate to 3 decimal places (so gives you 1.414). If I'm not mistaken for practical purposes, the Romans and Greeks both used the approximation 17/12 which is a very good approximation in the sense that it is unusually good a denominator of that size. This is related to the theory of continued fractions (the Greeks didn't have this and so calculated such fractions more or less by luck).

    Incidentally, the error here is still quite astounding and an explanation like Rashba's for what is going on really seems to make the most sense in this context.

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