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WIRED MAGAZINE -3/14/2013
It’s Pi Day! And in honor of this year’s celebration I decided to do a bit of historical research. While π—the ratio between a circle’s circumference and its diameter—has long been known and approximated, even since ancient times, only since the Eighteenth Century has it been proven to be an irrational number. Prior to that, various approximations were given, such as even the nice round number of 3.
I was recently reading the Wikipedia page on approximations of π and noticed that it included a note saying that Maimonides—the Jewish physician and scholar who lived nearly 1,000 years ago—seems to have alluded to its irrationality in his writing. The source took me to the book The Ancient Tradition of Geometric Problems and using Amazon’s helpful Look Inside feature, I traced this putative statement to the commentary of Maimonides on the Mishnah, a collection of Jewish Law which forms part of the Talmud (... Eruvin 1:5)...The ratio between a diameter of a circle and its distance around is not known. We cannot speak about it precisely…This ratio’s reality cannot be found, but it is known in approximation…we can place an approximation of this as one to three and a seventh…
Maimonides uses the approximation for π of 22/7 which is about 3.14 and is a good approximation. In addition, he does seem to imply that any value is necessarily an approximation and can’t be known precisely...
Nevertheless, these kinds of discussions are certainly intriguing and it is exciting that scholars of the Middle Ages were perhaps already intuiting that π could never be calculated as a ratio.

IF THIS IS SIMPLE MATH, WHY THEN THE NEED FOR A PROOF TEXT?!

It would therefore seem that the point of this Chazal is not to prove an equation (for which no text would be needed), rather it is to prove that Halacha would sanction a rounded figure!

Indeed, this seems to be what was 'bothering' Maimonides, which then caused him to say that the true value of Pi is an irrational one. For, why else would a verse be needed?

In other words, it was not Rambam/Maimonides, per se, who was the first to suggest this mathematical truth, rather he was the first to recognize this from the Talmud itself.

The Talmud was simply proving that due to its irrationality Halacha has no choice but to round it off.

Archive of Rav Kornfeld -linked from Rationalist Judaism blog 3/14/2013
QUESTION: The Gemara quotes the Mishnah (end of 13b) which says that the circumference of a circle is three times greater than its diameter... How do we reconcile this statement with the mathematical fact...?
ANSWERS:
(a) The TOSFOS HA'ROSH explains that the Gemara itself addresses this issue. The Gemara asks "from where do we learn" that the circumference of a circle is three times greater than the diameter (and then brings the source from Melachim mentioned in in #3 above MT). Why does the Gemara need a source to teach the ratio of the circumference of a circle to its diameter? We do not need a verse to teach us a mathematical, observable fact! It must be that the Gemara is asking for the source that teaches us that we may use a *slightly inexact* value to determine the circumference of a circle. The Gemara answers that the verse that describes the circumference of the pool that Shlomo ha'Melech constructed (the "Yam Shel Shlomo") as *three* times its diameter (Melachim I 7:23) teaches that for all Halachic purposes we may use the approximate ratio of three to one. Similarly, the RAMBAM (Perush ha'Mishnayos; see also Hilchos Tum'as Mes 12:7) points out that pi is an irrational number, and "the exact relationship of the diameter to its circumference cannot be known and it is not possible to speak of it... its actual value cannot be perceived." He writes that the value which is commonly used in calculations is 3 1/7 (3.142857...). The Tana'im of the Mishnah rounded this number and expressed it in terms of the nearest whole integer (3).
(SIC) (c) A fascinating insight regarding the value of pi is attributed to the Vilna Ga'on. (Actually, there is no source to substantiate the claim that the Vilna Ga'on said it. The actual source for the insight may be credited to Matityahu ha'Kohen Munk (Frankfurt-London), who published the thought in the journals "Sinai," Tamuz 1962, and "ha'Darom," 1967.) In the verse that the Gemara cites as the source for the ratio of the circumference to the diameter (Melachim I 7:23), there is a "Kri" and a "Kesiv" -- a word that is pronounced differently than it is spelled. The word in the verse is written "v'Kaveh" (with the letter "Heh" at the end), but it is pronounced "v'Kav" (with no "Heh" at the end). The Gematriya of the word "Kav" is 106, and the Gematriya of the word "Kaveh" is 111. The ratio of the Kesiv (111) to the Kri (106), or 111/106, is 1.0471698. This value represents the ratio of the value for pi to 3 (3.1415094/3 = 1.0471698).

Rabbi Y. Kaganoff:
"...The Tashbeitz, a rishon, wrote a lengthy responsum on the topic, in which he presents two ways to explain why Chazal used estimates that are not precisely accurate. His first approach reaches the same conclusion as we have already found in the later poskim, that these measurements are included within the halachos of shiurim that are part of the halachahlemoshe misinai.
The second approach of the Tashbeitz, however, differs with the above-mentioned halachic conclusion. In his second approach, he contends that all the above estimates were meant for pedagogic, but not halachic purposes. The rounding of pi to three and the diagonal of a square to 1.4 were provided to make the material easily comprehensible to all students, since every individual is required to know the entire Torah. Thus, when Chazal used these estimates in calculating the laws, their intent was to enable the average student to comprehend the halachic material, not to provide the most accurate interpretation. When an actual halachiccalculation is made, it must be totally accurate. Any halachic authority involved would realize that he must use a highly accurate mathematical computation and then round either upward or downward as necessary for the specific application. (A similar position is held by Chiddushim Uviurim, Ohalos 5:6.)..."

Shul Chronicles (#74), Ami Magazine, by this author, 2012:
“Behind Curtain Number Two…”
Everything is in Torah…even Game Theory?

Last week, in anticipation for Shavous and the Torah we come to celebrate, we discussed the famous statement of Ben Bag Bag found in Pirkei Avos “Turn it over, and turn it over for everything is contained in it (the Torah)”. We explained that Ben Bag Bag was a convert (based on Tosafos, Chagiga 9b) and how he was, according to some, the very same gentile who Hillel (see Shabbos 31a) impressed upon the importance of Torah Sh’bal Peh (oral law) through the Aleph Beis. We also touched upon another Mishnah in PirkeiAvos (Last Mishnah in the third chapter) “…astronomy and gematrios are the seasonings to wisdom”, and explained the term’ gematrios’ as the secrets and mysteries contained in the Aleph Beis as well as their numerology. In fact I have heard that Artscroll’s best selling book, outside of their translations, is ‘The Wisdom of the Aleph Beis’ by Rabbi Munk.

A quick example of the brilliance of the Aleph Beis: if one takes a closer look at how an Aleph is written in a Torah they will notice that it is constructed using a vav on a slant and two yuds on each end. The value of these letters is 26 the same value of the shem hamefuresh and contained in the letter signifying the Oneness of the Ribone Shel Olam!

However there is another interpretation of the term ‘gematrios’ in the above Mishnah. The Sforno and others point out that this word is of Greek origin and is related to the word ‘Geometry’! This should not be a surprise for the tanna who taught this Mishnah, Rav Eleizer ben Chisma, is the same man mentioned in the Gemera (Horiyos 10a) as being so proficient in math and statistics that he would be able to “…estimate the number of droplets in the ocean”!...

...It should also come as no great revelation that Torah contains mathematics and probabilities for they are at the heart of many halachos. One of the marvels of the Torah is how grounded it is, and forces us to be, in reality. Whereas other systems of faith may give rules and laws, and some may even discuss issues of monetary law, none are on par with real life scenarios as is secular law. The one exception is the Torah. In fact in the introduction to Rabbi Bleich’s recently published sixth volume in his ‘Contemporary Halacha’ series, where he delves into the dizzying minutia of halachos involving issues from tort law to laws of war to kashrus, he relates the following story: He was speaking at a conference and after he finished with his presentation a famous man in the world of academia approached him. “Rabbi Bleich you saved my life and brought me to Torah observance”. Rabbi Bleich was confused, as he had never met this person before. The man went on to explain that he had been searching for a long time and very much wanted to have faith and return to yiddeshkeit however it was not until someone had suggested his books on Halacha that he was exposed to the sheer brilliance and intellectualism of Torah.

In 1985 Prof. Auman, a professor of mathematics at Hebrew University and the winner of the 2005 Nobel prize in Economics wrote a paper in the prestigious Journal of Economic Theory (36 pp. 195- 213), and later in Jewish Law and Economics titled “Game Theory in the Talmud”. He begins:

“A passage from the Talmud whose explanation eluded commentators for two millennia is elucidated with the aid of principles suggested by modern mathematical Theory of Games.

A fascinating discussion of bankruptcy occurs in the Babylonian Talmud2 (Ketubot 93a). There are three creditors; the debts are 100, 200 and 300. Three cases are considered, corresponding to estates of 100, 200 and 300….When the estate is 100, it is divided equally; since 100 is the smallest debt, this makes good sense. The case in which the estate is 300 appears based on the different – and inconsistent – principle of proportional division. The figures for an estate of 200 look mysterious; but whatever they may mean, they do not fit any obvious extension of either equal or proportional division. A common rationale for all three cases is not apparent.Over a span of two millennia, this Mishnahas spawned a large literature. Many authorities (Rif, who follows a different ruling, writes ““My predecessors discussed this Mishna and its Gemara at length and were unable to make sense of it.”) disagree with it outright. Others (Shmuel, in the Gemara) attribute the figures to special circumstances, not made explicit in the Mishna. A fewhave attempted direct rationalizations of the figures as such, mostly with little success….” Without getting into the specifics, he offers a brilliant interpatation that can only work with a deep knowledge of modern Game Theory.

“Turn it over…” indeed!

Even in our daily halachik lives we are surrounded by the brilliance of Torah statistics and probabilities: batul b’shishim, rov, etc. In fact any of us who drank or ate any dairy today had, perforce, relied on Talmudic probibilities. Let me briefly explain: a certain percentage of cows are going to have treifos, for this reason we check them after slaughter. But what about milk –we can’t check slaughter and check the cow before milking it! We therefore rely on probability, in that we know that only a small minority of cows in group of 100 will be deemed treif. But this is only helpful when we get our milk from one or two cows, today however dairy farms utilize hundreds – some of them with certain treifos – and then all the milk is stored together (mixed) in a big tank.

So intrigued was I by all of the above that I decided I would give a class on the topic of probability in Halacha. My luck that a not-yet-frum statistician from UC Berkley was in town and had decided to attend the shiur. In truth having him there allowed the others to understand that what we were saying was not pie-in-the-sky outreach fluff, rather true science as seen through the study of Torah.

What sparked the interest of this statistician more than any example we brought was based on a famous debate we are all taught in school. The Mishnah (Yevomus 61) teaches that in order to fulfill the mitzvah of pru u’revu (be fruitful and multiply) one would need a minimum of a boy and a girl according to Beis Hillel, and two boys according to Beis Shammai.

Prof. Jonathan Rosenberg of the University of Maryland made an interesting discovery. Since no one can simply choose to fulfill one of these views, as it is not up to us and there is a 50-50% chance of either a boy or a girl, he wondered what the average boy girl percentage would be according to each view.

Here is his theorem for Beis Shammai:
Thus if X is the number of children, P(X=2)=1/4, P(X=3)=1/8+1/8=1/4, P(X=4)=1/16+1/16+1/16=3/16, etc.
Amazingly it will be the view of Beis Shammai who requires two boys that will result in the greater number of girls in klal Yisroel!

This Shavous let us celebrate the Torah for all we know it is and for the brilliance contained within it that we hope to discover.