Tag Archives: Maseches Eruvin

Eruvin 14 Science and Torah- Matters of PI

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Recent years have brought some extremely “vibrant” and often downright hostile discussions about the correct approach to apparent contradictions between the Torah, Chazal and modern scientific knowledge.

This applies across the board from astronomy, medicine, geography, physics, mathematics, and through archeology, even to history.

There are those like my good friend, the famous “Zoo Rabbi,” Rabbi Nathan Slifkin, who after his books touching on the subject were banned by various Chareidi authorities, has made it a major part of his life’s work to restore the popularity of “Rationalist Judaism.”

This, loosely speaking, encompasses the approach of many of the Geonim, the Rambam, and many other great early authorities that statements made by Chazal which appear to conflict with nature and science are not to be taken literally, and that when Chazal express their views on scientific matters, they are basing themselves on the accepted science of the time, and not on neo-prophetic revelation.

In contrast, Rabbi Moshe Meiselman, a renowned Chareidi Rosh Yeshiva who also has an academic background, has written his own work “Torah, Chazal, and Science” with the primary intention of both condemning and refuting this view, as well as attempting to prove that even the oft-quote protagonists amongst the Rishonim have been misunderstood.

In all humility, in a series of shiurim of my own on Agada, and a Hebrew analysis that I am still working on, I have brought numerous sources, including the introduction to the Talmud attributed to Rabbeinu Shmuel haNagid (printed at the back of most traditional versions of Masechta brachos,) which explicitly say that Agada is not comparable to halacha in its divine source and authority, but rather consists of Chazal’s own interpretations of the pesukim.

I also brought the words of the Ran (drasha 5) in his Derashos that seem to say the exact opposite and define anyone who does not believe that every word of Chazal’s Agadot are sourced at Sinai as a heretic.

As complex and sometimes aggressive this debate tends to be, it is exponentially more problematic when Chazal’s apparently out of date scientific knowledge forms the basis for practical halachik rulings.

In such cases, agreeing to disagree is not even an option, as huge areas of halacha are affected and a practical decision must made. What in one view invalidates a Sukkah or an Eruv, for example, can be essential to making it valid in the other. What renders a fish permitted according to one view, might render it forbidden according to another.

The above applies both ways, but although some authorities do indeed take into account discrepancies between modern scientific knowledge and that which Chazal were presumed to have, it is virtually always when it results in a stringency and not in a leniency.

It is not my intention to take sides in this longstanding and critical debate, but rather, as is my way in general, to examine the relevant sugyas on their own merit, together with the way the Rishonim interpret them, and see what we can learn from them

In our mishna, we are told that the minimum width of the pole used to “close” the open side of a מבוי is 1 handbreadth.

If the pole is round, we are to view it as if it is square and go by the width of its diameter.

As directly measuring the diameter of a solid cylinder is tricky, the Mishna advises us to measure its circumference, something far more practical and rely on a universal ratio between the circumference of a circle and its diameter to calculate the diameter.

The ratio given by the Mishna is the number three, according to the formula:

“כל שיש בהקיפו שלשה טפחים יש בו רוחב טפח”

“any (circle) whose circumference is 3 handbreadths has a diameter of one handbreadth.

As such, it follows that so long as the circumference of the round pole is at least 3 tefachim, we can assume that the diameter meets the minimum width of 1 tefach (handbreadth).

The same principle is employed (Sukkah 7b) to measure the diameter of a circular Sukkah to ensure it meets the minimum width of 4 amos. In the same sugya, the square-root of 2 is also assumed to be 1.4.

Every student of basic mathematics is immediately faced with the fact that the Mishna’s ration of 3 to 1 appears extremely inaccurate.

The universal ration between the circumference and diameter of any circle is of course the constant pi, a little more than 3.14, which has been known for some time already to be an irrational number.

Tosfos on our daf is so bothered by this apparent contradiction that after pointing out that it seems that our Gemara understood our Mishna’s ratio of 3 to be precise, based on the continuation of the sugya and another sugya in Bava Basra, he notes that this requires further investigation, since the mathematical experts hold that 3 is certainly not the precise ratio.

One should note that Tosfos leaves this question unanswered- he does not suggest explicitly that either Chazal or the contemporary mathematicians were wrong!

In contrast, both the Rambam and Tosfos haRosh on this Mishna are adamant that the Mishna is simply giving an estimation, and each have their own approaches as to why and how this is acceptable.

Whereas this approach certainly seems more logical, we obviously need to learn the sugya and its parallel sugyas properly to see if this fits into the flow of the Gemara.

Please join me on this exciting journey:

The Gemara opens its analysis on this part of the Mishna towards the bottom of the first side of today’s daf.

It asks the simple question: מנא הני מילי – from where are these words?

The very fact that the Gemara is looking for a verse to prove a mathematical reality that should be known to all is itself indicative of something deeper at work.

The Gemara answers that we derive this from the description of the circular ים (lit “sea” but probably referring to a water feature)) that Shlomo haMelech made, which had a diameter of 10 amos and a circumference of 30.

By describing the precise measurements of this circular feature, the passuk seems to be telling us that the ration of a circle’s circumference to its diameter is 3.

Once again, the fact that a verse is brought to teach us a simple mathematical fact seems very strange.

This question is strengthened by the fact that the ancient Greeks were very familiar with the concept of PI, and although they could not measure it precisely (though they might have suspected it was an irrational number,) it seems from my research that they certainly knew that it was more than 3, and could approximate it to at least 2 decimal points as around 3.14 .

It is hardly likely that Chazal, who took their mathematics very seriously, were unaware of this common knowledge of their time.

The Gemara then asks how we account for the width of the rim itself, which needs to be added to the actual diameter before working out the ratio with the circumference.

The Gemara responds that the possuk also tells us that the rim was extremely narrow (and thus negligible in the calculations.)

Seemingly unsatisfied by the assumption that the passuk was even nominally inaccurate in its workings, the Gemara points out that however narrow the rim was, it still would widen the exterior diameter slightly and effectively change the ratio accordingly.

The Gemara concludes that the circumference of 3 tefachim mentioned in the passuk was also measured from the inside, excluding the rim.

By now, it seems blatantly obvious that Chazal seem to take this measurement extremely precisely, and Tosfos’ observations to this affect are more than understandable.

It is harder to understand the Rambam’s approach, where he claims that any fraction that cannot be accurately measured is rounded off by Chazal.

If this is true, why were Chazal so bothered by the fact that the passuk could be doing exactly the same thing?

The Tosfos haRosh goes further and interprets the flow of the Gemara entirely differently in a way that he feels backs up his claim that we are dealing with approximations.

He understands that the Gemara’s original question, “from where are these words” is not referring to the precise value of PI but rather to the very rule that it is permissible to rely on approximations.

He understands that this leniency is sourced from the very passuk that described the properties of the circle in a way that is clearly an approximation, and quite a large one at that.

He does not say how large an approximation is needed, not under which circumstances it becomes acceptable- it could be that he agrees with Rambam who limits this to an irrational number, but is also possible that he would hold the same for other improper fractions that are hard to work with.

What remains is to understand how both the Rosh and the Rambam would explain the continuation of the sugya which certainly seems to be require precision rather than approximation.

Furthermore, even if we are able to reinterpret the rest of the flow of the sugya in a way that fits with this, or to distinguish between the approximations that are permitted and the one’s the Gemara adamantly seems to reject, we are faced with a very strong difficulty from another related sugya (Bava Basra 14b)

There, the Gemara describes how in addition to the tablets, a sefer Torah was also placed in the ark that rested in the holy of holies.

Based on the view that the circumference of a Torah needs to be 6 handbreadths, the Gemara uses our ratio to show how the 2 tefach wide Torah could fit into the 2 empty tefachim that remained in the Ark after the tablets where placed therein.

The Gemara then notes that an item with a precise width of 2 cannot fit into a precise space of 2 (presumably due to friction.)

It therefore concludes that the Torah was rolled in a way that was not precisely round (the last part was folded on top of the “cylinder”), and the width therefore was less than a third of its circumference.

It seems clear once again that the Gemara is assuming the value of PI to be precisely 3- after all, if it were more than three, a circumference of 6 would produce a width of less than 2 which would easily fit in the remaining space.

Hopefully to be continued

These posts are intended to raise issues and stimulate further research and discussion on contemporary topics related to the daf. They are not intended as psak halacha.

Eruvin 12-13 Tolerance, the origins of dispute, and “”אלו ואלו

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In the Mishna on 12b, there is a 3-way dispute between Beis Shamai, Beis Hillel, and Rabbi Eliezer, regarding what is required to mark the open fourth side of a מבוי closed on the other 3 sides.

Whereas we have been working until now under the correct assumption that the lenient view requiring only one post on either side is authoritative, we see that though this is the view of Beis Hillel, Beis Shamai require a post and a beam, and Rabbi Eliezer requires 2 posts, one on each side.

In an early post )Shabbos 130b), we discussed the way Rabbi Eliezer was referred to by the Gemara there as a שמותי , a term that the Yerushalmi quoted in the second explanation in Rashi there, as well as in Tosfos, understood to mean that he belonged to Beis Shamai.

However, as we also know )Avos 2/9) that Rabbi Eliezer was one of the greatest students of Rabban Yochanan ben Zakai, who received his tradition from both Hillel and Shamai, it is very possible that he had influence from the scions of both houses.

In this Mishna, it seems that Rabbi Eliezer was neither bound completely to either Beis Hillel or Beis Shamai, but fiercely independent.

This is somehow despite the fact that he took such pride in the fact that he never said anything that he never heard from his Rabbi (Sukkah 28a.)

This can be explained by the fact that his tradition came directly from Rabban Yochanan ben Zakai who learnt from both Hillel and Shamai, as mentioned above, and thus sometimes was in line with Beis Shamai and sometimes with Beis Hillel.

It can also be that what he meant was not that he heard every precise ruling from his Rebbe, but that he always followed the methodology he learnt from his Rebbe, no matter what conclusion it led to, thus ironically adding to his independence in the realm of practical halacha.

It is a common experience that independents, despite often being revered on both sides of the aisle, usually struggle for acceptance on either side.

Perhaps Rabbi Eliezer was both great and confident enough to have his feet in both worlds, but still able to interpret the Torah and rule according to his own view, even taking a separate stand from both strong “councils of sages.”

Even for the usually tolerant members of Beit Hillel, despite his having learnt from some of their teachers, he was never “Beis Hillel” enough, but a “שמותי”

This independence and consequent lack of acceptance came to a head in the case of the stove of Achnai (Bava Metzia 59b), where his refusal to accept the majority opinion of all the other sages resulted in his virtually unprecedented excommunication, which according to the first explanation in the above-quoted Rashi, was the reason for his being called a שמותי”” , from the word “שמתא” (excommunication.)

The truth is that lacking the safety in numbers that members of both main parties tend to have, independents often suffer the most, and are treated more harshly by members of the dominant party than members of the opposition are treated, even if their ideologies are somewhere in-between.

Whereas Rabbi Eliezer’s independence was virtually stamped out by his colleagues, who burnt all the things that he had declared pure, Beis Hillel are lauded towards the end of daf 13b for their respectful attitude to the views of Beis Shamai, not only quoting their views, but even mentioning them before their own, as illustrated in a Mishna in Sukkah brought by our Gemara.

My beloved son, Noam, asked me the other night, while learning mishnayos Shabbos together, why Beis Shamai are mentioned first so often in Mishnayos even though Beis Hillel are more authoritative.

I answered that this could very well be a reflection of this tolerant attitude first illustrated in the Mishna in sukkah quoted by our Gemara, which Rabbi Yehuda haNasi, a direct descendant of Hillel, carried on when he compiled the Mishna.

This tolerance of the other side, is given in our Gemara as the reason that the view of Beis Hillel became normatively accepted over that of Beis Shamai- In order for one’s view to be accepted, it seems important that one is open-minded, respectful, and confident enough to hear, consider, and even quote dissenting views. This shows that this view was acquired after fully considering all sides and without automatically putting down the other side and is thus a view worthy of acceptance.

Our Gemara points out that this preference given to Beis Hillel was not because the rulings of Shamai were not legitimate, but rather for the above reason- objectively speaking, “אלו ואלו דברי אלוקים חיים”-“these and those are the words of the living G-d!”

It might seem logically impossible that two seemingly contradictory views can both be considered objectively part of the divine Torah, but this seems to be precisely what the Gemara is saying.

The Torah is the word of the “living G-d” and thus constantly branching off into different explanations and possibilities.

So long as different views are all based on the “מסורה”, that living chain of transmission that goes back to Moshe at Sinai, it is not so much the actual conclusion that makes them legitimate, but the way that was achieved.

Not every alternative view has halachik legitimacy- only those that can be justified based on previous stages in the tradition.

Both Beis Hillel and Beis Shamai were required to back their views up with pessukim in the Torah, or oral traditions dating back to Moshe at Sinai, and the same applies to those that come after them.

It is not tolerance alone that gave Beis Hillel their authority, as reading this sugya in isolation might imply. In other places, their authority is derived on their greater numbers and on the “voice from heaven” )see Eruvin 7a for example) that proclaimed that the halacha is like them.

In fact, in the earlier sugya in this masechta, it seems that before this voice from heaven, one was permitted to choose which one of these great schools of Torah to follows, and that according to those who did not consider a voice from heaven to be authoritative, such as none other than Rabbi Yehoshua himself, this was permitted even after this voice from heaven, despite the other factors in Beis Hillel’s favor!

We have explained the idea of “אלו ואלו” with the understanding that in matters subject to debate, there is no objectively true answer, but both sides are legitimate “Torah”- the superiority of Beis Hillel is only practical, and as a result of the traits they possess, their superior numbers, and the voice from heaven.

Yet the Rambam appears to limit this idea significantly.

According to him (Mamrim 1/4), matters mentioned explicitly in the Torah or received orally through tradition tracing back to Moshe’s revelation at Sinai are never subject to debate.

Only matters that are derived though Chazal’s use of the rules for interpreting the Torah, can be subject to debate.

Even the later, were subject to the final ruling of the great court, so long as that court was still functioning, and debate was only legitimate until such a ruling was given.

Once it ceased to function, such matters that had not yet been resolved became subject to debate again.

Given that many or most of the disputes between Beis Shamai and Beis Hillel occurred at a time when the great court was still functioning, it follows that these disputes were subject to a final ruling by that court.

It is those rulings, according to the Rambam, not the tolerance of Beis Hillel or the voice from heaven, that were authoritative.

This fits well with the view that the main reason for their greater authority was their superior numbers, which would have allowed them to dominate the great court they were part of.

How the Rambam would explain the other reasons given for their authority, is subject to further analysis- it is possible, that as might often be his way, he simply regarded one source to be more authoritative in keeping with other accepted rules of halacha, and rules accordingly.

It is also possible that it is precisely that tolerance and extra willingness to engage in respectful debate that swung the majority of the Sanhedrin, including the “independents” towards their side, and that the “voice from heaven” was the מכה בפטיש (final blow) that brought them to this decision.

This also explains the harsh treatment meted out to Rabbi Eliezer by the members of Beis Hillel.

According to the Rambam above, even matters that were subject to dispute, were eventually concluded by the Sanhedrin while it functioned , and thereafter, no-one had the authority to act to the contrary.

Their opinions were still recorded out of respect, but they were now out of the realm of accepted halacha.

They might still have theoretical value in the study-halls, and even be considered “the words of the living G-d,” but the option for anyone to rule accordingly was now closed.

It follows that in post Sanhedrin times, debate in practical halacha is possible once again, and there is no threat of excommunication for those who follow their own or other minority views, but only in matters that had not already been decided by the Sanhedrin, and in matters that fit the Rambam’s strict criteria for debate.

If so, given that the Talmud itself was sealed by Ravina and Rav Ashi long after the Sanhedrin had ceased to function, how do we explain the universally accepted binding authority given to it by all followers of the מסורות?

We will have to leave this for a difference discussion, but a good place to start is the introduction of the Rambam himself to the Mishna Torah, the great masterwork we just quoted from.

Does everyone agree with the Rambam’s strict criteria regarding which matters are subject to debate, and does the huge collection of debates scattered throughout the Mishna and the Gemara back up this very strong claim? This too, will need to be left for a later discussion.

These posts are intended to raise issues and stimulate further research and discussion on contemporary topics related to the daf. They are not intended as psak halacha.

Shmuli Phillips Ari Kahn Johnny Solomon