A Halakhic–Mathematical Problem and Its Implications (Column 683)
In the previous column I discussed what constitutes greatness in Torah. Among other things, I argued that beyond knowledge, skill, and analytic ability, one also needs common sense and familiarity with the world and with other areas of knowledge. As an example there I brought a mathematical problem from Tractate Mikva’ot. The reason is that, in my view, a decisor or scholar need not be a professional mathematician, yet for problems of this sort a professional’s focused knowledge is sometimes required, not just general understanding. Whether in practice poskim consult a mathematician or physicist (I doubt it—see below in this column) is itself in question. Still, everything here looks fairly straightforward.
A pit of drawn water into which a channel flows
Many years ago at the Technion a doctoral student who was dealing with mathematics and halakhah asked me whether I had topics that merited mathematical analysis. I suggested he work on Mishnah Mikva’ot 3:3 (I mentioned this from a different angle in column 381):
“A pit that is full of drawn water, and a channel enters into it and goes out of it constantly—its invalidity remains until it is calculated that of the original [drawn water] no more than three log remain.”
We are dealing with a pit of drawn water that we wish to use as a mikveh. If it is entirely drawn water, the mikveh is invalid (biblically). But if a channel of natural water (an ama) flows slowly into the pit, the incoming water is gradually mixed into the contents of the pit; the percentage of drawn water decreases over time. The Mishnah rules that immersion becomes valid once less than three log of the original drawn water remain in the pit. How do we know when that happens? How do we compute it?
The standard assumption among the commentators is that water entering the pit mixes uniformly with what is already there. Practically, this means a slow process in which the inflow is sufficiently weak, allowing time for thorough mixing throughout the volume. If that is the case, the water exiting the pit will contain the same proportion of drawn water as the pit’s current composition.
The Beit Yosef (Yoreh De’ah end of the relevant siman) brings two interpretations in the name of the Ra’avad for how to apply the Mishnah’s phrase “until it is calculated”:
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Full mixing (proportional outflow): We assume perfect mixing. At any moment, the water exiting contains a fraction of drawn water equal to the fraction in the pit. Under this view, if initially the pit contains 40 se’ah of drawn water, the pit remains invalid “until it is calculated” that the proportion has dropped to less than three log. A common way this was computed yields a number like 12,760 se’ah that must pass through(!), since as water flows in and out, the original 40 se’ah are diluted until only 1/8 se’ah (= 3 log) remain.
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“Half-and-half” outflow: Alternatively, because the water that entered most recently is closer to the outlet, the outflow is taken as half from the drawn water and half from the natural water (a very arbitrary assumption). Under that assumption, after a certain moderate volume has entered and left, the remaining water is essentially from the channel and the mikveh is valid.
The first interpretation assumes perfect mixing. While not a realistic description, as a first-order approximation for slow processes it can be acceptable. One can even view it as an estimate of the average proportion of drawn water (early on the outflow contains a high percentage of drawn water; later very little).
A mathematical model
Modern calculus (differential and integral) was developed precisely for such questions. For simplicity, I continue to assume a slow inflow—i.e., thorough mixing in the pit. Without that assumption an explicit calculation becomes extremely complicated and would require simulations; still, as noted, this assumption is the one commonly used by the commentators and poskim (the first Ra’avad reading). Let us examine the quality of the Beit Yosef/Ra’avad calculation under this assumption.
Let the channel’s flow rate be V se’ah per hour. Denote by P(t) the amount of drawn water (in se’ah) in the pit at time t. The pit’s total volume is 40 se’ah. At the beginning (t = 0) the drawn water is P(0) = 40.
Over a tiny time interval dt, the amount that exits is V dt (equal to what enters, since the volume stays constant). Because mixing is uniform, the fraction of drawn water in the outflow equals P/40. Hence the drawn water lost during dt is (P/40)·V dt. Therefore:
dP = − (V/40) · P · dt, i.e., dP/dt = −(V/40)·P.
The solution is the familiar exponential:
P(t) = 40 · e^{−(V/40)·t}.
We ask for the time t₀ when the remaining original drawn water equals 3 log = 1/8 se’ah:
40 · e^{−(V/40)·t₀} = 1/8 ⇒ t₀ = (40/V) · ln(320) ≈ 230/V.
To get a sense of scale: 1 se’ah is ≈ 8.3 liters (R. Ḥayim Na’eh) or ≈ 14.3 liters (Ḥazon Ish). A reasonable channel flow might be about 1 se’ah/minute (like a bathtub tap). Then t₀ is about 230 minutes—just under four hours. In that time about 230 se’ah have entered and left. Notice this result (≈ 230) holds for any flow rate (with 40 se’ah initial volume), because the required inflow volume equals V·t₀ = 40·ln(320) ≈ 230. For a different initial amount P₀, the formula is Inflow = P₀ · ln(P₀ / (1/8)).
Comparison
We can now compare this precise calculation (under the same assumptions!) with the Ra’avad/Beit Yosef arithmetic used by many. The common “full mixing” calculation found there requires on the order of 12,760 se’ah to pass through, whereas the precise result is ≈ 230 se’ah. The gap is not 55%—nor 555%—but about 5,500% (a factor of ≈ 55!).
Since we do not usually know how much water has passed through, time is a more sensible yardstick. In terms of waiting time, the Ra’avad-type computation multiplies the required waiting by ≈ 55. With a slow channel, that becomes an astronomically long wait; the pit’s water might evaporate before then.
A meta-halakhic look: what follows?
We saw that even the “precise” computation rests on unrealistic assumptions (very slow flow). So the comparison here is not between an exact result and an approximation, but between two ways of approximating the same situation. Yet the gap is dramatic. Here the Ra’avad’s method happens to be stringent; but it could just as well have been lenient by a huge factor. A stance that mechanically follows his arithmetic would—by chance—invalidate immersions for the scrupulous in cases where in fact the mikveh is already valid; conversely, in other scenarios it could lead to unwarranted leniencies.
In practice, if someone immersed once the criterion was truly met (less than three log of the original drawn water remain), do we worry that, given reliance on a crude calculation, perhaps more than three log remained? Typically, halakhah does not demand laboratory-grade checks (e.g., insect inspection with microscopes). The law is built upon reasonable estimates, not retrospective metaphysical anxieties. Still, the question of how much “control” poskim have over the metaphysical status remains (what about “timtum ha-lev,” etc.?).
Every calculation rests on assumptions; halakhic rulings likewise rest on assumptions. That is not a defect (scientific computations are always assumption-laden too; one chooses workable assumptions and estimates the error). Our issue here, however, is not the assumptions but the technique of computation. We assumed slow flow and full mixing, as did the Ra’avad. The difference lies in the mathematics used to implement that very assumption. We are not tweaking π in its tenth decimal place; we are talking about a factor of ~55 in the bottom line—whether we must wait until the equivalent of six entire pits of water have flowed through, or merely a fraction of one pit. That is an enormous, manifest error, not a tolerable approximation. Is such an error halakhically acceptable?
A note on consulting experts
For the Ra’avad and Beit Yosef one can argue “the Almighty does not exact beyond one’s means”: given the tools of their day, this is the calculation expected. But today we have better mathematical and scientific tools. Is it not reasonable to expect a contemporary posek to reach a different answer here?
Elsewhere (see cols. 325–326) I discussed cases of p’sik reisha: dragging a bench on Shabbat when it is uncertain whether a furrow will form (that is aino mitkaven), versus closing a box when one is unsure whether a fly is inside (some Acharonim saw that as a “retroactive certainty” and a biblical doubt). I explained the difference between an ontic uncertainty (ambiguity in reality itself) and an epistemic one (lack of knowledge). Even if an expert could often tell in advance, halakhah typically does not require recourse to experts for every act. Still, there are many areas where we do routinely consult experts (medicine, electricity on Shabbat, etc.). In our case—determining the status of a mikveh—there seems no reason not to obtain an expert’s opinion. The mathematics needed here is modest; a first-year student can handle it. This is consultation, not outsourcing halakhah.
“Common-sense” (ba’al-batim) halakhah and its limits
I once wrote (end of col. 397) about “grama solutions” for Shabbat devices: even if internally the mechanism is engineered as grama, to the ordinary eye there is a switch that turns on a machine; halakhah is not determined by microscopic inner processes but by the reasonable layperson’s perspective. One could adopt that stance while still insisting that, for insects, we check under a microscope; or that for our mikveh dilution we compute with maximal scientific precision. I will not expand on this here, only note that the perspectives can diverge across contexts.
Two related anecdotes illustrate the value of cross-disciplinary literacy. Prof. Shimshon Frankenthal told us of a U.S. case: by federal law, when two states transmit something to each other via a third state, they must pay the transit state—if it passes through wires. State A sent electricity to State C via State B. State B sued, claiming the electricity passed “through cables.” The defense brought a physicist who explained (citing the Poynting vector) that the power flows around the wire in the electromagnetic fields, not inside the metal; hence, they argued, it did not pass “through wires.” This is nonsense: the Poynting formulation is a dual description equivalent to current in the conductors; legally, the statute clearly meant transmission by wire, not via empty space. Yet a judge could be swayed by expert jargon. (See related.)
Similarly, an avrech once asked me whether glass is a liquid or a solid, having read that physicists sometimes call glass a liquid. For the laws of cooking on Shabbat, glass is a solid (dry, not “wet”). Physicists label it “amorphous” (non-crystalline), but that scientific classification does not answer the halakhic question. Again, without some literacy, a posek might take an expert’s statement out of its proper context.
Even in our mikveh case one might argue halakhah asks for a “ba’al-batim” perspective rather than a professional scientific one. But even within the “common-sense” assumptions (full mixing, slow flow), the computation should be done with the appropriate mathematics; here modern calculus gives the correct implementation of those very assumptions.
What will poskim actually do?
Independent of what halakhah ideally demands, what will a typical posek do in practice? I suspect he will rely on the Ra’avad’s calculation—probably the first interpretation, since it is stringent—and will likely not think to consult a physicist or mathematician for a more accurate computation. “What was good for the Ra’avad is good for us.”
We can, of course, debate meta-halakhically what the Almighty expects. But does the posek even raise both sides and consciously choose one? I doubt it. My claim is not that the conservative or less-educated posek is wrong, but that his horizon of deliberation is narrow; he may not realize that he has made a meta-halakhic choice that changes outcomes by orders of magnitude.
What if the Ra’avad’s arithmetic had been plainly wrong even by the standards of his day? I suspect many would still follow it, on the premise that a Rishon “cannot be mistaken” (Spirit of Sanctity, Providence, and so on), and that we dare not impugn the early authorities. This seems far less plausible. The recurrent protests about “casting aspersions on the Rishonim” have always struck me as misplaced. We are required to work with what is accessible to us, just as they did with what was accessible to them. Should someone today be considered “coerced” because in the days of Ḥazal there were no cars? Adopting a Rishon’s halakhic premises is one question (and even that is hardly necessary in my view); adopting their mathematics and science is quite another.
We often assume that conservatism is “safer,” i.e., less prone to error. Here is a counterexample: conservatism leads to massive halakhic mistakes, where stringency becomes leniency and vice versa (see also col. 503 for a case where pluralism was more stringent than monism, again against common intuition).
Back to breadth of horizons
A posek who clings to the Ra’avad’s arithmetic in such questions reaches halakhic errors—not necessarily identical to being factually wrong, but in today’s context they are errors. Although at the outset I wrote that a posek or scholar need not be a professional in mathematics, and therefore this mikveh example is not a perfect illustration for the previous column, it nevertheless has bearing on it. One does not need advanced expertise here, but one does need breadth of horizons: acquaintance with mathematics, recognition of its role, awareness of the meta-halakhic questions raised, and the readiness (and ability) to consult experts.
References mentioned in passing: here; here; here; and cols. 325–326, …, and related.
Article Contents
With God’s help
A Mathematical Problem in Jewish Law and Its Implications
In the previous column I dealt with the question of what constitutes greatness in Torah learning. Among other things, I argued that familiarity with additional fields of knowledge is required, and I gave examples and implications of such knowledge for legal decision-making and for conceptual Talmudic analysis. Among other things, I thought of bringing there as an example a mathematical problem from tractate Mikvaot that I once dealt with, but I realized that this was not a very good example. The reason is that I do not think a decisor or conceptual analyst must be a mathematician, and problems of this kind require specific professional knowledge rather than general understanding. All in all, it seems quite obvious that in problems like these one ought to consult a mathematician or physicist (although I doubt whether decisors would actually do so in practice. See more on this later in the column).
A Cistern of Drawn Water into Which a Water Channel Enters
Many years ago, a man who was doing a doctorate at the Technion on mathematics and Jewish law came to me and asked whether I had any legal topics and questions worthy of mathematical analysis. Among other things, I suggested that he deal with Mishnah Mikvaot 3:3 (the issue was mentioned from another angle in column 381)[1]:
A pit that is full of drawn water, and the channel enters it and exits from it continually, remains invalid until it can be reckoned that fewer than three logs remain of the original water.
We are dealing with a cistern of drawn water that one wishes to use as a ritual bath. But when there are three logs of drawn water, they invalidate the ritual bath (and if all of it is drawn water, perhaps it is even invalid by Torah law). Through our cistern passes a channel of non-drawn water, which slowly mixes with the water in the cistern, and thus the percentage of drawn water in the cistern steadily decreases over time. The Mishnah rules that immersion in the cistern is effective from the moment when fewer than three logs of drawn water remain in it.
How is the calculation made? How do we know when one may immerse in this cistern? The accepted assumption among the commentators is that every quantity of water that enters the cistern mixes within it uniformly. In practice, this means that the process is slow, that is, the water channel creeps along at a sufficiently low flow rate. In such a state, every quantity of water that enters the cistern really does mix approximately uniformly throughout its volume, because it has time to do so. If that is indeed the case, then the stream leaving the cistern is composed so that the percentage of drawn water in it corresponds to the overall percentage of drawn water in the cistern.
In __Beit Yosef__ on Yoreh De’ah sec. 201, no. 20, he brings two interpretations of this Mishnah in the name of the Raavad. The first interpretation:
Since it teaches ‘until it can be reckoned’ and does not teach ‘until its full contents have gone out,’ as it teaches above, it implies that we do not regard the water floating and leaving there as all from the original water, as we did in the first clause; rather, what goes out does so according to the ratio between the water that was in the pit and the water descending into it.
According to this interpretation, we must assume that the water that left contains drawn water in the same proportion as the original water in the cistern bears to the total of that water plus the water entering from the channel. If we assume there were 40 se’ah of drawn water in the cistern, it follows that the cistern remains invalid until 12,760 se’ah of water from the channel have entered it. How did I arrive at this number? Because in that state the total quantity of water from the cistern and the channel together is 12,800 se’ah, which is 320 times the 40 se’ah of drawn water that were there at the beginning. Of course, the quantity of water that left the cistern during that time is also 12,760 se’ah (for it contains 40 se’ah the whole time until then). How much drawn water left with it? 1/320 of the total quantity, that is, 39.875 se’ah. This means that 1/8 se’ah of drawn water remained in the cistern, which is three logs (a se’ah is 24 logs).
The second interpretation brought there is the following:
Alternatively, half and half, because the water that descends into the pit last is closer to leaving than the earlier water that was in the pit.
According to this interpretation, what leaves is composed half of drawn water and half of non-drawn water, because the water in the upper layer leaves more quickly. Therefore, if 80 liters entered and left the cistern, the water remaining in the cistern is entirely from the channel and the cistern is valid.
The second interpretation makes a completely arbitrary assumption. By contrast, the first interpretation assumes perfect mixing, which admittedly is not the realistic situation, but it is an assumption that can be accepted as a first approximation, at least as long as the process is slow. Even in faster processes one may perhaps see this as an estimate not wholly detached from reality, because it does provide some estimate of the average percentage (since at the beginning a high percentage of drawn water left, and at the end a very low percentage of it did).
A Mathematical Model
In modern times we have a mathematical tool designed precisely to deal with such questions: infinitesimal calculus (differential and integral). For the sake of simplicity, I will continue to assume a slow process, that is, a low flow rate of the channel, so that the mixing of the channel water in the cistern is perfect. Without that assumption the calculation is very complicated (in fact there is no way to do it explicitly. It requires computer simulations). As noted, this assumption is not so unreasonable, but in any event, as we saw, it is the accepted assumption among the commentators on the Mishnah and the legal decisors (the first interpretation in __Beit Yosef__). Let us therefore examine the quality of the calculation of the __Beit Yosef__ and the Raavad under that assumption.
Let us assume that the flow rate of the water channel is V se’ah per hour. Let us denote the quantity of drawn water in the cistern at moment t by P(t). The entire cistern contains 40 se’ah of water at every given moment, and at the beginning of the process it contains P(0)=40 se’ah of drawn water. The change in the amount of drawn water in the cistern over a tiny interval of time dt is dP. Under the assumption of perfect mixing, the percentage of drawn water in the water that leaves is P/40. The total amount of water that leaves during the interval dt is Vdt se’ah. Hence the quantity of drawn water that leaves the cistern during dt is: V(P/40)dt.
Since this quantity is subtracted from the quantity of drawn water in the cistern at time t, we obtain the following relation (the mathematicians among us are asked to close their eyes, since I am dividing and multiplying by differentials. That is what they call ‘nonstandard analysis’):
dP = -(VP/40)dt
We have obtained the following differential equation (again assuming that everyone is keeping their eyes shut):
dP/dt = -(V/40)P
And its exponential solution is:[2]
P(t) = 40e^(-Vt/40)
The coefficient is determined by the initial condition (the quantity of drawn water in the cistern at the moment t=0 was 40 se’ah).
What is the moment t0 at which one may immerse in the ritual bath? This occurs when the quantity of drawn water drops below 3 logs (which are 1/8 of a se’ah). From the formula above we obtain:
40e^(-Vt0/40) = 1/8 ; t0 = 40ln(320)/V ≈ 230/V
To get some sense of reasonable orders of magnitude, note that a se’ah is 8.3 liters according to R. Chaim Naeh and 14.3 according to the Chazon Ish (40 se’ah is a cubit by a cubit with a height of three cubits of ritual-bath water). A reasonable order of magnitude for the flow rate of the channel is something like a se’ah per minute (something like a bathtub faucet), and the time t0 at which there will be less than three logs of drawn water in the cistern is about 230 minutes, that is, about four hours. During that time 230 se’ah entered the cistern from the channel. Incidentally, this result is correct for any value of the flow rate (assuming that the initial quantity of water in the cistern is 40 se’ah), since the quantity of water that entered the cistern during that time is t0 X V, and this always comes out to 230. With a different initial quantity of water (P0), the result is: P0 X Ln(320) = 5.77P0.
Comparison
We can now compare this to the first calculation of the Raavad, which as noted was made under the same assumptions. We have seen here that according to the exact calculation it is enough for 230 se’ah to enter in order to validate the cistern, as opposed to the 12,760 that emerge from the Raavad’s calculation. His result is 55 times (!!!) the exact result. Not 55%, and not even 555%, but 5,500 percent.
As observers from the outside we have no way of knowing how much water entered the cistern, and therefore a more sensible measure for us is the time one has to wait. As to the waiting time, we saw the result above. According to the Raavad this would be 12,760/V rather than 230/V. The ratio is of course fixed, but at sufficiently low flow rates of the channel we can arrive at truly astronomical waiting times. By then the water in the cistern will already have evaporated.
The Meaning of This: A Meta-Legal Perspective
We have seen that even the ‘exact’ calculation assumes assumptions that are not very realistic (a very slow process). So this is not a comparison between an approximation and an exact result, but between two ways of calculating a given approximation. And still, the gap between them cries out for explanation. True, here the Raavad’s calculation comes out stringently, but that is accidental. Just as well there could be situations in which his calculation came out 55 times lower than the exact answer. A position like that invalidates the immersion of someone who waited until 5,000 se’ah had passed through (because 13,000 have not yet passed), when in such a state the ritual bath has long since been valid according to even the most exacting standard. This can already become a ruling with lenient consequences as well.
In situations like these one can ask what we would say about a case in which, in reality, when we immersed, the amount of drawn water in the cistern was greater than three logs (because we relied on an imprecise calculation). One can say that the Torah was not given to angels. Just as it is accepted that we are not required to check fruit for worms or insects by sophisticated laboratory means, although it is possible that in reality we ate worms, so too here. Jewish law is based on rough lay calculations and not on the actual results (and what about spiritual dulling of the heart and the metaphysical consequences? Here we have entered the question of the extent to which legal decisors control metaphysical reality).
Indeed, every calculation rests on assumptions, and there is no escaping the fact that legal rulings will be made under some set of assumptions. That is not a defect in legal instruction (I will tell you in confidence that scientific calculations too are always made under assumptions, and usually they are not entirely realistic, but one chooses assumptions that allow a calculation to be performed and then tries somehow to estimate the error). But in our case the situation is somewhat different. The issue is not the assumptions of the calculation but the technique used to perform it. For we saw that we assumed slow flow and full mixing, an assumption that is not accurate in actual reality, but the Raavad assumed it too. The difference was only in the form of the calculation and not in its assumptions. We saw that with a more modern mathematical technique one gets an enormous correction to the result of the calculation. This is not a matter of correcting the value of pi in some digit after the decimal point, but of a result that is 55 times smaller than the proper approximation. The difference is whether we have to wait until an amount of water equal to six cisterns has entered from the channel into the cistern (6 times 40 se’ah is about 230 se’ah), or an amount equal to 330 cisterns. An enormous amount of time, of course. Do we also here have no problem with the error in the result? This is simply an obviously incorrect calculation, not merely a rough lay approximation or certain assumptions underlying the calculation. It is highly doubtful whether such an error is legally acceptable.
A Note on Consulting Experts
This brings us to a theoretical meta-legal question. Is this really correct? As far as the Raavad and the __Beit Yosef__ are concerned, it is clear to me that the Holy One, blessed be He, does not come with excessive demands upon His creatures, and if they had no other tools then this is the calculation they were expected to make and to act upon (not so with the values of pi, which were known fairly well even in their day). But today we have better mathematical and scientific tools. Is it not reasonable to expect that a decisor in our day would give a different answer to this question?
In columns 325 – 326 I dealt with doubtful cases of an inevitable consequence. We saw there that if I drag a bench across the ground on the Sabbath and it is unclear whether a furrow will be formed, this is a case of ‘unintended action,’ and therefore even if a furrow was in fact formed, I have not violated a prohibition. But if it is clear to me in advance that a furrow will be formed, that is an inevitable consequence and it is forbidden. By contrast, I brought there the opinions of later authorities (R. Akiva Eiger, against the __Taz__) according to which if I close a box and I am uncertain whether there is a fly in it, then if I closed the box and it turned out that there was indeed a fly in it, this is a doubtful Torah prohibition and I have committed a violation. What is the difference between the cases? After all, with the ground too I am uncertain whether a furrow will be formed, and with the box I am uncertain whether there is a fly in it. I explained there that in the box my uncertainty is due to lack of information on my part (an epistemic doubt), whereas with the furrow in the ground there is indeterminacy in reality itself (an ontic doubt). True, if we ask an expert he may be able to tell us whether a furrow will be formed or not (in our reality there is no true ontic doubt, except perhaps in quantum theory). But resorting to an expert is beyond the bounds of the legal estimate. When no one knows the answer in advance, this is an ontic doubt. When an expert knows it in advance, it is still an ontic doubt.
At the end of the second column I brought the words of R. Shlomo Zalman Auerbach, in the responsa __Minchat Shlomo__ second edition (vols. 2–3, sec. 63), where he deals with the prohibition against eating worms. He cites from the legal authorities that when a person eats a fruit and there is a worm inside it, he is considered an unwitting actor or one who acts unintentionally, and therefore according to that view there is no prohibition:
And likewise what Shivat Tzion wrote in sec. 28, and which was also brought in Imrei Binah, the laws of meat and milk, at the end of sec. 4, in the name of a certain gaon, and in Darkhei Teshuvah sec. 84, no. 28 this was likewise brought from the author of Beit Ephraim, that with respect to the worm, since his attention is not directed to it, this is considered only an unwitting act. And although unwitting involvement does not apply in the case of forbidden fats because one derives benefit, here it is different, because the benefit is only from the fruit and not from the worm.
He explains that eating the fruit with the worm is a doubtful inevitable consequence regarding a past fact (since the doubt is whether there is now a worm in the fruit or not, just like the question of flies in the box), and therefore seemingly this depends on the dispute among the later authorities that I mentioned.
R. Shlomo Zalman seems to hold that one should forbid this in accordance with the view of R. Akiva Eiger (it appears that he rules like him):
And although this is like a doubtful inevitable consequence regarding a past fact, which is not considered unintentional, as explained by R. Akiva Eiger, Yoreh De’ah sec. 87, par. 6.
In the end, however, he rejects this and writes:
Nevertheless, it appears that if the clarification can be achieved only through very great effort, this is considered an after-the-fact situation. In our case it is therefore properly regarded as something done only afterward, at the time of eating, through unwitting action and without intention, and it is permitted. For even in dragging a bed and the like one could also know in advance by means of a great expert, and nevertheless it is permitted. And since it is permitted for the restaurant owner, it is likewise permitted for others, as is known.
According to him, in a situation where checking whether there is a worm in the fruit is difficult and involves great effort, this is not considered a doubtful inevitable consequence regarding a past fact, but rather a doubtful case regarding the future. His proof is from dragging a bench, where by means of a great expert there is room to determine whether a furrow will be made (as I described above), and nevertheless this is treated as a doubtful inevitable consequence. If the Talmud sees this as a case that is not an inevitable consequence, even though it is a doubt about a preexisting fact, that is proof that when the check is difficult (requiring expertise — as with the ground, or opening up the fruit — as in the case of worms), it is considered a doubt about the future and therefore not an inevitable consequence.
If so, one might have argued here too that Jewish law does not obligate us to resort to an expert, and that the Raavad’s calculation is acceptable even today. But this is not plausible for several reasons. First, the Raavad too resorted to a calculation that is not trivial for the ordinary layman. So in what way is it preferable? Second, going to an expert before eating every fruit is not reasonable. But when one wants to determine the status of a ritual bath, there is no reason not to seek an expert opinion. And above all, the calculation technique I used is not very advanced mathematically. A good high-school student, and certainly a first-year university student, can do it without many problems. This is not really a matter of going for professional consultation.
There is of course room to discuss whether there is a basis for such a view that excludes experts from the field of legal discussion (the Torah was given only to ordinary people). Especially since there are many contexts in which we do rely on them, from the laws of electricity on the Sabbath, through consultation with physicians in various situations, and so on. But even if for the sake of argument we assume that there is room for such a view, as we have seen, in our case the situation is different.
On the other hand, one could argue that today we are expected to make an exact calculation. Not only the calculation I presented above, but even not to assume a slow process as the Raavad did, and instead to run a simulation that will give us exact results. In effect, this means solving on a computer the full equations of fluid dynamics. After all, these tools too are available to us today. But these are truly expert tools (and even there it is not clear to me to what extent this can be done exactly), which undermines the very assumption, accepted by many, that Jewish law was given to ordinary people.
At the end of column 397 I dealt briefly with indirect-causation solutions for the Sabbath. I argued there in favor of the position that such solutions do not really help, because even if the activation of the mobility scooter by the switch is carried out internally through a mechanism of electrical indirect causation, to ordinary eyes there is still a switch that activates a machine. Jewish law does not depend on the microscopic processes that occur inside, which are the concern of the professional, but on the way the ordinary reasonable person sees it. That is a different claim from the one I made here. One can adopt the position that indirect-causation solutions do not help (that is, that what determines the law is the lay perspective), and together with that argue that worms should be checked under a microscope or that the calculation of the quantity of drawn water should be done with maximum scientific precision (that is, that what determines the law is the professional scientific perspective). But I will not elaborate further on this here.
More Examples of Lay Reasoning in Jewish Law
In a course on electromagnetic field theory at Tel Aviv University, the lecturer, Prof. Samson Frankenthal, told us about a case that occurred in the United States. There is a federal law according to which if two states transfer something from one to the other through a third state, they must pay the transit state if the thing passes through wires. If it does not pass through wires, then not. It happened that State A transmitted electricity to State C through State B. State B sued them to pay it, since the electricity had passed through cables. The states brought a physicist who explained to the court that although there are wires, the electricity passes around the wires and not within them. What he meant was the Poynting theorem, which shows that one can calculate the power transmitted along an electric line by calculating the power of the electromagnetic fields surrounding the wires.
The truth is that this is, of course, nonsense. There is indeed such a theorem, but it merely speaks of equivalence: one can describe the transfer of power either through the current in the wires or through the fields around the wires. These are two equivalent ways of describing the process and calculating it. But there is no correct and incorrect answer to the question whether the electricity passes inside the wires or around them. These are just forms of description. Legally, it is quite clear that the legislator intended the case of electricity to count as transmission by means of wires. But a lay judge could have accepted this scientific argument, since it came explicitly from the mouth of an expert in the field, and who is he to disagree with him.
A similar example happened to me personally when a scholar in the Chazon Ish Kollel approached me, knowing that I am a physicist. He asked me whether glass is a liquid or a solid, because he had read that physicists treat glass as a liquid. I told him that if the question concerns the laws of cooking on the Sabbath, it is a solid and not a liquid (dry and not wet). Physicists treat glass as a liquid because its crystalline structure is not ordered (not periodic, unlike solid crystals). And again, a judge or decisor could have accepted the physicist’s claim that it is a liquid, since this is expert testimony.
These two are also examples connected to the discussion in the previous column. They demonstrate the importance of familiarity with other fields of knowledge, and the ability to understand what the expert is saying and what to ask him. I do not think every decisor has that ability. But for our purposes here it is the answer that matters to me, not the question. It is clear that Jewish law and American law both intend us to relate to these questions (whether this is a liquid and whether that passes through wires) on the lay plane and not as a professional-scientific question.
Even in the case of the ritual bath there is room for the claim that Jewish law requires us to adopt a lay and not a professional-scientific perspective. One can argue that from the legal point of view what matters is not how much drawn water there really is in the cistern (a professional calculation), but how much drawn water there is in it from the ordinary lay perspective. Therefore we are permitted to assume perfect mixing, even though this is not accurate. And still, as I argued above, even under the lay assumptions (that the mixing is perfect), one can do the calculation in the Raavad’s primitive way or in the more exact way of modern mathematics. Here I do not wish to argue for the superiority of the expert over the layman with respect to the assumptions, but for his superiority with respect to the calculation, that is, with respect to the implementation of the assumptions. The calculation ought to be done in the best possible way.
What Will the Decisors Actually Do?
I will now ask a different question. Quite apart from the question of what Jewish law really requires in such situations, what, in your estimation, will an ordinary decisor actually do in practice? I assume he will rely on the Raavad’s calculation, and will probably take the first calculation because it is stringent. Moreover, I suspect it will not even occur to him to turn to a physicist or mathematician and ask for a more exact calculation. He will say to himself that what was good enough for the Raavad is certainly good enough for us.
And again, one can of course argue on the meta-legal level about what the Holy One, blessed be He, really expects. But will a decisor even think about this question? Does he raise these two sides and decide in favor of one of them? I very much doubt it. My claim is not that the conservative and poorly educated decisor is mistaken, but that he has too narrow a horizon of discussion. He is not even aware that he has made here a meta-legal decision that changes the picture by orders of magnitude, to completely different results. He does not even ask himself this question.
Let me ask further: what would happen if the Raavad’s calculation were completely mistaken even by the tools available at the time? I assume that decisors would still follow him, because their assumption is that the Raavad (through divine inspiration, or divine providence) certainly did not err, and that one may not disagree with the medieval authorities. On the contrary, we would even be casting aspersions on the medieval authorities. All this already sounds far less plausible.
Arguments about casting aspersions on the medieval authorities, which arise not infrequently in discussions of this sort, have always seemed baffling to me. The medieval authorities were required to use what was accessible to them, and we are required to use what is accessible to us. If today a person can get to some place by car, would he still be considered under duress because in the days of the sages there were no cars? It is one thing to adopt the legal assumptions of the medieval authorities (even that is by no means necessary and in my view not even plausible), but why adopt their mathematics and science?
We usually assume that a conservative approach is safer, that is, more immune to mistakes. But here one sees an example in which conservatism leads to enormous legal errors, and its stringency becomes its leniency (see in this article and in column 503 an example of a case in which pluralism leads to stringency compared with legal monism, again contrary to the prevailing intuition).
Back to the Question of Breadth of Outlook
My claim is that a conservative decisor who clings to the rulings of the Raavad in questions like these arrives at legal errors (and not only factual errors. As noted, that is not necessarily the same thing). At this point the present column joins the previous one. Although in the opening I wrote that a decisor or conceptual analyst certainly does not need professional expertise in mathematics, and therefore the example of the ritual bath is not a good example for the previous column, it still seems to have an implication for that discussion as well.
What we have seen here is that mathematical expertise is indeed not required, but breadth of outlook in these subjects is. This can prevent mistakes and lead to completely different legal results. What is needed is breadth of outlook in the sense of familiarity with mathematics and recognition of mathematics, and also awareness of the need to formulate a position on the meta-legal questions I presented here, and of course awareness of the need and the ability to consult experts.
1.
Footnotes
- There I touched on the question whether a sheet on which the formulas that will appear here below are written must be set aside as sacred writing rather than discarded. This also has a practical implication for anyone who prints this column.
- For anyone who does not immediately see it: separate variables in the penultimate equation, that is, divide by P and integrate on both sides.
Discussion
Perhaps the Ra’avad also understands that one can dilute in stages. If the pit contains 40 se’ah, then every inflow of 40 se’ah dilutes the drawn water in the pit by half (this is also how you understand the Ra’avad), so in total 9 dilutions would be needed, i.e. 360 se’ah. And every inflow of 1 se’ah dilutes the drawn water in the pit by 1/40, so in total 234 dilutions would be needed, i.e. 234 se’ah. The differential equation performs continuous dilution, and in the Ra’avad you did a discrete dilution in one pulse, but perhaps the Ra’avad too (with simple arithmetic and with his reasoning that it leaves according to the calculation) could use discrete dilutions in several pulses. What do you think? By the way, it would be interesting to check whether with compound interest (for gentiles) they discussed the number of capitalization periods even before arriving at continuous compounding with an exponential.
Because it does not seem that here they were guided by the tradition they had received. This is a matter of applying common sense to a mathematical problem.
I take it you mean the Ra’avad’s second explanation? If so, the main point is missing from the book. In any case, this is of course a reasonable approximation the closer you get to continuity (the smaller the increments).
I didn’t understand the question about compound interest.
I’m talking about the first explanation, that it leaves according to the calculation. Only because you did it in a single pulse did you end up needing an astronomical quantity of water. But how to divide the incoming water into several pulses can be chosen however is convenient. To me that does not seem like a far-fetched reading of the Ra’avad. Like the isuryata debei Rabbi in Nedarim 39, where they understood the idea of taking a part and then a part of that part.
As for compound interest, I was just wondering. Compounding interest in several periods (how many periods to use in calculating and accumulating the annual interest) is apparently a parallel problem, since there too in each period one multiplies by a fixed ratio (one plus the annual interest divided by the number of periods), and increasing the number of periods (and correspondingly decreasing the interest in each period) accelerates the growth until at the continuous limit one gets an exponential. And so too here: in each period one multiplies by a fixed ratio (the dilution ratio), and increasing the number of periods (and correspondingly decreasing the amount of water in each period) accelerates the decline until at the continuous limit one gets an exponential. But matters like interest are much more general and widespread than dealing with a channel flowing into a mikveh, so perhaps it would be possible to assess the level of understanding in this area through the question of interest.
The connection to compound interest is clear; I just didn’t understand the question. What is there to understand here? It is obvious that compound interest in the limit is an exponential, and the smaller the step, the closer you get to that. What is there to investigate?
Just to check historically, in the general world and in the Jewish world, what the computational proficiency in this matter was.
Seemingly this is a question that could be tested empirically even in the period of the Rishonim, without calculation.
They would mix dye into water in the ratio of 3 login to 40 se’ah in a cup.
They would take colored water and put it in a small pit (a hole in the table). They would simulate a water channel and wait until the color of the water became like the color of the water in the cup.
No?
Thank you very much for the wonderful column. I am in the middle of studying Mikvaot, and precisely today I started this very sugya (in my language: divine providence). But I did not understand how you know that the Ra’avad did not mean what you wrote. The Ra’avad writes that we assume the water leaves according to the calculation, but he never said that the way to calculate is as you put in his mouth. His whole intent is only to describe the gradual process and that the waters mix uniformly, which you too assume, but certainly one should always calculate as you wrote. In truth there is nothing new here, and I do not know why you charge at all the decisors as though it would never have occurred to them that the Ra’avad’s calculation was incorrect and that they would not ask an expert, etc., etc., when in fact the decisors definitely addressed this. The Shakh, Yoreh De’ah 201:56, indeed understands the Ra’avad as you wrote, but the Chazon Ish, Mikvaot Tinyana 5:4-5, wrote that his words are very puzzling, see there, and he concedes with bowed head: “We do not know the precise calculation.” He certainly would have been willing to hear from an expert what he thought (and one cannot object that if so, what benefit do we gain from the Ra’avad’s words if we do not know the precise calculation, because there are things that can be determined clearly—for example, when there were 40 se’ah of drawn water in the pit and 40 entered, then certainly most of the drawn water left, as the Chazon Ish writes there.)
More than that: it seems to me that you conflated two things. A. That the Ra’avad’s calculation (in your view) is incorrect. B. So what is the correct calculation? Anyone with a head on his shoulders understands that one cannot calculate as the Shakh says, because it is obvious that more of the water already in the pit will leave; rather, we do not know the precise calculation, as the Chazon Ish wrote, and for that we need your mathematical model. Therefore, according to your understanding of the Ra’avad, he really did err in knowledge that was already known in his own time.
(By the way, perhaps you could explain to me what the Chazon Ish meant in the diagram he drew there?)
I did not get into the commentators, since this was only an example. Nor am I familiar with what they say about this. Now, to your points.
This is disingenuous. If the Ra’avad had meant that, he should have given details on how to carry out the calculation. A similar suggestion about the Ra’avad’s intent was raised above by Tirgitz; see there.
I did not write that all decisors would accept the Ra’avad, only that many of them would.
I was speaking about decisors of our own time, not about the Shakh and the like.
The entire calculation is clearly incorrect. Therefore there will always be objections to the calculation. The question is how one performs the calculation given the assumptions. Therefore objections to the manner of calculation are not relevant.
And from your own premise: did the Chazon Ish himself consult a mathematician?
Maybe he did, maybe he didn’t. I do not know whether he wrote the second edition on Mikvaot in Kossova or in Bnei Brak, and whether he had an opportunity to consult an expert. As someone who does esteem the Chazon Ish, I can ask you why in fact he did not consult an expert. It is not that he thought the Ra’avad was right and what was good enough for the Ra’avad is good enough for us; after all, he quite clearly agrees with what you wrote, that one should not calculate as the Shakh says (and that his words are very puzzling). Rather, he wrote that he has no way of knowing the precise calculation, so why did he not ask someone who does know the precise calculation? Did it not occur to him that there are mathematicians in the world? Or did he think mathematics was at the same stage it had been in the days of Yashar of Candia? (from whom the Chazon Ish drew his mathematical knowledge).
That is exactly what I am crying over. I can raise hypotheses, but the fact remains.
The Chazon Ish’s method in Torah study was like this even among the commentators on the Gemara themselves—that is, after he studied the Gemara, Rambam, and some well-known Rishonim and Acharonim, he would raise difficulties and often remain with the matter unresolved, or resolve it by his own reasoning; but he did not “turn to experts in Talmud” and halakhah who had often already explained in their books what had been difficult for the Chazon Ish [the complete opposite of Rav Ovadia Yosef’s method, for example]—and he noted this in his letters, that it was not his way to rummage through the archives searching for answers. All this applies mainly to interpretations that did not bear on practical halakhah in a case before him.
Therefore it is not a question why he did not turn to a mathematician to explain the Ra’avad or the Mishnah to him, because in his view this was not required of him as part of Torah study, and therefore he did not do it.
It was different with questions that actually came before him; if I am not mistaken, the Chazon Ish did consult doctors and the like.
Other decisors, such as R. Shlomo Zalman Auerbach, as far as I know did consult experts on electricity and the like even when the matter did not pertain to an immediate practical ruling.
Personally, the path suggested by Rav Michi seems right to me even in study that is not practical—if one would preface sugyot with current research and learning, it would add understanding and clarity to the sugya. Even if the conclusion remains as explained by the Rishonim, who did not proceed by innovative methods [as I saw regarding DNA testing, where many decisors hold that one should not permit questions of mamzerut on that basis, nor establish mamzerut and the like, even though according to the doctors it is 99% accurate—because we have only the words of Hazal], at least the view would be broader.
But the Chazon Ish’s path is not like that—not because of “external wisdoms,” but because of what is included in the obligation of the commandment of Torah study.
Your words are correct—that he did not turn to an expert because he was not issuing a practical ruling here. But I did not understand the whole “consistent with his general approach” point from the fact that it was not his way to search responsa, unlike R. Ovadia Yosef, etc. The reason he did not search responsa was that he was not so interested in what they thought (how does Michi say it? It’s worth seeing what they say, just to make sure I didn’t miss something), and the Chazon Ish’s way was not to gather every opinion, count them up, and decide which view has the largest numerical majority (including Netai Gavriel and Piskei Teshuvot). What connection is there between that and the fact that he did not clarify with an expert what the Ra’avad meant by saying that one calculates according to a calculation, when he agrees that the mode of calculation written by the Shakh is incorrect?
To the best of my understanding, the Ra’avad’s words were not properly understood here, and this can be proven from the second approach that the Ra’avad himself wrote.
The Ra’avad writes in the second approach that the water leaves half and half, because the water that arrived last is closer to leaving the pit. This implies that according to that approach, the water entering from the channel has a greater share in the water leaving the pit than the water already in the pit. By contrast, according to the first approach, where the water is mixed, the water entering from the channel has no greater share in the outgoing water. On that basis, it should follow that according to the second approach, a much larger quantity of water would need to enter the pit than according to the first approach. But according to the calculation you presented, in the second approach one needs far less water to enter from the channel—which does not sound at all like the Ra’avad’s words!
Therefore it seems clear to me that the Ra’avad meant a calculation divided into small units of water, not a calculation of all the water at once. If we calculate that way, even without resorting to differential calculus, we will conclude that in the first approach the amount will be much smaller than what you presented in his words, and in the second approach we will need more water coming from the channel than in the second approach, because each small unit of water that enters the pit—half of it leaves and half of it mixes with the pit water, so that in the next unit of water that leaves the pit, half the water will be from the channel and half from the pit water that has already begun mixing with the channel water, and so on.
I hope my words are understood properly, because in my opinion this is the straightforward understanding of the Ra’avad’s words.
Thank you for the enlightening columns, and Shabbat shalom!
Hello. This suggestion was already raised above by Tirgitz (regarding compound interest), and I wrote that I do not think this is what the Ra’avad meant.
Have a good week!
I do indeed see that this suggestion was raised, but I think I fairly well proved it from the second approach the Ra’avad proposed. You did not address the proof I brought: if the Ra’avad really did not mean what I said, then it becomes very difficult to understand how it could be that in the second approach, where the Ra’avad writes that the water coming from the channel is closer to going out, we would require far less water from the channel than in the first approach.
In any case, thank you for the response.
I understand your argument from the wording, but the expression “half and half” is very clear, and if he meant something else he should have said so. What you are proposing is much more complex than Tirgitz’s proposal, because according to your view, each portion that leaves consists half of channel water and half of pit water already mixed with channel water. All of that is absent from his words.
One could say that you forced the Ra’avad’s words into your interpretation. There is no necessity at all to understand the Ra’avad as you explained him. He merely gives two basic assumptions for the quantity that leaves the pit—a proportional quantity or an equal quantity—but he gives no method of calculation, only uses the Mishnah’s expression “until it is reckoned.” Everything else is your interpretation, and it seems mistaken.
I now saw that in Higayon 4 (Studies in the Modes of Thought of Hazal, Jerusalem 1997), p. 113ff., there was an article claiming that the Ra’avad meant as you wrote [the Shakh’s understanding], and that the correct calculation is the differential calculation. And in Higayon 5 (Jerusalem 2001), p. 151, Eliyahu Beller wrote in response: “In truth, there is no basis at all for this interpretation […] of the Ra’avad’s first approach, neither from the Ra’avad’s language nor from common sense. It is clear that the ratio between the amount of the original drawn water and the rainwater from the channel changes at every moment, and therefore the ratio between the types of water that leave must also change accordingly, in the language of the Chazon Ish (about the Ra’avad’s words): ‘And if we estimate according to a calculation … at every moment the valid water increases and the invalid water decreases.’ Moreover, from the continuation of the Ra’avad’s words there is a decisive refutation of the H”K’s understanding. (See there.) […] Rather, certainly the Ra’avad’s ‘according to a calculation’ does take into account the changing ratio at every moment, just as the H”K do in their ‘novel interpretation’ with the help of a differential equation. Of course, in the Ra’avad’s time differential calculus had not yet been developed, and therefore it stands to reason that the Ra’avad meant a discrete calculation with units of water that are small relative to the amount of water in the pit. For example: […]” See there.
Many thanks. I had already forgotten.
Why does it not seem reasonable to you to adopt the halakhic assumptions of the Rishonim? You wrote in the comments on the previous column that because of their proximity to the source they have an advantage over us.