A Halakhic–Mathematical Problem and Its Implications (Column 683)

With God’s help

Disclaimer: This post was translated from Hebrew using AI (ChatGPT 5 Thinking), so there may be inaccuracies or nuances lost. If something seems unclear, please refer to the Hebrew original or contact us for clarification.

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”:

1. 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.

2. “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.