Lesson 42: Balak

From the book Mida Tova: Articles on the Principles of Halakhic Thinking by Rabbi Michael Avraham. Translated from Hebrew using gpt-5.4 (reasoning_effort=high, batch API).

---

With God’s help

Concepts

Process and instantaneous state.

A discrete point on a continuous axis.

Two kinds of impossibility: an event of probability 0, and absolute impossibility.

Abstract

In this week’s essay we begin with the aggadah (rabbinic narrative) brought in the Talmud about the solitary instant each day in which God is angry — an instant that only Balaam knew how to identify. From there we discuss the nature of the time axis as a continuum. The modern mathematical conception of the continuum sees it as built from infinitesimals, not from discrete points. We bring Torah sources that view the matter similarly, and therefore treat a discrete point on the time axis as something that is not really time.

We point to several distinctions between a mathematical description of a continuous axis and its essence as such, and argue that a discrete point on a continuous axis nevertheless has significance. In the course of our discussion we distinguish between a process, as opposed to a state, which can exist only over a stretch of time, and an instantaneous state; and between a point, whose dimension is 0, and an infinitesimal, whose dimension is 1. We then discuss the paradox of the arrow in flight presented by Zeno of Elea, and connect it as well to these distinctions.

Within this discussion we briefly address the issue known as “exactness is impossible,” and distinguish between several meanings of this inability, as well as between the disagreements of the medieval authorities regarding them.

We conclude with a brief discussion — neither definitive nor exhaustive — of the boundaries of the concept “Torah,” and of the distinction between Torah and wisdom.

What Is a “Moment”?

A Look at the Nature of the Time Axis

Introduction

This week we will discuss a topic that is not distinctly halakhic (pertaining to Jewish law): the nature of the time axis as a continuum. We will mention halakhic ramifications in connection with the issue of “exactness is impossible,” but that will not be our main concern. Even so, clarifying the concepts and ambiguities that bear on our conception of the time axis can be helpful in studying quite a number of halakhic and other issues.

A. Probability, Dimensions, and the Nature of Time

Introduction: Two Interpretations of “Moment”

The Babylonian Talmud, Berakhot 7a, relates that each day there is a moment in which the Holy One, blessed be He, is angry. It continues with an exposition of the verses in our Torah portion and says that only Balaam knew how to identify that moment:

As it was taught: “God is angry every day” (Psalms 7). How long does His anger last? A moment. And how long is a moment? One fifty-eight-thousand-eight-hundred-and-eighty-eighth part of an hour. That is a moment, and no creature can determine that time exactly, except for wicked Balaam, of whom it is written: “and knows the knowledge of the Most High” (Numbers 24). Now then — he did not even know the mind of his own animal, yet he knew the knowledge of the Most High? Rather, this teaches that he knew how to determine the exact hour when the Holy One, blessed be He, is angry. And this is what the prophet said to Israel: “My people, please remember what Balak king of Moab devised…” (Micah 6).

The “moment” spoken of here is a very short span of time. It is not, however, a discrete point in time, but a short interval. Immediately afterward a parallel exposition is brought, based on a verse in Micah in which God tells us of the kindness He did for us in not becoming angry during the days when Balaam came to curse Israel:

What is the meaning of “that you may know the righteous acts of the Lord” (Micah 6)? Rabbi Elazar said: The Holy One, blessed be He, said to Israel: Know how many acts of righteousness I did for you, in that I did not grow angry in the days of wicked Balaam. For had I grown angry, there would not have remained among Israel a survivor or escapee. And this is what Balaam said to Balak: “How shall I curse whom God has not cursed, and how shall I denounce when the Lord has not denounced?” (Numbers 23). This teaches that throughout all those days He did not grow angry.

In Tosafot, s.v. “For had I,” on that passage, it seems that at this stage of the Talmud they still assume that this moment is an interval of finite length, though a short one:

“For had I grown angry, there would not have remained…” — And if you say: what could he have said in so brief a moment? One may answer: “Destroy them all.” Alternatively, once he began his curse at that moment, it would do harm even afterward.

In the second answer, however, Tosafot may already be adopting the position that the moment is a point in time, and not a short interval. Indeed, at the end of the Talmudic passage such a view is cited:

And how long does His anger last? A moment. And how long is a moment? Rabbi Avin — and some say Rabbi Avina — said: a moment is as long as its utterance. And how do we know that He is angry for a moment? As it is said: “For His anger is but for a moment; life is in His favor” (Psalms 30). And if you wish, say from here: “Hide yourself for a brief moment until the wrath passes” (Isaiah 26).

That is, Rabbi Avin or Rabbi Avina says that the moment is a discrete point in time.

According to the first interpretation, the inability to identify the moment stems merely from lack of knowledge. According to the second, that lack of knowledge results from the inability to hit exactly a moment that is a discrete point in time.

Midnight

A similar hesitation appears with respect to midnight. In Exodus 11:4, God informs us:

Moses said, “Thus says the Lord: about midnight I will go out into the midst of Egypt.”

The expression “about midnight” calls for interpretation. And indeed, the Sages noticed that there is a parallel verse, Exodus 12:29, that says:

And it came to pass at midnight that the Lord struck every firstborn in the land of Egypt, from the firstborn of Pharaoh who sat on his throne to the firstborn of the captive in the dungeon, and every firstborn of beast.

Here the expression is “at midnight,” which is more precise. Mekhilta de-Rabbi Ishmael, Tractate Pisha, parasha 13, expounds as follows:

“And it came to pass at midnight” — why is this stated? Because it says, “Thus says the Lord: about midnight I will go out…” (Exodus 11:4). A human being cannot determine the exact midpoint of the night, but its Creator divided it. Rabbi Yehudah ben Beteira says: the One who knows its hours and its seasons divided it.

That is, God Himself can determine the precise instant of midnight, and therefore the verse that reports the divine act itself uses the language “at midnight.” But Moses cannot determine that instant, and therefore in his words it says “about midnight.”

The same appears in Mekhilta de-Rabbi Shimon bar Yochai on Exodus 12:29.¹

But when Moses said to Pharaoh, “Thus says the Lord: about midnight…” he said to him: the matter is balanced on the dividing of the night — whether by a hair’s breadth above it or by a hair’s breadth below it. He sits upon the stone of the hours and determines the hour to a hair’s breadth, for one kingdom does not encroach upon another even by a hair’s breadth. Rather, when the time comes for a kingdom to fall — if by day, it falls by day; if by night, it falls by night. So it says, “Noph shall have adversaries by day” (Ezekiel 30:16), and “At Tahpanhes the day shall grow dark” (ibid. 18), and “That night Belshazzar the Chaldean king was slain” (Daniel 5:30). And when the time came for our fathers too to fall, what was said of them? “Woe to us, for the day has declined…” (Jeremiah 6:4).

This midrash (rabbinic interpretive text) implies that God determines the hours with hairbreadth precision, although here the idea is applied to the fall of kingdoms.

The Babylonian Talmud, Berakhot 3b, discusses David’s rising at midnight to play music and study Torah, and continues the same line by bringing a parallel discussion that compares King David to Moses our master:

Did David know when midnight was? Now even Moses our master did not know, as it is written: “About midnight I will go out into the midst of Egypt” (Exodus 11). What does “about midnight” mean? If you say that the Holy One, blessed be He, said to him “about midnight” — is there any doubt before Heaven? Rather, He said to him: tomorrow at midnight, at a time like this present time; and Moses came and said: “about midnight.” This implies that he was uncertain. And David knew? David had a sign, for Rav Aha bar Bizna said in the name of Rabbi Shimon Hasida: A lyre hung above David’s bed, and when midnight arrived a north wind came and blew upon it, and it played by itself. Immediately he would rise and engage in Torah study until dawn broke.

The conclusion is that Moses indeed did not know, whereas David knew by virtue of a special sign given to him from above. However, later in that same passage, on Berakhot 4a, a view is brought that Moses too knew — and this is the opinion Rashi cites in his commentary on the Torah; see also Torah Shelemah, section 16:

Rabbi Zeira said: Moses always knew, and David too knew. If David knew, why did he need the lyre? To awaken him from sleep. And if Moses knew, why did he say “about midnight”? Moses thought: perhaps Pharaoh’s astrologers will err and say that Moses is a liar. As the master said: teach your tongue to say “I do not know,” lest you be disproved and caught out.

The Comments of Ohel Yehoshua

Rabbi Shimon Moshe Diskin, author of Ohel Yehoshua, in his commentary on the second verse we cited, Exodus 12:29, raises the following difficulty:²

It should be noted that this time of “midnight” is impossible to locate. For when the night is divided into two, the first half is before midnight and the second half is after midnight, but midnight itself occupies no time at all and is not a state whatsoever, for it is only a name for the division of the night. Any middle point is itself divisible into two, half to this side and half to that side. If so, how can there be such a thing as midnight?

That is, he is troubled by how we can speak of the time of midnight, if in fact there is no such time at all. It is the point that divides the night into two, but it has no existence of its own. Up to it, this is the first half of the night; from it onward, this is the second half. It has no existence at all, like a point that divides two parts of a line: in itself it has no length and no existence in the terms of the time axis.

He then proposes the following ingenious answer:

It likewise should be noted that death too is not a matter of time. Before that point a person is alive, and now he is dead. It follows that there is no “time of death” at all, for so long as he has not yet died he is alive, and once he has ceased to live he is dead. Thus there is no time of death at all.

Clearly this is correct, and thus the original difficulty is resolved. So too in the plague of the firstborn: in the first half they were alive, and in the second half they had already ceased to live and were dead. Indeed, midnight is not a time, and the time of death is not a temporal entity. Rather, the transition from life to death stood in the same relation as the transition from the first half of the night to the second. This is the meaning of “and it came to pass at midnight”: the intention is not the time of midnight, but that midnight, just as it divided the night, divided between life and death for the Egyptian firstborn.

He begins by pointing out that death too does not occur at any time. Death — that is, the departure of the soul — is nothing but the transition between life and death, and therefore in itself it does not occur on the time axis. It divides that axis into two parts, each of which has temporal significance. The two descriptions thus complement one another: indeed, there is no moment called “midnight,” and there is no moment in which death occurs. The moment of midnight is the point at which the first segment, in which the firstborn were alive, changes into the second segment, in which the firstborn are dead. Death itself is not a state at all, and it does not last any length of time.

Two Kinds of Zero Probability

Usually people understand that when some event has probability 0, it is impossible. But that is not correct. The probability that some particular integer will be chosen in a lottery among all integers is 0, and yet one integer will in fact be chosen anyway. That means that in such a case it is certain that an event of probability 0 will occur. Sometimes there are impossible events, such as the immediate disappearance of an object into nothingness, and from this we infer that their probability is 0. In such cases the statement is not probabilistic but of another kind. After reaching the conclusion that the event is impossible, we attach to it probability 0. In the first case, the probabilistic calculation is what underlies the matter. Because there are infinitely many integers, the probability that a random trial will yield one specific number is 0, but the event is not impossible. In the second case, the impossibility is the basic determination, and assigning probability 0 is a consequence of it.

The conclusion is that the claim that some event is impossible is not a probabilistic claim. The claim that the probability of an event is 0 does not mean that it is impossible for it to occur. We will now present an example that sharpens this point.

Illustration: Creation of the World as Spontaneous Formation

Many people seek to prove that the claim that the world came into being on its own, without a Creator, is impossible. A significant part of the arguments for this are based on various calculations of the probability that some entity or reality could arise by chance — for example, a calculation of the probability that a single organic molecule would form by a random process.

But this argument is mistaken and worthless. Such calculations cannot refer to a quantity called “the probability that some thing will arise by chance.” At most, they can calculate the rate of formation, or the probability per unit time. Obviously, the probability that a single organic molecule will form by a random process depends on how much time is allotted to the process. Over a billion years the probability is greater than over a single year. Hence such calculations can yield only the probability that an organic molecule will be formed randomly per unit time. For example, if such a calculation gives a result of 1/1000, the units of that result are molecules per year — that is, on average 0.001 molecules will be formed each year, the average rate of formation. The probability that a molecule will be formed in one year is small, but if we wait a thousand years, that is, the inverse of the average rate, then the average number of molecules formed randomly will be 1.

Thus, ironically, every calculation of this kind actually proves — or assumes — that a process of spontaneous formation is indeed possible; one merely has to give it enough time. So how can one prove that such a process is impossible altogether? Certainly not by probabilistic tools. The claim that such a process is impossible is not the result of a calculation showing that its probability is 0. On the contrary: a philosophical-intuitive argument would have to show that it is not possible, and only then would we say that its probability is 0.

In fact, no probabilistic calculation of this type ever yields 0. There is no way to arrive probabilistically at the conclusion that some process has probability 0. The result of a probabilistic calculation of the chances of a process is always finite, between 0 and 1, since it is composed of products of the probabilities of intermediate stages and sums over the possibilities of different processes.² Therefore, whenever we speak of probability 0, that determination cannot be grounded in probabilistic reasoning. Only the determination that a probability is very small can be the result of a probabilistic calculation.

A probabilistic claim that some event has probability 0 can be made only by pointing out that there are infinitely many equiprobable possible outcomes of the process, and therefore the probability that one specific outcome will occur is 0. But this is not really a mathematical calculation of the probability of a process. It is only a formal mathematical formulation of a simple intuition. Everyone understands that in a fair lottery, the probability of obtaining one particular result out of infinitely many is 0. This claim does not require calculation or probabilistic sophistication.

Moreover, if such a claim does emerge from a probabilistic calculation, that itself is a sign that the claim is merely that the probability is 0, not that the process is impossible. The determination that a process is categorically and absolutely impossible — the essential kind of impossibility defined above — can never be the result of any probabilistic calculation.

Returning to Balaam’s Ability

The impossibility of hitting the moment in which God is angry can likewise be understood in two senses:

1. It is the result of lack of knowledge. It is hard to hit one tiny moment out of the countless moments within a day. The probability of doing so is 0. This is a conception that sees the time axis as composed of discrete moments, except that each such moment is infinitely small, and therefore it is hard to hit precisely that one — its probability is 0.

2. From the description in Ohel Yehoshua it seems that hitting this moment is impossible. The reason is that there is in reality no moment called “midnight,” just as there is no other discrete moment. Midnight is not a particular time but a transition between two stretches of time. Of course midnight is not unique in this respect, and in precisely the same way one could conclude that every moment of time is only a transition between intervals. If so, it seems that hitting the moment in which God is angry is impossible, and not merely an event of probability 0. God can be angry for a moment, whatever exactly that may mean, but the term “moment” does not reflect any reality, and therefore it is impossible to hit it. Here we are not dealing with extreme difficulty that has a probabilistic expression, but with an event that belongs to the category of the impossible.

The distinction between these two suggestions follows from two different conceptions of the time axis. The first sees the time axis as composed of discrete moments packed densely side by side. According to the second, the time axis is not composed of moments. One may perhaps mark discrete moments on the axis, but they are not actual parts of it.

Comparison to Ohel Yehoshua

From the words of the author of Ohel Yehoshua it emerges that death is a process and not a state. It is a transition from a state of life to a state of death. Both states endure over continuous stretches of time, but the transition between them lacks ontological status — that is, the status of an entity. If so, his description does explain how death can occur at one discrete point in time. A moment does not truly exist, but it does reflect a transition between stretches of time. Therefore events that express processes rather than states can occur at a discrete moment of time.

But the wrath of God is a state and not a process. God is angry for one discrete moment only, and no more, and therefore it cannot be grasped. But how can a state exist at one discrete moment of time? Here the description of Ohel Yehoshua can no longer be applied. At first glance, the conclusion is that with respect to the wrath of God we are dealing with probability 0 and not with actual impossibility. But above we saw views according to which these are indeed impossibilities — that is, the moment has no duration, and therefore it is altogether impossible to hit it.

Dimensions and Measure

In modern mathematics, a continuous line is treated as something not composed only of discrete points.³ One can indeed mark such points on it, but the elementary particles that make up the continuum are infinitesimals. The infinitesimal is an arbitrarily small segment whose length is 0 — or, in the language of mathematics, whose length “tends” to 0. What is the essential difference between a point and an infinitesimal, if both have length 0?

The simplest formulation of this distinction points to a difference of dimension. Their measure is indeed the same — the length of both is 0 — but their dimension is different: the dimension of a point is 0, whereas the dimension of an infinitesimal is 1, since it is a one-dimensional segment and not a point. It is therefore more accurate to say that a point has no length, rather than that its length is 0, for length characterizes only segments whose dimension is 1. An infinitesimal, however, does have length, since its dimension is 1, except that this length is 0.

The conclusion that emerges from this picture is that the status of the moment is complex. On the one hand, it has no length, and therefore it is not a duration of time. On the other hand, it has some kind of existence, and is not merely a transition between stretches of time. If time is an “entity” of dimension 1, the moment is an “entity” of dimension 0. The conclusion is that the moment certainly does exist. It is not only a process, or a marker of transition between “entities.” It is itself an entity of lower dimension.

Application to the Divine Moment of Wrath

We can now try to apply the description of Ohel Yehoshua to instantaneous states — such as God’s momentary wrath — and not only to processes like death. Indeed, there is no time over which God is angry, since a state is characterized by an interval of time and not by a point. On the other hand, it is not correct to say that the moment has no existence at all. A discrete point on the time axis is also a kind of time, but it has no duration. For this reason it is impossible to hit it, since every human action takes place over an interval of time, and thus a human being cannot coincide exactly with an instantaneous state.

There is room to wonder whether this impossibility is an absolute impossibility, or whether only its probability is 0. At first glance, it is only the probability that is 0, for there is a moment on the time axis in which God is angry. But for the moment we have described it as an impossibility, since human actions always extend over a stretch of time, however short, and therefore cannot coincide exactly with a discrete moment.

Are Process and Instantaneous State Really Two Different Things?

On further reflection, one can say more. From our discussion it emerges that death too is a state, and not only a transition or process. It is a state that endures for only one moment of time. The author of Ohel Yehoshua does not treat a discrete moment as a genuine temporal reality, because its dimension is 0. But as we described above, the moment too has reality in some sense, and therefore one can speak about it in terms of an instantaneous state and not only in terms of a process. It is true that a human being cannot himself create a state whose duration is only one discrete moment. Our conclusion is that the second explanation — which sees the moment as a reality of lower dimension — may be correct in both cases.

B. Implications for the Question Whether Exact Precision Is Possible

Introduction

Several of the distinctions we have made here are relevant to understanding the halakhic principle that exactness is impossible, as discussed in the Mishnah, Bekhorot 17a, in the Talmud there, and in parallel passages. This principle states that two things that occur together cannot happen at precisely the same instant. Likewise, two things equal in measure cannot be perfectly equal.

The very fact that the same principle is expressed in two different contexts — time and measure — indicates that it is probably rooted in the problem of the continuum with which we dealt in the previous section. This problem appears both when we ask whether two events can occur at one and the same discrete instant, and when we ask whether two things can have one and the same exact measure, since measures too usually take continuous values. In this section we will say a bit about that principle and about the application of our discussion thus far to it.

Exactness Is Impossible

The Torah establishes that the firstborn of a pure animal goes to the priest as one of the priestly gifts. What happens when two firstborn are born at the same time? On this matter the tannaim disagree in the Mishnah in the above passage of Bekhorot:

If a ewe that had never before given birth delivered two males, and their two heads emerged simultaneously, Rabbi Yosei the Galilean says: both belong to the priest, as it is said, “the males belong to the Lord” (Exodus 13). And the Rabbis say: exactness is impossible; rather, one is his and one belongs to the priest.

According to Rabbi Yosei the Galilean, both go to the priest. According to the Rabbis, it cannot be that both emerged at once, and therefore only one of them is the firstborn; only one must be given to the priest. Later in the Mishnah other tannaim disagree about how exactly this is to be implemented. The Mishnah itself states that the disagreement turns on whether exactness is possible.

Now Rashi on that passage explains:

“Exactness is impossible” — it is impossible that their two heads emerged at the same time; rather, one emerged first, but we do not know which one.

It is explicit from his language that the possibility that both emerged at once is impossible. One of them must have emerged first, and therefore he is the firstborn. Clearly the intent is not a physical limitation — that two offspring cannot emerge from the womb at the same time — for the case assumes that both in fact emerge in parallel. Moreover, if this were a principle specific to the laws of the firstborn, there would be no reason to formulate it as a sweeping principle applying throughout halakha. If so, Rashi explains the problem here as a problem of the continuum: two events cannot occur at the same discrete instant.

It seems quite clear that we are not dealing here with principled impossibility, since there is no reason to assume that such a thing is impossible at the level of principle. A determination of principled impossibility belongs not to probability but to physics, or physiology. And since we showed above that this is not a claim from physiology, the Mishnah must mean that this is an event whose probability is 0. In other words, the probability that the two events will occur at exactly the same discrete instant is 0.

This is also implied by the language of Maimonides, who rules like the Rabbis and writes in Mishneh Torah, Laws of Firstborn Animals 5:1 — see also Shulchan Arukh, Yoreh De’ah 318:1, which rules likewise:

If a ewe that had not yet given birth bore two males, then even if their two heads emerged at one time, it is impossible that one did not precede the other. Since it is not known which of them emerged first, the priest takes the weaker one, and the second is a doubtful firstborn. If one of them dies, the priest receives nothing, for the surviving one is in doubt, and the burden of proof rests on the claimant. Likewise, if she bore a male and a female, the male is doubtful, for perhaps the female emerged first; therefore the priest receives nothing, for the burden of proof rests on the claimant.

Thus Maimonides too sees here a difficulty connected with the continuum: the probability that two events will occur at the same discrete instant is 0.⁴

The View of Tosafot

The Tosafists on that passage, Bekhorot 17b, s.v. “Is exactness possible?”, disagree. They hold that the thing is possible, but we cannot determine whether they emerged equally or whether one preceded the other, and if so which. Therefore only one is given to the priest, because the burden of proof rests on the claimant. In other words, the Tosafists hold that the thing is indeed possible, but we can never know it.

At first glance, this seems to be a principled dispute. According to Tosafot, this is a possible event, and the doubt exists only from our standpoint, whereas according to Rashi and Maimonides the event is impossible.

But in light of what we established above, it seems that there is no principled dispute here. All agree that this is an event whose probability of occurrence is 0, but it is clear that in principle it can occur; it is not absolutely impossible. In fact, in this matter the outcome must necessarily be one whose probability is 0, for every moment at which the firstborn might emerge has probability 0, since there are infinitely many moments at which it might have emerged. If so, it seems that Rashi and Maimonides are not in substantive disagreement with Tosafot, and in the opinion of all of them this is an event that can occur, though with probability 0, but we cannot know it.

Even so, there does seem to be a halakhic disagreement, not a probabilistic one. According to Rashi and Maimonides, two events that occur at the same discrete instant are treated halakhically as impossibilities. An occurrence whose probability is 0 is not taken into account, not even as creating a doubt. According to Tosafot, by contrast, the possibility that the two events truly occurred simultaneously is a possibility that must be taken into account, and it therefore creates doubt.

Exactness by Heaven and by Man

The Talmud there, on Bekhorot 17b, says:

The school of Rabbi Yannai said: with regard to Rabbi Yosei the Galilean, we have heard him say that exactness is possible at the hands of Heaven — all the more so at the hands of man. As for the Rabbis: at the hands of Heaven exactness is impossible; what about at the hands of man?

And Rashi there explains:

“Exactness is possible at the hands of Heaven” — as in this case of birth, even though no one is taking care to achieve exactness.

“All the more so at the hands of man” — when people intentionally try to make some measure or some other matter exact, exactness is certainly possible.

“At the hands of man” — when people intentionally measure in order to be exact, is exactness possible or not?

The Talmud assumes that exactness by Heaven is harder than exactness by man. The reason is that exactness by Heaven is a random event, and therefore the probability of exact coincidence is 0. But exactness by man is a directed event: a person is trying to attain simultaneity, and therefore there is more room for the claim that exactness can be achieved.

One of the examples the Talmud brings for exactness by man is the scarlet thread that had to be placed at the middle of the altar, to distinguish between bloods applied above and below. When the thread is fastened to the altar, there is an intentional attempt to reach exactly the middle, and therefore there is a greater chance of hitting the midpoint precisely. Since the event is not random, there may be room to say that calculating the chance of hitting the exact middle does not yield probability 0.⁵

Exactness by Man: A Problem in the View of Tosafot

At first glance, the distinction between exactness at the hands of Heaven and exactness at the hands of man can be made only according to the view that we are dealing with impossibility in reality itself. Then one can say that if one aims to achieve exactness, its likelihood increases. But according to Tosafot, who hold that this is only a human ambiguity and not an impossibility in reality itself, what room is there to distinguish between exactness by Heaven and exactness by man? If a human being cannot discern an exact coincidence, then exactness is impossible — whether at the hands of Heaven or at the hands of man.

And indeed, in Tosafot there, s.v. “Is exactness possible?”, after citing Rashi’s words, they explain the matter in several ways with respect to several kinds of cases. For example, in exactness by Heaven a person has no leisure to try to align things, and therefore does not succeed. But if the exactness is at human hands, then the person is intentionally trying to make it exact, and in such a case he may perhaps succeed, and so forth.

The Halakhic Ruling and Its Meaning

From our discussion it emerges that according to Tosafot we are truly not dealing here with essential impossibility. It seems that they do not see in the problem of exactness an aspect of the problem of the continuum, but only a technical difficulty that is hard to overcome. According to Rashi and Maimonides, by contrast, this seems to be an expression of the problem of the continuum, and therefore exactness is impossible — that is, its probability is 0.

And indeed, the halakha is ruled that at the hands of Heaven exactness is impossible, as in the Maimonides and Shulchan Arukh cited above. As for exactness at the hands of man, the medieval authorities disagree: according to the Tosafot just mentioned, and in the parallel passage in Eruvin 5a, the ruling is that exactness is possible. Maimonides, by contrast, in Mishneh Torah, Laws of Murderer and Preservation of Life 9:8, rules that exactness by man is impossible.

It is quite plausible that they are following their own consistent methods. Maimonides, who sees here a principled problem of impossibility in reality — an expression of the problem of the continuum — obviously holds that even at the hands of man exactness is impossible. Tosafot, however, who see this as a technical difficulty from the human side, rule that at the hands of man exactness is possible.

C. Zeno’s Arrow

Introduction

We will conclude the essay with a brief discussion of one of the paradoxes presented by Zeno of Elea, intended to undermine the concept of motion. We will propose a solution to it, and then briefly consider several aspects in which that solution has implications.

The Paradox of the Flying Arrow

The Greek philosopher Zeno of Elea raised the following question:⁷ Let us consider an arrow in flight. At every single moment, if we observe it, we see it standing still. In other words, at every moment of time it is at rest, each time in a different location. So when — that is, at which moment — does it change its location?

It is customary to see the solution to the paradox in the infinitesimal picture. That is, the assumption that the time axis is composed of discrete moments is treated as the stumbling block that generates the paradox. If we adopt the infinitesimal picture — namely, that the time axis is composed of a collection of infinitesimals — then the question disappears of itself. One can no longer speak of the arrow’s condition at one moment of time, because there are no discrete moments of time.

The impression made by this solution is that it bypasses the obstacle, but does not really remove it. Clearly there are discrete moments on the time axis. One can mark such points on any continuous axis. Therefore, even if we cannot concretely relate to a discrete moment, we can still ask the theoretical question of the arrow’s condition at some discrete moment. The fact that infinitesimal language forbids asking the question does not constitute a solution.

If the question concerned our inability to discern some state, then there would be a principled path to a solution, since human reference always concerns a stretch of time and never a discrete moment. That is one of the possibilities we raised regarding the impossibility of exactness. But here the question is theoretical, and therefore that answer does not solve it.

Solving the Paradox

The correct solution to this paradox lies in the distinction between “being in a place” and “standing still in a place.” It is true that at every moment in time the arrow is found in a different place, but it is certainly not true that it is standing still in those places. It is found in some place, but at the same time it is also moving. Therefore, the answer to the question when the arrow changes its location — that is, moves — is: at that very moment at which we observe it. It is both located somewhere and moving to another place.

The Root of the Confusion

What underlies Zeno’s confusion is the intuition that one cannot speak about velocity at a point in time. The feeling is that velocity exists and is defined only over a stretch of time, and also of space, but not at a discrete point of time and space. Velocity means change of place, and a change of place cannot occur across one point, whether of time or of space. From this it would follow that at a discrete point of time a body cannot have velocity, and it must be standing still — that is, at rest, with velocity 0.⁶

But this assumption is altogether mistaken. Velocity is not change of place. Change of place is a result of the fact that a body has velocity. Bodies that have velocity other than 0 change their place, but one must not identify the concept of velocity with change of place. Velocity is a potential for change of place, and therefore even though change of place is not carried out within one discrete moment, the potential for change can be defined even at a discrete moment of time.

Physics defines velocity at a point in time — that is, there is velocity at every discrete moment of time — though the way this quantity is calculated is by dividing a spatial interval by a temporal interval, meaning: how long it takes to traverse the spatial interval. That is the form of calculation, an operative definition, but not an essential definition. Essentially, a body has velocity at every discrete point in time. As stated, calculating that velocity requires expanding to a small interval around the point under discussion, but that is only a computational constraint, not an essential one.

Returning to Exactness

One may see in this distinction a further aspect of the distinction we presented above. Velocity is a quantity that exists at a single discrete point in time, but we as human beings cannot relate to a discrete point in time; we can relate only to an interval of time. That is what causes us to confuse the operative definition of velocity, which refers to intervals of time and space, with the essential definition, which refers to discrete points. In this formulation we can say that Zeno’s paradox is created by the fact that exactness is impossible.

The treatment in Ohel Yehoshua of death as an event that cannot be captured — that is, an event that does not exist on the time axis, a process rather than a state — stems from the fact that we cannot relate to a discrete point in time. But as we suggested above, this event does exist on the time axis; it is simply a momentary event. The same applies to God’s momentary wrath.

Conclusion

In this concluding section we will raise a number of reflections. Most of them will end with a question mark and will not lead to a clear conclusion. One may well wonder what the significance is of the inquiry we undertook this week. Is it Torah study? Or is it the use of Torah sources in order to study physics — for understanding the meaning of the time axis ordinarily belongs to the domain of physics? Is it the study of aggadah, or actual halakhic study, since this topic has quite a few ramifications in the areas of halakha?

The question can be broadened by examining the boundaries of the concept “Torah.” Is the study of Guide of the Perplexed, with its philosophical sections — even when drawn from external sources — included in “Torah”? If so, how are modern mathematics and physics different from Aristotelian physics? Why did Aristotelian physics, once incorporated into Maimonides’ Laws of the Foundations of the Torah, receive the status of Torah study, whereas the study of concepts and insights from modern science does not receive that standing? The same question may be asked about modern philosophy.

There are quite a few situations in which mathematics, or some science, is needed in order to understand a halakhic discussion or to decide a halakhic question. In such cases, engagement with mathematics or science seems to be in the category of preparations for a mitzvah (commandment), that is, a preparatory means for the mitzvah of Torah study. But here the situation is the opposite: the goal was not to understand a distinctly halakhic or Torah topic, but to use Torah and halakhic sources in order to understand the nature of the time axis. Is that Torah study? If it is, the conclusion is that it stands at a higher level, since it is Torah proper and not merely a preparation for the mitzvah of Torah study.

Are abstract legal theories that explain the laws of neighbors or the laws of evidence considered Torah, whereas insights into the nature of the time axis, space, human nature, and the like are not Torah? Is this a question of sources — from where we draw our conclusions — or of content? Maimonides himself, in Mishneh Torah, Laws of the Foundations of the Torah 4:10-11,⁷ states that Ma’aseh Bereishit and Ma’aseh Merkavah — which the Talmud refers to as a “great matter,” in contrast to the discussions of Abaye and Rava, which are a “small matter” — are physics and metaphysics. He therefore devotes several chapters of his Laws of the Foundations of the Torah to these subjects. If we learn from him, it seems that the required conclusion is that this too is included within the category of Torah. Of course, one may then ask: what is not called Torah? Is every engagement with some wisdom included in Torah? If so, what is the difference between wisdom and Torah?⁸

Footnotes