Time is change itself.

So I don’t understand.

First of all, one thing is clear: the equation in your words, time = change, is nonsense. There is no identity at all between those two. Time is some abstract axis; change is a process that things undergo.
So what is true? As I explained, time is the axis on the basis of which we describe changes—and not that time is itself change, as you wrote. From that it follows that in a world where there is no time, there is no reason to assume there cannot be changes. Rather, the inhabitants of such a world would simply describe change differently; they would not use terms of time. It’s like someone who doesn’t use spatial terms is not thereby denying the existence of a world. It just means that the conceptual system he uses to describe the world is not spatial, but different. That is certainly what follows from Kant’s assumption that space and time are only forms of our intuition and not things that exist in the objective world in itself. By analogy: someone who does not wear glasses cannot see, but that does not mean there is nothing to see.

He is probably relying on the Aristotelian definition of time, which appears in many medieval Jewish books. In that view, time, like space, is not a medium with independent existence, but only our abstraction from changes—we place the states of the world, especially of the outer sphere that rotates regularly and causes all the other motion in the world, on an axis, and we call that axis time. In a world without change, according to Aristotle, there is no time at all.

I don’t understand.
After all, it is impossible to describe change without time not because we have some deficiency in description,
but because only through time can there be change.

That is, without the concept of time the world would be stuck in its “initial” state, and therefore it makes no sense to claim that the world was created by a Creator. Because if there were a Creator who created the world, time would have to begin just before creation in order to allow for the change in the world’s state.

From this it follows not only that there is proof that a Creator isn’t needed, but that there cannot be a Creator. The absence of time prevents Him from changing…

We’ve exhausted this.

Yishai—yes, actually that’s what I mean.
Good thing there’s someone who understands me…. 🙂

In short, maybe the Rabbi can explain why what Yishai says is not correct?
Instead of saying, “We’ve exhausted this.”

I think Yishai also didn’t understand you; he only explained what you meant. I understood it too, and as I explained, it is simply a mistake.
I’ll try one last time. In a world where there is no time, a different conceptual system prevails from ours, one that I myself don’t really manage to understand. Everything that happens there may be similar to what happens here, except that in that conceptual framework it would be explained in different terms. If there is no time, change would be described in a different conceptual system, but that does not mean there would be no change. You are taking the absence of time to mean being frozen in place—there is only one static time-point. But that is not so. It simply means the time-axis is irrelevant. Think of a creature that has no perception of time or space. Would the world not exist for it? It would exist, but would be described in a different conceptual system.
That’s it. I don’t know how to explain it better. All the best.

In my humble opinion, the very fact that scientists look for an answer to the question, “What was before the Big Bang? And what caused it?” shows that everyone assumes a chain of causality even before the time-axis.

The Rabbi is turning the concept of time into a personal concept.
That is, we feel time, not that it is an axis that really exists.

That’s why you give the example of a creature without a sense of time.
I agree that it could live in our world, but because of its lack of stupidity it wouldn’t understand why.

But in reality, in that creature’s world time would still prevail…

I just can’t understand how there can be change without time.
Of course that would be a freeze in place. There might be other concepts within the freeze, like width and length and so on.
But it is a freeze with no possibility of change.

A creature without a sense of time is an example of ourselves in a world where there is no time. A creature without a sense of time is not necessarily stupid, but rather a creature that organizes phenomena in terms of a conceptual system different from ours. If a person has no eyes but perceives the world around him through another sense unknown to us, he is not more stupid, just different. That was an example showing that in a world without time there can certainly be changes, and we would perceive them the way a creature lacking a sense of time would perceive events in our world.

I still don’t understand why this isn’t a freeze.

But I think what I’m saying is not so logical.
Because otherwise, if the absence of time is a freeze, then it cannot be that the universe underwent a change from a timeless state in which there was nothing to a state in which there is a universe.
After all, that change of state—from before the Big Bang to after it—is a change. And it happened and began even before time.
That is a sign that change can occur without time.

What does the Rabbi say about that? Maybe he has another explanation?

Hello Kobi. If time was created with the creation of the universe, then your question is not fully well-defined. At the moment of creation, time too came into being. This is the mechanism of “they come simultaneously” in the Talmudic text.
But this whole discussion is unnecessary, as I already explained, because the concept of change does not depend on time. So I don’t know what other explanation you are looking for.
On this mechanism, from a similar perspective, read my article here (Midah Tovah, Balak 5767):

What Is a ‘Moment’?
A Look at the Nature of the Time Axis

Introduction
In this week’s article we will deal with a topic that is not distinctly halakhic: the nature of the time-axis as a continuum. We will indeed mention halakhic implications in the topic of “exact coincidence is impossible,” but that will not be our main subject. Even so, clarifying the concepts and ambiguities that relate to our perception of the time-axis can be useful in studying quite a few halakhic topics, and others as well.

A. Probability, Dimensions, and the Essence of Time

Introduction: Two Interpretations of a ‘Moment’
The Talmudic text in tractate Berakhot 7a says that every day there is a moment in which the Holy One, blessed be He, is angry. It goes on to expound the verses in our Torah portion and says that only Balaam knew how to pinpoint that moment:
As it was taught: “God is angry every day” (Psalms 7). And how long does His anger last? A moment. And how long is a moment? One fifty-eight-thousand, eight-hundred and eighty-eighth of an hour, and that is a moment. No creature can precisely determine that hour except wicked Balaam, of whom it is written: “And he knows the knowledge of the Most High” (Numbers 24). Now then, the knowledge of his own animal he did not know—would he know the knowledge of the Most High? Rather, this teaches that he knew how to determine the hour at which the Holy One, blessed be He, is angry. And this is what the prophet said to Israel: “My people, remember now what Balak king of Moab counseled…” (Micah 6).
The “moment” spoken of here is a very short segment of time. True, it is not a single time-point but a short segment. Immediately afterward a parallel exposition is brought based on a verse in Micah, in which the Holy One, blessed be He, 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 performed for you in not becoming angry in the days of wicked Balaam. For had I become angry, not one remnant or survivor would have remained of the enemies of Israel. And that is what Balaam said to Balak: “How shall I curse whom God has not cursed? And how shall I rage when the Lord has not raged?” (Numbers 23). This teaches that all those days He did not rage.
In Tosafot, s.v. “for had I,” ad loc., they apparently understood that at that stage of the Talmudic discussion they still held that this “moment” is a segment with finite length, though short:
“For had I become angry, none would have remained…” And if you say: what could he have said in the span of a moment? One may answer: “Destroy them.” Or alternatively, once he began his curse at that hour, he could have caused harm even afterward.
Though in the second answer Tosafot may already adopt the position that the “moment” is a time-point and not a short segment. And indeed at the end of the discussion such an approach is brought:
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 saying the word “moment.” And from where do we know that He rages 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/Avina says that a moment is an isolated time-point.
According to the first interpretation, the inability to hit the moment comes merely from lack of knowledge. According to the second interpretation, the lack of knowledge results from inability to hit exactly a moment that is a discrete point in time.

Midnight
We find a similar hesitation regarding midnight. In Exodus 11:4, the Holy One, blessed be He, informs us:
And 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 cattle.
Here the expression is “at midnight,” which is more precise. The Mekhilta expounds this as follows (Masekhta de-Pischa, parasha 13):
“And it came to pass at midnight”—why is this said? Because it says, “And Moses said: Thus says the Lord: About midnight I will go out…” For it is impossible for flesh and blood to determine exactly the middle of the night, but its Creator divided it. Rabbi Yehuda ben Betera says: the One who knows its hours and seasons divided it.
That is, the Holy One Himself can determine the exact moment of midnight, and therefore the verse that reports God’s own word speaks in the language of “at midnight,” but Moses cannot determine that exact moment, and so with him it says “about midnight.”
And similarly in Mekhilta de-Rabbi Shimon bar Yochai:
But when Moses said to Pharaoh, what did he say? “Thus says the Lord: About midnight…” He said to him: the matter is balanced for when the night will split—whether by a hair above or by a hair below. He sits upon a sundial and determines the hour to a hairsbreadth, for one kingdom does not encroach upon another even by a hair’s breadth; rather, when the time of a kingdom comes to fall, it falls by day if by day, by night if by night…
This midrash implies that the Holy One determines the hours exactly to a hairsbreadth, though here it is applied to the downfall of various kingdoms.
The Talmudic text in Berakhot 3b deals with David rising at midnight to play music and study Torah, and it continues the line we saw and brings a parallel discussion comparing King David to Moses our rabbi:
And did David know when midnight was? Now Moses our rabbi did not know, as it is written: “About midnight I will go out into the midst of Egypt.” What does “about midnight” mean? If you say that the Holy One, blessed be He, said to him “about midnight”—is there uncertainty before Heaven? Rather, He told him “at midnight,” and he came and said “about midnight,” implying he was uncertain. And David knew? David had a sign, as Rav Aha bar Bizna said in the name of Rabbi Shimon Hasida: A harp hung above David’s bed, and once midnight arrived a north wind would blow upon it and it would play on its own. Immediately he would rise and study Torah until dawn.
The conclusion is that Moses indeed did not know, while David knew by means of a special sign he received from above. Though later there (4a) an opinion is brought that Moses knew as well—and this is the opinion Rashi cites on the Torah, see also Torah Shelemah there:
Rabbi Zeira said: Moses certainly knew, and David also knew. And since David knew, why did he need the harp? To awaken him from sleep. And since Moses knew, why did he say “about midnight”? Moses thought: perhaps Pharaoh’s astrologers will err and say Moses is a liar. As the Master said: Train your tongue to say, “I do not know,” lest you be exposed as false and be caught.

The Words 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 question:
It should be noted that this time of midnight cannot be found, for when one divides the night 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 at all, for it is only a name for the division of the night. Every middle time is itself also divided into two, half to this side and half to that side. So how can there be such a thing as midnight?
That is, he asks how one can speak of “the time of midnight,” since there is really no such time. It is the point that divides the night into two, but it itself has no independent existence. Up to it is the first half of the night, and after it is the second half. It itself has no existence at all, like a point dividing two parts of a line, which in itself has no length and no existence in terms of the time-axis.
To this he proposes the following sharp answer:
It likewise ought to be noted that death is also not a matter of time, for beforehand he is alive and now he is dead. It turns out there is no “time of death” at all, for as long as he has not yet died he is alive, and once he stops living he is dead. So there is no time of death at all.
And plainly this is indeed true, and therefore the first question is settled. So too it was in the plague of the firstborn: they were alive in the first half-midnight, and in the second half-midnight they had already ceased living and were dead. Indeed midnight is not a time, and the hour of death is not a temporal matter. Rather, the transition from death to life was parallel to the transition from the first half of midnight to the second half. And this is the meaning of “and it came to pass at midnight”: it does not mean the time of midnight, but that midnight, just as it split the night, split between death and life for the firstborn of Egypt.
He first explains that death too does not occur at any time. Death—the departure of the soul—is only the transition between life and death, and therefore it itself does not occur on the time-axis. It divides the axis into two parts, each of which has temporal meaning. These two descriptions complement one another: indeed there is no instant called “midnight,” and there is no instant in which death occurs. The “instant” 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 lasts no time.

Two Kinds of Probability 0
Usually people understand that when an event has probability 0, it is impossible. But that is not true. The chance that some particular integer will come up in a lottery over all integers is 0, yet in the end one integer will in fact come up. That means that in such a case it is certain that an event whose probability is 0 will occur. Sometimes there are impossible events, such as the immediate disappearance of an object into nothingness, and from that we infer that their probability is 0. In these situations, however, the claim is not probabilistic but something else. Once we have concluded that the event is impossible, we assign it probability 0. In the first case, the probabilistic calculation is what underlies the matter. Since there are infinitely many integers, the chance that a random trial yields one particular number is 0, but the event is not impossible. In the second case, impossibility is the basic determination, and assigning probability 0 is only its result.
The conclusion is that the claim that some event is impossible is not itself a probabilistic claim. Saying that the probability of an event is 0 does not mean it is impossible for it to happen. Let us now present an example that sharpens this point.

Demonstration: Creation of the World as Spontaneous Formation
Many are inclined to prove that the claim that the world came into being by itself without a Creator is impossible. A considerable part of the proofs for this claim are based on various calculations of the probability that some object or reality could arise by chance—for example, a calculation of the likelihood that a single organic molecule could form in a random process.
But this argument is invalid and void. Such calculations cannot relate to a quantity called “the probability that something will form by chance.” At most one can calculate the rate of formation, or the probability per unit time. Clearly, the probability that an organic molecule will form in a random process depends on the amount of time allowed for the process. Over a billion years the probability is higher than over one year. So such calculations can only yield the probability that an organic molecule will form 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—meaning that on average 0.001 molecules are formed each year, the average rate of formation. The probability that a molecule will form in one year is small, but if we wait a thousand years, that is, one over the average rate, the average number of molecules formed randomly will be 1.
So ironically, every calculation of this kind actually proves—or assumes—that a process of spontaneous formation is certainly possible; it just needs enough time. So how can one prove that such a process is not possible at all? Certainly not with 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 show that it is impossible, and only then would we say that its probability is 0.
In fact, no probabilistic calculation of this sort 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 any process is always finite, between 0 and 1, since it consists of products of the probabilities of intermediate stages and sums over possibilities for different processes. Therefore, whenever we speak of a chance of 0, that determination necessarily does not rest on a probabilistic consideration. Only the claim that the chance is very small can be the result of a probabilistic calculation.
A probabilistic determination that an event has probability 0 can be made only by pointing out that there are infinitely many equally weighted possible outcomes of the process, and therefore the chance that one particular outcome will occur is 0. But that is not really a mathematical calculation of the probability of a process; rather, it is a mathematical-formal formulation of a simple intuition. Everyone understands that in a fair lottery, the chance of getting one particular outcome out of infinitely many is 0. This does not require calculations or expertise in probability.
Moreover, if such a claim arises from a probabilistic calculation, that itself is a sign that we are talking about the event having probability 0, not about the process being impossible. The claim that a process is categorically and absolutely impossible—the substantive kind of impossibility defined above—is never the result of any probabilistic calculation whatsoever.

Back to Balaam’s Ability
The impossibility of hitting the moment when the Holy One, blessed be He, is angry can also be interpreted in two ways:
1. It is the result of lack of knowledge. It is hard to hit one tiny moment out of the countless moments in every night. Its probability is 0. This is an approach that sees the time-axis as composed of discrete moments packed densely side by side. According to it, each such moment is infinitely small, and therefore it is hard to hit that one in particular; the probability is 0.
2. From the description in Ohel Yehoshua, it appears that hitting this moment is impossible. The reason is that there is no actual thing in reality called “midnight,” just as with any other discrete moment. Midnight is not a specific time, but a transition between two segments of time. Of course midnight is not unique in this, and in exactly the same way one can conclude that every moment of time is only a transition between segments. If so, it seems that hitting the moment when God is angry is impossible, and not merely something whose probability is 0. Here we are not talking about a huge difficulty that can be expressed probabilistically, but about something impossible in principle.
The distinction between these two suggestions arises from two different conceptions of the time-axis: the first sees the time-axis as composed of discrete moments lying densely side by side; according to the second, the time-axis is not made of moments. One may perhaps mark discrete moments on the axis, but they are not a real part 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 the state of life to the state of death. The two states endure for continuous stretches of time, but the transition between them has no ontological status. If so, his description indeed explains how death can occur at one discrete time-point. A moment does not really exist, but it reflects a transition between time-segments. Therefore, events that express processes rather than states can occur at a discrete time-point.
But God’s anger is a state, not a process. God is angry for only one discrete moment and no more, and therefore one cannot grasp it. But how can a state exist over a single discrete moment of time? Here the description of Ohel Yehoshua can no longer be applied. Seemingly the conclusion should be that with respect to God’s anger we are dealing with probability 0 and not with real impossibility. Yet above we saw opinions that this too is impossible—meaning that this moment has no duration, and therefore it is not possible at all to hit it.

Dimensions and Measure
In modern mathematics, a continuous line is not treated as something composed solely of discrete points. One can indeed mark such points on it, but the elementary constituents of the continuum are infinitesimals. An infinitesimal is a segment as small as one likes, whose length is 0—or in mathematical language, tends to 0. What is the principled difference between a point and an infinitesimal, since both have length 0?
The simplest formulation of the distinction points to a difference in dimension. Though the measure of the two is the same—their length is 0—their dimensions differ: a point has dimension 0, whereas an infinitesimal has dimension 1, since it is a one-dimensional segment and not a point. Therefore it is more accurate to say that a point has no length, rather than that its length is 0, because length characterizes only segments of dimension 1. And an infinitesimal does have length, because it is dimension 1, except that this length is 0.
The conclusion that emerges from this picture is that the status of a moment is complex. On the one hand, it has no length and therefore is not a duration of time. On the other hand, it has some kind of existence and is not merely a transition between time-segments. If time is an entity of dimension 1, then a moment is an entity of dimension 0. The conclusion is that a moment certainly does exist. It is not merely a process or a marker of transition between entities. It is an entity of lower dimension.

Application to God’s Anger
Now we can try to apply the description of Ohel Yehoshua to momentary states like God’s momentary anger, and not only to processes like death. Indeed there is no duration in which God is angry, since a state is characterized by a segment of time and not by a point. On the other hand, it is incorrect to say that a moment has no existence at all. A discrete point on the time-axis is also a kind of time, but one with no duration. For that reason it is impossible to hit it, since every human action takes place over a stretch of time, and therefore a person cannot coincide exactly with a momentary state.
There is room to wonder whether this impossibility is one of absolute impossibility, or only that its probability is 0. Seemingly, only the probability is 0, since there is indeed a moment on the time-axis when God is angry. But at the moment we have described it as impossible, since human actions always extend over a segment of time, however short, and therefore cannot exactly overlap with a discrete moment.

Are a Process and a Momentary State Really Two Different Things?
On further reflection, one can say more. From our discussion it follows that death too is a state, not merely a transition or process. It is a state that lasts only one single moment of time. The author of Ohel Yehoshua does not treat a discrete moment as a temporal reality, since its dimension is 0. But as we described above, a moment too has some kind of reality, and therefore it is possible to speak of it in terms of a momentary state and not only in terms of a process. True, a person 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 apply to both cases.

B. Implications for the Question Whether Exact Coincidence Is Possible

Introduction
Several of the distinctions we made here are relevant to understanding the halakhic principle that “exact coincidence is impossible” (see Mishnah Bekhorot 17a, the Talmudic discussion there, and parallels). This principle states that two things that occur simultaneously cannot occur at precisely the same time. Likewise, two things equal in measure cannot be completely identical.
The very fact that the same principle appears in two different contexts—time and measure—indicates that it is probably rooted in the problematic nature of continuity that we dealt with in the previous section. This problem appears both when we ask whether two events can occur at the same discrete moment, and when we ask whether two things can have exactly the same measure, since measurements too generally take continuous values. In this section we will deal a bit with that principle and with applying what we have said so far to it.

Exact Coincidence Is Impossible
The Torah states that the firstborn of a kosher animal goes to the priest as one of the priestly gifts. What happens when two firstborns are born at once? On this point the tannaim dispute in the Mishnah in the above topic in Bekhorot:
If a ewe that had not previously given birth delivered two males, and their two heads emerged at once, Rabbi Yosei HaGelili says: both belong to the priest, as it says, “the males shall be the Lord’s” (Exodus 13). But the Sages say: exact coincidence is impossible; rather, one belongs to him and one to the priest.
According to Rabbi Yosei, both go to the priest. According to the Sages, it cannot be that both emerged together, and therefore only one of them is the firstborn, so only one is given to the priest. The Mishnah itself says that the dispute is over whether exact coincidence is possible.
Now Rashi there explains:
“Exact coincidence is impossible”—that their two heads emerged together; rather, one emerged first, but we do not know which one.
It is explicit from his language that the possibility of both emerging together is impossible. Necessarily one of them emerged first, and therefore he is the firstborn. Clearly this does not refer to a physical limitation—that two newborns cannot emerge from the womb at once—for the case is that both really do 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 Jewish law. So Rashi explains that the issue here is the problem of continuity: it is impossible for two events to happen at the same discrete moment itself.
It is fairly clear that this is not absolute impossibility in principle, for there is no reason to assume the thing is impossible at that level. 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 the realm of physiology, we are forced to conclude that the Mishnah means that this is an event whose probability is 0. That is, the chance that two events occur at exactly the same discrete moment in time is 0.
So too it appears from the language of Maimonides, who rules in accordance with the Sages and writes (Laws of Firstborns 5:1; see also Shulchan Arukh, Yoreh De’ah 318:1):
If a ewe that had not previously given birth delivered two males—even if their two heads emerged together—it is impossible that one did not precede the other. Since it is not known which emerged first, the priest takes the weaker one, and the second is a doubtful firstborn. If one of them died, the priest receives nothing, for the living one is only doubtful, and the burden of proof rests on the claimant. Likewise, if she gave birth to a male and a female, the male is doubtful, perhaps the female emerged first; therefore the priest receives nothing, for the burden of proof rests on the claimant.
So Maimonides too sees here a problem related to continuity: the chance that two events occur at exactly the same discrete moment is 0.

The Approach of Tosafot
The Tosafists there (17b, s.v. “exact coincidence is possible”) disagree and hold that the thing is possible, but we cannot determine whether they emerged exactly equally or whether one preceded the other, and which one, and therefore only one is given to the priest, because the burden of proof rests on the claimant. That is, the Tosafists hold that this is indeed possible, but we can never know it.
At first glance this seems like a principled dispute: according to Tosafot this is a possible event and the doubt is only from our perspective, whereas according to Rashi and Maimonides the event is impossible.
But in light of what we established above, it seems there is no principled dispute here. Everyone agrees that this is an event whose chance of occurring is 0, though clearly in principle it can happen—it is not absolutely impossible. In fact, in this matter the result must in any case be one whose probability is 0, since every moment at which the firstborn emerged has probability 0, because there are infinitely many moments at which it could have emerged. If so, it seems that Rashi and Maimonides do not fundamentally disagree with Tosafot. According to everyone, this is an event that can happen, with probability 0, but we cannot know it.
Still, there does appear to be a halakhic dispute here, not a probabilistic one: according to Rashi and Maimonides, two events occurring at the same discrete moment are treated halakhically as impossible. An event whose probability is 0 is not taken into account even as creating doubt. By contrast, according to Tosafot, the possibility that the two events truly occurred simultaneously must be taken into account and does create doubt.

Exact Coincidence by Heaven and by Human Agency
The Talmudic text there on 17b says:
The school of Rabbi Yannai said: According to Rabbi Yosei HaGelili, we hear that he says exact coincidence is possible by Heaven—and all the more so by human agency. And according to the Rabbis, by Heaven exact coincidence is impossible. What about by human agency?
And Rashi there explains:
“Possible by Heaven”—such as this birth, even though no special care is taken about it.
“And all the more so by human agency”—for they intentionally try to match some measure or some matter, so certainly exact coincidence is possible.
“By human agency”—where they intend and measure in order to match exactly; what then: is exact coincidence possible or not?
The Talmud assumes that exact coincidence by Heaven is harder than by human agency. The reason is that exact coincidence by Heaven is random, and therefore its probability is 0. But exact coincidence by human agency is intentional; a person is trying to reach simultaneity, and therefore there is more room to claim that exact coincidence is possible.
One of the examples brought in the Talmud for exact coincidence by human agency is the red line that was supposed to be in the middle of the altar, distinguishing between bloods applied above and below. When the line is tied to the altar, there is an intentional attempt to place it exactly in the middle, and so there is a higher probability of hitting the midpoint. Since the event is not random, perhaps one can say that the calculation of hitting exactly the middle does not yield probability 0.

Exact Coincidence by Human Agency: A Problem in Tosafot’s View
Seemingly the distinction between exact coincidence by Heaven and by human agency can be said only according to the view that this is a matter of impossibility in reality itself. Then one can say that when aiming for exact coincidence, the probability increases. But according to Tosafot, for whom this is only a matter of human ambiguity and not impossibility in reality itself, what room is there to distinguish between exact coincidence by Heaven and by human agency? If a person cannot distinguish an exact match, then exact coincidence is impossible in both cases.
And indeed, in Tosafot there, after bringing Rashi’s words, they explain in several ways regarding several kinds of cases. For example, in exact coincidence by Heaven a person has no opportunity to try to align precisely, and therefore he does not succeed. But if the exact coincidence is by human agency, then the person is aiming to align exactly, and in such a situation perhaps he can succeed.

The Halakhic Ruling and Its Meaning
From our discussion it follows that according to Tosafot this is not a matter of substantive impossibility. It seems they do not see the problem of exact coincidence as an aspect of the problem of continuity, but only as a technical problem that is difficult to overcome. Whereas according to Rashi and Maimonides, this appears to be an expression of the problem of continuity, and therefore exact coincidence is impossible—that is, its probability is 0.
Now as practical Jewish law, it is ruled that by Heaven exact coincidence is impossible, as in Maimonides and the Shulchan Arukh cited above. Regarding exact coincidence by human agency, the medieval authorities (Rishonim) disagree: according to the Tosafot there and in the parallel passage in Eruvin 5, exact coincidence is possible; whereas Maimonides, see Laws of Murderer 9:8, rules that exact coincidence is impossible even by human agency.
It is entirely plausible that they follow their own general approaches here: Maimonides sees this as a principled problem of impossibility in reality, an expression of the problem of continuity, and therefore of course exact coincidence is impossible even by human agency. Whereas Tosafot see it as a technical human difficulty, and therefore rule that by human agency exact coincidence is possible.

C. Zeno’s Arrow

Introduction
We will conclude the article with a brief discussion of one of the paradoxes presented by Zeno of Elea in order to undermine the concept of motion. We will propose a solution, and then briefly discuss several aspects in which this solution has implications.

The Paradox of the Flying Arrow
The Greek philosopher Zeno of Elea raised the following question: Let us look at a flying arrow. At every single moment, if we observe it, we see that it stands still. That is, at every moment of time it is stationary, each time in a different place. So when—at which moment—does it change its location?
It is customary to see the solution to this paradox in the infinitesimal picture. That is, the assumption that the time-axis is made of discrete moments is considered the pitfall that leads to the paradox. If we adopt the infinitesimal picture—namely, that the time-axis is composed of infinitesimals—then the question disappears on its own. One cannot speak of the state of the arrow at one moment of time, because there are no discrete moments of time.
The impression made by this solution is that we have gone around the obstacle, but not really removed it. Clearly there are discrete moments on the time-axis. One can mark such points on any continuous axis. Therefore, even if we cannot meaningfully refer to a discrete moment, one can still ask the theoretical question about the state of the arrow at some discrete moment. The fact that infinitesimal language forbids raising the question does not constitute a solution.
If the question concerned our inability to discern a given state, then this might be a principled solution, because human reference always concerns an interval of time and never a discrete moment. This was one of the possibilities we raised regarding the impossibility of exact coincidence. 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 indeed true that at every moment the arrow is in a different place, but it is definitely not true that it is standing still in those places. It is in some place, but at the very same time it is also moving. So the answer to the question when the arrow changes its place—that is, moves—is: at that very moment at which we observe it. It is both in some place and moving to another place.

The Root of the Confusion
What underlies Zeno’s confusion is the intuition that one cannot speak of velocity at a time-point. The feeling is that velocity exists and is defined only over an interval of time—and of space—but not at a discrete point of time and space. Velocity means change of place, and a change of place cannot occur over one point of time or space. From this it follows that at a discrete time-point, the body cannot have velocity and must be standing still—that is, at rest, with velocity 0.
But this assumption is completely mistaken. Velocity is not change of place. Change of place is the result of the fact that the body has velocity. Bodies with nonzero velocity change their place, but one must not identify the concept of velocity with change of place. Velocity is a potential for changing place, and therefore, even though change of place is not carried out in one discrete moment, the potential for change can be defined even at a discrete moment of time.
Physics defines velocity at a time-point—that is, there is velocity at every discrete moment of time—but the way this quantity is calculated is by dividing a spatial segment by a time segment, namely how much time it takes to traverse the spatial segment. That is the mode of calculation, an operational definition, but not the essential definition. Essentially, a body has velocity at every discrete time-point. As stated, calculating that velocity requires widening to a small interval around the point in question, but that is merely a computational necessity, not an essential one.

Back to Exact Coincidence
This distinction can be seen as another aspect of the distinction we presented above. Velocity is a quantity that exists at one discrete time-point, but we as human beings cannot relate to a discrete time-point, only to a time interval. That is what causes us to confuse the operational 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 arises because exact coincidence is impossible.
The reference of Ohel Yehoshua to death as an event that cannot be captured—that is, an event that does not exist on the time-axis—namely as a process and not a state, stems from the fact that we cannot relate to a discrete time-point. But as we suggested above, such an event does exist on the time-axis; it is simply a momentary event. The same applies to God’s momentary anger.

Summary
In this concluding section I will raise several reflections. Most of them will end in a question mark and will not lead to a clear conclusion. One may wonder what the significance is of the analysis we undertook this week. Is this Torah study? Or is it the use of Torah sources in order to learn physics, since understanding the meaning of the time-axis generally belongs to physics? Is this the study of aggadic literature, or actual halakhic study, since this topic has quite a few implications in the realm of Jewish law as well?
One can broaden the question and examine the boundaries of the concept “Torah.” Is the study of Guide for the Perplexed, with its philosophical parts—even from foreign sources—within the category of Torah? If so, then how is modern mathematics and physics different from Aristotelian science? Why did Aristotelian physics, which entered the laws of the foundations of the Torah in Maimonides, receive the status of Torah study, while learning concepts and insights from modern science does not receive that status? The same can be asked regarding modern philosophy.
There are quite a few situations in which mathematics, or some science, is needed in order to understand a halakhic topic or to decide a halakhic question. In such cases it seems that engaging in mathematics or science is in the category of preparation for a commandment—a preparatory means for the commandment of Torah study. But here the situation is the reverse: the goal was not understanding a distinctly halakhic or Torah subject, but using Torah and halakhic sources in order to understand the nature of the time-axis. Is that Torah study? If so, the conclusion is that this is on an even higher level, since it is truly Torah itself and not merely preparation for Torah study.
Are abstract legal reasonings that explain the laws of neighbors or the laws of evidence Torah, whereas insights into the nature of the time-axis, space, human nature, and the like are not Torah? Is this a matter of sources—from where we draw our conclusions—or of content? Maimonides himself, in the Mishneh Torah, Laws of the Foundations of the Torah 4:10–11, determines that the Account of Creation and the Account of the Chariot—to which the Talmud refers as a “great matter,” as opposed to the discussions of Abaye and Rava, which are a “small matter”—are physics and metaphysics, and he therefore devotes several chapters in those laws to those subjects. If we learn from him, it would seem that the conclusion called for is that this too is included within Torah. At that point, of course, one can ask what is not called Torah. Is every engagement with any kind of wisdom included in Torah? If so, what is the difference between wisdom and Torah?

To Kovach (Kobi?)
It seems to me that one can view “difference” as change without time.