Logical and Spiritual REFLECTIONS
Book 5. Zen Judaism
Appendix 1. Round numbers in Torah statistics
List of numbers. The following concerns the near ubiquity of round numbers in Torah population statistics, not to mention other contexts. This is a possibly non-exhaustive listing, drawn from Exodus and Numbers.
The point of the present listing is to show that round numbers are the rule (with one exception) in enumerations of people in the Torah. That is to say, the numbers here listed (49 cases, including totals given in the text) usually end in hundreds of thousands (2 cases), thousands (5 cases), or hundreds (31 cases), or fifties (6 cases), or other tens (4 cases), and only one case in units.
This near ubiquity of round numbers is very surprising, not to say suspicious. It suggests the numbers are not empirical, but guesses or rough estimates or deliberately rounded figures or sheer fabrications. I have not to date found the question asked or an explanation offered in the commentaries. Such failure to notice or to comment is itself problematic.
Rabbinical commentaries are also often in round numbers, but these are usually openly intended as approximations. However, in most the statistics here listed exact enumeration is apparently intended. Traditional commentaries so interpret them, and insist that this shows God is interested in each and every individual. It is therefore difficult to suppose that Moses wrote down approximate numbers for some reason (unless we abandon such commentaries).
In Ex. 12:37 – parashah Bo:
“Men on foot, besides children”.
This is explicitly stated as a rough number: “about” (ke-). It is not stated how this number was arrived at.
A methodology is given in Ex. 30:11-15 for the subsequent, more precise censuses. Each individual to be numbered would donate half a shekel (or ‘beka’), then the total receipt would be multiplied by two. This methodology is confirmed by actual practice in Ex. 38:25-26, where the total receipt in silver is specified.
It is not stated how long it took to carry out such a census. It could conceivably be done in a day or less, if well organized. Every man to be counted could hand his coin to the head of his small group (say, of one hundred men), then each of these heads could hand his collected coins to the head of a larger group (say, of one thousand men), who in turn would take the coins to the central collection point. There they would declare the total coins collected under their responsibility, and a grand total would be calculated. If this is indeed how the coins were collected, this total could be expected to be precisely correct.
In Ex. 32:28 – parashah Ki tissa:
“… fell of the people that day” at the hands of the Levites (after the golden calf episode):
This is explicitly stated as a rough number: “about” (ki-). Further on, it is written “and the L-rd smote the people, because they made the calf…” (Ex. 32:35), but no number is specified, and the commentators (e.g. Nachmanides) are not sure this meant more people were killed. Rashi on Ex. 31:18 implies that these events occurred on the 17th of Tammuz or soon thereafter.
In Ex. 38:26 – parashah Pekudei:
Men “from 20 years old and upward”.
The Torah gives us this total number without breakdown into tribes as in the book of Numbers, note; perhaps this suggests no tribal distinctions were made, only the total being sought out. Anyway, the total is confirmed within the text in the previous verse (v. 25), where it says “the silver of them that were numbered of the congregation” was 100 talents (kikar) and 1775 shekels (of the sanctuary).
The commentator Rashi tells us contextually that 1 talent is equal to 3000 such shekels; thus, the amount of silver corresponded to 603’550 half-shekel contributors. This calculation, note well, strongly confirms the idea that the total number of men given here is intended as exact, since it is unthinkable that the amount of silver was not accurately reported. It is thus understandable that Rashi offered no comment on the roundness of the number: he evidently regarded it as exact. This is the significance of his comment to Num. 1:1 (shown next), that this census was a demonstration of God’s love for Israel – God wanted to show his interest in each and every Jew.
It is worth here quoting Rashi’s comment to Num 1:1 in full –
Because they [the Jews] are precious before Him [Hashem], He counts them all the time. When they went out of Egypt He counted them [Ex. 12:37]; and when they fell because of the golden calf He counted them to know the number of those who remained [Ex. 32:28]; when He came to cause His Presence to rest upon them He counted them [Ex. 38:26];. On the first day of Nissan the Mishkan was set up, and [a month later] on the first day of Iyar He counted them [Num. 1:1].
Brackets mine. Note however that the actual number of survivors immediately after the sin of the golden calf is not given in Ex. 32. A Sifte Hakhamim comment explains this by pointing out Rashi’s wording to have been “the number of those who remained”, implying the number to be calculable (by subtracting about 3’000 at least from about 600’000) rather than known by enumeration.
In Num. 1:20-47 (confirmed 2:1-33) – parashah Bemidbar:
“Every male from 20 years old and upward, all that were able to go to war”, “but the Levites… were not numbered among” them.
Children of Reuven
Children of Simeon
Children of Gad
Children of Judah
Children of Issachar
Children of Zebulun
Children of Ephraim
Children of Manasseh
Children of Benjamin
Children of Dan
Children of Asher
Children of Naphtali
Children of Israel except Levites.
Note the exceptional ending in 50 for the Gadites, all other tribes ending in multiples of 100. This census occurred, on “the first day of the second month, in the second year after they were come out of the land of Egypt” (Num. 1:1). According to Ibn Ezra and Sforno, the purpose of the census was “to make arrangements for the encampments and the marching of the people”; they also specify that “these arrangements had to be completed by the twentieth of the same month, the day on which they left…” (Soncino Chumash, p. 793).
Notice the equality between the 603’550 total here, and that given in Ex. 38:26. This is surprising in that some time (over a month) seemingly elapsed between them. Rashi and other commentators noticed this, and claimed it meant that (miraculously) no one died in between.
However, as one perspicacious reader, Eddie Kerem-Sadeh, has pointed out to me, this explanation of the equality does not account for men who were under 20 years during the Ex. 38:26 census and then entered the 20+ age group in time for the first Numbers census. Moreover, he pointed out, the census surely did not happen instantaneously, but must have taken some time to carry out – and during such time more changes may have occurred in the statistics.
I wonder if Rashi and other commentators thought of this important objection, which makes the miracle claim logically inadequate, for though people might miraculously be kept alive, they cannot be prevented from getting older (i.e. time keeps passing). The best that could be said is that the number of those who died in the interim was equal to that of those who came of age during that time.
Another and simpler explanation might be to regard these two censuses as one and the same. It could be argued that the Torah merely mentions the total number in Ex. 38, because it is there mainly concerned with detailing what was eventually done with the silver collected in the census; whereas in Num. 1, the Torah returns to this same census and so as to give us more statistical details. But this theory is not easy to defend..
In Num. 3:14-39 – parashah Bemidbar:
Levites “from a month old and upward”.
by addition (not in text).
This total includes, according to commentaries:
mentioned in text
inferred by commentators.
The motive for numbering the Levites was to replace the first-born of Israel (traditionally in charge of religious duties) with the Levites (Num. 3:11-13). The 300 first-born Levites couldn’t replace first-born Israelites, being themselves already subject to the duties of first-born. How the Levites were counted is not clarified in the text. According to Rashi (v. 16), the Levites (or at least their underage children) were not actually counted, but their numbers were revealed to Moses by God. These numbers are obviously intended as exact, since the total of 22’000 is thereafter used in a precise calculation (see next).
In Num. 3:40-43 – parashah Bemidbar:
“First-born males of the Children of Israel from one month old and upward” (excluding Levites).
to be redeemed by 22’000 Levites.
Note this is the only non-round number so far listed. Thus, 273 first-born Israelites could not be redeemed by Levites, but had to pay 5 shekels each (total Sh. 1’365), according to Num. 3:44-51.
The mismatch between the numbers of first-born Israelites and Levites to redeem them is significant, in that it makes improbable the hypothesis that God willed the round numbers for some purpose. Unless we assume He wanted some first-born not to be redeemable by Levites for some reason (perhaps just to make them pay 5 shekels each).
In Num. 4:1-49 – parashah Naso:
Levites 30-50 years old “that entered upon the service”.
It is not stated how these numbers were arrived at or whether they are meant as exact. This census of Levites was apparently (in view of the age group it concerns) motivated by the assignment of religious duties to the three family groups.
In Num. 11:21 – parashah Behaalotekha:
“Men on foot”.
It is reasonable to assume this is intended by Moses as a rough number, based on the last census.
In Num. 16:2 – parashah Korach:
“Certain of the Children of Israel” who joined the rebellion.
It is not stated how this number was arrived at or whether it is meant as exact.
In Num. 17:14 – parashah Korach:
“They that died by the plague” in this episode of rebellion.
It is not stated how this number was arrived at or whether it is meant as exact.
In Num. 25:9 – parashah Balak:
“Those that died by the plague” in this lustful and idolatrous episode.
It is not stated how this number was arrived at or whether it is meant as exact.
Note additionally that in a commentary on this episode by Rashi (specifically to Num. 25:5, quoting Sanhedrin 18a), it is stated that there were 78’000 judges each of whom killed 2 of his subjects, a total of 156’000 people.
In Num. 26:1-53 – parashah Pinchas:
All “the Children of Israel, from 20 years old and upward… all that are able to go forth to war” except Levites.
Children of Reuven
Children of Simeon
Children of Gad
Children of Judah
Children of Issachar
Children of Zebulun
Children of Manasseh
Children of Ephraim
Children of Benjamin
Children of Dan
Children of Asher
Children of Naphtali
Children of Israel except Levites.
Note the exceptional ending in 30 for the Reuvenites, all other tribes ending in multiples of 100. The text makes clear this is the new generation, about to enter the Promised Land, so it is not much use comparing this census to the earlier one. However, it is significant that the numbers are just as round here as there.
Note too that a comment by the Ramban (Nachmanides, mentioned in the Soncino Chumash, p. 940) states that the men numbered were between 20 and 60 years old. Although this is not (to my knowledge) specifically stated in this and the previous census(es), it is presumably applicable to them all, being apparently inferred from the specification that the men numbered were “able to go to war”. We could similarly suggest that the “able to go to war” specification would exclude men 20-59 years old with permanent physical or mental disabilities.
In Num. 26:62 – parashah Pinchas:
Levites, “every male a month old and upward”.
Same comment as the preceding: not to compare past and present populations.
In Num. 31:40 – parashah Mattoth:
“Persons” (virgin women) taken captive in the war against Midian.
This may be intended as an exact number, since the priestly “tribute” from half this number was exactly 32 persons. However, this 1/500th tribute (as well as the 1/50th levitical tribute from the other half) might be deemed only applicable to the nearest round numbers, up or down. Note in passing that the animals captured are also listed in round numbers.
Discussion. Now, the extremely low mathematical probabilities of the numerical coincidences noted here should be elucidated. The chances that a number end in 00 rather than in 01, 02, 03… or any other pair of last digits is simply 1 in 100; this is nothing special, since each ending has an equal chance. However, the chances that the 00 ending occurs in two separate statistics simultaneously are 1 in 100 times 100, i.e. 1/10’000. For three statistics, the chances are 1 in 100×100×100 or 1003 (100 cubed or 106). And so forth. Thus, for a conjunction of eleven numbers ending in 00, as above, the chances are 1 in 10011 (100 to the 11th power or 1022), clearly extremely slim. For this conjunction to be repeated in another set of eleven numbers, the chances are 10022 (1044).
When I put this problem to a local rabbi, he argued that this was simply “a miracle, like the splitting of the sea” during the Exodus from Egypt. I replied that this was not a convincing argument to my mind, because whereas the splitting of the sea had an obvious purpose, viz. to allow the Children of Israel to pass through, the conjunction of so many round numbers is inexplicable. Why would God bother making sure the numbers of Jews was round at the time of counting? Was it a love of symmetry, perhaps? To claim a miracle, one has to conceive of a reasonable purpose for it.
However, I later (during my next meditation) realized a purpose can indeed be proposed for these round numbers. Perhaps God performed this miracle simply to signal His presence, i.e. to tell us the numbers involved were not fortuitous but of His own making! That would constitute a worthwhile purpose. In this manner, the low probability of a peculiar conjunction of statistics can be turned from a source of skepticism to a source of added faith.
An objection might be raised to such proposed interpretation by pointing out that the number of Gadites ends in 50 in Num. 1:24 and that of Reuvenites ends in 30 in Num. 26:7. Similarly, the number of Kohathites ends in 50 in Num. 4:36 and that of Gershonites ends in 30 in Num. 4:40. Why such endings instead of 00 as in all other cases?
Skeptics would argue that these irregularities are a feeble attempt by the inventor of all the numbers to make them seem a bit more realistic. The attempt is feeble, because while more variation would have been credible, such rare exceptions are not too convincing.
Those who favor the theory that Moses wrote down numbers to the nearest hundred could explain the occurrence of a 50 ending, by saying such an exact number cannot reasonably be increased or decreased to the nearest 00, since it is precisely halfway between. But they could not similarly explain the occurrence of a 30 ending; besides, why precisely 30 twice?
However, the proponents of the thesis that God is signaling His presence can reply that God inserted these slight irregularities in order to make room for skepticism. For, they would say, He desires us to believe in Him and His Torah through some measure of faith, rather than exclusively through proofs.
What is the logical upshot of all the above considerations?
· Looking at the highly improbable conjunction of numbers in certain passages of the Torah, one is inclined to a negative conclusion concerning them – i.e. to view the censuses they report as of very doubtful authenticity, if these are intended as exact. If they are taken as approximations or rough estimates, their negative impact is of course thoroughly dissolved, note well.
· Moreover, if the stated passages of the Torah are put in doubt, then to some extent so is the Torah as a whole; at least in the sense that it cannot readily be claimed entirely true. And to the extent that we base our faith in God’s existence on the Torah, as many people do, such faith is in turn somewhat shaken – even if logically belief in God is quite possible without belief in the Torah.
· However, these skeptical conclusions remain inductive, because the opposing view is able to muster an alternative hypothesis in its defense. That is to say, the improbable set of exact numbers may conceivably be explained as an intentional creation of God to indicate His presence to the faithful. The mere fact that a counter-argument is possible suffices to ensure that a skeptical conclusion is not deductively necessary.
· But of course the skeptical conclusion remains inductively very strong. That an alternative hypothesis is remotely possible does not make belief in it necessary. It just provides a logical escape route, however farfetched. It leaves a little room for continued faith in the Torah, and thence God, even if mathematics suggests improbability. Such belief is very improbable, but not quite impossible.
This overall conclusion is in accord with our general thesis that belief in God can neither be proved nor disproved. In this instance, it is not directly belief in God that is at stake, anyway, but the Torah or just a part thereof. We have above shown that the latter, although weakened considerably by certain numerical improbabilities in it, cannot be decisively discredited by them.
Now, let us examine certain implications of the above figures of 600’000 plus males in more detail. We know that the first set of 603’550 males 20 or more years old all (except for Joshua and Caleb) died in the 40 or so years until the second count of 601’730. Thus, the latter set consists of males who were 0-19 years old at the time of the earlier census, plus males born in the first 20 years of the intervening period. Males born in the second tranche of 20 years or so of the intervening period being under 20 years old are not included in the latter number.
At the time of the earlier census, there was no doubt some males over 60 years old who not being fit for war were not included in the figure of 603’550. But at the time of the later census, there were no males over 60 years old (with the said two exceptions), since all the earlier generation over 20 years old died off. (Presumably, the female Israelites over 20 years old of the earlier generation died off too.) Therefore, the count of 601’730 is actually the number of all adult males, and we do not have to consider how many might be over 60 years old.
If we wish to estimate the total population that was poised to inherit the Promised Land, we must thus consider only three tranches of males: 0-19, 20-39 and 40-59. We know the latter two age groups add up to 601’730. If we assume (as a first guess, in view of the equality between the latter total and the earlier) these three tranches to be about equal, we can estimate the total male Israelite population at that time at 900’000. If we assume there were an equal number of females, the total Israelite population would have been 1’800’000.
These figures all exclude Levites. We know there were 23’000 males of all age groups (over one month old). This figure may include many over 60 years old, since the older generations of Levites did not all die off (as implied by Rashi’s comment to Num. 1:49). If we assume here again an equal number of females, the total Levite population would be 46’000. This presumably includes the priests (kohanim). Add this to the Israelite total, and we get a grand total of 1’850’000 or so (in round numbers).
This calculation explains the traditional figure of 2-3 million. The two million figure is a reasonable minimum – given the Torah numbers – and the higher three million figure is based on somewhat larger reproductive assumptions. Some people have suggested even larger figures, in the 4-5 million range, but this seems wildly exaggerated to me. I am not a demographer, but it seems to me the above assumptions are more reasonable.
As already mentioned, the figures given in the Torah can be doubted on simple grounds of mathematical improbability that so many round numbers would occur together. But modern historians have come up with much more serious cause for doubt – namely the fact that a population of the magnitude proposed (two million or more) was far too large for the time and place concerned. They estimate Egyptian population at that time was of the same order of magnitude, and Canaanite population (including all ethnic groups) was probably far less (though the Hittites and Amorites, it is now known, were spread out well beyond Canaan).
I personally tend to believe them. I would if necessary accept arbitrarily dividing all the figures in Numbers by ten, say, on the ground that accounts given in ancient times were often hyperbolic. The reason I am so willing to compromise on the numbers is that logically this makes little difference to the essence of the story of the Exodus. I still believe there was a mass exodus of slaves from Egypt, who then proceeded to conquer and settle Canaan. The reason I believe this is that I ask the question: why would a people invent the story that it had been enslaved abroad and that it escaped and conquered its land from other peoples? There would be no sense in inventing such a disadvantageous story.
That such a people existed there and then is phenomenologically indubitable. They are mentioned throughout subsequent history by many people and have left many archeological and existential traces. This people still exists today (and I know for a fact my family has been part of it from way back). Some of the historians who bring up the said numerical doubts have a hidden religious or political agenda. Their goal is either to debunk religion or to de-legitimatize the presence of Jews in the Holy Land today. They do not speak as historians, but as anti-Judaic, anti-Jewish, anti-Israeli, anti-Zionist or ‘post-Zionist’ advocates.
But the point I wish to make here is that arguing that the numbers given in the Torah are incredibly high does not prove their point. It does not imply that there was no exodus and no ancient Jewish presence on the land. They cannot logically infer from the doubtfulness of some historical or scientific claims in the Torah that all such claims in the Torah are false. That would be an invalid generalization, since certainly many of the claims made in the Torah have turned out to be true. It is important to remain lucid and impartial in this matter as in all others.
Postscript. After writing most of the above article, I received from Israel a copy of an article written (in Hebrew) by Prof. Ely Merzbach of Bar Ilan University years ago on the same topic: “The Census of Israeli Tribes in the Torah” (Higayon, vol. 5 ). It is clear from that article that Prof. Merzbach noticed the issues here raised long before me, and moreover that so did some notable rabbinical commentators which he mentions. I shall first consider the latter.
· The first mentioned is R. Moshe Sofer (1762-1839), known as the Chatam Sofer. In his work Torat Moshe, he notes that the census numbers end in 10, 50, 100 or 1000, and explains the fact somewhat by pointing out that these multiples correspond to groupings of people under the responsibility of the judges. He also points out that the Rosh (another important commentator) considers approximation common in the Torah (as for example 49 days of the counting of the omer are called 50 days).
Thus, though this commentator seems to imply and accept that the census figures are approximate, he does not apparently go more deeply into the issues such admission raise.
· Next mentioned is R. Meir Simha HaCohen (1843-1926). He explicitly admits that numbers were rounded (“assu mispar hakolel”) and suggests such approximations were made downwards, towards the lesser round number. He explains all this with reference to military musters (as indeed is justified in view of the repeated mention of “men able to go to war”). This reason was not applicable to Levites (who were not actually counted) or to the first-born (who were counted precisely).
However, it appears this commentator too did not take stock of the difficulty posed by a couple of numbers ending exceptionally in 50 or 30 instead of 00.
· Thirdly mentioned is R. Aharon David Goldberg (presumably a more recent commentator), who (in his Shirat David) rightly expresses amazement (“ze davar pele”) that units are not included in the censuses. He tells us of a book called Shaarei Aharon that mentions another book called Imrei Noam, in which census numbers are said to be rounded upwards, to the next greater 100 (note disagreement with preceding view). But he apparently favors the theory that the numbers were rounded to the nearest 100, whether up or down. In the case of the tribe of Gad, whose total ends in 50, the number must have ended exactly in 50, so could not be rounded either way. The numbers of Levites were rounded to the nearest 10, because Levite figures were relatively low (in the thousands, rather than tens of thousands).
However, this commentator does not seemingly notice or explain the fact that the number of Reuvenites ends in 30. Moreover, he does not realize that the number of Levites (given in Num. 3) must be taken as exact, to justify the calculations made with it regarding the first-born.
· A fourth commentator, R. Yaakov Kamenetsky (1891–1986), does attempt to address the problem posed by the number of Reuvenites. He considers that in all matters military numbers end in 100 or in 50, thus explaining all other Israelite census numbers. He suggests that if after all such groupings say 45 men were left – they would be counted as another “50”, by putting 49 men in five groups of (almost) “50”. This could explain the number of Gadites ending in 50. For the number of Reuvenites ending in 30, however, he offers a very unsatisfactory solution. According to him, the number was deliberately not rounded to the nearest 50, so as to signal that the rebels Dathan and Abiram came from this tribe.
This solution seems patently absurd to me, considering that the Korach rebellion episode referred to occurred after the census concerned. This would imply that the Torah was first written with a number ending in 100 or 50 for the Reuvenites, and then was modified ex post facto to stigmatize this tribe. Moreover, why only this tribe? There were 250 rebels (not to mention 14’700 sympathizers), and even if most came from the tribe of Reuven, many came from other tribes. How does this commentator know how many came from each tribe and their respective degrees of responsibility? It is not specified anywhere. Clearly, the proposed explanation is not at all convincing.
Thus, to conclude this overview, contrary to what I initially assumed (and was told by some Torah scholars I queried), some relatively recent rabbis did notice and comment upon the fact of the round numbers. However, some of these commentators did not realize the full extent of the problems it raises, and moreover the solutions they did propose were not sufficient to solve all these problems.
It is worth mentioning, additionally, that these commentators, who all admit of approximation, are de facto at loggerheads with Rashi and other earlier commentators, who effectively take the figures in question as exact, as the Torah (as we have shown in some instances above) seems to imply them to be. Noteworthy, too, is the fact that these commentators considerably disagree with each other as to how approximation occurred.
Prof. Merzbach seems to reach the same general conclusion, viz. that these four commentators did not succeed in actually solving the enigma(s) at hand, and goes on to propose some possible solutions of his own.
To my mind, his most important and relevant insight is that the tribal totals might be approximations that cancel each other out, so that the national total may be taken as an exact number. This idea is brilliant, because it allows for a reconciliation between the apparent exactitude of the 603’550 figure in Ex. 38:26 (since it corresponds to the silver collected) and thence of the same figure in Num. 1:46 (granting Rashi’s claim), and the need to admit that the improbable round numbers for the various tribes were approximate.
Prof. Merzbach points out that, assuming each number was rounded to the nearest hundred up or down, the most that each tribal total would differ from reality would be 49, and the most that the national total would differ from reality would be twelve times that, i.e. only 588. But, he adds, it would be very unlikely that the deviation would be uniformly so large – and not inconceivable that it would happen to be nil due to rounding up and down of numbers of various magnitudes.
In the first census in Numbers, the total for the tribe of Gad ends in 50. This, as already explained, would be explicable as an exact number, which being midway could not be rounded up or down. Thus, here, only eleven rounding of numbers would have to cancel each other out, to yield an exact grand total.
However, what of the total for the tribe of Reuven in the second census in Numbers, which ends in 30? He suggests that this too may have been an exact number, arguing that in a list of ten (or even, in our case, twelve) numbers it would not be improbable for one of the numbers to end in 0. He then supposes that this number (ending in 30) was not rounded to the nearest hundred below, because it seemed round enough as it was. He points out that the practice of rounding is historically a relatively recent mathematical artifice, dating perhaps from the Middle Ages.
While it is conceivable that the people who made that census reasoned as here suggested, it is not of course very logical. They should either have rounded all tribal totals to the nearest 10 or to the nearest 100. If, as the commentators earlier mentioned suggested, they chose to round numbers to the nearest 100 for military purposes, there would have been no reason for them to make an exception in the one case ending in 30. Of course, people sometimes do weird things, but this argument is not very convincing.
Unless! Unless we suppose that they consciously ended that tribal total in such exceptional manner, to make sure that the national total of 601’730, which they knew to be an exact number, could be calculated from the tribal totals. All the other numbers were rounded, and their approximations canceled each other out, but this number could not be rounded in the same way (i.e. to the nearest 100) without distorting the grand total, so it was kept with a 30 ending. It was thus quite intentional and not illogical.
Upon further reflection, it would be acceptable to suppose that the number ending in 30 was itself rounded to the nearest ten, provided what was added or subtracted from it was balanced by some other number(s) in the list (we could in that case accept R. Kamenetsky’s explanation of the choice of the Reuvenite number rather than another tribe’s for that purpose). Also, similar reasoning can be applied to the previous census, if necessary; i.e. we can imagine the number ending in 50 (for the Gadites) to have likewise been rounded so as to arrive at the required exact grand total, if the need ever arise.
Thus, Prof. Merzbach’s analysis together with past commentaries provide us with some conceivable and reasonably credible solutions to the problems involved for the Israelite numbers. His thesis restores the consistency of the Torah passages concerned with mathematical principles of probability, and so makes the Torah claim to literal truth in this matter more credible. This of course does not prove the factuality (i.e. historicity) of its numerical claims, but it at least removes some of the possible sources of doubt.
 As we shall see further on (as I found out after writing most of this article and publishing it on the Web), the question has in fact been asked before, both by rabbis and academics, and various answers have been proposed, which I shall present and evaluate.
 See Rashi comment to Num. 1:1, further on. Also, e.g. Hiddushei Harim.
 Note that this method was applicable only to tribes other than Levi, since the latter was not subject to such monetary contributions and moreover even children in it were counted. See more on this further down.
 A question that comes to mind here is: where did all these half-shekel silver coins come from? Were they minted in Sinai before these censuses, or were they brought over from Egypt – and in the latter case, who minted them there and in what context? This question affects the credibility of the narrative somewhat. We could further ask whether coins at all existed at the time of the Exodus (traditionally, 2448 BCE). According to the findings of historians so far, Sumerians and Egyptians used silver and gold bars of set weight as money already in the fourth millennium BCE, and the shekel as a measure of weight existed in Mesopotamia already in about 3000 BCE – but the first stamped coins in the Mediterranean region date from about 650 BCE in Lydia, though there may have been earlier coinage in India or China.
 If the age group counted was not literally 20+ but 20-59, as seems intended by the “able to go to war” specification, then we must take into consideration people who were under 60 (and so counted) in the first of these censuses and over 60 (and so not counted) in the second. The equation then is that the net sum of entries due to coming of age and of exits due to aging or death is zero. In that case, Rashi’s scenario is conceivable – i.e. there may have been no deaths, provided the number who reached maturity (20) equaled to the number who became too old (60).
 To uphold this speculation, the two enumerations must somehow be conflated. Ex. 38 concerns the preparations for erection of the sanctuary (mishkan), which presumably required the silver collected in the census; while Num. 1 concerns the preparations for departure from the wilderness, after the sanctuary was completed. Could it be that the actual erection of the sanctuary occurred after the census that started on or after the 1st of Iyar? If they indeed left that place within 20 days, as already mentioned, would they have had time take the census and then to erect and take down the sanctuary (and do all they did in between)? It seems difficult to uphold…
 Eliahu Beller, of Bar Ilan U., Math. Dept., has argued (in “The Problem of the First Born”, Higayon No. 2 ) that the numbers of first-born Israelites and Levites given in the Torah “seems astonishingly low” and on the basis of a mathematical model suggests that “the Torah counted only those first-born who were born in the year between the Exodus and the census”, concluding that “the total number of first-born was many times higher”. The questions to ask here are: (a) why does the Torah not specify this, but instead give the impression it is referring to complete enumerations of first-born, with precision; and (b) how come the Rabbis never raised this issue?
 If we accept these figures as credible (personally, I hesitate to, considering that such a massive 20% population cull would have merited explicit mention in the Torah text) – we can infer the total adult male population at the time to have been 780’000 (if the judges of tens were included in their minyans) or perhaps 858’000 (if the judges were not included, which I am told is the case). This total, I guess (but do not know), would include all Israelite males aged over 13, since youths under 20 were also legally responsible.
 Some interesting possible explanations of the round numbers have been suggested to me by the already mentioned reader, Eddie Kerem-Sadeh. One is that supposing the half-shekel coins were not counted but summed up by weighing large quantities of them together, and (as seems likely) the coins in use at the time were not all exactly equal in weight, the resulting total could only be approximate. Another is that the collecting and counting (or weighing) of coins must have taken considerable time, during which time there were age changes, as well as deaths; in that case, the numbers at the end of the process had perforce to be rounded, so as not to give a wrong impression that they were exact. Finally, he adds, if the results were audited, and found to vary somewhat, it would have been natural to record round number estimates, to express the margin of error involved. These seem to me excellent proposals on the whole. One objection I can think of is that, though from a secular point of view the coins might be deemed probably of unequal weight in view of technology then available, such coinage would be morally unacceptable according to Deut. 25:13. Moreover, if we look at Ex. 38:25-26, we see that the weight of silver and number of people inferred are precisely related.
 The following reflections on population were stimulated in me by the feedback of another reader, who asked not to be named.
 This in itself proves nothing, since in a slave economy the slaves may eventually outnumber the freemen. I recently read, for example, that at one point in time the black and white populations of the U.S. South were about equal.
 At most 100’000, according to one article (see http://en.wikipedia.org/wiki/The_Exodus and the references given there). The Egyptian army is there estimated as 20’000 at the most. These figures are significant, since the Torah claims the Hebrews could muster some 600’000 men of fighting age (though it does not say where their weapons would come from). The implications being that though it is conceivable that these men (just escaped from slavery) could be afraid of the Egyptian host, it is less credible that they would fear a Canaanite population of far inferior size.
 However, the division by 100 proposed by Prof. Avraham Malamat of Hebrew U. (according to the aforementioned Wikipedia article) seems an exaggeration in the opposite direction. Considering my comment in the previous footnote, the figure of 60’000 fighting men would be more credible – in proportion to the Egyptian and Canaanite forces they faced.
 Or more precisely, reconquer and resettle the land, since their ancestors Abraham, Isaac and Jacob had already lived there until some 400 years earlier, according to the Torah (less, according to commentators). There is no reason to think this people might have originated in Egypt, or elsewhere than (as they claimed) in Canaan. They were ethnically clearly Semites and not Hamites.
 Please note that in view of the poverty of my knowledge of Hebrew, my interpretation of this article (albeit with the help of a more knowledgeable friend) may not be fully accurate. However, I did send Prof. Merzbach copy of my comments by e-mail, asking him to correct me if I misread him, and he replied to me that my reading was “grosso modo” accurate.
 As mentioned earlier, in the name of Rashi.
 One reflection that comes to my mind after reading these commentaries is that the groupings of men, whether in 10s, 50s, 100s or 1000s, for whatever purpose, could not have been as easy in practice as it looks on paper. Such groups must have been in constant flux, as men died or became ineligible due to aging or other disabilities, or became newly eligible for inclusion. When a place became vacant in a group, it could not necessarily be instantly refilled; and conversely, when a youth became old enough for inclusion, he might not immediately find a spare place. Moreover, the people in a group would have to live in the same vicinity of the camp, for practical reasons; so groups would not interchange populations at will across the whole nation or even the tribe. All this suggests that groups would not at any given time necessarily have their defining number of members.
 Note lastly that Prof. Merzbach suggests additionally that the subtotals given for the four “standards” (degalim) under which the tribes were grouped in threes (under Reuven, Judah, Ephraim and Dan – see Num. 2:1-33) can, with the same hypothesis, be explained as exact numbers that sum up approximations. For, he argues, why else would the Torah spell out subtotals that any reader could easily calculate? He thus reinforces his thesis by making it more useful still. However, in my view this additional hypothesis has disadvantages that far outweigh its advantage. First, because it considerably limits the scope for approximation in the individual tribal numbers, since now every three grouped under the same banner have to add up to an exact number. And second, because it restores the issue of exponential improbability, since it claims the subtotals (like the grand total) to be exact numbers, albeit round (one ending in 50 and three in 00). Since this subsidiary thesis is inessential to the main thesis, I would eschew it.
 Note however that some difficulties remain with regard to the numbers of Levites. Namely how come the numbers for their three families given in Num. 3:14-39 all end in 00, even though no military motive can be adduced (the total must also be considered exact, since as earlier stated, it is used in calculations relating to the first-born); and similarly why the numbers given in Num. 4:1-49 all end in 0 (and coincidentally one ends in 50 and one in 30 – those same numbers again!), though in the latter case, groupings by tens would be explicable with reference to work teams.