a fifteenth-century law of large numbers

A Fifteenth-Century Law of Large NumbersAuthor(s): Nachum L. RabinovitchSource: Isis, Vol. 65, No. 2 (Jun., 1974), pp. 229-238Published by: The University of Chicago Press on behalf of The History of Science SocietyStable URL: http://www.jstor.org/stable/229372 .

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A Fifteenth-Century Law

of Large Numbers

By Nachumn L. Rabinovitch*

I

THE CORNERSTONE OF A RATIONAL CONCEPTION of the universe is the idea of natural law. Amidst the manifold and often conflicting changes constantly

taking place, observant man early noticed certain regularities of rhythm or pattern and seized upon these as characteristic of nature's operations. It became the object of scientific enquiry to encompass more and more of the observed phenomena within a framework of explainable changes by searching for regularities where none had been noticed before. Where such could be discovered, "natural causes" were recognized to be at work.

Thus Aristotle, summing up the wisdom of his predecessors and formulating guide- lines for many centuries of successors wrote:

For those things are natural which, by a continuous movement originated from an internal principle, arrive at some completion: the same completion is not reached from every principle; nor any chance completion, but always the tendency in each is towards the same end, if there is no impedinment.... In natural products the sequence is invari- able, if there is no impediment.'

Aristotle felt that no matter how extensive might be one's knowledge of "natural causes," there would still be some things which would remain unexplained but must be attributed to "chance."

We observe that some things always come to pass in the same way, and others for the most part. It is clearly of neither of these that chance is said to be the cause, nor can the 'effect of chance' be identified with any of the things that come to pass by necessity and always, or for the most part. But as there is a third class of events besides these two events which all say are 'by chance'-it is plain that there is such a thing as chance and spontaneity.2

The philosophers of the Middle Ages received this doctrine and elaborated it. The term "law of nature" was understood in an Aristotelian sense as expressing the function of nature as a cause. The usage of the term law in this connection apparently

Received Sept. 1972: revised/accepted Mar. 1973. *Jews' College, Montagu Square, London

W1H 2BA, England. Except where a translation is cited, the render-

ing from Hebrew into English is my own. As usual, brackets indicate explanatory additions to

the text. l Physica II 8, 199b15. All quotations from

Aristotle are from the Oxford translation, ed David Ross, 12 vols. (Oxford: Clarendon Press, 1908-1952).

2 Ibid., 11 5, 196b1O.

229



230 NACHUM L. RABINOVITCH

goes back to the Bible where the order of nature is described as Divine legislation: "thus saith the Lord, which giveth the sun for a light by day and the ordinances of the moon and of the stars."3

In the rabbinic tradition the Biblical concept of God as the legislator served also to provide a rationale for the persistence of natural law. The regularity of natural pro- cesses as observed in the past can be assumed to continue in the future because God willed it so. This too has its root in the same Biblical source, which continues: "If these ordinances depart from before me, saith the Lord, then the seed of Israel also shall cease from being a nation before me forever."4

This rabbinic view that physical law is ordained by God seems to have been introduced into Western thought by Philo (born c. 20 B.C.E.). In his De opificio mundi (XIX) he wrote about the "operations of nature ... which are invariably carried out under ordinances and laws which God laid down in this universe as unalterable."5 This remained the accepted outlook among medieval Jewish philosophers. In his classic work The Guidefor the Perplexed, Moses Maimonides (1135-1204) wrote: "We believe that what exists is eternal a parte post and will last forever with that nature which He, may He be exalted, has willed: that nothinig in it will be changed in any respect unless it be in some particular of it miraculously."6

A troublesome difficulty was that Aristotelian thinkers had to admit the existence of chance, for as Aristotle had already noted, regularity is not absolutely universal. Moreover, even things subject to natural law sometimes behave anomalously: "nmistakes are possible in the operations of nature also ... and monstrosities will be failures in the purposive effort."7 Now Aristotle maintained that the accidental or chance event cannot be subsumed under any law, nor can there be any true knowledge of it:

The accidental, then, is what occurs, but not always nor of necessity, nor for the most part.... It is obvious why there is no science of such a thing: for all science is of that which is always or for the most part, but the accidental is in neither of these classes.8

The same thought is expressed in different contexts. Perhaps the following passage makes the point most emphatically:

... chance is a thing contrary to rule.... For 'rule' applies to what is always true or true for the most part, whereas chance belongs to a third type of event. Hence, to conclude, since causes of this kind are indefinite, chance too is indefinite.9

It is clear that as long as the concept of law was rooted in the nature of individual things or events, chance could not but be outside any law.

3 Jeremiah 31: 35. 4 Ibid., 36. 5 P1ilo, English trans. F. H. Colson and G. H.

Whitaker (London: Heinemann, 1929), Vol. 1, p. 47. On this point cf. J. S. Mill, A System of Logic (8th ed., New York, 1888), Bk. III, Ch. 4, Sec. 1, p. 230. See also E. Zilsel, "The Genesis of the Concept of Physical Law," Philosophical Review, 1942, 51:245-279.

6 Moses Maimonides, Daldlat al-ha'irin (The Guide for the Perplexed), ed. S. Munk, 3 vols. (Paris, 1856-1866); The Guide of the Perplexed, trans. Shlomo Pines (Chicago: University of Chicago Press, 1963), p. 346.

7 Physica 11 8, 199a35. 8 AMetaphysica XI 8, 1064b30. 9Physica II 5, 197al7; see also Metaphysica

VI 2, 3.



15TH-CENTURY LAW OF LARGE NUMBERS 231

II

Now there was a tradition in Jewish thought going back to early times, according to which classes or collections of objects or events were considered statistically rather than individually.'0 Apparently basing his thought on this conception, Maimonides proposed a concept of Divine edict that applies to an entire group though not to every single individual in it nor to any particular one. In this sense he understood the existence of certain social and economic laws as well. Thus Maimonides attempted to resolve the apparent contradiction between certain prophetic statements and the principle of free will. He interpreted the prophecy as a statistical forecast, which still permits the individual his freedom:

[God] told Abraham "They shall serve them and they shall afflict them.""1 Some say that having decreed that the Egyptians do evil to the seed of Abraham, why did He punish them? They were after all compelled to enslave them as He had decreed.... The answer to this question is that this is as if He, may He be exalted, had announced that among those to be born in the future some will be rebellious and some obedient, some righteous and some wicked. That is true; yet it does not follow because of it that so-and-so who is wicked must in any case be wicked or that so-and-so who is righteous must inevitably be righteous. Rather, whoever is wicked is so by his will and had he willed to be righteous it would be within his power and there would be no restraint. For the announcement does not refer to every individual in which case he might say "It has been decreed for me." Rather it is a general one, and every man retains his own will in accordance with nature. Similarly every single Egyptian who harmed and troubled Israel was able to choose, had he wanted to, not to harm them, for the decree was not on any specific person to do evil.12

Over two centuries later R. Isaac Aramah13 discussed this concept in greater detail:

God's intent in a general matter does not imply the individual details with respect to persons. Every one innocently does as he wishes and desires, and if these should refrain, this does not prevent the attainment of the end by others ... Now Aristotle claims that if one were compelled to the overall object of his deeds he would therefore be compelled in all their details, except that he has explained that in fact fate does not compel a man either to general ends or to partial means.14 Nonetheless, the truth is that there can be no disagreement or uncertainty on our conclusion, namely, that the general Divine decree does not compel the partial acts of men, for this is a true proposition without doubt.... As for the nature of the Divine will, the general end is necessary but the particular means leading to it are contingent, for all of them can possibly come about in many ways and no

10 See N. L. Rabinovitch, "Probability in the Talmud," Biometrika, 1969, 56:437-441. A study of numerous texts in this Talmudic tradition is the subject of my book Probability and Statistical Inference in Ancient and Medieval Jewish Litera- ture (Toronto: University of Toronto Press, 1973).

11 Genesis 15:13. 12 Moses Maimonides, Commentary on the

Mishnah, Introduction to Tractate Avot, Ch. 8, ed. J. Kapah (Jerusalem: Mossad Harav Kook, 1964), Vol. III, pp. 400-401.

13 Isaac ben Moses Aramuah (1420-1494), rabbi, philosopher, and preacher, lived in Spain until the expulsion of the Jews in 1492, when he

settled in Naples, where he died. His major work is Aqedat Yitzhak, which is written in the form of philosophical homilies on the Pentateuch. It consists of 105 chapters (called "Gates" or "Portals") and in it Aramah develops his views on the major philosophical problems of his time. His work is in the anti-Aristotelian trend introduced by Crescas (see my paper "Rabbi Hasdai Crescas on Numerical Infinities," Isis, 1970, 61: 224-230).

14 Aristotle, Ethica Nicomnachea I1I, I 11 lOb8: "But if someone were to say that pleasant and noble objects have a compelling power, forcing us from without, all acts would be for him com- pulsory."




one is determined. That is certainly so when we consider matters of greater generality, for example, the prosperity of a city or its destruction. For that is possible to be achieved in exceedingly many aspects so that the inhabitants of the city are not constrained to any one of them, and it is even more so that it is not determined for any single one of its dwellers. It goes without saying that if we consider the totality of a whole nation or of all the nations together so that the means are numberless, it is impossible to estimate what predetermination at all applies to any one people or to the will of each individual person. It is like a man to whom is assigned the entire realm as his prison, in that although he is prevented from leaving it completely, he is free to travel wherever he chooses throughout the length and breadth of the land.15

Aramah proceeded to apply the same idea to biological law. He considered the instinct for self-preservation to be part of the nature of all living things, yet he saw it as being a statistical law which is operative only in sufficiently large assemblages of individuals.

The law of nature implies that no being will sever itself from its life. It is a covenant with every living thing that it will not destroy itself. Though it is conceivable that "one in a City"16 in the multitude of days because of a deteriorated imagination will transgress that law and destroy himself, in truth, this possibility cannot be conjectured for all the individuals of a species, as the Talmud says: "his attitude can be disregarded among all meni' . ... the more so will the nature of this matter be strengthened and be firmly established forever if the individuals in the covenant are very great in number.... ... It is as if it was conjectured that one day all men should choose to strangle themselves. Now that which is not in the nature of man to do is considered an absolute natural impossibility. Similarly it is a covenant with the world that its population will not cease by choosing to desist from procreation, and like matters, although it is possible that "one in a city" will so choose.18

Ararnah goes much further. Not only does he posit the existence of a law that applies to classes rather than individuals with the consequent proviso that such a law is established when the individuals constituting the class "are very great in number," but using the same idea, he proposes a rule that applies to chance as well.

In order to understand the context of his argument, we must digress for a moment to explain the Biblical and Talmudic background of the rite of the scapegoat. The Bible prescribes that on the annual Day of Atonement the high priest shall offer two he- goats.

And he shall take from the congregation of the people of Israel two male goats for a sin offering ... and set them before the Lord ... and Aaron shall cast lots upon the two goats, one lot 'for the Lord' and the other lot 'for Azazel'. And Aaron shall present the goat on which fell the lot 'for the Lord', and offer it as a sin offering; but the goat on which fell the lot 'for Azazel' shall be presented alive before the Lord to make atonement over it, that it may be sent away into the wilderness to Azazel.19

From the detailed Talmudic description we quote: There was an urn there, and in it two lots. They were of box-wood until Ben Gamla made them of gold.... The lots may be of any material ... but they must be alike; they must not be one of gold and the other of silver, one big and the other small.20

"I Aramah, Aqedat Yitzlhak, ed. H. J. Pollack (Pressburg, 1849), Ch. 28.

16 The phrase is from Jeremiah 3:14 and is used here to convey the sense of a very small minority.

17 Berakhot 35b. 18 Aramah, Aqedat Yitzhak, Ch. 99. 19 Leviticus 6: 3-10. 20 Yoma37a.




The high priest inserted both hands into the urn, bringing up one lot in each hand, and placed them upon the two animals, the one in his right hand upon the animal at his right and similarly for the left. Since it was regarded as an auspicious omen if the lot "for the Lord" came up in the right hand, precautions were taken to prevent tampering with the lots.

He shook the urn and brought up the two lots.... Why did he shake the urn? In order to prevent choosing one intentionally.... The capacity of the urn ... was just great enough to contain no more than his two hands in order that he should not take one intentionally.21

It is described as a miracle due to the special merit of the high priest Shimon the Righteous that "during the forty years that he served, the lot 'for the Lord' came up in the right hand; thereafter, sometimes it came up in the right, sometimes in the left."22

Later Bible commentators found in the rite of the lots for the scapegoat occasion to discuss the meaning and role of chance. R. Isaac Aramah comments:

That which is by chance is equal for every one of the sides. While it is true that when Israel did God's will or when they had a worthy minister such as Shimon the Righteous, a miracle occurred with these lots in that the lot "for the Lord" continually came up in the right.... this was by special providence and by the mouth of God as it is written "The lot is cast into the lap, but the decision is wholly from the Lord".23 However later they were like ordinary lots that are due to chance without any greater tendency to one side rather than the other. Therefore, even if the lot "for the Lord" were to come up in the right it would not be a 'sign', for matters of this kind are not established unless they are found to be so many times....24

He is troubled by a seeming difficulty presented by the Biblical record in which it is reported that when the ship in which Jonah fled came up upon a great tempest the mariners said, "Come, let us cast lots, that we may know on whose account this evil has come upon US."25 Aramah comments:

Apparently this was foolish counsel. For it is impossible for it to be otherwise than that the lot should fall on one of them whether he be innocent or guilty.... However the meaning of their statement "let us cast lots" is to cast lots many times. Therefore the plural-goralot-is used rather than [the singular] goral.... They did so and cast lots many times and every time the lot fell on Jonah and consequently the matter was verified for them. It follows then that the casting of a lot indicates primarily a reference to chance.26

The idea was taken up by other writers. Thus Rabbi Isaac Abarbanel in his commentary on Jonah amplifies Aramah's explanation:

The third question concerns the words of the sailors on the ship who said to him [Jonah], "Come, let us cast lots, that we may know on whose account this great tempest has come upon us," .... How can a lot give true judgement in this matter? Perhaps it is not because of the sin of any of them; nonetheless a lot must of necessity fall on one of them although his heart be innocent and his hands clean ... Perhaps the storm was not due to the sin of man altogether and what do they want with a lot ? . . . They made tests to prove that that storm was not natural, but providential.... The second test was that they did not cast lots once only, for that might be a coincidence

21 Ibid., 39a. 22 Ibid. 23 Proverbs 16:33.

24 Aramah, Aqedat Yitzhak, Ch. 63. 25 Jonah 1:7. 26 Aramah, Aqedat Yitzhak, Ch. 63.




Rather they cast lots many times and different kinds of lots, and always the outcome of each of them was that the lot fell on Jonah. When they saw the multiple repetition of the thing they were convinced that the matter issued from God and that it was not by chance. For this there is support from the text, in that they said 'Come, let us cast Goralot [lots]' and then it is written 'So they cast goralot'. Now the singular form Goral refers to one casting as in 'the land shall be divided by goral' [Numbers 26:55] and 'according to goral' [ibid., 56]. However, the plural form goralot applies to repeated castings and different kinds of lots. If all of them coincide it is known that 'The lot is cast into the lap but the decision is wholly from the Lord' [Proverbs 16:33]. Thus it is clear that the sailors did not act rashly but with wisdom and knowledge.27

Aramah makes it clear that where the outcome is due to chance alone, in many trials

all possibilities should be realized in equal proportion. If, however, one outcome

predominates, the result must be attributed to Divine intervention rather than chance.

Aramah assumes that in the casting of lots the mutually exclusive alternatives are

indeed all equally probable. In practice this assumption underlies some of the early

rabbinic computations of probability. Those occur mainly in legal contexts, and I have

discussed them elsewhere.28 Given this assumption, Aramah formulates a law of

chance-perhaps the first such law known: as the number of trials increases, the

relative frequency of each of all the equally probable outcomes will approach that

probability.

A closer examination will reveal that although Aramah's formulation is the most

explicit, one can trace back the kernel of his thought-that is, the idea that in the

operations of chance, all possibilities will be realized in equal measure. Moreover,

highly significant applications of this "law of chance" were made much earlier, at

least among Jewish thinkers. We shall take up only one instance.

III

Maimonides found the intellectual world of his time dominated on the one hand

by the Moslem theological school of Kalam and on the other by the philosophical

teachings of Aristotelianism. He devoted his remarkable talents of systematization to

expound the tenets of both of these theories in order to lay bare their weaknesses,

which he proposed to discard, and to expose their strengths, which he proceeded to

incorporate into his own system.

The Kalam is concerned with establishing the truths of religion, mainly that God

created the universe and sustains it. To this end it postulated an extreme kind of atomic

theory. Not only all material objects but even time consists of atoms. Atoms of matter

exist for an instant (= atom of time) and no more. If things persist it is because they

are re-created anew every instant. Thus the concept of nature is illusory. There are no

27 Commentarius Rabbi Isaac Abarbanel (Hebrew with Latin title-page) (Amsterdam, 1642). Isaac ben Judah Abarbanel (1437-1508) successfully combined the careers of statesman, philosopher, and Biblical exegete. Exiled from Spain when the Jews were expelled in 1492, Abarbanel went to Naples and spent the rest of his life in Italy. He wrote commentaries on the Pentateuch and the Prophets as well as numerous other works.

28 See n. 10. Dr. Oscar B. Sheynin (Moscow) in

a private communication points out that Giro- lamo Cardano (1501-1576) seems to have had some awareness of a rudimentary law of large numbers (see Oystein Ore, Cardano the Gambling Scholar, Princeton:Princeton University Press, 1953, p. 177). It seems hardly likely that there was any connection, though, between Aramah and Cardano. The celebrated "law of large numbers" was given as a theorem by Jacob Bernoulli in his Ars conjectandi published posthumously in Basle in 1713.




fixed laws or patterns of events, and there are no enduring phenomena; what appear to be such are in fact completely new creations. 29

Leucippus and Democritus had made their atomic theory support a cosmogony of chance: the random aggregation and separation of the atoms brought all things into being.30 Kalam dispensed with both chance and natural law at one stroke. Nothing happens except by Divine fiat, and every passing instant requires a new act of God.

The line of argument is developed on the premise that everything conceivable is possible; yet not everything conceivable exists. Thus for example it is possible for a particular flower or star to exist of not to exist, and moreover it is possible for it to be round or square or octagonal, blue or white or green, and so on for all its attributes. But what determined that it did come into existence with the particular qualities which it in fact possesses? The Kalam theologians answer: a purposive agent must have been at work to bring about this particular result. This is the Kalam "principle of particularization."31

Maimonides finds grave fault with this reasoning. He rejects with ridicule the concept that nothing endures but for an instant and that everything conceivable is possible. Surely we observe the reality of natural law-that is, permanence and regularity in most phenomena; it does not make sense to speak of new creation all the time. Rather the cause of these phenomena resides in nature.

Nonetheless, Maimonides finds a very valuable insight in the "principle of particularization." The crux of the issue is in the definition of the "possible." For a flower growing from a given seed, it is determined within certain limits what its shape, size, and color will be. This follows from the nature of the seed, which is different from that of a seed of another species. While it is possible for nature to make a "mistake" and for a monstrosity to appear, that possibility is not at all comparable to the possibility which accords with nature. Therefore, in most cases nature will prevail; exceptions may be thought to be fortuitous. There is here an implicit identification of "possible" with "probable." Probability is quantified-from zero through greater probability to certainty-and the range of the "possible" is correspondingly broad. In Maimonides' own words: "For among contingent things some are very likely, other possibilities are very remote, and yet others are intermediate. The possible is very wide." 32

Maimonides recasts the "principle of particularization" by restricting its applica- bility. This restriction is a most fundamental one, for he limits himiself to cases of equal possibilities, or, what amounts to the same thing, equal probabilities. In such cases nature cannot be supposed to be predilected to the one or the other. The class of regular phenomena is thus excluded from consideration, but in the class of nonregular events (or things) it becomes possible to discriminate between those that are due to chance and those that are a result of intelligent design. He phrases his own "principle of particularization" thus:

There can be giving of preponderance and particularization only with respect to a particular existent that is equally receptive of two contraries or of two different things.

29 The Guide for the Perplexed, I, 73 (Pines, pp. 194-214).

30 Maimonides attributes this view to Epicurus. See Guide, II, 14 (Pines, p. 285) and III, 17 (Pines, p. 464).

31 Guide, I, 74; "The fifth method" (Pines, p. 218).

32 The Book of the Commandments, Negative Command 290, English trans. Charles B. Chavel, 2 vols. (London: Soncino Press, 1967), Vol. II, p. 270.




Accordingly it can be said of that inasmuch as we have found it in a certain state and not in another, there is proof of the existence of an artificer possessing purpose.33

Two points need to be stressed. The first has already been noted, namely, the pro- vision that one is dealing only with "a particular existent that is equally receptive of two contraries." The second is the premise that "we have found it in a certain state and not in another." The full meaning of this condition appears when we compare it with Maimonides' statement elsewhere in the Guide:

This is the nature of the possible, for it is certain that one of the possibilities will come to pass. And no question should be put why one particular possibility and not another comes to pass, for a similar question would become necessary if another possibility in- stead of this particular one had come to pass. Know this notion and grasp it.34

Here Maimonides points out that a single random event cannot but have only one of all the possible outcomes, and consequently it need not on that account alone be attributed to a cause other than chance. However, in a series of similar events (or things) each with equally possible mutually exclusive outcomes, we can expect all of the possibilities to be realized if chance alone were operative. If, however, this does not turn out to be true, one may infer the existence of an agent other than chance that determines the actual outcome. Now this agent cannot be "nature," for nature would give a regular outcome. The principle of particularization, according to Maimonides, states that the particularizing agent is purposeful.

IV

What Maimonides is saying can be rephrased as follows: if two possibilities are mutually exclusive and both are equally likely, then chance alone would make for both possibilities being realized ultimately, so that if only one outcome is actually observed (always or preponderantly), then we may conclude that a purposive cause is at work rather than chance.

On the basis of this principle Maimonides reasons that the universe is the handiwork of an intelligent, purposive creator. One of his arguments is an elaboration of one proposed earlier by R. Abraham Ibn Ezra (d. 1167): "Why is it that one place in the [celestial] sphere is full of stars and another is not so, though some are large and some small, also there are white ones and red ones all in the same sphere ?"35 Maimonides states that there are three possible hypotheses to explain the observed distribution of the stars in the heavens. 36

33 Guide, I, 74; Pines, p. 220. In a legal context, Maimonides actually applied a generalized form of this principle to computations of probability. Examples appear in my papers "Combinations and Probability in Rabbinic Literature," Bio- metrika, 1970, 57: 204; "On Probability Theory in Rabbinic Literature," Tarbiz, 1972, 42: 79-89.

34 Guide, 111, 26; Pines, p. 509. 35 Abraham Ibn Ezra, Commentary on Exoduts

(ed. Krinski, reprint Bnei Brak: Horev, 5721 A.M.

[1960]), 23:20. It is unlikely that Maimonides drew upon Ibn Ezra. For somewhat similar

though imnprecise arguments in earlier writers see H. A. Wolfson, "Hallevi and Maimonides on De- sign, Chance and Necessity," Proceedings of the American Academy for Jewish Research, 1941, 11. Wolfson (p. 119) traces the argument to al-Ghaz- zali. Of course, Maimondes himself refers to the Kalam origins of this type of reasoning. However, Pines in his introduction to the Guide (p. cxxvii) is not certain that Maimonides knew al-Ghazzali.

36 Guide, II, 19; Pines, pp. 309-310. 1 have paraphrased much of the exposition, quoting verbatim only the most salient passages.



15TH-CENTURY LAW OF LARGE NUJMBERS 237

1. The distribution is due to chance. 2. The Aristotelian view that this "proceeded obligatorily and of necessity from the

Deity" in some unknown manner, implies the existence of unsuspected properties of nature, which account for the observed irregular and asymmetric distribution.37

3. It is due to design. As for the first alternative Maimonides says:

It is even stranger that there should exist the numerous stars that are in the eighth sphere all of which are globes, some of them small and some big, one star being here and another at a cubits' distance according to what seems to the eye, or ten stars being crowded and assembled together while there may be a very great stretch in which nothing is to be found. What is the cause that has particularized one stretch in such a way that ten stars should be found in it and has particularized another stretch in such a way that no star should be found in it? . . . ... How then can one who uses his intellect imagine that the positions, measures and numbers of the stars ... are fortuitous ?

Aristotle had argued that regularity and symmetry preclude chance and indicate natural law,38 but here we have neither regularity nor random distribution.

If chance is rejected, there remains either unknown necessity or design for some unknown purpose. Now teleological design is known as an efficacious cause for irregularities:

In fact you know that the veins and nerves of any individual dog or ass have not happened fortuitously, nor are their measures fortuitous. Neither is it by chance that one vein is thick and another thin, that one nerve has many ramifications and another is not thus ramified, that one descends straight down and another is bent. All this is as it is with a view to useful effects whose necessity is known.

Is it reasonable then to postulate unknown natural necessity to explain the strange distribution of the stars, which is clearly neither raindom nor regular, when there is a familiar principle at work in the world which can explain the irregularity? That would be "very remote indeed from being conceivable." On the other hand, though,

If it is believed that all this came about in virtue of the purpose of one who purposed who made it thus, that opinion would not be accompanied by a feeling of astonishment and would not be at all unlikely. And there would remain no other point to be investigated except what is the cause for this having been purposed ?39

87 Maimonides remarks that Aristotle does not explicitly state this, but it is a consequence of his system as expounded by the followers of his school (Guide, II, 13; Pines, p. 284).

38Physica 11, 4, 196bIO. 3 The interpretation of this chapter is crucial.

Is Maimonides arguing for the elimination of chance as responsible for the stellar distribution, or is he rather only contrasting Aristotelian ""necessity" with design? In reply to the comment of an anonymous referee, who suggested that perhaps the references to chance or fortuity are just casual and are meant to convey only "pur- poselessness," the following considerations are offered.

a. At the end of the chapter, Maimonides sums up: "There must of necessity be something that particularizes." He goes on to remark, "This examination has thus conducted us to the investi- gation of two problems, one of which may be stated as follows: Is it of necessity obligatory or not, considering the existence of these differences, that these should be due to the purpose of one who purposed and not due to necessity?.... In the following chapters I shall begin to treat of these two problems." What does he consider to have been accomplished in this chapter just con- cluded (incidentally, one of the longer ones in the Guide)? Clearly Maimonides feels that his "principle of particularization" effectively elim-




I am indebted to Dr. Sheynin, who drew my attention to the importance assigned to the argument for design by the exponents of natural theology. 40 In a passage redolent of Maimonides, Sir Isaac Newton compares the structures of the heavens with that of living organisms.

... it's unphilosophical to.... pretend that it might arise out of a chaos by the mere Laws of Nature ... blind fate could never make all the planets move one and the same way.... Such a wonderful Uniformity in the planetary system must be allowed the Effect of Choice. And so must the Uniformity in the Bodies of Animals. . . Also the first contrivance of those very artificial parts of animals, the Eyes, Ears ... can be the effect of nothing else than the wisdom and skill of a powerful ever-living Agent.41

Although the nature of the probabilistic reasons advanced differ, it may not be entirely farfetched to suppose that there is some connection between Maimonides and Newton, for Max Jammer has marshalled considerable evidence to demonstrate that Newton's conception of absolute space was strongly influenced by Jewish sources.42

Substantially the identical Maimonidean argument from the distribution of the stars is still a live issue. On the supposition of random distribution one can try to compute the probability of getting a constellation like the Pleiades where six stars cluster together. R. S. Fisher took up this problem and reached the conclusion that one must "exclude at a high level of significance any theory involving a random distribution."43

inates chance, and he claims that it decides in favor of design. However, a partisan Aristotelian might suggest an alternative way of eliminating chance which will decide in favor of "necessity." The decision between these two approaches remains to be made. In fact in the very next chapter this question is taken up and it begins with Aristotle's approach to chance.

b. The commentators that I have seen are not very helpful on the matter of stellar distribution. Yet the following passage from the fifteenth- century commentary on the Guide by Isaac Abarbanel is relevant. It refers to Maimonides' words, ". . . the numerous stars that are in the eighth sphere, all of which are globes ... ," which Abarbanel explains: "The stars are all globes. However, since they are at rest [fixed in the sphere] and a globular shape is caused by rotatory motion, what is the reason for the shape of the stars? We cannot say that this happened

fortuitously and by chance, for if that were so not all the stars would be globes, for that which is by chance is infrequent."

If the argument from the globular shape of the stars is meant to eliminate chance, why should we attribute quite a different sense to the argument from the distribution of the stars? They are both mentioned in the same breath!

40 0. B. Sheynin, "Newton and the Classical Theory of Probability," Archive for History of Exact Sciences, 1971, 7: 217-243.

41From Query 31 of the Optics. See Isaac Newton, Opticks (reprinted from the 4th ed., London: G. Bell and Sons, 1931), pp. 402-403.

42 Max Jammer, Concepts of Space (Cam- bridge, Mass.: Harvard University Press, 1954), pp. 108ff.

43 R. S. Fisher, Statistical Methods and Scien- tific Inference (London: Oliver and Boyd, 1956), p. 39.



a fifteenth-century law of large numbers

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