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battery, or the electric organs of fish. Almost all the comparisons that we have drawn of material objects, in order to make intellectual matters sen­sible, are basically identities.

I hope that the preceding thoughts, however imperfect they may be, may draw the attention of philosophical observers to the laws of the sensorium or of the intellectual world, for it is important that we examine these laws as thoroughly as those of the physical world. Hypotheses that are nearly the same have been devised to explain the phenomena of these two worlds. But the foundations of these hypotheses escaping all our means of obser­vation and calculation, one may say regarding them with Montaigne [60] that ignomnce and incuriosity are a soft and sweet pillow for resting a well-balanced mind. U

On various approaches to certainty

Induction, analogy, hypotheses founded on facts and continually corrected by new observations, a fortunate intuition, given by nature and strength­ened by numerous comparisons of the things it suggests with experience -these are the principal means of arriving at the truth [1].

H one attentively considers a sequence of objects of the same kind, one begins to see between them and in their changes, resemblances a which become more and more obvious as the sequence is prolonged, and which, continually being extended and generalized, finally lead to the principle Gfrom which they {i.e. the resemblances} proceed.G But often {3 these re­semblances are enshrouded in so many foreign circumstances, that great sagacity is required to uncover them, and to work back to that principle. It is in this that the true genius of science consists. Analysis and natural phi­losophy owe their most important discoveries to this fecund method which is called induction. Newton was indebted to it for his binomial theorem and the principle of universal gravitation. It is difficult to estimate the proba­bility of btheb results cof induction, whichc are based on this: the simplest relationships 'Y are the most common. This is verified in the formulae of Analysis, and recognized in natural phenomena, in crystallization and in chemical combination. This simplicity of 6 relationships appears in no whit astonishing when one considers that all natural consequences are only the mathematical results of a small number of immutable laws.

a and laws a-a on which they {Le. the resemblances and laws} depend. (3 these laws and b-b its c-c They ., and laws 6 laws and

A. I. Dale, Philosophical Essay on Probabilities© Springer-Verlag New York, Inc. 1995

On various approaches to certainty 113

But induction, although disclosing the general principles of science, is not sufficient to establish them rigorously. It is always necessary to cor­roborate them with proofs or by conclusive experiments, for the history of science shows us that induction has sometimes led to inaccurate results. I shall cite, for example, a theorem by Fermat on prime numbers [2]. This great mathematician, having thought deeply about the theory {of primes}, tried to find a formula that, containing only prime numbers, would directly give a prime number greater than any given number. Induction led him to think that 22n +1 would always be prime. Thus 221 +1=5 and 222 +1=17, both primes. He found that this still held true for 28 +1 and 216+1, and this induction, reinforced by several arithmetic considerations, led him to regard this as a general result. However he acknowledged that he had not yet proved it rigorously. Indeed, Euler discovered that it failed for 232+1, which is 4,294,967,297, a number that is divisible by 641 [3].

dWe conclude by induction that if various events - motions, for exam­ple - appear constantly and have been connected for a very long time by a simple relationship, they will continue constantly to be subject to it, and from this we may conclude, by probability theory [4], that this relationship is due, not to chance, but to a regular cause. Thus the regularity of the mo­tions of the rotation and revolution of the moon, that of the motions of the nodes of the lunar orbit and equator, and the coincidence of these nodes; the singular relationship between the motions of the first three satellites of Jupiter {i.e. 10, Europa and Ganymede, or I, II and III}, according to which the mean longitude of the first satellite, minus thrice that of the second, plus twice that of the third, is equal to 180 0 [5]; the equality of the interval between the tides to that of the passage of the moon to the meridian; the recurrence of the highest tides with the syzygies, and the lowest with the quadratures; all these things, which have remained in force since they were first observed, indicate with an exceptionally high likelihood the existence of constant causes which mathematicians have successfully {or fortunately} managed to associate with the law of universal gravitation, and the knowl­edge of these causes ensures the perpetuity of these relationships. d

Francis Bacon, the Lord Chancellor and so eloquent promoter of the true scientific method, has misused induction in a very curious way in proving the immovability of the Earth [6]. This is how he reasons in the Novum Organum, his greatest work: the further the heavenly bodies are from the Earth, the swifter is their motion from east to west. This motion is most rapid in the case of the {fixed} stars: it decreases a little in Sat­urn's case, and a little more in Jupiter's, and so on until we reach the moon and the nearest comets. It is still perceptible in the atmosphere, especially within the tropics, because of the great circles that the air molecules de­scribe there. Finally it is almost imperceptible in the oceans, and it is of course zero in the case of the Earth. But this induction proves only that eSaturn ande the heavenly bodies 'that are inferior to {lower than} it' have their own motions, in a direction contrary to the real or apparent

114 Philosophical Essay on Probabilities

motion that carries the whole celestial sphere from the east to the west, and that these motions appear slower in the more distant heavenly bodies - which is consistent with the laws of optics. Bacon ought to have been struck by the inconceivable swiftness that, 9if9 the Earth is immovable, it was necessary to suppose that the heavenly bodies have in order to carry out their diurnal revolution, and by the extreme simplicity with which its {i.e. the Earth's} rotation explains how bodies as distant from each other as the stars, hthe sun,h the planets, iand the mooni, all seem subject to this revolution. AB for the oceans and the atmosphere, he should not have compared their motion to that of the heavenly bodies, which are separate from the Earth; whereas the air and the sea, being part of the terrestrial globe, ought to share in its motion or its rest. It is curious that iBaconi , whose genius produced kgreatk insight, was not won over by the majestic idea of the universe afforded by the Copernican system. He was however able to find strong analogies in favour of this system in Galilei's discover­ies, which were well-known to him. He has given the precept, and not the example, for the search for truth. But Ion insisting with all the force' of argument and oratory, on the necessity for abandoning the niggling niceties of mscholastic philosophym, in order to apply oneself to observation and experiment, Rand by indicating the correct method of rising to the gen­eral causes of phenomena, this great philosopherR has contributed to the immense progress °thatO the human mind "has made" during the glorious century in which qhis career was brought to a close. q

Analogy is based on the probability that similar things have the same kind of causes and produce the same effects. The more perfect the similar­ity, the greater this probability. Thus we may imagine, without any doubt, that creatures having the same organs and doing the same things, E ex­perience the same sensations and are prompted by the same desires. The probability that animals that have organs like ours have feelings analo­gous to ours, although a little inferior to those of individuals of our species, is still extremely large, and it has taken all the influence of religious prejudice to make some philosophers think that the animals

9-9 under the hypothesis that h-h and j - j this philosopher /c-/c the greatest I-I although the sciences are not indebted to him for any discovery, as he has

ceaselessly insisted with all the authority m-m the schoolmen n-n he 0-0 of q-q he lived. • and communicating with each other,

On various approaches to certainty 115

are , mere automata [7]. The probability of the existence of consciousness {or feeling} decreases as the similarity of the organs to ours decreases, but it is always very great, even in the case of insects. On seeing those of one and the same species carrying out very complicated tasks in exactly the same manner for generation upon generation and without any teaching, one is led to believe that they behave by a kind of affinity, analogous to that which draws crystalline molecules together, but which, blending itself with a collective animal consciousness, produces many most singular combina­tions with the regularity of chemical combination. One may perhaps call this mixture of elective affinities and consciousness {or sentiment}, animal magnetism [8]. Although there exists a strong analogy between the organi­zation of plants and that of animals, it does not however seem sufficient to me to allow the extension of the perceptive faculty to plants: rbutr nothing stops us from refusing it to them.

Since the sun, by the salutary action of its light and heat, gives life to the anjmals and plants that cover the Earth, we may analogously suppose that it produces similar effects on other planets. For it is unnatural to think that matter from which we see activity developing in so many ways should be barren on so large a planet as Jupiter which, like the Earth, has its days, its nights, and its years, and on which observations indicate changes that imply very active forces. 8However8 it would be stretching analogy too far to conclude from this that the inhabitants of the planets would be similar to those of the Earth. Man, created for the temperature he enjoys and the element {viz. the air} he breathes, would not be able, according to all ap­pearances, to live on other planets. But should there not be an infinity of organisms corresponding to the various constitutions of the spheres of this universe? IT the mere difference in the elements and the climate leads to so great a variety of terrestrial products, how much more different ought those of the various planets and their satellites to bel The most active imagina­tion can form no idea of them; but it is very likely that they exist.

By a strong analogy we are led to regard the stars as so many suns, endowed, like our own, with an attractive power {directly} proportional to their masses and inversely proportional to the square of their distances. For since this power has been demonstrated by all bodies of the solar system and by their smallest molecules, it seems to belong to all matter. Already the motions of the small stars, called double because of their proximity to each other, seem to indicate this. A century at most of precise observation will, by verifying their revolution about each other, put their reciprocal attractions beyond doubt [9].

The analogy leading us to regard each star as the centre of a planetary

, only r-r as a-a But

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system is not nearly as strong as the preceding one, but it gains in likeli­hood under the hypothesis that we have proposed on the formation of the stars and the sun. For since each star, under this hypothesis, was originally surrounded by a widespread atmosphere, as was the sun, it is natural to attribute the same effects to this atmosphere as to the solar atmosphere, and to suppose that, in condensing, it has produced planets and satellites.

t A large number of scientific discoveries are due to analogy. I shall men­tion as one of the most remarkable the discovery of atmospheric electricity [101, to which one was led by the analogy between electrical phenomena and the effects of thunderbolts. t

The surest method to guide us in the search for truth consists in pro­ceeding "inductively" from "phenomena to laws, and from laws to forces [111. Laws are relationships that connect particular phenomena to one an­other; when the general principle of the forces from which they are derived has been made known by them, it may be verified", either by direct ex­periments, whenever that is possible, or by our investigating whether it agrees with known phenomena. And if by a rigorous analysis one finds that they all follow from this wprinciplew, down to the smallest detail, and if moreover they are very varied and very numerous, then science attains the highest degree of certainty Zand of perfectionz possible. This is what has happened to astronomy by the discovery of universal gravitation. tlThetl history of science shows that inventors have not always followed this slow and laborious course of induction. The imagination, impatient to reach the causes, takes pleasure in creating hypotheses; and it often distorts the facts to suit its own ends. In such a case the hypotheses are dangerous. But when one regards them only as the means of connecting the phenomena together to discover their laws, and when by avoiding the attribution of any reality to them one continually corrects them by new observations, then they may lead to the real causes - or at least put us in a position to conclude from observed phenomena what should happen under given circumstances.

If we could test all the hypotheses that could be formed on the causes of phenomena, we would arrive at the truth by a process of elimination. This method has been successfully used: zsometimesz one arrived at several hypotheses which explained all the known facts equally well, and between which scientists were divided until decisive observations revealed the true one. In such a case it is interesting, for the history of the human mind, to return to these hypotheses, to see how they succeeded in explaining a

u-u by way of induction v-v particular phenomena to more and more extensive relationships, until

we finally arrive at the general law from which they are derived. Then this law is verified

w-w law 11-11 But the z-z But sometimes

Historical note on the probability calculus 117

large number of facts, and to investigate the changes that they ought to undergo to be naturally resurrected. It is thus that the Ptolemaic system, which is only the realization of celestial appearances, is transformed into the hypothesis that the planets move about the sun, when we make the circles and epicycles that Ptolemy described annually and whose magni­tude he left undetermined, equal and parallel to the solar orbit. In order to change this hypothesis into that of the true system of the world, it is then sufficient to transfer, in the contrary sense, the apparent motion of the sun to the Earth.

It is almost always impossible to handle, by the calculus, the probabil­ity of the results obtained by these various means: this is also the case with historical facts. But the ensemble of phenomena accounted for, or of testimony, is sometimes such that, without being able to estimate the prob­ability, one cannot reasonably have any doubt about them. In other cases, it is prudent to entertain them only with extreme caution.

Historical note on the probability calculus

The ratios of favourable to unfavourable chances for players in the sim­plest games have been known for a long time [1]. The stakes and the bets were settled by these ratios. But before Pascal and Fermat, the principles and the methods for calculating such things were not known, and rather complicated questions of this kind had not been solved. Thus it is to these two great mathematicians that we must attribute the fundamentals of the science of probabilities, the discovery of which may be ranked among the remarkable things that have made the 17th century famous - the century that has brought the greatest credit to the human mind. The main problem that they a resolved, by different means, consists, as we have already seen [2], in distributing the stakes fairly between players who are of equal ability and who have agreed to cry quits before the game is finished. The condition of play is that he who first reaches a given number of points a (different for each of the players)a, will win the game. It is clear that the distribution ought to be made proportionally to the players' respective probabilities of winning this game, probabilities that depend on the numbers of points each of them still lacks. Pascal's method is very ingenious; it is basically only f3 the partial difference equations bofb this problem capplied to the determin­ination ofc the successive probabilities of the players, working one's way from the smallest numbers upwards. This method is restricted to the case

a both together f3 the use of b-b pertaining to c-c to determine