The aim and structure of physical theory

by Pierre Duhem

Excerpt 1

The Greeks were acquainted, properly speaking, with only one physical theory, the theory of celestial motions; that is why, in dealing with systems of cosmography, they expressed and developed their conception of physical theory. Moreover, other theories that they carried to a certain degree of perfection, and that today emerge again in physics — namely, the theories of equilibrium of the lever and hydrostatics — rested on principles whose nature could not be subject to any doubt. The axioms or demands of Archimedes were plainly propositions of experimental origin which generalization had transformed; the agreement of their consequences with the facts summarized and ordered the latter without explaining them.

The Greeks clearly distinguished, in the discussion of a theory about the motion of the stars, what belongs to the physicist — we should say today the metaphysician — and to the astronomer. It belonged to the physicist to decide, by reasons drawn from cosmology, what the real motions of the stars are. The astronomer, on the other hand, must not be concerned whether the motions he represented were real or fictitious; their sole object was to represent exactly the relative displacements of the heavenly bodies.

In his beautiful research on the cosmographic systems of the Greeks, Schiaparelli has brought to light a very remarkable passage concerning this distinction between astronomy and physics. The passage is from Posidonius, was summarized or quoted by Geminus, and has been preserved for us by Simplicius.

Here it is: "In an absolute way it does not belong to the astronomer to know what is fixed by nature and what is in motion; but among the hypotheses relative to what is stationary and to what is moving, be inquires as to which ones correspond to the heavenly phenomena. For the principles he has to refer to the physicist."

These ideas, expressing pure Aristotelian doctrine, inspired many a passage by the astronomers of old; Scholasticism has formally adopted them. It is up to physics — that is, to cosmology — to give the reasons for the astronomical appearances by going back to the causes themselves; astronomy deals only with the observation of phenomena and with conclusions that geometry can deduce from them.

Saint Thomas, in commenting on Aristotle’s Physics, said: "Astronomy has some conclusions in common with physics. But as it is not purely physics, it demonstrates them by other means. Thus the physicist demonstrates that the earth is spherical by the procedure of a physicist, for example, by saying its parts tend equally in every direction towards the center; the astronomer, on the contrary, does this by relying on the shape of the moon in eclipses or the fact that the stars are not seen to be the same from different parts of the world."

It is by furtherance of this conception of the role of astronomy that Saint Thomas, in his commentary on Aristotle’s De Caelo, expressed himself in the following manner on the subject of the motion of the planets: "Astronomers have tried in diverse ways to explain this motion. But it is not necessary that the hypotheses they have imagined be true, for it may be that the appearances the stars present might be due to some other mode of motion yet unknown by men. Aristotle, however, used such hypotheses relative to the nature of motion as if they were true."

In a passage from the Summa Theologiae (I, 32), Saint Thomas showed even more clearly the incapacity of physical method to grasp an explanation that is certain: "We may give reasons for a thing in two ways. The first consists in proving a certain principle in a sufficient way; thus, in cosmology (scientia naturalis) we give a sufficient reason to prove that the motion of the heavens is uniform, In the second way, we do not bring in a reason which proves the principle sufficiently, but the principle being posited in advance, we show that its consequences agree with the facts; thus, in astronomy, we posit the hypothesis of epicycles and eccentrics because, by making this hypothesis, the sensible appearances of the heavenly motions can be preserved; but that is not a sufficiently probative reason, for they might perhaps be preserved by another hypothesis."

This opinion concerning the role and nature of astronomical hypotheses agrees very easily with a good number of passages in Copernicus and his commentator Rheticus. Copernicus, notably in his Commentariolus de hypothesibus motuum caelestium a se canstitutis, simply presents the fixity of the sun and the mobility of the earth as postulates which he asks that he be granted: Si nobis aliquae petitiones . . . concedentur. It is proper to add that in certain passages of his De revolutionibus caelestibus libri sex, he professes an opinion concerning the reality of his hypotheses which is less reserved than the doctrine inherited from Scholasticism and expounded in the Commentariolus.

This last doctrine is formally enunciated in the famous preface, which Osiander wrote for Copernicus’ book De revolutionibus caelestibus libri sex. Osiander expresses himself thus: "Neque enim necesse est eas hypotheses esse veras, imo, ne verisimiles quidem; sed sufficit hoc unum, si calculum observationibus congruentam exhibeant.* And he ends his preface with these words: "Neque quisquam, quod ad hypotheses attinet, quicquam certi ab Astronomia expectet, cum nihil tale praestare queat." Such a doctrine concerning astronomical hypotheses aroused Kepler’s indignation. In his oldest writing, he said: "Never have I been able to assent to the opinion of those people who cite to you the example of some accidental demonstration in which from false premises a strict syllogism deduces some true conclusion, and who try to prove that the hypotheses admitted by Copernicus may be false and that, nevertheless, true phenomena may be deduced from them as from their proper principles. . . . I do not hesitate to declare that everything that Copernicus gathered a posteriori and proved by observation could without any embarrassment have been demonstrated a priori by means of geometrical axioms, to an extent that would be a delightful spectacle to Aristotle, were he living."

5 vols. (Paris, 1913-1917), Vol. ii, Part i, Chs. x and xi, pp. 50179. 10 We have borrowed several of the informative items which follow in the text from a very important article by P. Mansion, "Note sur le caractère géomètrique de 1’ancienne Astronomie," Abhandlungen zur Geschichte der Mathematik, ix (Leipzig). See also P. Mansion, Sur les principes fondamentaux de la Géométrie, de la Mecanique et de l’Astronomie (Paris, 1903).

*Translator’s note: "Nor is it, to be sure, necessary that these hypotheses true, or even probable; but this one thing suffices, namely, whether the calcutions show agreement with the observations.

*Translator’s note: "Nor should anyone, because he holds fast to hypotheses expect certainty from astronomy, as it cannot be responsible for anything like that." For the English translation of the Commentariolus and a brief discussion of Duhem’s attitude to Copernicus' view of astronomical hypotheses, see E. Rosen, Three Copernican Treatises (New York, 1939), pp. 57-90 and p. 33 respectively.

11 In 1597, Nicolas Raimarus Ursus published a book in Prague entitled hypothesibus astronomicis, in which he upheld the opinions of Osiander to exaggerated extent. Three years later, hence in 1600 or 1601, Kepler answered with the following: Joannis Kepleri "Apologia Tychonis contra Nicalaum Raymarum Ursum"; this work remained in manuscript in a very incomplete state and was published only in 1858 by Frisch (Joannis Kepleri astronomi "Opera omnia" [Frankfort-on-the-Main and Erlangen], s, 215). This work contains lively refutations of Osiander’s ideas.

12 Prodromus dissertationum cosmographicarum, continens mysterium cosmographicum . . . a M. Joanne Keplero Wirtembergio (Georgius Gruppenbachius, 1591); see Joannis Kepleri astronomi "Opera omnia," i, 112-153.

This enthusiastic and somewhat naive confidence in the boundless power of the physical method is prominent among the great discoverers who inaugurated the seventeenth century. Galileo did indeed distinguish between the point of view of astronomy, whose hypotheses have no other sanction than agreement with experience, and the point of view of natural philosophy, which grasps realities.

When he defended the earth’s motion he claimed to be talking only as an astronomer and not to be giving hypotheses as truths, but these distinctions are in his case only loopholes created in order to avoid the censures of the church; his judges did not consider them sincere, and if they had regarded them as such, these judges would have shown very little insight. If they had thought that Galileo sincerely spoke as an astronomer and not as a natural philosopher or, in their idiom, "physicist," if they had regarded his theories as a system suited to represent celestial motions and not as an affirmative doctrine about the real nature of astronomical phenomena, they would not have censured his ideas.

We are assured of this by a letter which Galileo’s principal adversary, Cardinal Bellarmin, wrote to Foscarini on April 12, 1615: "Your Fatherhood and the honorable Galileo will act prudently by contenting yourselves to speak hypothetically, ex suppositione, and not absolutely, as Copernicus has always done, I believe; in fact, to say that by supposing the earth mobile and the sun stationary we give a better account of the appearances than we could with eccentrics and epicycles, is to speak very well; there is no danger in that, and it is sufficient for the mathematician."’ In this passage Bellarmin maintained the distinction, familiar to the Scholastics, between the physical method and the metaphysical method, a distinction which to Galileo was no more than a subterfuge.

The one who contributed most to break down the barrier between physical method and metaphysical method, and to confound their domains, so clearly distinguished in the Aristotelian philosophy, was surely Descartes. Descartes’ method calls into doubt the principles of all our knowledge and leaves them suspended on this methodological doubt until it can reach the point of demonstrating the legitimacy of principles by a long chain of deductions stemming from the famous Cogito, ergo sum.

Nothing is more contrary than such a method to the Aristotelian conception, according to which a science, such as physics, rests on self-evident principles whose nature is investigated by a metaphysics which cannot increase their certainty.

13 H. Grisar, Calileistudien: Historische-theologische Untersuchungen uber die Urtheile der romischen Congregationen in Galileiprocess (Regensburg, 1882), Appendix, ix.

The first proposition in physics that Descartes established, in pursuing his method, grasps and expresses the very essence of matter: "The nature of body consists only in the fact that it is a substance having extension in length, width, and depth."14 The essence of matter thus being known, we shall be able, through the procedures of geometry, to deduce from it the explanation of all natural phenomena.

Summarizing the method by which he claimed he dealt with the science of physics, Descartes said: "I accept no principles of physics which are not also accepted in mathematics, for the sake of being able to prove by demonstration everything that I shall deduce from them, and these principles are sufficient, so long as all the phenomena of nature may be explained by means of them."

Such is the audacious formula of Cartesian cosmology: man knows the very essence of matter, namely, extension; he may then logically deduce all the properties of matter from it. The distinction between physics, which studies phenomena and their laws, and metaphysics, which seeks to know the essence of matter insofar as it is the cause of phenomena and the basis of laws, is deprived of any foundation.

The mind does not start from the knowledge of phenomena to rise to the knowledge of matter; what it can know from the start is the very nature of matter, and thence the explanation of phenomena. Descartes pushed this proud principle to its extreme consequences. He was not content with asserting that the explanation of all natural phenomena may be derived completely from this single proposition: "The essence of matter is extension;" he tried to give this explanation in detail. He investigated the question of constructing the world with shape and motion by starting with this definition.

And when he reached the end of his work, he stopped to contemplate it, and declared that nothing was missing in it: "That there is no phenomenon in nature not included in what has been explained in this treatise" — so runs the title of one of the last paragraphs of the Principia Philosophiae.’

Sometimes Descartes seemed for a moment to have been frightened by the boldness of his cosmological doctrine and to have wished to assimilate it to the Aristotelian doctrine.

That is what happens in one of the sections of the Principia; let us quote this section in its entirety, for it touches closely on the object of our study: "It may still be retorted to this that, although I may have imagined causes capable of producing effects similar to those we see, we should not for that reason conclude that those we see are produced by these causes; because, as an industrious watchmaker may make two watches indicating the hour in the same way and without any difference between them in their external appearance, yet without anything similar in the composition of their wheels, so it is certain that God works in an infinity of diverse ways, each of which enables him to make everything in the world appear as it does without making it possible for the human mind to know which of all these ways he has willed to employ. I have no difficulty in agreeing with this.

"And I believe I shall have done enough if the causes that I have explained are such that all the effects they may produce are similar to those we see in the world without being informed whether there are other ways in which they are produced. I even believe that it is as useful in life to know the causes thus imagined as if we had knowledge of the true causes, for medicine, mechanics, and generally all the arts served by a knowledge of physics, aim only to apply certain observable bodies to one another in such a manner that certain observable effects are produced by a series of natural causes.

"This could be accomplished just as well by considering the series of a few causes thus imagined, however false they may be, as if they were the true ones, since this series is supposed to be the same so far as the observable effects go. And in order that it may not be imagined that Aristotle ever claimed to do more than that, he himself said, at the beginning of the seventh book of his Meteors, that ‘concerning things not manifest to the senses, they are sufficiently demonstrated, as much as may be reasonably desired, if it can only be shown that they may be such as are explained.’"

But this sort of concession to the ideas of the schoolmen is manifestly in disagreement with the very method of Descartes. It is simply one of those precautions against any censure by the holy office that the great philosopher took, very much disturbed, as we know, by the condemnation of Galileo. Moreover, it seems that Descartes himself feared that his circumspection might be taken too seriously, for be followed the section we have just quoted by two others entitled "That nevertheless we have a moral certainty that all the things in this world are the same as what is demonstrated here they may be" and "And even that we have more than a moral certainty about them."

14 B. Descartes, Principia Philosophiae (Amsterdam, 1644), Part iii 4. ~ ibid., Part iv, 199. 16 ibid., Part iv, 204.

The words "moral certainty" do not suffice, indeed, to express the boundless faith Descartes professed in his method. Not only did he believe he had given a satisfactory explanation of all natural phenomena, but he thought he had furnished the only possible explanation for them, and could demonstrate it mathematically.

On March 16, 1640 he wrote to Mersenne: "As to physics, I should think I knew nothing about it if I could only say how things may be without demonstrating that they cannot be otherwise; for having reduced physics to the laws of mathematics, I know it is possible, and I believe I can do it for all the little knowledge I believe I have; although I did not do it in my Essais because I did not want to give my principles there, and I still do not see anything which invites me to give them in the future."

This proud confidence in the boundless power of the metaphysical method was just the thing to cause Pascal to smile disdainfully; when you but admit that matter is nothing but extension in three dimensions, how foolish it is to wish to draw the detailed explanation of the world: ‘We must say crudely: that is done by shape and motion, for that is true. But to tell more, and to compose the machine — that is ridiculous, for that is useless, and uncertain, and painful."

Pascal’s famous rival, Christian Huygens, was not so harsh about the method which claims to derive the explanation of natural phenomena. Of course, Descartes’ explanations are untenable on more than one point; but that is because his cosmology which reduces matter to extension is not the sound philosophy of nature, namely, the physics of the atomists. From the latter we may hope to deduce, though with great difficulties, the explanation of natural phenomena: "Descartes has recognized, better than those before him, that we should never understand anything important in physics except what might be related to principles not going beyond our mind’s reach, such as the principles which depend on bodies, considered devoid of qualities, and on their motions. But as the greatest difficulty consists in showing how so many diverse things are brought about by these principles alone, in that respect he has not succeeded in several particular subjects he proposed to examine; one of them, among others, is the subject of weight. This may be judged by the remarks I make in several places about what he has written, to which I could have added others. And yet, I confess that his essays and his insights, though false, have helped me to discover the road to the discoveries I have myself made on the same sub ject.

"I do not offer it as being exempt from all doubt, nor one to which no objections can be made. It is too hard to go that far in investigations of this nature. Still, I believe that if the principal hypothesis, which I take as basic, is not the true one, there is little expectation that it can be found while staying within the limits of true and sound philosophy." 19

Between the time Huygens communicated to the Académie des Sciences of Paris his Essay on the Cause of Weight and the time he had it published, there appeared the immortal work of Newton: Philosophiae naturalis principia mathematica. This work transformed celestial mechanics, and inaugurated opinions on the subject of the nature of physical theories altogether opposed to those of Descartes and Huygens.

Newton expressed clearly what he thought about the construction of physical theories in several passages in his works. The attentive study of phenomena and their laws permits the physicist to discover by the inductive method appropriate to his science some of the very general principles from which experimental laws may be deduced; thus the laws of all celestial phenomena are found condensed in the principle of universal gravitation.

Such a condensed representation is not an explanation; the mutual attraction that celestial mechanics imagines between any two parts whatsoever of matter permits us to submit all celestial movements to calculation, but the cause itself of this attraction is not laid bare because of that. Must we see in it a primary and irreducible quality of matter? Must we regard it as the result of impulses produced by a certain ether, as Newton was to judge probable at certain times in his life? These are difficult questions whose solution can only be obtained later, In any case, this problem is the task of the philosopher and not of the physicist; whatever the answer may be, the representative theory constructed by the physicist will keep its full value.

17 R. Descartes, Correspondance, ed. P. Tannery and C. Adam, iii, 39. 18 B. Pascal, Pensées, ed. Havet, Art. 24. This thought is preceded by these words: "To write against those who go too deeply in the sciences: Descartes." 19 Christian Huygens, Discours de la cause de la Pesanteur (Leyden, 1690).

Here is the doctrine stated in a few words in the "General Scholium" with which the Philosophiae naturalis principia mathematica ends; "And now we might add something concerning a certain most subtle spirit which pervades and lies hidden in all gross bodies. By the force and action of this spirit the particles of bodies attract one another at near distances, and cohere, if contiguous; electric bodies operate to greater distances, as well repelling as attracting the neighbouring corpuscles; and light is emitted, reflected, refracted, inflected, and heats bodies.

"All sensation is excited and the members of animal bodies move at the command of the will, namely, by the vibrations of this spirit, mutually propagated along the solid filaments of the nerves, from the outward organs of sense to the brain, and from the brain into the muscles. But these are things that cannot be explained in few words, nor are we furnished with that sufficiency of experiments which is required for an accurate determination and demonstration of the laws by which this electric and elastic spirit operates."

Later, in the famous Query XXXI at the end (the fourth paragraph from the last) of the second edition of his Optis, Newton enunciated with great precision his opinion concerning physical theories; he assigned to them as their object the economic condensation of phenomena: "To tell us that every species of things is endowed with an occult specific quality by which it acts and produces manifest effects, is to tell us nothing; but to derive two or three general principles of motion from phenomena, and afterwards to tell us how the properties and actions of all corporeal things follow from those manifest principles, would be a very great step in philosophy, though the causes of those principles were not yet discovered; and therefore I scruple not to propose the principles of motion above mentioned, they being of very large extent, and leave their causes to be found out." Those who shared the proud confidence of the Cartesians or atomists could not allow such modest limits to be imposed on the claims of theoretical physics. To limit one’s self to giving a geometric representation of phenomena was to their mind not to advance in the knowledge of nature. Those who were content with such vain progress deserved scarcely anything but sarcasm. One Cartesian said: "Before making use of the principles we have just established, I believe it will not be inappropriate to examine those Mr. Newton used as the foundation of his system. This new philosopher, already distinguished by the rare knowledge he had drawn from geometry, suffered impatiently because a nation foreign to his own could take such advantage of the position it had as to teach other nations and serve as a model for them. Moved by a noble pride and guided by his superior genius, he thought only of freeing his country from the necessity it felt of borrowing from us the art of throwing light on the processes of nature and of following her in her operations.

"That was still not enough for him. Opposed to all restraint, and feeling that physics would constantly embarrass him, he banished it from his philosophy; and for fear of being compelled to solicit its aid sometimes, he took the trouble to construct the intimate causes of each particular phenomenon in primordial laws; whence every difficulty was reduced to one level. His work did not bear on any subjects except those that could be treated by means of the calculations he knew how to make; a geometrically analyzed subject became an explained phenomenon for him. Thus, this distinguished rival of Descartes soon experienced the singular satisfaction of being a great philosopher by sole virtue of his being a great mathematician.

"I therefore return to what I first advanced, and I conclude that by following the method of this great geometer, we can with the greatest of ease develop the mechanism of nature. Do you wish to give an account of a complicated phenomenon? Expound it geometrically, and you will have done everything; whatever remains embarrassing to the physicist will depend, most certainly, either on a fundamental law or on some particular determination."

21 Newton’s disciples, however, did not all adhere to the prudent reserve of their master; several could not remain in the narrow confines assigned to them by his method in physics. Crossing these limits, they asserted, as metaphysicians, that mutual attractions were the real and primary qualities of matter and that a phenomenon reduced to these attractions was truly explained.

This was the opinion expressed by Roger Cotes in the famous preface he wrote at the head of the second edition of Newton’s Principia. This was also the doctrine developed by Boscovich that the Leibnizian metaphysics often inspired. However, several of Newton’s followers, and not the least distinguished ones, adhered to the method that their illustrious predecessor had so well defined. Laplace professed utmost confidence in the power of the principle of attraction. This confidence, however, is not a blind one; in some places in the Exposition du systeme du Monde, Laplace indicated that this universal attraction, which in the form of gravity or of molecular attraction coordinates all natural phenomena, is not perhaps the ultimate explanation, and that it may itself depend on a higher cause.

This cause, it is true, seems to have been relegated by Laplace to an unknowable domain. In any case, he recognized with Newton that the quest for this cause, if at all possible, constitutes a problem distinct from the one which physical and astronomical theories solve. He asked: "Is this principle a fundamental law of nature? Is it only a general effect of an unknown cause? Here, we are stopped by our ignorance of the intimate properties of matter, depriving us of any hope of answering these questions satisfactorily."22

Again, he said: "Is the principle of universal gravity a fundamental law of nature or but the general effect of an unknown cause? May we not reduce the attractions to this principle?

Newton, more circumspect than several of his disciples, did not pronounce judgment on these matters where our ignorance of the properties of matter does not permit us to give any satisfactory answer."

Ampére, a more profound philosopher than Laplace, saw with perfect clarity the importance of regarding a physical theory as independent of any metaphysical explanation; in fact, that is the way to keep out of physics the divisive quarrels of the diverse cosmological schools.

At the same time, physics remains acceptable to minds that profess incompatible philosophical opinions; and yet, very far from blocking the inquiries of those who would lay claim to giving an explanation of phenomena, we expedite their task. We condense in a small number of very general propositions the countless laws they are to explain, so that it suffices for them to explain these few propositions in order to get at anything mysteriously contained in that enormous collection of laws: "The chief importance of the formulas which are thus immediately concluded from some general facts, given by a number of observations sufficient to make their certainty incontestable, is that they remain independent both of the hypotheses used by their authors in the search for these formulas and of those hypotheses which may be substituted subsequently.

The expression of universal attraction deduced from Kepler’s laws does not depend on the hypotheses that a few authors have ventured concerning a mechanical cause they wished to assign to it. The theory of heat really rests on general facts immediately given to observation; and the equation deduced from these facts being confirmed by the agreement of the results drawn from the equation with those given by experience, should be regarded as expressing the true laws of the propagation of heat, both by those who attribute heat to a radiation of calorific moleculs as well as by those who explain the same phenomenon by having recourse to the vibrations of a fluid pervading space.

But it is necessary the former show how the equation in question results from their way of looking at things, and that the latter deduce it from the general formula of vibratory motions, not for the sake of adding anything to the certainty of this equation but to maintain their own respective hypotheses.

The physicist who has not taken sides in this regard accepts this equation as the exact representation of the facts without worrying about the way it may result from either one of the above explanations.

²³ 25 E. S. de Gamaches, Principes généraux de la Nature appliqués au méchanisme astronomique et comparés aux principes de la Philosophie de M.Newton (Paris, 1740), p. 67. 21 ibid., p. 81. 22 P. S. Laplace, Exposition du système . . . i, iv, Gh. xvii. 23 ibid., i, v, Ch. v.