The Mechanical Universe.

Does the mechanism of the universe
require periodic maintenance?
From Charlie Chaplin's 1936 film Modern Times

Newton (1642-1727), Descartes (1596-1650), Leibniz (1646-1716) and all of the other major players in the 17th and 18th century development of mechanics were devoutly religious, and as was the custom of the time, they framed their scientific views in harmony with their religious views. These in turn influenced other natural philosophers. Much of the debate betwen proponents of the mechanical theories of Newton and Descartes had religious overtones.

Newton, to his credit, did not explicitly employ religious arguments to support his physics. Others were not so reluctant to mix the two.

Two interpretations of physics arose. Both saw the universe as a vast machine operating by mechanical laws of interaction between particles. This machine was created by God at some time in the past.

Newton felt that this machine would, if unattended, operate somewhat imperfectly. So the hand of God was continually required to "tweak the mechanism" now and then and provide maintenance of the machinery. Since Newton had not accounted for all the observed details of the motions of planets, he assumed that these slight "discrepancies" were in fact defects of the machinery, requiring continual adjustment by God. [Later, others were able to show that these "defects" were simply due to the gravitational forces between the planets, and far from being a deficiency of Newton's laws, they were in fact a beautiful confirmation of those laws.] Newton could not accept the notion that once the creation happened, then God was thenceforth superfluous.

The opposing view held that Newton's "slightly imperfect machine" was inconsistent with a perfect creator. Surely God would not have made anything imperfect. God made the universe machine, set it running, and it would run perfectly forever after without adjustment or maintenance. This view was held by Gottfried Wilhelm von Leibniz. (1646-1716), and is the source of the phrase "Ours is the best of all possible worlds". Leibniz could not accept that God was a cosmic maintenance-man. God, Leibniz thought, had considered an infinity of possible worlds, and had chosen to create the one he knew was "best".

When an idea becomes so pervasive, influencing others, it's often a good idea to go back and find out exactly what its originator said, for it's sometimes more interesting than later interpretations of it by others.

…if we were able to understand sufficiently well the order of the universe, we should find that it surpasses all the desires of the wisest of us, and that it is impossible to render it better than it is, not only for all in general, but also for each one of us in particular… — Leibniz (from The Monadology, 1714)

Here Leibniz seems to be saying that if the world seems imperfect to us, it is that we do not "understand sufficiently". And also he's suggestiong that if we do understand it we will see that it is impossible to make a better one. It's hard to escape the interpretation that Leibniz considered the world to perfect in every way, in spite of appearances to the contary.

Of course this raises the question of why we, supposedly a creation of God, are not perfect enough to "understand sufficiently" the perfection of everything else. If everything was created perfectly, including us, we should understand everything perfectly. This whole debate looks a bit silly from our modern perspective, but at the time it was consdered to be of great importance.

The French philosopher/encyclopedist Denis Diderot (1713-1784) called Leibniz the "father of optimism", which contrasts with the fact that Leibniz suffered professional misfortunes and lack of recognition of his many accomplisments. His funeral was attended only by his former secretary. The "best of all possible worlds" phrase is often taken as characteristic of 18th century "enlightened" attitudes, and was ridiculed and satirized by Volatire (1694-1778) in his novel Candide. Voltaire's fictional character, Dr. Pangloss, is very likely a parody of the ideas of Leibniz. Voltaire asked: How can anyone say that this corrupt and imperfect world was the best God could do? The ideas of Leibniz, originally applied to physics, had become extended beyond physics to all of human experience.

Newton's and Leibniz's views clashed again over the question of space and time. Newton had declared space and time as real and absolute entities, providing a framework in which material things move and interact with each other. Both were absolute and unchanging, independent of any experiment we might perform. In an early edition of Newton's Optiks Newton speculated that space was a "sensorium" (sensory organ) of God, allowing God to keep track of what things were doing (as if God didn't know already!). Apparently realizing that this was a dumb idea, Newton tried to recall copies of the book so that speculation could be replaced with other text. But one of the early copies found its way to Leibniz, who was outraged by the idea that God needed sensory organs to perceive.

Leibniz's views of space and time were quite different, and even today we recognize how perceptive he was. He considered space and time as merely an ordering of events, space being an "order or coexistences" and time an "ordering if successions" They were not something "real" in the same sense as matter is real. Space and time are only our concepts, handy for labeling the relationship of events and their successions. This question of "What concepts are real and which are no more than convenient accounting schemes?" and the companion question "Is there an absolute framework for events, independent of what we do?" would recur many times, and such questions were at the heart of the theory of relativity.

We see the issue of religious assumptions raised again when scientists were trying to understand the passage of light through transparent materials, and reflected from curved mirror surfaces. Maupertuis thought that the path taken by light was always the path that would get it from point A to point B in the shortest time. This, he said, was because god does things "in the best way", and presumably the best way from A to B is to take the least time to get there. Such principles were applied to more than light, even to the paths of cannonballs, and the problem of something rolling down a curved track. In many cases these "least time" principles do seem to work, if you make just the right auxiliary assumptions. This convinced some that "God is economical." Critics said that was the same as saying "God is lazy." But as the "least time" or "minimal" principles were refined, cases were found where the quantity that is minimized is not always time, but something else: action, the product of position and momentum. Thus arose "principles of least action" that are still useful even today for analyzing mechanics problems. Euler, Lagrange and Hamilton were instrumental in developing these principles into useful methods, so useful that they are part of mechanics courses required of engineers.

As for "least time", that idea was killed when someone noticed that the path light takes when reflecting from a curved mirror is not the path of least time at all, but the path of relatively greatest time, compared to adjacent paths. So the "least time" principles had to be modified to be principles of "relative maximum or minimum", which is, in mathematics, called an "extremum" principle. One of my students (back in the 1960s) quipped, on hearing this account of the history: "Oh, so that means that God is not a minimalist after all. God is really an extremist." Of course the moral of all of this is that you can get into all sorts of phantom philosophical difficulties when you try to mix theology with physics, and try to base physics principles on principles of religion. Of course, scientists ought to have learned that lesson much earlier. But sometime during the 18th century the lesson seems to have finally sunk in, and since then we no longer see reputable scientists mixing religion with physics.

An extremely significant issue underlies this search for a theory of refraction of light. In the 17th and 18th century no one knew the speed of light, nor did anyone know for sure how the speed of light might depend on the material light moved through. Did light move faster in glass than in air, or vice versa? No one had a clue, though some thought that it ought to go more slowly through "denser" transparent media. But was this ordinary mass density (mass/volume) or some other measure of "optical density"? Descartes, in his Dioptric tried to explain reflection of light by an analogy with mechanical reflection of bodies (balls bouncing from a solid wall). That worked rather well, but when he tried a similar analogy for refraction (a ball tearing through a flimsy net) he got into serious trouble, as one always does sooner or later with analogies. But, to make the result come out right Descartes had to assume that the speed of the light increases when passing from air into water (or into glass), just the opposite thing one would expect from the mechanical analogy. For this, and other, reasons, many could not accept Descartes' model.

But Descartes' model did give the right result. That result was Snell's law (actually discovered by Johannes Kepler), which relates the incident and emergent angles of a reflacted light ray by the equation: n1sin(q1) = n2sin(q2). The subscripts 1 and 2 label the two media, the n's are the indices of refraction (properties of the materials) and the q's are the angles the ray makes to the surface normal.

Pierre Fermat was one of those who did not accept Descartes' refraction model. In fact, he went farther, and doubted even the result (Fermat's law, experimentally established). He accepted that the Fermat law might be approximately correct, but that it's true equation might be slightly different. Fermat started with a principle of "economy" that had long been accepted by natural philosophers, and he was disappointed to arrive at the sine law as a result (for he doubted the sine law). Also, Fermat used an assumption exactly contrary to that of Descartes. Descartes' convoluted explanation required that light move faster in glass than air. Fermat's method required light to move slower in glass than air. Both hypotheses, of course, were (at that time) impossible for the purpose of experimentally determining the velocity.

Fermat thought (as did many natural philosophers) that contrary hypotheses could never lead to identical results. Surely one hypothesis must be true and the other false, so how could a false hypotheses lead to the same result as a true one? Surely some mistake had been made in the mathematics of one or the other models. He spent quite a while checking the mathematics, and found no error. Fermat was astounded:

The reward for my effort has been the most extraordiary, the most unforseen and the happiest that ever was. For after having gone through all the equations, multiplications, antitheses and other operations of my method, and having finally concluded the problem which you will see in a separate paper, I have found that my principle gave exactly and precisely the same proportion for refractions as that which M. Descartes has laid down.

I have been so surprised by this unexpected result that I find it difficult to recover from my astonismnent. I have repeated my algebraic calcuations several times and the result has always been the same.

Why was Fermat so surprised? He, along with most other investigators assumed that wrong hypotheses must lead to wrong results. Perhaps the results were "only a little bit" wrong, but careful experimentation ought to reveal their deficiciency. But here two contradictory hypotheses were leading to exactly the same result, a result already confirmed by experiment. The experiments that had been done could not possibly determine which hypothesis about the speed of light was right! Putting it another way, a wrong hypothesis could lead to a correct result.

This is an important lesson that science learned, and one that is too often forgotten by students (and even by some scientists) today. The correctness of a result does not guarantee the correctness of all of the assumptions on which that result was founded.

References for additional reading.

A very good account of the conroversies surrounding Newton, Descartes and Laplace can be found in Hal Hellman's Great Feuds in Science, Wiley, 1998.

In checking the historical details of the history of minimal principles, I found A. I Sabra's Theories of Light From Descartes to Newton (Oldbourne, 1967) to be very helpful. Unfortunately this book is not readily available in most libraries. It's not easy going for students, for it's a book that tells you far more detail than you may have cared to know.

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    —Donald E. Simanek, 2005, 2009