Warning: long post!

Abstract: Neil deGrasse Tyson has argued that Isaac Newton’s religious views stymied his science, preventing him from discovering what Laplace showed a century later – that the planetary orbits are stable against perturbation. This conclusion is highly dubious. Newton did develop perturbation theory, and applied it to the moon’s orbit. His lack of progress is explainable in terms of his inferior geometrical, rather than algebraic, approach. Laplace built on the important work of Clairaut, Euler, d’Alembert and Lagrange, which was not available to Newton. Laplace’s discovery was not definitive – computer simulations have showed that the Solar system is chaotic. And finally, Newton does not give up on science and invoke God at the first sight of ignorance, saying rather “I frame no hypothesis”. His “Reformation” of the Solar System is plausibly not supposed to be miraculous. I conclude that scientists (myself included) are terrible at history.

Introduction

I’ve got a lot of time for Neil deGrasse Tyson, who is doing a wonderful job of bring the excitement and importance of science to the general public and to the next generation in particular. Despite its seeming disconnect from everyday life, astronomy is an important way to get people into science. Plenty of people who are now solving all manner of important problems in our society got interested in science via astronomy.

(Incidentally, I was a dinosaur nerd first. Put me in a science museum and I’m going straight for the fossils.)

I have a problem, however, with this clip (It’s long; I’ll quote the relevant bits below). In it, Tyson discusses a famous story about a conversation between physicist Pierre-Simon Laplace and Napoléon Bonaparte in 1802. Here is a passage from A Budget of Paradoxes by Augustus De Morgan (1872, p. 249-50), which is the earliest account that I can find.

The following anecdote is well known in Paris, but has never been printed entire. Laplace once went in form to present some edition of his ‘Systeme du Monde’ to the First Consul, or Emperor. Napoleon, whom some wags had told that this book contained no mention of the name of God, and who was fond of putting embarrassing questions, received it with — ‘M. Laplace, they tell me you have written this large book on the system of the universe, and have never even mentioned its Creator.’ Laplace, who, though the most supple of politicians, was as stiff as a martyr on every point of his philosophy or religion … drew himself up, and answered bluntly, ‘Je n’avais pas besoin de cette hypothese-la.’ (I had no need of that hypothesis.) Napoleon, greatly amused, told this reply to Lagrange, who exclaimed, ‘Ah! c’est one belle hypothes ; ca explique beaucoup de choses.’ (Ah, it is a fine hypothesis; it explains many things.)

Did it Happen?

No eyewitness reports Laplace’s zinger. We don’t get this story from Napoleon, Laplace or Lagrange. The only person in the room whose account we have is the British astronomer William Herschel, and he does not record the story above, but rather notes that “Mons. De la Place wished to shew that a chain of natural causes would account for the construction and preservation of the wonderful system. This the first Consul rather opposed.”

There is even evidence that Laplace opposed this anecdote, demanding its deletion from a forthcoming publication shortly before his death in 1827. (Source. p. 111.)

So the story is at least suspect. Laplace’s agnosticism needed a pithy parable, it seems, and someone obliged. The story should not be told without at least some sort of caveat.

Tyson’s Version

Napoleon was into the physics, the engineering and the material science of war. And so he immediately summoned up the five-volume production of Laplace and read it through, cover to cover. He called in Laplace and – I have the exact quote here – asked him what role God played in the construction and regulation of the heavens. That’s what Newton would ask. Laplace replies ‘Sir, I had no need for that hypothesis.’

A few details to note. Tyson says: “I have the exact quote here”. No, he doesn’t because no one does. The story is at best hearsay.

Even our dubious version of the story from De Morgan has Napoleon being informed about Laplace’s book “by some wags” and saying to Laplace “they tell me” that the book doesn’t mention God. So Tyson’s “cover to cover” detail is doubtful.

Note also Tyson’s version of Napoleon’s question. The earliest versions of the story have Napoleon asking about the absence of God from the book, not God’s role in the whole scheme of things. This exaggerates the scope of Laplace’s supposed answer. Stephen Hawking appreciates this point: “I don’t think that Laplace was claiming that God does not exist. It’s just that he doesn’t intervene, to break the laws of Science.”

Tyson’s Analysis

Let’s assume for the moment that the story is at least reflective of some conversation between Laplace and Napoleon. I’m particularly interested in the moral that Tyson draws from this episode. Tyson claims that Newton (1642-1727) should have discovered what Laplace (1749-1827) did – that that the combined pull of the planets on each other do not destabilise their orbits – but was hamstrung by his theism.

What concerns me is, even if you’re as brilliant as Newton, you reach a point where you start basking in the majesty of God, and then your discovery stops. It just stops. You’re no good any more for advancing that frontier. You’re waiting for someone to come behind you who doesn’t have God on the brain and who says “that’s a really cool problem, I want to solve it.” And they come in and solve it. But look at the time delay – this was a hundred-year time delay. And the math that’s in perturbation theory is like crumbs for Newton. He could have come up with that. The guy invented calculus just on a dare, practically. When someone asked him why planets orbit in ellipses and not some other shape, and he couldn’t answer that, he goes home for two months and comes back: out comes integral and differential calculus, because he needed that to answer that question. This is the kind of mind we’re dealing with in Newton. He could have gone there but he didn’t. His religiosity stopped him.

Could Newton have anticipated Laplace?

You should be immediately suspicious of Tyson’s account for this reason: Newton and Laplace weren’t the only two physicists on the face of the planet in the 17th and 18th century. Even if Newton was held back, what’s everyone else’s excuse? Did everyone catch Newton’s God-bothering disease, and only Laplace found the cure?

Hardly. Here’s a few relevant historical details.

A. Newton did develop a theory of perturbations.

Tyson’s “he could have gone there but didn’t because of religion” is immediately derailed by the fact that Newton went there. Here is historian William L. Harper, quoting Newton:

… Newton developed this method in an effort to deal with the extreme complexity of solar system motions. … The passage continues with the following characterization of the extraordinary complexity of these resulting motions. “By reason of the deviation of the Sun from the center of gravity, the centripetal force does not always tend to that immobile center, and hence the planets neither move exactly in ellipses nor revolve twice in the same orbit. There are as many orbits of a planet as it has revolutions, as in the motion of the Moon, and the orbit of any one planet depends on the combined motion of all the planets, not to mention the action of all these on each other. But to consider simultaneously all these causes of motion and to define these motions by exact laws admitting of easy calculation exceeds, if I am not mistaken, the force of any human mind.” (Wilson 1989b, 253) It appears that shortly after articulating this daunting complexity problem, Newton was hard at work developing resources for responding to it with successive approximations. The development and applications of perturbation theory, from Newton through Laplace at the turn of the nineteenth century and on through much of the work of Simon Newcomb at the turn of the twentieth, led to successive, increasingly accurate corrections of Keplerian planetary orbital motions. [emphasis added]

Indeed, Newton developed two perturbation methods, one of which “corresponds to the variation of orbital parameters method first developed in 1753 by Euler and afterwards by Lagrange and Laplace.”

B. Newton didn’t have the right tools

Why did Newton not achieve what Laplace did a century later? We have seen that it is not from want of trying. He was primarily interested in calculating the moon’s orbit, which is unavoidably a three-body problem: one cannot meaningfully simplify the problem by considering only the Moon and the Earth. Newton applied his method to the Moon, but not successfully. The first edition of the Principia notes: “These computations, however, excessively complicated and clogged with approximations as they are, and insufficiently accurate, we have not seen fit to set out.” Later editions remove this comment entirely. Newton was obviously dissatisfied with his calculation.

Why was Newton’s calculation unsuccessful? Was he too busy “basking in the majesty”? Historians have a more mundane explanation.

The first successful derivation of the Moon’s apsidal motion (or rather, of most of it) was announced some sixty years later, by Alexis-Claude Clairaut, in May 1749. Euler obtained a derivation in good agreement with Clairaut’s by mid-1751. … Jean le Rond d’Alembert published a more perspicuous derivation, with the degree of approximation made explicit, in 1754. Success came for Newton’s successors only with a new approach, different from any he had envisaged: algorithmic and global. The Continental mathematicians began with the differential equation, the bequest of Leibniz.

Further,

From Newton to d’Alembert, the essential theoretical advance in lunar theory consisted in the decision to start from a set of differential equations, while relinquishing the demand for direct geometrical insight into the particularities of the lunar motions.

Chris Smeenk and Eric Schliesser (highly recommended!) conclude:

Newton also faced a more general obstacle: within his geometric approach it was not possible to enumerate all of the perturbations at a given level of approximation, as one could later enumerate all of the terms at a given order in an analytic expansion. It was only with a more sophisticated mathematics that astronomers could fully realize the advantages of approaching the complexities of the moon’s motion via a series of approximations.

This is one of the most surprising things to the modern physicist about Newton’s Principia: having invented calculus, Newton doesn’t really use it. He thought that geometry was more insightful, more fundamental. This prohibited Newton from developing the analytic tools needed to incorporate the perturbations of the other bodies in the Solar System into his model, and – crucially – to evaluate the accuracy of his approximations. Moreover, Newton’s version of calculus is actually rather clunky compared to Leibniz’s, which was being used on the Continent.

Obviously, if Newton’s approach is floundering on the three-body problem (Sun, Earth, Moon) and a few centuries of observational data, the problem of the stability of all the planets in the Solar System into the indefinite future cannot be attacked with much confidence.

C. Laplace had some help

The idea that Newton could have come to the conclusions that Laplace did is extremely doubtful. We have already seen that his methods are not quite up to the task. Further, note the mathematicians who worked on the problem of perturbations to planetary orbits before Laplace: Clairaut, Euler, d’Alembert, and Lagrange. These are the greatest mathematicians of their age; Leonard Euler is arguably the greatest mathematician of all time: “Read Euler, read Euler, he is the master of us all.” That quote, incidentally, is from Laplace. Euler was a devout Christian and a Lutheran Saint. Apparently, having “God on the brain” didn’t prevent him – as it didn’t prevent Newton – from working on this scientific problem.

So, I think we can safely say that if Leonard Euler attempts to solve a mathematical problem and fails, the problem is a difficult one. And he took the problem very seriously: when Clairaut successfully applied perturbation theory to the Moon’s orbit, Euler described “this discovery as the most important and profound which has ever been made in mathematics.”

But these mathematicians didn’t merely make failed attempts; they laid the foundations for Laplace’s work. Joseph-Louis Lagrange, in particular, is crucial:

Though traditionally given credit for establishing the stability of the solar system, it is only after Lagrange’s work that Laplace made his first major contribution to the theory of the stability of the solar system.

Jacques Laskar gives Lagrange equal credit, referring to the “Laplace-Lagrange stability of the Solar System“:

Laplace and Lagrange, whose work converged on this point, calculated secular variations, in other words long-term variations in the planets’ semi-major axes under the effects of perturbations by the other planets. Their calculations showed that, up to first order in the masses of the planets, these variations vanish.

Newton, of course, was a mathematical genius. But we can hardly blame him for not being smarter than Clairaut, Euler, d’Alembert, Lagrange and Laplace combined.

D. The Solar System and the General Scholium

Did Newton ignore the problem of the stability of the Solar System so that he could call upon God as an explanation? Well, we have already seen that he did not ignore the problem at all.

Further, Newton developed his perturbation methods in 1685-6, according to Michael Nauenberg. The definitive statement of his theological conclusions drawn from physics comes in the General Scholium, an essay appended to the end of the second and third editions of the Principia in 1713 and 1726 respectively. With 40 years of reflection on the problem of the perturbations of the orbits of the planets and their theological implications, what does Newton have to say about God’s intervention in the universe?

Nothing. Nada. Zilch.

Newton states that “This most beautiful System of the Sun, Planets, and Comets, could only proceed from the counsel and dominion of an intelligent and powerful being.” This is about the creation of the whole physical order in the first place, not about God intervening in the laws of nature to perform a miracle: “In him are all things contained and moved; yet neither affects the other: God suffers nothing from the motion of bodies; bodies find no resistance from the omnipresence of God.”

In the Scholium, Newton notes a major gap in scientific knowledge:

Hitherto we have explain’d the phænomena of the heavens and of our sea, by the power of Gravity, but have not yet assign’d the cause of this power.

Here is a golden opportunity for Newton, facing a scientific gap in our knowledge, to invoke God as the very power of gravity itself. His famous response:

But hitherto I have not been able to discover the cause of those properties of gravity from phænomena, and I frame no hypotheses [hypotheses non fingo]. For whatever is not deduc’d from the phænomena, is to be called an hypothesis; and hypotheses, whether metaphysical or physical, whether of occult qualities or mechanical, have no place in experimental philosophy.

For a fellow who is supposed to be looking for any gap in scientific knowledge as an excuse to insert the direct intervention of God, Newton’s doing it all wrong.

Interestingly, Newton seemingly appeals to the intervention of God in a single sentence in Opticks, published in 1704.

For it became [God] who created them to set them in order. And if he did so, it’s unphilosophical to seek for any other Origin of the World, or to pretend that it might arise out of a Chaos by the mere Laws of Nature; though being once form’d, it may continue by those Laws for many Ages. For while Comets move in very excentrick Orbs in all manner of Positions, blind Fate could never make all the Planets move one and the same way in Orbs concentrick, some inconsiderable Irregularities excepted, which may have risen from the mutual Actions of Comets and Planets upon one another, and which will be apt to increase, till this System wants a Reformation.

This is a puzzling passage, for a number of reasons. Firstly, it seems out of place – the context is about God’s sustaining of the “wonderful Uniformity in the Planetary System”. Mentioning “inconsiderable Irregularities” somewhat undermines Newton’s point.

Secondly, nowhere in Newton’s corpus can we find the calculations to sustain this claim, even though (as we noted above) he had pioneered perturbation theory. How did Newton convince himself that the mutual attractions of the planets increase the irregularities of the Solar System? We don’t know.

Thirdly, why is this not mentioned in the General Scholium? If this is Newton’s great scientific proof of God’s intervention in the world, why does it not appear in his most famous essay on God, at the conclusion of his scientific magnum opus? In light of its absence from the Principia, it is difficult to know how much weight Newton placed on this particular argument.

Finally, it is difficult to know exactly what Newton was thinking. It is, after all, a single sentence with no further comment. Leibniz, for example, responded to this passage in November 1715:

According to [Newton’s] Doctrine, God Almighty wants to wind up his Watch from Time to Time: Otherwise it would cease to move. He had not, it seems, sufficient Foresight to make it a perpetual Motion. Nay, the Machine of God’s making, is so imperfect, according to these Gentlemen; that he is obliged to clean it now and then by an extraordinary Concourse. … I hold, that when God works Miracles, he does not do it in order to supply the Wants of Nature, but those of Grace.

In response to Leibniz, Samuel Clarke argues in 1716 that Newton is not appealing to a miracle or a suspension of the laws of nature:

… the word correction, or amendment, is to be understood, not with regard to God, but to us only. … But this amendment is only relative, with regard to our conception. In reality, and with regard to God; the present frame, and the consequent disorder, and the following renovation, are all equally parts of the design framed in God’s original perfect idea.

In other words, Newton’s “reformation” would be entirely natural, part of God’s orderly sustaining of the universe, rather than a violation of its laws. (I recommend this article for more details). Newton is not committing the God of the gaps fallacy, because he does not see a gap.

E. The Solar System is not stable

Finally, Laplace and Lagrange’s demonstration of the stability of the Solar System was shown by later scientists to be inconclusive. Henri Poincaré established that it was impossible to produce exact solutions to the equations of motion in the n-body problem, where n is bigger than 2: approximate solutions by means of infinite series are the only viable solutions. Moreover, these series generally diverge, making them useless for prediction over infinite time. Laplace and Lagrange’s calculation is informative but not decisive.

Since Poincaré, computer simulations have shown that the orbits of the Solar System are chaotic over timescales of a few billion years. So the “Laplace solves it” part of Tyson’s story has a problem: Laplace didn’t solve it.

Conclusion: Scientists suck at History

What historians do is read primary sources, in the original languages as much as possible, consider all the characters involved, trying to understand their context, their influences, their personal lives and their professional motivation. Nuance, nuance, nuance.

What amateurs do – myself included – is read secondary sources, skim for interesting sections, pick favourites, judge anachronistically, and hope that an amusing anecdote or two can summarise an entire cultural milieu. (Huxley vs Wilberforce is a great example.) We want simple stories of progress, pithy quotes and heroes who look like us.

The historian Steven Shapin, reviewing Steven Weinberg’s recent book “To Explain the World”, gives the view from his discipline of scientists who attempt to write history.

There’s a story told about a distinguished cardiac surgeon who, about to retire, decided he’d like to take up the history of medicine. He sought out a historian friend and asked her if she had any tips for him. The historian said she’d be happy to help but first asked the surgeon a reciprocal favor: “As it happens, I’m about to retire too, and I’m thinking of taking up heart surgery. Do you have any tips for me?”

To illustrate the point, we can retell our story to make Newton the hero. Inspired by God’s providence, he argued correctly that the Solar System is ultimately unstable and was beautifully vindicated by modern computer simulations. Laplace (cue the villainous music), desperately seeking to avoid God, promotes the idea that the Solar System is perfectly stable, allowing his agnosticism to hold back the progress of science and delay the discovery of long-term chaos among the planets by Poincaré and modern physicists.

We can even make Lagrange the hero, since he is at least as important as Laplace in the scientific study of the Solar System. Lagrange is there with Napoleon – so the story goes – to counter Laplace’s myopic inference that, since God doesn’t poke the planets moment by moment, He is not needed. Shall we immortalise Lagrange’s answer to Napoleon, rather than Laplace’s?

This is whig history – the heroes of the past are the people who, for whatever reason, believe something like what I believe now. Why does Tyson venerate Laplace, the agnostic? Because Tyson is an agnostic. That’s all the story proves.

Neil deGrasse Tyson on Newton (Part 2) is here.