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The Limitations of Machine Learning in Finding the Laws of Nature

I have heard people claim that a machine learning program could have figured out, as Newton did, the force relation between two masses and the distance between them, that is Newton’s inverse square law of gravitation: F(m,M) = GmM/r^2. They imagine that Newton had taken a bunch of masses, put them at various distances from each other, and then measured the force between them with some sort of force-measurer, and then run a regression via some ML program.

That is not it at all, as Alfred J Prufrock said a different context.

Let me explain how it was done.

Newton didn’t figure out the inverse square law in isolated experiments. He did no experiments. He couldn’t measure the force between a variety of masses. And he couldn’t measure how particles accelerated under forces. Instead, he looked at Kepler’s famous and amazing laws, extracted over 40 years after many mistakes and with much difficulty and intuition, which described various regularities about the motion of the planets:

  1. planets move about the sun in ellipses;
  2. the line between a single planet and the sun (there is no such line, it’s in Kepler’s imagination) sweeps out equal areas in equal times; and
  3. comparing different planetary orbits, the square of the period is proportional to the cube of the distance.

And then Newton showed that these amazing laws/patterns followed from his astonishing plucked-from-his-head theories. That convinced everyone (who counted).

To do this, he didn’t simply”observe” the inverse square law. And even if he had, the inverse square law alone doesn’t tell you anything about how planets move, because you also need to know how that force will make the planets revolve. To do that you need not just the force laws butalsoNewton’s Law of Motion, or one of them: F= ma, which tells you how the planet m will respond to the pull of the sun M. Put the two of them together, the inverse square law and the law of motion (and also invent calculus on the side), and then you can derive the orbits of the planets and show that they satisfy Kepler’s laws, which convinced everyone.

So, for machine learning to do something comparable, it would have to be given Kepler’s laws and then from them, by regression,

  1. come up with a force law and force of motion, and then
  2. work out their consequences and
  3. show that they explain Kepler’s laws, all by regression.

Newton didn’t solve one puzzle; he didn’t experiment with masses at different distances (impossible to measure the small gravitational force between small masses on earth at that time); he didn’t shoot projectiles out of a catapult and show that they satisfied F = ma by experiment. He came up with two separate theories and a method of solving them and put it all together to explain what Kepler had first intuited and then demonstrated.

Read Keynes on Newton. He writes:

In the eighteenth century and since, Newton came to be thought of as the first and greatest of the modern age of scientists, a rationalist, one who taught us to think on the lines of cold and untinctured reason.

I do not see him in this light. I do not think that any one who has pored over the contents of that box which he packed up when he finally left Cambridge in 1696 and which, though partly dispersed, have come down to us, can see him like that. Newton was not the first of the age of reason. He was the last of the magicians, the last of the Babylonians and Sumerians, the last great mind which looked out on the visible and intellectual world with the same eyes as those who began to build our intellectual inheritance rather less than 10,000 years ago. Isaac Newton, a posthumous child bom with no father on Christmas Day, 1642, was the last wonderchild to whom the Magi could do sincere and appropriate homage.

For in vulgar modern terms Newton was profoundly neurotic of a not unfamiliar type, but – I should say from the records – a most extreme example. His deepest instincts were occult, esoteric, semantic-with profound shrinking from the world, a paralyzing fear of exposing his thoughts, his beliefs, his discoveries in all nakedness to the inspection and criticism of the world. ‘Of the most fearful, cautious and suspicious temper that I ever knew’, said Whiston, his successor in the Lucasian Chair. The too well-known conflicts and ignoble quarrels with Hooke, Flamsteed, Leibniz are only too clear an evidence of this. Like all his type he was wholly aloof from women. He parted with and published nothing except under the extreme pressure of friends. Until the second phase of his life, he was a wrapt, consecrated solitary, pursuing his studies by intense introspection with a mental endurance perhaps never equalled.

I believe that the clue to his mind is to be found in his unusual powers of continuous concentrated introspection. A case can be made out, as it also can with Descartes, for regarding him as an accomplished experimentalist. Nothing can be more charming than the tales of his mechanical contrivances when he was a boy. There are his telescopes and his optical experiments, These were essential accomplishments, part of his unequalled all-round technique, but not, I am sure, his peculiar gift, especially amongst his contemporaries. His peculiar gift was the power of holding continuously in his mind a purely mental problem until he had seen straight through it. I fancy his pre-eminence is due to his muscles of intuition being the strongest and most enduring with which a man has ever been gifted. Anyone who has ever attempted pure scientific or philosophical thought knows how one can hold a problem momentarily in one’s mind and apply all one’s powers of concentration to piercing through it, and how it will dissolve and escape and you find that what you are surveying is a blank. I believe that Newton could hold a problem in his mind for hours and days and weeks until it surrendered to him its secret. Then being a supreme mathematical technician he could dress it up, how you will, for purposes of exposition, but it was his intuition which was pre-eminently extraordinary – ‘so happy in his conjectures’, said De Morgan, ‘as to seem to know more than he could possibly have any means of proving’. The proofs, for what they are worth, were, as I have said, dressed up afterwards – they were not the instrument of discovery.

There is the story of how he informed Halley of one of his most fundamental discoveries of planetary motion. ‘Yes,’ replied Halley, ‘but how do you know that? Have you proved it?’ Newton was taken aback – ‘Why, I’ve known it for years’, he replied. ‘If you’ll give me a few days, I’ll certainly find you a proof of it’ – as in due course he did.

ML is great, and may even be useful in medicine or something other than click patterns, and that’s good, but finding Newton’s Laws or Maxwell’s Equationsor other causal laws of nature via regression is still a wild fantasy.


Published in writing