On “The Curse of Knowledge”

January 2015
Ivan Obolensky

Many have heard the story of Newton and the apple.

Isaac Newton (1642-1726) was sitting beneath a tree as an apple fell to the ground. In some versions of the tale, it hit him on the head. In others, it fell beside him. Regardless, the falling apple inspired him to come up with the idea of gravity, but few know how the story started, how the apple captivated him, or how it changed the world.

The apple story is usually thought to be apocryphal but is in fact true. It was originated by an Anglican clergyman by the name of William Stukeley (1687-1765). A scientist in his own right, Stukeley pioneered archeology by investigating prehistoric monuments at Stonehenge and at Avebury, in the southwest of England. Stukeley was a friend of Isaac Newton as well as his first biographer. He wrote his memoirs in 1752 in which he noted:

“After dinner, the weather being warm, we went into the garden and drank thea, under the shade of some apple trees…he told me, he was just in the same situation, as when formerly, the notion of gravitation came into his mind. It was occasion’d by the fall of an apple, as he sat in contemplative mood. Why should that apple always descend perpendicularly to the ground, thought he to himself…”¹

Why, indeed.

It is difficult to imagine, but before Newton, there was no concept of gravity (the idea that masses attract each other) as we know it today*. There wasn’t even the concept of force to describe the motion. (Newton formulated the idea of forces as well as gravitation in his Principia** which was published in 1687.) The concept existed only as “falling”. Things dropped when let go. Everyone knew that. They did not gravitate. They descended.

So how did a falling apple lead Sir Isaac to the idea of gravity?

One of the difficulties for those that live in later times is to come to grips with the thinking and ideas of those of the past. One might ask: how could those people not know what is so obvious today? This is an example of a cognitive bias called the Curse of Knowledge.²

An example of this bias would be knowing that the US stock market tanked in October of 1987 and then analyzing the charts of that time period. One sees what appears to be obvious signs in the patterns that a crash was imminent and therefore one feels certain one would have avoided it and any large market drops in the future. One might even feel that those who were surprised by the crash and lost money were simply stupid. This is the Curse of Knowledge in action. It is more common than we think.

When one “knows”, it is hard to take the point of view of “not knowing”. Even in modern physics texts the derivation of the formula for gravity as put forward by Newton rarely involves the process of thought that assumes no notion of gravity prior to the start.

So how did he do it?

Newton liked “Thought experiments”.

A thought experiment is an intellectual device in which one looks at the possible consequences of an action. They are theoretical excursions in the imagination. The most famous thought experiment was Einstein’s chasing a light beam which led to the theory of special relativity. Thought experiments often combine real world data taken to extremes.

An empirical result which Newton knew before he began his thought experiment, and one that I confess I stumbled upon in my youth, had to do with how objects travelled when tossed out a window several stories above the ground. My brother and I discovered after a great deal of experimentation (and much to the dismay of those below), that it took an object the same amount of time to hit the ground regardless of whether it was simply let go, or thrown horizontally. This meant that only a small horizontal force could send a water balloon to the other side of the street where detection was lessened. (Who looks across the street if something happens to hit one from above?) In the parlance of mechanics, one could say we discovered that the horizontal component of the motion was independent of the vertical one.

Artillery people know all about this. A bullet fired from a gun falls vertically at the same rate as a bullet simply dropped from one’s fingers. The bullet travels so fast that it takes only a fraction of a second to hit the target. The bullet drops only a small distance during that time, but it falls nonetheless.

Newton wondered why the apple fell perpendicular to the ground. He knew all objects did so whether they dropped in Oxford, London, or in China. He knew the Earth was spherical, and objects fell towards its center.

Here is a thought experiment:

Imagine a cannon 5 meters high that fires an unlimited number of projectiles horizontally, or parallel to the ground. We ignore all air friction, buildings, and mountains. We can do that because it is a thought experiment, and we are using our imagination.

Each successive firing sends a projectile out of the muzzle of our cannon at a greater and greater speed. Let’s say, the first projectile travels 100 meters. (Note: the flight time is one second; the same amount of time it takes to drop an apple from the cannon’s height.) The next projectile comes out faster and travels farther before it hits the ground. We now imagine each shell travelling greater and greater distances during its one second flight time. Eventually the projectile travels so far it passes beyond the horizon. What happens now?

Firstly, the projectile continues to travel with the same speed that it was shot from the cannon initially. Secondly, it continues to “fall” toward the center of the earth, but the ground is now getting out of the way because it is part of a curved spherical surface (the Earth). As the speed with which the projectile leaves our imaginary cannon increases, the projectiles will “fall” farther around the Earth’s surface.

Eventually the forward motion and the “falling” of the projectile toward the Earth’s center will just balance with the curvature of the Earth’s surface and the projectile will circle the planet entirely. The cannon ball would in effect be in orbit even if only 5 meters above the ground.

The thought experiment concludes with the idea that a projectile travelling at sufficient speed could “fall” around the Earth if it travelled fast enough.

Note: gravity was not mentioned in the above thought experiment. Only a few facts were used: The earth is a sphere and objects fall perpendicular to the ground toward the center.

As to what happens next we can also imagine: Newton sees the moon and, being the genius that he was, wonders if it, too, is “falling around the Earth” just like the bullet of his thought experiment.

What Newton wanted to know was how were the “falling” of the apple, the “falling” of the imaginary projectile, and the “falling” of the moon related?

He worked out some specifics. If it took the apple one second to fall the 5 meters from halfway up the tree, and one second for the superfast bullet to fall below the horizon; how far would the moon “fall” beneath the horizon during that same second? Newton made several observations and found the moon “fell” 1.37 mm, or about one sixteenth of an inch during that time.

(Newton, being Newton, also invented a curved mirror telescope configuration called, as you might imagine, the Newtonian. It is still in use today.)

This 1.37 mm was far less than the 5,000 mm the apple or the projectile fell, but the moon was a great deal farther away. This meant that the “falling” tendency was proportional to distance but inversely. Objects fell less strongly, farther away from the Earth. This idea was known to Galileo, but that was as far as anyone had gotten. Science was still for the most part qualitative rather than quantitative. Newton wanted to know how this tendency changed exactly, using mathematics.

To do this he used proportions, or the common mathematical tool called a ratio.

A ratio is the comparison of two numbers expressed as a fraction.

What he wanted to know was quantitatively how the “falling” of objects toward the earth’s surface varied with the distance from it.

He used the ratio of the distance the moon “fell” in one second (1.37 mm) to the distance the imaginary projectile “fell” in one second (5000 mm) to express the strength of the moon’s and the projectile’s falling.

This ratio came to 1/3600 (1.37mm/5000mm)

For the distance ratio, he used the Earth’s radius (6378 km) as the distance measure for the projectile (it was travelling 5 meters above the surface) and the radius of the moon’s orbit (384,000 km) for the distance of the moon to the Earth.

This ratio came to 1/60 (6378km/384000km)

Comparing the two ratios, he found that one was the square of the other, a powerful result.

(1/60)² = 1/3600

From this relationship, Newton was able to generalize and assert that the gravitational attraction between any two bodies is inversely proportional to the square of the distance between them. This is known today as simply the “inverse square law” and is found to be applicable in many areas of physics from the intensity of light and electromagnetic radiation, to acoustics and magnetism.³

But what is so significant about that and the apple?

The significance was that Newton was able to relate the apple and the moon together using the thought experiment projectile, not just conceptually, but mathematically, using real numbers. No one had done that before. What was even more spectacular was that it worked not only on Earth but universally with a precision that could not be denied. It was a giant step.

How big a step? It would be comparable today to someone announcing that interstellar travel was not only possible, but could be done by anyone for the price of a small car. It was that big.

Because we suffer from the Curse of knowledge, we can barely imagine the impact of just this one discovery.

To give it some context, Galileo had been sentenced by the Inquisition some fifty years earlier for writing that the Earth revolved around the sun. It was a common belief that the heavens, including the moon and the planets, were the sole province of the Almighty with man relegated below to his own planet. The idea that physical laws invented by man applied in Heaven as well as on Earth was still a huge mental leap.

What was thought to drive the universe before Newton formulated his laws of motion were, for the most part, only vague ideas and concepts. Now for the first time, there were mathematical laws never before imagined that illuminated many areas of knowledge with a precision hardly to be believed. Everything changed. It was as if someone kicked open a door to a vast treasure house.

It threw philosophy on its head and transformed it into what we know today as science.

Newton’s mechanics and mathematics would dominate scientific thought for the next 250 years.

It is no wonder that Alexander Pope wrote in his epitaph:

“Nature and Nature’s laws lay hid in night:

God said, Let Newton be! and all was light.”⁴

We forget how far we have come and how big a change can be made by a single man and a handful of ideas.

Knowledge seems so readily available today on the Internet in the form of web pages that contain information, mathematical tools of extraordinary power, and courses from leading universities such as MIT. The availability of knowledge has to a degree deflated its worth. Young people question whether they should learn at all because all they need to do is Google it. Why take the time to learn when I can search for it in a micro-second?

Why, indeed.

It is the curse of knowledge.

*We won’t be taking up gravity as envisaged by Einstein’s General Relativity which is a very different story.

**Philosophiæ Naturalis Principia Mathematica, Latin for “Mathematical Principles of Natural Philosophy”, often referred to as simply the Principia.

Gefter, A. (2010). Newton’s apple: The real story. Culturelab. Retrieved January 7, 2015 from http://www.newscientist.com/blogs/culturelab/2010/01/newtons-apple-the-real-story.html
Heath C., Heath D. (2006) The Curse of Knowledge, Harvard Business Review. Retrieved January 7, 2015 from https://hbr.org/2006/12/the-curse-of-knowledge
Collier, P. (2014). A Most Incomprehensible Thing: Notes Towards a Very Gentle Introduction to the Mathematics of Relativity.K., Incomprehensible Books.
Pope, A. (1728) Epitaph on Sir Isaac Newton. Retrieved January 7, 2015 from http://www.bartleby.com/297/154.html