# Imagination and Mathematics: A Tale about Knowledge – 2/4

How can we know what gravity is? We know it, because it is part of us.

This is the second article in our series about imagination and mathematics. You can find part 1 here.

What do I mean by writing that gravity is a part of us? We humans have developed into what we are now over millions of years. Gravity has been with us the whole time. We have raised ourselves upright and become bipedal despite gravity – no, I should not write “despite” here. Without gravity, we would not exist, but that is a different story altogether.

Gravity is part of what defines us as humans.

Now, even though a grown-up person has some sort of understanding for what gravity is, we still have to assume that every child grows through his or her own experiences, and, to be frank, that process does not really seem to be too boring either. My daughter, at any rate, laughs heartily when daddy puts the lego-giraffe back onto the table in front of her. It will be down on the floor again after about two seconds. She knows that. There have been many experiments. The same thing has happened every single time. That is just the way it is.

That is just the way of science, too, really. One examines and investigates. If the same thing happens often enough, one begins to become more confident about having learned something. So this process is not foreign to us. I may even go a step further and state that, actually, we seem to be born scientists. Children come out of their mothers and begin almost at once to try to find out about things, don’t they?

Fortunately, some of us manage to stay scientists all our lives. No, this is not meant to be sarcastic at all! For where would we be without those who keep their curiosity throughout their lives, despite life not always rewarding them for their curiosity? What would we do without the Teslas and daVincis of this world?

There are innumerable ways for curiosity to manifest itself. The world is filled to the brim with enthusiastic biologists, painters, geologists, architects and many others. What I want to focus on here, however, is astronomy, since it is the science which, at least traditionally, concerns the universe itself.

Astronomy may very well be the oldest of all the disciplines of science. In the beginning, however, the skies and what could be seen there were examined under a premise which is rather different from what we understand to be science today.

Do not misunderstand me now. The movements of the Sun, the fixed stars and the wandering stars[1] like Jupiter, Saturn, Mars and even the rarely appearing Mercury, were known already to the Egyptians and other cultures 6000 years ago. Image 1 shows the tomb of Ramses VI. There, archaeologists found what is most likely a star map. Yet, and this may not be all that surprising, the movements of the objects in the sky were correlated with stories which guided people in their daily lives. It was all about the blessing of a deity.

Perhaps, it was also a little bit about power, getting it or keeping it. For example, it was said that the pharaohs had to help the Sun god Ra in his underworld fights against the basilisk every night. Without the pharaoh, the Sun would not rise the next morning. What a useful thing to take credit for.

Then there were the Greeks. The Hellenic culture, which spread over big parts of the Mediterranean for a while, took religion less and less seriously with time. People slowly gained knowledge of what we summarize as geometry today. The measurements of distance was formalized, trigonometry was invented (or should we say, “discovered”?), so that ideas could be correlated to one another as long as it was possible to express them in terms of geometry. This lead to discussions of ideas, which themselves could no longer be heralded by a priest with “secret knowledge”.

A good example for this is Klaudios Ptolemaios of Alexandria, Ptolemy for short (Image 2). His scientific endeavors are amongst the most interesting, in that no one has ever been remembered as selectively as he is. Parts of his work survived about 1400 years in the form of dogma!

Ptolemy conducted his first experiments in the year 127 BC and his last in 151 BC. During that time, there was no developed theory of how bodies moved under the influence of forces, which we call mechanics today. So, philosophers very often had to come to their own conclusions with respect to the simplest things – or the most difficult things, if one wants to look at it that way. The amazingly accurate answer to life, the universe, and everything, was, therefore, a matter of opinion, and discussed with rather more gusto.

The movement of the bodies in the sky turned out to be especially challenging. What does one, in fact, do without knowledge of mechanics here? One thinks beauty, simple as that. What is beautiful has to be true. Ptolemy thought this to be true, and many others did, too. It simply had to be possible to explain the movement of the heavenly bodies with the help of perfect geometrical shapes, such as circles and spheres – whatever it was that made a circle or a sphere “perfect” to a Hellenic philosopher.

Ptolemy actually managed to do this. He found a way to explain all movement in the sky, as he knew them, with the help of circles and spheres.

But let us start, as it were, at the beginning. The first observation Ptolemy made was that Earth had to be a sphere. Why? He noticed the same things we know and see today, i.e. the horizon and how objects close to it behave as well as the change in the star map as one moves in North-South direction[2]. It is worth noting, that the spherical Earth is something people did not believe in for 1400 years afterwards.

The next problem then became how such a sphere would be fastened in space. What was it be fastened to? Ptolemy’s answer was rather ingenious: No matter where one went, things always fell down. If Earth was a sphere that meant that everything fell towards a common center – inwards, if you will. But, if that was the case, then why would Earth have to be fastened to anything at all? If a bit of Earth were ripped apart, the only thing it would do is fall back. He concluded, on the basis of experiments and observation, that Earth hovered freely in space with sky on all sides around it. Let the gravity (yes, that pun is intended) of this conclusion in the second century BC sink in before we continue …

Ptolemy observed the stars as well. What he saw lead him to believe that they were fixed to the inside of another sphere and that the Earth was the center of this other sphere; but, since he had also found out that the stars shone with the same intensity at different places on Earth, he concluded that Earth’s dimensions had to be minuscule when compared to those of the heavenly sphere. Now, realizing that Ptolemy came as far as this in his conclusions, the next step seems at first to be rather surprising. He concluded, that it had to be the heavenly sphere which rotated, not the Earth! Think about this yourselves for a moment. Which is more likely to be the one rotating, the Earth or the immense heavenly sphere? What does physics, or indeed common sense, say about these distant stars as they rotate once per day around 360 degrees?

The stars are very, very fast. That is what it says. Much too fast, in fact. As far as we understand, Ptolemy knew about this rather well, too. It is possible to start asking why he stood by a non-rotating Earth, even convincing others of his idea. It is said that he could not find a satisfactory answer to the question as to why the air would rotate with the planet. He thought it could not[3].

On to the Sun, the moon, and the planets. Sun and moon were rather easy to explain. They moved in circles around the Earth. The planets, however, were a different story. They seemed, at times, to move backwards. It is quite a feature of Ptolemy’s, that he managed to explain this motion with the help of circles. Take a look at image 3. Ptolemy postulated that the planets moved in epicycles, i.e. described circular movements on top of a circular movement.

Perfect, beautiful epicycles. The way it was supposed to be. What is beautiful had to be true. But does it help to understand? It took around 1400 years before circles had to give way as forms with which to describe planetary motion; but, when this first happened, other beautiful ideas had to be dropped, too.

(With thanks to Sebastian Holmgård. To be continued.)

#### Now read this:  Imagination and Mathematics: A Tale of Knowledge – 3/4

[1] This is an expression some great Hellenic philosophers used.

[2] Yes, pictures of the Earth taken by astronauts and space probes do help a little, too. It is worth noting that these pictures are not necessary to draw a conclusion here.

[3] Such a statement is much too easy to disprove, really, for Ptolemy to believe it per se. Read, what is written at the link and think “knight on horseback casting a body into the air” instead of balloon. In this way you will get an example that fits well into Ptolemy’s time.

Categories:

1. I like your opening point about growing up with gravity. Of course, all land animals have to cope with gravity in one way or another (not sure about fish). What is unique about humans is that when we rose up off of four feet several million years ago, we became the only creature walking on two feet (without a tail or other support). Doing that contributed a lot to our putting our hands to new uses, getting a better view of the environment for safety, and developing our brains.

Thanks for the post.

Brock

Liked by 1 person

1. Alexander Biebricher says:

Absolutely. And it is rather inspiring to think about what those brains of ours are able to do.

Like