And they agreed with these earlier assumptions. The idea that space should be like that comes from the principle of sufficient reason , which seems rather obvious at first glance:. The principle is at least as old as Archimedes , and it allows us to explain things in the world around us.
Here's an example: How do we know that a lever with equal weights at equal distances from the fulcrum must balance? Well, why not? There is no reason for the lever to go down on either side: so it must go down on neither; therefore it balances. The greatest advocate of sufficient reason was the 17th century mathematician Gottfried Wilhelm Leibniz , who believed that God used the principle, along with the laws of logic, in making the universe.
And since God made it rationally, we humans can figure it out. A striking example of how the universe is transparent to human reason is the discovery of Newton's first Law of Motion , found fifty years before Newton independently by Descartes and Pierre Gassendi. Here's how they figured it out.
The law says that a body in motion with no forces acting on it continues in a straight line at a constant speed. The body continues in a straight line because it has no reason to turn away in some other direction, all the directions are the same. It goes at a constant speed because it has no reason to speed up or slow down: all the points in space are the same, so there's no reason the body should prefer being at one point to another.
Similar arguments make a body at rest with no forces acting on it stay right where it is. A parallelogram of forces. The two solid blue arrows represent two forces acting on a particle that sits at their tails. The lengths of the arrows represent the velocities the two forces could produce in the particle by acting on it for a unit of time, and their directions indicate the directions of the forces. The red arrow indicates the net effect of both forces acting together.
Sufficient reason is a powerful principle, and it seems to suggest that the space we live in is just like the space of Euclid's geometry.
It's no surprise, then, that 17th and 18th century thinking was Euclidean through and through. Newton's physics, for example, implicitly relied on Euclid's 5th postulate. It needed those parallelograms of forces you might have met at school. Proving the properties of parallelograms requires Euclid's theory of parallels and thus the 5th postulate. This is why mathematicians of the 18th century cared so much about proving the 5th postulate.
It was hugely important; not just geometry, but all of science rested on it. An interesting example comes from the mathematician Joseph-Louis Lagrange : so impressed was he with the power of sufficient reason that he tried to use it to prove the 5th postulate.
His argument was flawed, but it's a significant fact that such an important mathematician was willing to get up in public and link space being Euclidean with the principle of sufficient reason. Philosophy was equally permeated by Euclid's ideas. A super-influential philosopher, Immanuel Kant , said that space is something that exists in our minds, and we each have the same unique "space" in our minds.
And it turns out that, for Kant, this space had to be Euclidean. To argue that we can come to know complex truths about non-material things, Kant used Euclid's proof that the sum of the angles of a triangle is two right angles. The proof uses geometrical constructions. Where do you make these constructions? Not on paper — geometry isn't about physical triangles or lines.
You make them, Kant says, in the space within your mind. It turns out that Euclid's proof requires the 5th postulate. In this regard, he is understood to be the Father of Geometry since he paved the way for so many future thinkers to expand upon his organized ideas. Other thinkers have even used his geometric method as a foil, in which they have expanded thought in a completely different direction. Today, we understand Ancient Greek culture as classical, in which thought, discussion, mathematics, sciences, and the arts developed and flourished as never before within Greece.
Euclid was a part of that culture. Euclid existed around BC, and he was a prominent figure in Greek culture at the time as a thinker and scholar. He was part of a new tradition of questioning thought, understanding the changing world, and developing ideas so that we could better understand the patterns in the world around us. He, among other Ancient Greek scholars, has left a legacy of thought that many scholars and academics today continue to follow.
He was in no way documented biographically, at least in terms of detail and depth. He was mentioned by several other Greek thinkers, such as Pappas of Alexandria.
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