Math: The Basics: Part 1

Part 1: Setting the Scene

In 1687, Isaac Newton published the PhilosophiƦ Naturalis Principia Mathematica or the Principles of Natural Philosophy. In this work, Newton shows that every body in the universe follows the same universal law of gravitation. This work gave a concrete formula that could be used to predict the motion of not just everything under the sun, but everything including the sun. There was just one problem. It wasn’t correct; at least not quite. Among other problems, scientists had noted over time that Mercury was not behaving exactly as it should under Newtonian laws.

Newtonian physics predicts that the orbit of a planet around the Sun should take on the shape of an ellipse, and that much was fine. The problem was how the ellipse was moving. As a planet orbits around the Sun, the orbit itself also rotates. See the image below:

Precession of Mercury’s Perihelion. Everything has been exaggerated to make it easier to see.

This is caused by the gravitational pull of other planets and celestial bodies in the area and is normal under Newtonian physics. The problem was that when scientists went and calculated how the ellipse should be moving, their calculations were off by about 7%. Now, getting a calculation like this does not necessarily mean that there is something wrong with the formula. After all, everything in the solar system besides the Sun is dark, and, therefore, hard to see in the empty vastness of space.

The thought was that maybe there was another planet between Mercury and the Sun that was affecting its orbit. This theory was not just a shot in the dark. The planet Neptune had in fact been found in a very similar way by analyzing the orbit of Uranus and noting that it was not behaving exactly as it should be. Further giving credence to the idea was just how close Mercury was to the Sun. If there were another planet between Mercury and the Sun, then it would be hard to make out since the Sun’s light drowns out any competitors. The idea so captivated scientists of the time that the new planet was given a name, Vulcan, and a few scientists even claimed to have observed it. None of the observations could be verified, however, and as time went on, it became more and more unlikely that there was a planet escaping detection. Something else must have been going on.

Enter Albert Einstein. In 1915, he released his general theory of relativity on the world which encapsulated his special relativity from a decade before and generalized it to work in contexts where gravity was significant. This new theory worked in every place that Newton’s old theories of gravitation worked, but it also predicted that the orbit of Mercury was just as it was.

What are we to make of this? In some sense, it feels like we should say that Newton’s equations were wrong. They make definite predictions about the world, and, yes, sometimes they are remarkably accurate and useful, but if you plug certain conditions into them (or really any conditions), they just give you “the wrong answer”. Now, make no mistake. I am not proposing here that we should be ultimately frightened or unnerved by this. This is exactly how science is supposed to work. We try to find equations and formulas that model how the real world works. Then, we try to find scenerios to test the limits of our formulas and understanding. Then, we see if we can come up with better equations and formulas that cover the old cases and the new ones.

Mathematics, on the other hand, works differently. It would be a much more serious issue if we thought that a^2 + b^2 = c^2 for all Euclidean right triangles and then someone found a particular Euclidean right triangle that it doesn’t work for, much more serious.

Part 2: The Structure of Mathematics