The History of "Using Equations and Stuff"

As a kindergartner falling madly in love with science and science fiction, the chalkboard had a symbolic mysticism to it. It was a place where brilliant people made discoveries and had eureka moments. A person could walk up to a chalkboard knowing only a little bit about something and after writing some symbols down, “A-HA!”, walk away knowing a great deal more about that something. I wished that someday I too could wield the power of this symbolic magic, and that I too would have eureka moments. It took more than a decade for me to get my kindergartner's wish, but it did come true.

Today I want to share the history of humans wielding such magic. Let’s explore an example:

What happens to the liquid level when a floating freshwater cube melts in a glass of heavily salted water?

Your instinct might be to run to the kitchen and give it a try, but don’t; we can get an answer much faster with only a pencil and paper. I didn't know the answer to this question when I first stumbled upon it, so did what any trained scientist would do: I sat down, drew some diagrams, did some algebra, and got an answer in a few minutes.

DISCLAIMER: It’s not at all important that you understand the following bits of math. Today I'm only interested in the historical and philosophical significance.

The water level will go up. To folks trained as scientists this will be trivial, but to most folks this will be indistinguishable from sorcery.

Briefly, Δh is the height of the water level. If it’s positive, the level went up; if it’s negative, the level went down. My goal was to figure out whether Δh is greater or less than zero based on the other things I could hypothetically determine if I were actually doing the experiment. In my final evaluation, all I needed to know was that salt water is denser than freshwater and that densities, masses, and areas can’t be negative.

Let’s take a step back to think about this. I didn't know the answer when I sat down… yet I was able to figure it out with nothing more than pencil and paper. Who was the first person to solve a problem in this way? Did they live in Ancient Greece? Rome? Maybe Classical India? Perhaps the Golden Age of Islam? Or Medieval Europe? Nope, nope, nope, nope, and… nope. This achievement certainly came after 1687. Why? Because I used two of Newton’s (1642–1727) three laws of motion, and 1687 is the year these laws were published. This fashion of solving physics problems came even later because Newton actually published his work in the tradition of Ancient Greek geometry; algebraic versions came later.

Newton’s forerunners were Galileo (1564–1642) and Kepler (1571-1630). Galileo’s model was a truly (perhaps the first) theoretical model because he showed that a simple rule like “constant gravitational acceleration” manifests in an observable phenomenon; namely, the parabolic motions of projectiles. Kepler's work on the other hand, while is often referred to as his “Laws of Planetary Motion” is more correctly called an empirical model; namely that planets follow elliptical trajectories around the sun.

An empirical model falls short of a theoretical model because it only proposes that data follow some mathematical shape without offering any reason why. Seventy years after Kepler published his model, Newton determined a theoretical basis in a new theoretical model called the "Law of Universal Gravitation" (which states that the gravitational force between two bodies is directly proportional to the masses of those bodies and inversely proportional to the square of the distance between them). Galileo’s work turned out to be a subset of this law. As for why Newton’s Law of Gravitation works, scientists are still trying to figure that out.

There was no such thing as a theoretical or empirical model (at least not relating to motion) before Galileo and Kepler. Before this there were variations of Aristotlean motion such as the theory of impetus. These ideas, in addition to being outright wrong, had no mathematical basis whatsoever. There is, however, something to be said about models relating to geometry: A theoretical example is Eratosthenes’s 240 BCE determination of the circumference of the earth, and an empirical example is Bhāskara I's 629 CE method of approximating sines; neither of these models characterize motion.

That mathematical models could be used to describe reality (and motion in particular) was the key to unleashing the floodgates of the Scientific Revolution, which began in 16th-century Europe and continues to this day.

The discovery of being able to use symbolic math to characterize reality is, in my opinion, the greatest intellectual achievement ever made by humans. Virtually every technological advancement of the last 300 years owes thanks to the practice of applying algebra to the physical world.