Science and Math

When I was in college, it blew my mind to discover that there are exactly two effective systems for distilling legible[1] truth. Science starts with observations and then summarizes them into theories. Math starts with axioms and then generates theorems via a series of proofs. Most of the classes I took were science and math classes.

I took two philosophy classes. One of them was taught by an anthropologist who lost his sanity in Sierra Leone. His class was highly entertaining but not-at-all “legible”. The other one was taught by a traditional professor of philosophy. Her class covered the Greek classics: Socrates and Aristotle. I don’t feel like I learned anything from Socrates and Aristotle. The problem with philosophy is there’s no system for resolving disputes. Any two scientists who disagree can (in theory) conduct an experiment to determine who is correct. Any two mathematicians can (in practice) resolve any mathematical dispute from inside a locked room with nothing but paper and pencils. Philosophers don’t discuss things which can be falsified.

So what have philosophers been arguing about for thousands of years? Not talking precisely about observable reality. Consider the Arian controversy, a disagreement about whether the God the Father (God) was composed of the same substance as Jesus Christ. What does that even mean? How would you measure it? When philosophers lack dispute-resolution mechanism they can get away with saying anything, as long as its sufficiently meaningless.

I’m not saying philosophy isn’t useful. Philosophy is useful for lawyers, politicians and anyone else in the business of persuasion who needs to be good at rhetoric. But persuasion and truth-finding are different things.

So are science and math. How can science and math—two very different epistemologies—both lead us to truth? The trick is to notice that the word “truth” is pointing to two slightly different ideas. Scientific truth and mathematical truth aren’t the same thing. Science cares whether a fact is observable. Math cares whether a fact is tautological.

For example, consider the Banach–Tarski paradox. Scientifically, the Banach–Tarski paradox is nonsense. You can’t disassemble a ball and then reassemble the pieces into two balls identical to the original. Mathematically, you can (if you take the axiom of choice, that is).

Does this mean math is better than science? Of course not. Mathematics cannot tell you how to cure smallpox or whether the Moon orbits the Earth.

If math and science are so different then why do they converge so frequently? Why is a physics class mostly math? Because Rationalism defines truth as that which is observable, legible and tautologically consistent. Science is useful for the “observable” part. Math is useful for the “consistency” part. Both are necessarily legible, since without well-defined terms you end up back in the quagmire of meaningless philosophy.

I must emphasize that there is value beyond that which is “observable, legible and tautologically consistent”. Ethics is “legible” and “tautologically consistent” but not observable. Zen is “observable” and “tautologically consistent” but not legible. Daoist philosophy is “observable” and “legible” but not (practically-speaking) tautologically consistent. Visual art is “observable” but neither legible nor tautologically consistent. Poetry is “legible” but neither observable nor tautologically consistent. Vague, abstract philosophy is “tautologically consistent” but neither observable nor legible.

All of the above: ethics, Zen, Daoism, painting, poetry and philosophy are attempts to explore what is “true”. They all have their uses, but none of them come remotely close to the power of rationality = science + math. That’s because the most useful truth is that which is observable, legible and tautologically consistent.

  • “Observability” is necessary because it ties your statements to the real world. If a statement is not tied to observation then that statement is not about reality.

  • “Legibility” is necessary because the reliability of a statement is limited by how precisely it is stated. (The more precisely an idea is stated, the less it leaks.)

  • “Tautological consistency” is a prerequisite to universal applicability. (Or, at the very least, carefully-circumscribed domains of applicability.)

Grand Theories

For most of human history, rationality has not been a especially useful tool. If you lived as a subsistence farmer in ancient Mesopotamia then the best way to survive would usually be to do whatever everyone else is doing and to not think about it too hard. Rationality is useful for adapting to a rapidly-changing technological civilization where ① you have access to vast information supplies and ② you can’t rely on traditional strategies because the rules are constantly being rewritten via technological advance.

Vast information supplies are important because science + math need to compound for a while before they can supplant traditional ways of doing things. Consider chemistry. For most of history, alchemy was magic. Theories existed for why such-and-such reaction produced such-and-such result, but they were garbage. Before 1600, only fourteen elements had been isolated: copper, lead, gold, silver, iron, carbon, tin, sulfur, mercury, zinc, platinum, arsenic, antimony and bismuth. Twelve elements were discovered between 1750 and 1800. Cerium, osmium and iridium were discovered in just one year (1803). Ten elements were discovered in 1808. In 1869, Dmitri Mendeleev created the Periodic Table of the Elements.

In the year 1000, there just wasn’t enough information available to create a comprehensive theory of chemistry. If you attempted to do so you’d probably end up with a classical element (fire/​earth/​water/​air) theory like phlogiston theory. Which would be wrong.

The power of rationality comes from universally-applicable theories like chemistry, physics and evolution. These theories are based off of a mountain of data. Without that data, the theories wouldn’t exist.

  1. ↩︎

    By “legible”, I refer to that which is described using well-defined words.