# 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.

• 27 Nov 2022 11:40 UTC
13 points
0 ∶ 0

Re: Aristotle. Large part of what Aristotle wrote is science and math. If you felt you didn’t learn anything from Aristotle, that’s because only non-science and non-math part of Aristotle is usually taught, because science and math are usually taught in ahistorical manner.

Barbara, Celarent, Darii, Ferio is good math. It is just that first-order logic and Venn diagrams are better, so we don’t teach Barbara etc. One thing lost by this ahistorical teaching is how much better first-order logic is, and how difficult advance it was when it was first done by Frege.

• 27 Nov 2022 17:48 UTC
11 points
0 ∶ 0

Philosophers don’t discuss things which can be falsified.

Sometimes in life, one simply faces questions whose answers can’t be falsified, such as “What should we do about things which can’t be falsified?” If you’re proposing to avoid discussing them, well aren’t you discussing one of them now? And why should we trust you, without discussing it ourselves?

I think you had the bad luck of taking a couple of philosophy classes that taught things that were outdated or “insane”. (Socrates and Aristotle may have been very confused, but consider, how did we, i.e., humanity, get to our current relatively less confused state, without doing more philosophy?) Personally I took a philosophy of physics class in college that I really liked, which led me to learn about other areas of philosophy.

I wrote more about my guess of what philosophy is and what philosophers do at Some Thoughts on Metaphilosophy, which you may find interesting since we both come from a math/​science background (computer science in my case).

• Science starts with observations and then summarizes them into theories. Math starts with axioms and then generates theorems via a series of proofs.

Nice. And close to how I defined these for myself a long time ago:

• Math is everything valid (in the strict sense if being derived by a consistency-preserving process).

• Physics is everything useful (in the strict sense of tracing back observable reality)

Hm, translating to English this doesn’t come out too well. But anyway, I agree that it is possible to carve truth in different ways. One with a focus on consistency and one on observability.

I don’t feel like I learned anything from Socrates and Aristotle.

My guess would be because you already knew all of the facts. We don’t learn philosophy to learn true facts. You have to read philosophy backwards: How did people arrive at truth?

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 think you are misreading what philosophy does (or at least did fir most of the time): Struggling to build words that describe difficult to describe things. Creating legibility out of nothing.

Yes, you need legibility, but where does it come from? Once you have words like “observation” or “particle” (aka atom) or force you can do stuff with it. But reality is not legibile enough to provide these for free. Reality has joints but they are obscursed by a lot of messy flesh that is in the way. You have to throw words around that you gave an intuition that may or may not be right and see what sticks. And you don’t do this alone because it is too easy to spot fake patterns. Other people will try to catch yiu. But you can’t exchange intuitions directly. You have only words with all their imprecision. You can try definitions—but they must be circular because nothing is grounded—yet. That’s philosophy: Creating legibility out of nothing. At least that’s some idealized view of philosophy. Reals philosophy is not like that. But ideal math is also not like real math.

Anyway, I think you can observe this process of creating legibility about something in real time with consciousness. I think progress is being made with recent posts by Scott Alexander and others as well as real studies—putting a meditator into a CT and measuring what goes on.

• The highly educated are more willing to defer to experts on mathematical or scientific questions than we are on philosophical questions. It seems to me that the reverse is true for less educated people, though I am not confident about this.

Part of philosophy is making a legible model to explain the way things “seem to us,” to capture our intuitions. If our intuitions are not coherent or consistent, then insofar as philosophy attempts to reflect those intuitions and take them seriously, it will either fail to be coherent or become counterintuitive. Philosophy that does focus on topics more distant from human intuition, and that is grounded in physics or axioms, seems to achieve greater durability and consensus.

• 27 Nov 2022 16:39 UTC
−1 points
0 ∶ 0

Not talking precisely about observable reality.

Talking about what observable reality means. For instance , there is a set of observations and formulate called quantum mechanics. These might mean that we have free will, that we do n ot, that there is a parallel universe where the allies lost WWII, and so on. What it actually means can’t settled by further observations, or more maths, so something else is needed...unless you give up on the question of meaning, and settle for instrumentalism.