First Principles

Since I began studying first principles seriously, I've read three books that are directly relevant: Descartes's Discourse on the Method, Plato's writing on the Theory of Forms, and Russell's Introduction to Mathematical Philosophy. I'm also completely hooked on videos about sets, and on how Bertrand Russell's work ties into all of this.

This piece is my attempt to wander through first principles and set theory and see where they meet. I want to figure out what we can actually learn about real problems and about the world, because I think there's a real break between how you thought before something clicks and how you think after.

You become fascinated and completely lost at the same time, because everything stops making sense the way it used to. Descartes had to relearn how to think, and going through his work does something similar to you. You start questioning things the way school trained you not to. In class, whenever something got hard, the teacher would say "that's too complex to explain right now, we'll cover it later." Fair enough at the time. Looking back, it was a mistake. A kid who hears that enough times stops being able to explain something as simple as 2 + 2 = 4, or why it's true. It isn't simple. Life isn't supposed to be.

I'd guess someone who can master something genuinely hard has more capacity to pick up something easy than the other way around. They already did the hard part. Sometimes what's missing is just the basic principle underneath, and that's exactly what first principles and issue trees are for.

My point, in short: first-principles thinking isn't something our education system makes easy, because relying on analogy is so much simpler.

First principles as a way to solve very large problems

First principles is probably the clearest way to take a problem apart and find the smaller, self-contained pieces inside it, pieces you can solve independently and package up as value.

It also works on existing solutions, not just new problems. Like the five whys, it drills down to the root cause and figures out, atom by atom, how to actually fix something.

The alternative is chasing the most immediate pain, which is usually just a symptom, and patching that alone. Cutting to the root instead means fewer downstream problems and more value from however you end up designing the fix.

Long term, anything that keeps generating new problems has a negative net present value. Spending more time on the real pain up front cuts the cascade it would otherwise cause. Here's a rough hypothesis: most people's job, at bottom, is solving problems. You get hired and paid to fix things and create more value than you cost.

The typical problem-solver is also a little lazy. Instead of digging for the root cause, they take the quick, superficial fix.

That said, 80/20 thinking still has its place. We probably over-apply the Pareto framing, assuming any given scenario cleanly fits it. Say I think 20% of my time gets me 80% of the outcome. How sure am I, really? Certainty multiplies with expected return and effort, so the math shifts under you. If I'm 80% sure about the return and 80% sure about the effort, my actual expected assured return is 0.8 x 0.8 = 64% for that same 20% of effort. Still solid. But now compare it to spending a bit more, say 25% instead of 20%, at full certainty: 80% of the result for 25%; better ratios. Whether that's worth it comes down to completeness. You can only know your first choice was right once you understand the alternative, and most of the time we don't even look at the second option because it doesn't seem worth the extra effort. The 80/20 path usually wins once you've actually weighed it.

In practice this is a 2x2: essential, principle-based fix, or a quick patch that skips the core. Almost nobody stops to make that choice deliberately, near the edges of the Pareto curve. And because almost nobody looks, that's a market with very little competition, potentially a large one.

That might sound like a stretch, but it rests on something plain: most people aren't that attentive to detail. Problems big enough to need more than one person to solve tend to sit exactly where nobody's looking.

Palantir is a good example. It makes money on connectivity, taking decentralized systems and turning them into fully integrated ones instead of loosely, lazily linked ones. Companies often have multiple data sources that never really talk to each other, connected at best through weak links that don't deliver the full value. Whoever built that first pass probably followed the same 80/20 logic: connect two APIs, get 80% of the return for 20% of the effort. But the remaining 20% is still enormous at that scale. Even an imperfect fix for a slice of a big enough problem can produce a huge number.

Once you trust your own problem-solving and have real frameworks to work from, you start building in terms of jobs to be done, treating the product as something that scales. You build reusable blocks for the next client. Say 60% of the work carries over: you get roughly 15% of the return for 20% of the effort on the next project, then 17.5% for 5% more on the one after that. These ratios stack. Eventually the thing you built becomes generic enough to sell across industries and clients. That's basically the whole idea of a product.

A detour into set theory

Most of what gets solved by chasing an 80/20 return is data: what belongs to what, what belongs to whom, what's related to what, who's related to whom. That's set theory's actual subject: membership, cardinality, self-containment. The clearest walkthrough I've found is a video that builds naive set theory rule by rule and then shows exactly where it breaks. What follows is that argument, in my own words.

A set is a group defined by membership. "NBA all-time leading scorer" is a set: it defines who belongs to it, currently one person, LeBron James. If someone else matched his total, the set would have two members. A set can have one member. It can also have zero, the empty set, which is still a valid set because "nothing fits this description" is a valid description.

The rules

Naive set theory, the version most people use without naming it, runs on a small number of rules.

Rule one: any description defines a set. "Even numbers," "people who have read Discourse on the Method," "things that are yellow." Say the words, you have a set.

Rule two: a set's identity is fixed entirely by its members. Two sets with the same members are the same set, regardless of how each was described or built.

Everything else follows from those two.

Order doesn't matter.

Repetition doesn't matter either. Neither does the description used to build the set: "the set of prime numbers under 10" and "" are the same object, because rule two only checks membership, not phrasing.

A few more rules build on the first two. The union of any two sets is a set. Any subset of a set is a set. A set can have exactly one member, which sounds odd until you remember it's defined by membership, not size.

Sets of sets

A set of sets is still a set, by rule one. This is rule ten. It raises an immediate question: can a set contain itself?

Nothing in rules one and two forbids it. "The set of all sets describable in English" would have to include itself, since that description is in English. For a while, this was treated as rule eleven: sets can contain themselves.

Russell's paradox

Bertrand Russell found the problem with rule eleven in 1901, while examining the logical foundations Gottlob Frege had spent years building. The clearest version of the argument isn't about sets. It's about predicates.

Take any predicate and ask whether it applies to itself.

"Is a cat" is not itself a cat: not true of itself. "Is a predicate" is itself a predicate: true of itself. Every predicate sorts into one of the two categories.

Now define one more predicate: "is not true of itself." Which category does it belong to?

Either answer produces the other. Translated into set language: let R be the set of all sets that do not contain themselves. Is R a member of R? If yes, R fails its own definition. If no, R satisfies it and should be included. The contradiction holds regardless of which answer you pick.

The fix

The problem wasn't rule eleven specifically. It was rule one, unrestricted composition: not every description gets to define a set. Modern axiomatic set theory (Zermelo-Fraenkel is the standard version) restricts construction instead. Sets are built step by step from sets you already have, and an axiom of foundation rules out self-membership directly. The system gives up some freedom in exchange for consistency.

The rule everyone assumed was safest, the one nobody thought to question, was the one that broke the entire system. That's the same pattern as the 2 + 2 = 4 example from earlier: the assumptions treated as too basic to interrogate are usually the ones worth interrogating.

Biology has the same layered structure without the paradox. Atoms form molecules, molecules form organelles, organelles form cells, cells form tissue, tissue forms organs, organs form systems, systems form organisms, organisms form populations, populations form communities, communities form ecosystems. Each level belongs to the one above it, with no self-membership anywhere in the chain, which may be exactly why it never produces its own version of Russell's paradox.

Full derivation, chalkboard version: watch the video.