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🧱Philosophy8 min read

First Principles

Every argument you have ever made rests on something you never proved. Aristotle noticed this 2,300 years ago and it has been quietly breaking philosophy, mathematics and your own reasoning ever since.

In plain English

Ask someone why they believe something. They give a reason. Ask why they believe that reason. They give another. Keep going.

Three things can happen. The reasons go on forever, in which case nothing is ever actually established. Or they eventually loop back on themselves, in which case the argument is circular. Or you arrive at something the person cannot justify by anything else, and simply has to accept.

There is no fourth option. This is sometimes called Agrippa's trilemma, and it is roughly 1,800 years old.

That third option, the stopping point, is what a first principle is. It is a starting claim that everything else gets built on top of, and which is not itself built on anything. Aristotle called it an archē, meaning origin or starting point. In his Posterior Analytics he argued that genuine knowledge has to be demonstrated from premises that are true, primary, and better known than the conclusion they support. Otherwise you have opinion dressed as proof.

The appeal is obvious. If you can find the bedrock and build carefully upward, everything above should be as solid as the foundation.

The problem is equally obvious once you look at it directly. How do you know the foundation is solid? Not by deducing it from something else, because then it would not be a first principle. You know it by intuition, or self-evidence, or because denying it seems absurd.

Which means the most important part of the structure is the part with no proof under it.

Five things to file under "wait, what?"

  • The most famous "self-evident" truth in history turned out to be optional. Euclid built all of geometry on five postulates. Four are short and obvious. The fifth, the parallel postulate, is wordy and awkward, and for 2,000 years mathematicians tried to prove it from the other four. They all failed. In the 1820s Lobachevsky and Bolyai independently tried assuming it was false and found the result was not a contradiction but a whole coherent geometry. Einstein later used that non-Euclidean geometry to describe actual physical spacetime. The postulate everyone thought was too obvious to question was simply a choice.

  • Descartes' famous line is a first principle chosen for a specific technical reason. Cogito ergo sum, "I think therefore I am", is often read as a bit of clever wordplay. It was doing precise work. Descartes wanted a claim that survived maximum doubt, including the possibility that a deceiving demon was falsifying all his perceptions. The one thing a deceiver cannot fake is that there is something being deceived. It was picked not because it is important but because it is unfalsifiable from the inside.

  • Mathematics tried to formalise this completely, and it failed on purpose. In 1900 David Hilbert proposed proving that all of mathematics could be derived from a consistent, complete set of axioms. In 1931 Kurt Gödel proved that any formal system powerful enough to describe arithmetic contains true statements it cannot prove, and cannot demonstrate its own consistency. The foundationalist dream was not abandoned because it was hard. It was shown to be impossible.

  • Reasoning from first principles is usually the wrong tool. It is expensive, slow, and discards accumulated knowledge you cannot personally verify. Most of the time, reasoning by analogy and precedent gets you a good answer far cheaper. G. K. Chesterton's fence makes the point: before removing a fence whose purpose you cannot see, find out why it was put there. The person who reasons everything from scratch reliably rediscovers old mistakes.

  • Aristotle thought you reached first principles by induction, not deduction. This is the part usually skipped. He was clear that the starting points of a demonstration cannot themselves be demonstrated, and argued they are reached through nous, a kind of direct rational insight built up from repeated experience of particulars. So the foundation of his rigorous deductive system is, by his own account, arrived at non-deductively.

The full story

Aristotle and the regress

The problem Aristotle was solving was not abstract. Plato's dialogues are full of arguments that bottom out in someone simply conceding a point, and Aristotle wanted to know what makes a chain of reasoning genuinely terminate rather than merely stop when everyone gets tired.

His answer in the Posterior Analytics is that demonstrative knowledge requires premises that are true, primary, immediate, and prior to and explanatory of the conclusion. "Immediate" is doing heavy lifting: it means not derived from anything more basic. Without such premises, he argues, either the chain is infinite and nothing is known, or it circles and nothing is established.

He then faces the obvious objection. If first principles cannot be demonstrated, how are they known at all? His answer is that repeated perception of particular cases builds up, through memory and experience, into a grasp of the universal. The mind eventually sees the principle. This faculty he calls nous.

Modern readers often find this unsatisfying. It is worth noting that nobody has done substantially better since.

Euclid and the power of few assumptions

Around 300 BC, Euclid demonstrated what the method could actually do. The Elements begins with definitions, five postulates and five common notions, and from that small base derives 465 propositions across thirteen books, including a proof that there are infinitely many prime numbers.

It is arguably the most influential textbook ever written, and its influence is as much about form as content. Spinoza wrote his Ethics in the geometric style, with definitions, axioms and propositions. Newton's Principia is structured the same way. The American Declaration of Independence opens by holding certain truths to be self-evident, borrowing the vocabulary directly.

The fifth postulate is where it comes apart. Euclid himself seems to have been uneasy with it, avoiding its use for the first 28 propositions. The eventual discovery that consistent geometries exist in which it is false did not just add new mathematics. It severed the link between "self-evident" and "true", which was the assumption holding the whole enterprise together.

Descartes and starting from nothing

Descartes' Meditations (1641) attempts the most radical version: throw out everything that can possibly be doubted and see what survives.

The senses can mislead, so sensory beliefs go. Dreams are indistinguishable from waking while you are in them, so the existence of the external world goes. Mathematical reasoning could conceivably be corrupted by a powerful deceiver, so even arithmetic goes.

What remains is that the doubting is happening. There is a thinker.

From this single point Descartes tries to rebuild, arguing outward to God's existence and then to the reliability of clear and distinct perception. Most philosophers think the reconstruction fails, and that the argument for God is circular. But the demolition survives as one of the most effective pieces of philosophical writing ever produced, and the method, strip to bedrock then rebuild, escaped philosophy entirely.

The modern engineering version

The phrase has had a second life in technology and business, where it usually means something narrower: ignore what things conventionally cost or how they are conventionally done, break the problem into physical components, and reason up from there.

The standard illustration is battery packs. If everyone says packs cost around $600 per kilowatt-hour and always will, the first-principles move is to ask what a battery is physically made of, price those raw materials on the commodity market, and discover the material cost is a small fraction of the market price. The gap is not physics. It is manufacturing, supply chains and convention, all of which are changeable.

This is genuinely useful, and it is also a much weaker claim than Aristotle's. Nobody is asserting that commodity prices are indubitable foundations of knowledge. They are just a level of description more fundamental than the one everybody was arguing at.

Both senses share the underlying move: stop arguing at the level where the argument is stuck, and go down a level.

What survives

The strong programme is dead. Gödel killed the mathematical version, and non-Euclidean geometry killed the assumption that self-evidence tracks truth. There is no set of unquestionable foundations from which the rest of knowledge can be safely derived.

What survives is the discipline. Asking what you are actually assuming, noticing when a chain of reasoning has quietly become circular, distinguishing what you have verified from what you have inherited, and being willing to go one level lower when an argument stalls.

That is less than Aristotle wanted. It is still more than most reasoning does.

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