Pioneers of Probability: Gottfried Wilhelm Leibniz

You already know two things about Gottfried Wilhelm Leibniz, born in Germany in 1646 — even if you don't know you know them.

The first: he co-invented calculus in 1676, simultaneously and independently of Isaac Newton, igniting one of the most bitter priority disputes in the history of science.

The second: he believed we live in the best of all possible worlds. A claim so optimistic that French Enlightenment Philosopher & Satirist Voltaire spent an entire novel mocking it. But here is what that optimism actually meant to Leibniz: this is the best world — and reason, logic, and probability are the tools we use to navigate it more wisely. His faith in the world and his faith in rational method were the same faith.

But there is a third Leibniz that history has largely overlooked. A legal theorist. A philosopher of evidence. One of the first to argue systematically that probability wasn't just a tool for counting dice — it was a language for measuring belief itself.

When you roll a die, you can run the experiment a thousand times and watch the frequencies emerge. The mathematics we have seen in previous episodes from Cardano, Galileo, Pascal, Fermat, and Huygens built was designed for exactly that: repeatable events with countable outcomes.

But what about a murder trial? You cannot run a murder trial a thousand times and observe the frequency of guilt. The event happened once. The evidence is partial. The witnesses are unreliable. And yet a judge must still reach a verdict — must still make a decision about what probably happened, based on incomplete information.

Leibniz argued that this kind of reasoning — legal reasoning, historical reasoning, reasoning about any singular unrepeatable event — could, in principle, be given a mathematical structure. And he spent decades trying to build it.

Welcome to Pioneers of Probability.

Leibniz trained as a lawyer before he became a mathematician. He spent years working as a diplomat and legal advisor, navigating the messy human business of courts, contracts, testimony, and judgment. And in that world — the world of law rather than the world of games — he encountered a kind of uncertainty that the existing mathematics of probability couldn't handle.

His key insight was to reframe what probability actually measures.

For Cardano, whom we met in episode 2, and the gamblers, probability was a property of the world — the die has six faces, each equally likely, end of story. Leibniz argued that probability is also a property of the mind — a measure of how certain a rational person should be, given the evidence available to them.

In 1665, he described what we might call degrees of certainty. A spectrum — from impossible at one end, through degrees of likelihood in the middle, to certain at the other. Every claim, every piece of testimony, every inference sits somewhere on that line. The job of rational reasoning is to locate it as precisely as possible — and to update its position as new evidence arrives.

 This is shown as a spectrum on our coin. Zero to one. Impossible to certain. Scales of justice balanced above it — because this was, for Leibniz, fundamentally a legal idea before it was a mathematical one.

He was moving toward what we now call subjective probability — the idea that a probability is not just a frequency observed in repeated trials, but a rational agent's degree of belief in a proposition, given their current evidence. It is the philosophical foundation of Bayesian reasoning, which would later be formalized by Thomas Bayes and later Pierre-Simon Laplace, and which today underpins much of how sophisticated investors and researchers think about uncertainty.

The frequentist and the Bayesian approaches both use probabilities on a 0-to-1 scale — though they interpret them differently. That tension would later emerge as one of the deepest debates in the history of statistics, and it has never fully been resolved. But the spectrum itself — the idea that uncertainty can be measured on a continuous scale from impossible to certain — was something Leibniz was among the first to clearly articulate as a principle of rational reasoning.

Leibniz never completed the formal legal probability theory he envisioned. His ideas on the subject were scattered across letters, manuscripts, and unpublished fragments — a pattern that, as we have seen repeatedly in this series, was almost the rule rather than the exception. The mathematics of probability was built largely in private, published late, recognized slowly.

The notion that probability could describe degrees of rational belief — not just frequencies of physical events — became one of the most fertile concepts in the history of science. It is the thread that connects Leibniz to Bayes to Laplace who we will meet in later episodes - to the modern statisticians and decision theorists who gave us the tools we use today.

There is something worth reflecting on here. Every figure in this series so far — Fibonacci, Cardano, Galileo, Fermat, Pascal, Huygens — was working with probabilities derived from known, repeatable processes. Leibniz was among the first to clearly articulate what probability means when the process cannot be repeated. That is a different and deeper question. And it is the question that governs most of the important decisions we actually face.

For investors, the Leibniz frame is clarifying — and uncomfortable. (finger click to present)

Most investment decisions are not like dice rolls. They are more like legal cases. You are asking: given what I know right now, how confident should I be that this asset will appreciate, that this strategy will outperform, that this manager's track record reflects skill rather than luck? These are questions about degrees of certainty — questions that cannot be fully resolved by repeated trials alone.

The appropriate response to that kind of uncertainty is not a confident prediction. It is a calibrated degree of belief — held firmly enough to act on, loosely enough to revise when new evidence arrives.

Leibniz would have recognized the overconfident investor immediately. The person who is certain a stock will rise, certain a fund manager has an edge, certain the market is about to turn — that person is not reasoning probabilistically. They have collapsed the spectrum into a binary. They have stopped doing legal reasoning and started pronouncing verdicts without a trial.

The evidence-based investor does the opposite. They hold beliefs as probabilities, update them when the evidence changes, and build portfolios that don't depend on any single conviction being exactly right. That is degrees of certainty in practice.

We're seven steps into an 800-year story. There are eleven more to go.

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Sources

Leibniz, G. W. (1989). Philosophical papers and letters (L. Loemker, Ed. & Trans.). Kluwer. (Original works 1665–1716) Hacking, I. (1975). The emergence of probability. Cambridge University Press. Franklin, J. (2001). The science of conjecture: Evidence and probability before Pascal. Johns Hopkins University Press.

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