This Day in Philosophy — 2 September: two deaths in search of a foundation

What holds a structure up?

The question can be asked in mathematics and in politics alike. Two people who died on 2 September asked it separately.

The Irish mathematician, astronomer and physicist William Rowan Hamilton died on 2 September 1865. The German jurist and political theorist Franz Leopold Neumann died on 2 September 1954.

Hamilton: algebra is the science of pure time#

We know Hamilton today mostly from physics: the most elegant formulation of classical mechanics carries his name — Hamiltonian mechanics — as does the Hamiltonian operator at the centre of quantum mechanics.

But Hamilton took himself to be more than a mathematician. He had read Coleridge, worked on Kant and engaged seriously with philosophy.

And from the mid-1830s he defended an extraordinary thesis about algebra: algebra is the science of pure time.

The argument comes from Kant. For Kant, space and time are not data arriving from the world but the pure forms of intuition. Space is the form of outer sense; time, of inner sense.

Hamilton builds a symmetry from there: if geometry is the science of space — as it is for Kant — then algebra must be the science of time.

He has a reason. The concepts "past", "present" and "future" arise in consciousness before spatial ones. The number series is also an idea of order: 1, then 2, then 3. Counting is succession; and succession is time.

From 1835 Hamilton no longer saw algebra as an art, a language or the science of quantity, but as the science of order in certain sequences.

Why did he labour at this?#

Hamilton's motivation was the most embarrassing mathematical fact of his day: negative and complex numbers worked, but nobody could say what they were.

What does "minus one apple" mean? Which magnitude has −1 for its square?

Grounding algebra in time gave those numbers a foundation: as forward and backward orientations in time. Complex numbers Hamilton defined as pairs of real numbers — the approach that is standard today.

Then he wrestled with triples and stalled: no three-dimensional multiplication could be defined.

On 16 October 1843, walking along the Royal Canal in Dublin, he found the answer: not triples but quadruples. A four-dimensional algebra without commutativity — quaternions.

He carved the formula into the stone of Broom Bridge with a penknife. The carving wore away; the plaque that replaced it is still there.

That quaternions do not commute — that ab differs from ba — is a threshold in the history of mathematics. Until then the order of multiplication had been assumed irrelevant. Hamilton showed that a useful algebra could break that rule.

Quaternions are used today in computer graphics and in the attitude calculations of spacecraft.

The philosophical balance sheet#

Hamilton's Kantian grounding did not take among mathematicians. Algebra was built axiomatically in the following century; "pure time" was not required.

But what the story teaches still stands: what led Hamilton to quaternions was a philosophical conviction about what algebra is. A discovery can grow out of a metaphysics that turns out to be wrong.

Neumann: Behemoth, not Leviathan#

Franz Neumann was born in 1900. He practised as a lawyer for the trade unions in Weimar Germany and worked on labour law. When the Nazis took power in 1933 he had to leave the country, as a Jewish and socialist jurist. He went to London, then New York, and joined the Frankfurt School's Institute for Social Research.

In 1942 he wrote the most original book in his field: Behemoth: The Structure and Practice of National Socialism.

The thesis in the title#

In Hobbes, Leviathan is the state that secures order: the absolute sovereign who ends the war of all against all. Behemoth is the title of Hobbes's book on civil war — the monster of disorder.

Neumann's choice carries a thesis: Nazi Germany was not a state.

That ran against the received view of the period. The Nazis were taken to have built an absolutely centralised, single-handedly governed totalitarian state. Their own propaganda claimed as much.

Neumann examined the inside with a lawyer's precision and saw the opposite: there was no coherent legal order, no predictable machinery of administration, no single centre of decision.

Instead four great blocs of power competed with one another: the party, the army, the bureaucracy and big industry. What regulated their relations was not law but continuous bargaining and rivalry. Jurisdictions overlapped, agencies multiplied, decisions became arbitrary.

Neumann's conclusion: this is a condition of statelessness. The regime had put direct relations of force in the place of law.

Why it still matters#

Neumann's analysis lodged a lasting objection against theories of totalitarianism.

The common picture imagines authoritarian regimes as excessively ordered structures: a centre that sees and controls everything. Neumann showed that this might be an illusion — indeed an illusion the regime itself produces.

The real danger may not be excess order but the disappearance of law. When rules become unpredictable, everyone becomes dependent on the whim of the powerful.

That touches a basic point in the philosophy of law: before it means having good laws, the rule of law means laws operating predictably. Our recent piece on Scott Shapiro on law and code starts from a neighbouring place: law is a technology of planning.

Neumann worked for the US Office of Strategic Services during the war and contributed to the preparation of the Nuremberg trials. Afterwards he became professor of political science at Columbia University. He died in a road accident in Switzerland in 1954.

What they share#

Hamilton looked for the ground of number and found it in time — in the wrong place, but a productive wrong.

Neumann looked for the ground of order and did not find it in Nazi Germany — and that failure to find became his book's thesis.

Both did the same work: taking apart a structure that appears simply given, and asking what it stands on.

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Each day in this column we take up an event from the history of philosophy that falls on today's date.