Bertrand Russell · Mathematics
Russell’s great program in <em>Principia Mathematica</em>: the claim that mathematics is not a separate science built on its own axioms but a branch of logic, so that every truth of arithmetic can be derived from purely logical principles and definitions.
For most of history, logic and mathematics looked like different subjects: logic was a set of rules for valid argument handed down from Aristotle, while mathematics was the science of number and space. Bertrand Russell set out to prove that this division is an illusion. His thesis, which came to be called logicism, is that mathematics is nothing but logic carried far enough - that the concepts of mathematics can be defined using only logical ideas, and that all the theorems of mathematics can be deduced from a handful of purely logical axioms. If this could be shown, then there would be no special mathematical intuition, no separate mathematical subject-matter: arithmetic would be revealed as a vast, elaborate consequence of the laws of thought themselves. Russell put the point with a famous image about how far the two subjects had converged.
Between 1910 and 1913 Russell and his former teacher Alfred North Whitehead published Principia Mathematica in three enormous volumes: the most sustained attempt ever made to carry out the logicist program. Starting from a small set of logical axioms and definitions, they built up the whole apparatus of classes, relations, and numbers by rigorous deduction, in a dense symbolic notation that left nothing to intuition. The scale of the undertaking became legendary. So carefully was everything constructed from the ground up that the proposition 1+1=2 is not reached until deep into the first volume, at proposition *54.43, where the authors note only that from it, once addition has been defined, the familiar equation will follow. The point was not perversity but honesty: to show that even the most obvious arithmetical fact rests on nothing but logic, every hidden assumption had to be dragged into the light.
The heart of the program is the definition of number. Following Frege, Russell defined the number of a class as the class of all classes that can be put in one-to-one correspondence with it - so the number 3 is the class of all trios, the number 2 the class of all couples, and so on. Two collections have the same number precisely when their members can be paired off exactly, one for one, a purely logical relation that needs no counting. From this, 0 can be defined as the class whose only member is the empty class, and the successor of a number defined logically, so that the entire number series is generated. Russell insisted that this is all built from logical constants alone. Indeed his 1903 Principles of Mathematics had already opened with a startling definition of the whole subject as a matter of pure logical form.
This is the opening of the lesson. The rest — the dialogue, the primary source, and the recall — is in the app.
You learned that logicism is the thesis that mathematics reduces to logic: its concepts can be defined using only logical notions, and its theorems derived from logical axioms. Explain the two halves of that claim, and why Principia Mathematica needed such an enormous scaffolding to build arithmetic.
Leads to Gottlob Frege.
Begin this lesson →epoché — a humanities education that remembers you.