Concept
Hume's principle
Overview
In the foundations of mathematics, Hume's principle (or HP) says that, given two collections of objects
F
{\displaystyle {\mathcal {F}}}
and
G
{\displaystyle {\mathcal {G}}}
with properties
F
{\displaystyle F}
and
G
{\displaystyle G}
respectively, the number of objects with property
F
{\displaystyle F}
is equal to the number of objects with property
G
{\displaystyle G}
if and only if there is a one-to-one correspondence (a bijection) between
F
{\displaystyle {\mathcal {F}}}
and
G
{\displaystyle {\mathcal {G}}}
. In other words, that bijections are the "correct" way of measuring size.
HP can be stated formally in systems of second-order logic. It is named for the Scottish philosopher David Hume and was coined by George Boolos. The principle plays a central role in Gottlob Frege's philosophy of mathematics. Frege shows that HP and suitable definitions of arithmetical notions entail all axioms of what we now call second-order arithmetic. This result is known as Frege's theorem, which is the foundation for a philosophy of mathematics known as neo-logicism.
Overview adapted from Wikipedia (CC BY-SA 4.0; see the article history for authors).
Possible passages
Found by search and not yet checked by a person; they may only mention the idea in passing.
In modern expositions of the axioms of geometry certain conditions are laid down which the relation of congruence between segments is to satisfy. It is supposed that we have a complete theory of points, straight lines, planes, and the order of points on planes--in fact, a complete theory of non-metrical geometry. We then enquire about congruence and lay down the set of conditions--or axioms as they are called--which…
Alfred North Whitehead, The Concept of Nature The Tarner Lectures Delivered in Trinity College, November 1919
States what Hume's principle is.
Congruence was to refer to geometrical bodies, with none of the properties of ordinary bodies except impenetrability (Erdmann, Axiome der Geometrie, p. 26). A straight line is defined as determined by two of its points[14], and a plane as determined by three. These premisses, with a slight exception as to the straight line, we shall hereafter find essential to every Geometry. I have drawn attention to them, as it is…
Bertrand Russell, An essay on the foundations of geometry
States what Hume's principle is.
What further defines them is their similarity to sensations. This only applies strictly to simple images; complex ones may occur without a prototype, though all their parts will have prototypes among sensations. Such, at least, is Hume’s principle, and on the whole it seems to be true. It must not, however, be pressed beyond a point. As a rule, an image is more or less vague, and has a number of similar sensations…
Bertrand Russell, Philosophy
States what Hume's principle is.
Philosophers who discussed Hume's principle
Related ideas
- Georg Cantor
- Foundations of mathematics
- Transfinite number
- Geometry
- Number
- Euclid's Elements
- Metaphysics (Aristotle)
- A Treatise of Human Nature
- Set theory
- Aristotle
- Arithmetic
- George Boolos
- Second-order logic
- Logicism
- Second-order arithmetic
- Logical consequence
- Frege's theorem
- Gottlob Frege
- Cardinal number
- Quantity
- David Hume
- Proportionality (mathematics)
- Euclid
Place in the tradition
Part of British philosophy.