Hume's principle
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Hume's Principle, or HP—the terms were coined by Geroge Boolos—says that the number of F's is equal to the number of G's if there is a one-to-one correspondence (a bijection) between the F's and the G's. HP can be stated formally in systems of second-order logic.
HP 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.
Hume's Principle is so-called because, in introducing it in his Foundations of Arithmetic, Frege quotes from chapter III of the first book of David Hume's A Treatise of Human Nature, wherein Hume examines what he takes to be the seven fundamental relations between ideas. Concerning one of these, proportion in quantity or number, Hume argues that our reasoning about proportion in quantity, as represented by geometry, can never achieve "perfect precision and exactness", since its principles are derived from sense-appearance. He contrasts this with reasoning about number or arithmetic, in which such a precision can be attained:
"Algebra and arithmetic [are] the only sciences, in which we can carry on a chain of reasoning to any degree of intricacy, and yet preserve a perfect exactness and certainty. We are possessed of a precise standard, by which we can judge of the equality and proportion of numbers; and according as they correspond or not to that standard, we determine their relations, without any possibility of error. When two numbers are so combined, as that the one has always a unit answering to every unit of the other, we pronounce them equal; and it is for want of such a standard of equality in [spatial] extension, that geometry can scarce be esteemed a perfect and infallible science." (I. III. I.)
Note Hume's use of the word number in the ancient sense, to mean a set or collection of things rather than the modern notion of "positive integer". The ancient Greek notion of number (arithmos) is of a finite plurality composed of units. See Aristotle, Metaphysics, 1020a14 and Euclid, Elements, Book VII, Definition 1 and 2. The contrast between the old and modern conception of number is discussed in detail in a book by John P. Mayberry (The Foundations of Mathematics in the Theory of Sets, Cambridge, 2000, available in part online [1]). The credit Frege tries to give to Hume is therefore probably not deserved, and Hume certainly would have rejected at least some of the consequences Frege draws from HP, in particular, the consequence that there are infinite numbers.
The principle that cardinal number was to be characterized in terms of one-to-one correspondence had previously been used to great effect by Georg Cantor, whose writings Frege knew. Frege criticized Cantor, however, on the ground that Cantor defines cardinal numbers in terms of ordinal numbers, whereas Frege wanted (at least) to give a characterization of cardinals that was independent of the ordinals. Cantor's point of view, however, is the one embedded in modern theories of transfinite number, as developed in set theory.