Hume's principle

Hume's principle or HP—the terms were coined by George Boolos—says that the number of Fs is equal to the number of Gs if and only if there is a one-to-one correspondence (a bijection) between the Fs and the Gs. HP can be stated formally in systems of second-order logic. Hume's principle is named for the Scottish philosopher David Hume.

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 appears in Frege's Foundations of Arithmetic (§73), which quotes from Part III of Book I of David Hume's A Treatise of Human Nature (1740). Hume there sets out 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.)

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