Origins
Hume's Principle appears in Frege's Foundations of Arithmetic, which quotes from Part III of Book I of David Hume's A Treatise of Human Nature. 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 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 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 common 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 Mayberry (2000). 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 sets.
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