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 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. George Stephen Boolos (September 4, 1940, New York City - May 27, 1996) was a philosopher and a mathematical logician. ... In mathematics, a bijection, bijective function, or one-to-one correspondence is a function that is both injective (one-to-one) and surjective (onto), and therefore bijections are also called one-to-one and onto. ... A bijective function. ... In mathematical logic, second-order logic is an extension of first-order logic, which itself is an extension of propositional 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. Friedrich Ludwig Gottlob Frege (8 November 1848, Wismar – 26 July 1925, IPA: ) was a German mathematician who became a logician and philosopher. ... In mathematical logic, second order arithmetic is a stronger version of Peano arithmetic that allows quantification over subsets of the integers, rather than just over integers. ... Freges theorem states that the axioms of second-order arithmetic can be derived in second-order logic from Humes principle. ... Logicism is one of the schools of thought in the philosophy of mathematics, putting forth the theory that mathematics is an extension of logic and therefore some or all mathematics is reducible to logic. ...

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: For other persons named David Hume, see David Hume (disambiguation). ... A Treatise of Human Nature is a book by philosopher David Hume, published in 1739–1740. ... This article is about proportionality, the mathematical relation. ... Quantity is a kind of property which exists as magnitude or multitude. ... For other uses, see Number (disambiguation). ... For other uses, see Geometry (disambiguation). ... Arithmetic tables for children, Lausanne, 1835 Arithmetic or arithmetics (from the Greek word αριθμός = number) is the oldest and most elementary branch of mathematics, used by almost everyone, for tasks ranging from simple day-to-day counting to advanced science and business calculations. ...

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 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 numbers. For other uses, see Number (disambiguation). ... For other uses, see Aristotle (disambiguation). ... Metaphysics is one of the principal works of Aristotle and the first major work of the branch of philosophy with the same name. ... For other uses, see Euclid (disambiguation). ... For other uses, see Infinity (disambiguation). ...

Influence on set theory

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. The suggestion has therefore been made that Hume's Principle ought better be called "Cantor's Principle". But Frege criticized Cantor on the ground that Cantor defines cardinal numbers in terms of ordinal numbers, whereas Frege wanted to give a characterization of cardinals that was independent of the ordinals. Cantor's point of view, however, is the one embedded in contemporary theories of transfinite numbers, as developed in axiomatic set theory. Aleph-0, the smallest infinite cardinal In mathematics, cardinal numbers, or cardinals for short, are generalized numbers used to measure the cardinality (size) of sets. ... In mathematics, a bijection, bijective function, or one-to-one correspondence is a function that is both injective (one-to-one) and surjective (onto), and therefore bijections are also called one-to-one and onto. ... Georg Ferdinand Ludwig Philipp Cantor (March 3, 1845[1] – January 6, 1918) was a German mathematician. ... Friedrich Ludwig Gottlob Frege (8 November 1848, Wismar – 26 July 1925, IPA: ) was a German mathematician who became a logician and philosopher. ... Aleph-0, the smallest infinite cardinal In mathematics, cardinal numbers, or cardinals for short, are generalized numbers used to measure the cardinality (size) of sets. ... In set theory, ordinal, ordinal number, and transfinite ordinal number refer to a type of number introduced by Georg Cantor in 1897, to accommodate infinite sequences and to classify sets with certain kinds of order structures on them. ... Transfinite numbers, also known as infinite numbers, are numbers that are not finite. ... Set theory is the mathematical theory of sets, which represent collections of abstract objects. ...

References

• Anderson, D., and Edward Zalta (2004) "Frege, Boolos, and Logical Objects," Journal of Philosophical Logic 33: 1-26.
• George Boolos, 1998. Logic, Logic, and Logic. Harvard Univ. Press. Especially section II, "Frege Studies."
• Burgess, John, 2005. Fixing Frege. Princeton Univ. Press.
• Gottlob Frege, Foundations of Arithmetic.
• David Hume, . A Treatise of Human Nature.
• Mayberry, John P., 2000. The Foundations of Mathematics in the Theory of Sets. Cambridge. Online excerpts.

Edward N. Zalta is a Senior Research Scholar at the Center for the Study of Language and Information. ... George Stephen Boolos (September 4, 1940, New York City - May 27, 1996) was a philosopher and a mathematical logician. ... Friedrich Ludwig Gottlob Frege (8 November 1848, Wismar – 26 July 1925, IPA: ) was a German mathematician who became a logician and philosopher. ... For other persons named David Hume, see David Hume (disambiguation). ... A Treatise of Human Nature is a book by philosopher David Hume, published in 1739–1740. ...

External links

• Stanford Encyclopedia of Philosophy: "Frege's Logic, Theorem, and Foundations for Arithmetic" -- by Edward Zalta.
• "The Logical and Metaphysical Foundations of Classical Mathematics,"
• Arche: The Centre for Philosophy of Logic, Language, Mathematics and Mind at St. Andrew's University.

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