Hume's principle - Wikipedia Hume's principle From Wikipedia, the free encyclopedia Jump to navigation Jump to search This article includes a list of references, related reading or external links, but its sources remain unclear because it lacks inline citations. Please help to improve this article by introducing more precise citations. (May 2020) (Learn how and when to remove this template message) 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. Contents 1 Origins 2 Influence on set theory 3 References 4 External links Origins[edit] 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.) 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). Influence on set theory[edit] 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" or "The Hume-Cantor 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. References[edit] Anderson, D., and Edward Zalta (2004) "Frege, Boolos, and Logical Objects," Journal of Philosophical Logic 33: 1–26. George Boolos, "The Standard of Equality of Numbers" in George Boolos (ed.), Meaning and Method: Essays in Honour of Hilary Putnam (Cambridge Eng.: Cambridge University Press, 1990), pp. 261–277. 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, The Foundations of Arithmetic. David Hume. A Treatise of Human Nature. Mayberry, John P., 2000. The Foundations of Mathematics in the Theory of Sets. Cambridge. External links[edit] 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. v t e David Hume Books A Treatise of Human Nature An Enquiry Concerning Human Understanding An Enquiry Concerning the Principles of Morals Essays, Moral, Political, and Literary Four Dissertations Dialogues Concerning Natural Religion The History of England The History of Great Britain Criticism Argument for the existence of God from design Problem of induction Is–ought problem Philosophy Hume's principle Hume's law Hume's fork The Missing Shade of Blue "Of Miracles" Scottish Enlightenment Empiricism Price–specie flow mechanism Related Hume Studies A Treatise of Human Nature (Abstract) Retrieved from "https://en.wikipedia.org/w/index.php?title=Hume%27s_principle&oldid=955917564" Categories: Set theory Philosophy of mathematics Mathematical principles Concepts in logic Hidden categories: Articles lacking in-text citations from May 2020 All articles lacking in-text citations Navigation menu Personal tools Not logged in Talk Contributions Create account Log in Namespaces Article Talk Variants Views Read Edit View history More Search Navigation Main page Contents Current events Random article About Wikipedia Contact us Donate Contribute Help Learn to edit Community portal Recent changes Upload file Tools What links here Related changes Upload file Special pages Permanent link Page information Cite this page Wikidata item Print/export Download as PDF Printable version Languages Deutsch Español Français 日本語 Edit links This page was last edited on 10 May 2020, at 14:36 (UTC). 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