- Hume's principle
**Hume's Principle**, or "HP"—the terms were coined byGeorge Boolos —says that the number of "F"s is equal to the number of "G"s if there is aone-to-one correspondence (abijection ) between the "F"s and the "G"s. HP can be stated formally in systems ofsecond-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 callsecond-order arithmetic . This result is known asFrege's theorem , which is the foundation for a philosophy of mathematics known as neo-logicism.**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 inquantity ornumber , Hume argues that our reasoning about proportion in quantity, as represented bygeometry , can never achieve "perfect precision and exactness", since its principles are derived from sense-appearance. He contrasts this with reasoning about number orarithmetic , 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. SeeAristotle , "Metaphysics", 1020a14 andEuclid , "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.**Influence on set theory**The principle that

cardinal number was to be characterized in terms ofone-to-one correspondence had previously been used to great effect byGeorg 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 definescardinal number s in terms ofordinal number s, 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 axiomaticset theory .**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. [*http://www.maths.bris.ac.uk/~majpm/list.html Online excerpts.*]**External links***

Stanford Encyclopedia of Philosophy : " [*http://plato.stanford.edu/entries/frege-logic/ Frege's Logic, Theorem, and Foundations for Arithmetic*] " -- byEdward Zalta .

* " [*http://www.st-andrews.ac.uk/~arche/pages/projects/mathsproject.html The Logical and Metaphysical Foundations of Classical Mathematics,*] "

* [*http://www.st-andrews.ac.uk/~arche/pages/home.html Arche: The Centre for Philosophy of Logic, Language, Mathematics and Mind at St. Andrew's University.*]

*Wikimedia Foundation.
2010.*

### Look at other dictionaries:

**Hume's principle**— The principle that the number of things with the property F equal the number of things with the property G if and only if there is a one to one correspondence between those that are F and those that are G. The attribution to Hume derives from the … Philosophy dictionary**Hume: moral and political philosophy**— Rosalind Hursthouse INTRODUCTION Hume’s moral and political philosophy, like his epistemology and meta physics, originally appeared in A Treatise of Human Nature, (henceforth [7.1]), Book III of which, ‘Of Morals’, was published in 1740. He… … History of philosophy**Hume on human understanding**— David Hume on human understanding Anne Jaap Jacobson David Hume’s A Treatise of Human Nature1 was published before he was 30 years old. It is often said to be the greatest philosophical work written in English. Bold and ambitious, it is designed… … History of philosophy**Hume, David**— born May 7, 1711, Edinburgh, Scot. died Aug. 25, 1776, Edinburgh Scottish philosopher, historian, and economist. He conceived of philosophy as the inductive, experimental science of human nature. His first major work, A Treatise of Human Nature… … Universalium**Hume, David**— (1711 76) by Cliff Stagoll David Hume was a Scottish philosopher, historian, economist and religious theorist, and perhaps the best known of the philosophers commonly designated empiricists . Although Hume s grouping with such thinkers as… … The Deleuze dictionary**Hume, David**— (1711 76) by Cliff Stagoll David Hume was a Scottish philosopher, historian, economist and religious theorist, and perhaps the best known of the philosophers commonly designated empiricists . Although Hume s grouping with such thinkers as… … The Deleuze dictionary**Hume, David**— (1711 76) Although his first book, A Treatise of Human Nature, received a bad reception on its publication between 1739 and 1740, Hume did not let this prevent him from publishing An Enquiry Concerning Human Understanding in 1748. This… … Christian Philosophy**Principle of individuation**— The Principle of Individuation is a criterion which supposedly individuates or numerically distinguishes the members of the kind for which it is given, i.e. by which we can supposedly determine, regarding any kind of thing, when we have more than … Wikipedia**HUME, DAVID**— philosopher and historian, born in Edinburgh, the younger son of a Berwickshire laird; after trial of law and mercantile life gave himself up to study and speculation; spent much of his life in France, and fraternised with the sceptical… … The Nuttall Encyclopaedia**David Hume**— For other people named David Hume, see David Hume (disambiguation). David Hume David Hume Born 7 May 1711(1711 05 07) Edinburgh, Scotland Died 25 August 1776( … Wikipedia