Editing Hume's principle

{{Short description|Logical principle}}
{{more footnotes|date=May 2020}}

'''Hume's principle''' or '''HP''' says that, given two collections of objects <math>\mathcal F</math> and <math>\mathcal G </math> with properties <math>F</math> and <math>G</math> respectively, the number of objects with property <math>F</math> is equal to the number of objects with property <math>G</math> if and only if there is a [[one-to-one correspondence]] (a bijection) between <math>\mathcal F</math> and <math>\mathcal G</math>. In other words, that bijections are the "correct" way of measuring size.

'''HP''' can be stated formally in systems of [[second-order logic]]. It is named for the Scottish philosopher [[David Hume]] and was coined by [[George Boolos]]. The principle plays a central role in [[Gottlob Frege]]'s philosophy of mathematics. Frege shows that HP and suitable definitions of arithmetical notions [[logical consequence|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 [[Logicism#Neo-logicism|neo-logicism]].

== Origins ==

Hume's Principle appears in Frege's ''Foundations of Arithmetic'' (§63),<ref>{{cite book |chapter=IV. Der Begriff der Anzahl § 63. Die Möglichkeit der eindeutigen Zuordnung als solches. Logisches Bedenken, dass die Gleichheit für diesen Fall besonders erklärt wird |chapter-url=https://gutenberg.org/cache/epub/48312/pg48312-images.html#para_63 |title={{harvnb|Frege|1884}} |via=Project Gutenberg |quote=§63. Ein solches Mittel nennt schon Hume: »Wenn zwei Zahlen so combinirt werden, dass die eine immer eine Einheit hat, die jeder Einheit der andern entspricht, so geben wir sie als gleich an.« }}</ref> which quotes from Part III of Book I of [[David Hume]]'s ''[[A Treatise of Human Nature]]'' (1740).

In the treatise, Hume sets out seven fundamental relations between ideas, in particular concerning [[Proportionality (mathematics)|''proportion'']] in [[quantity]] or [[number]]. He 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:

<blockquote>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.)<ref>

{{cite book |chapter=Part III. Of Knowledge and Probability: Sect. I. Of Knowledge |chapter-url=https://gutenberg.org/cache/epub/4705/pg4705-images.html#link2H_4_0021 |via=Project Gutenberg |title={{harvnb|Hume|1739–1740}}}}

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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 (Aristotle)|Metaphysics]]'', 1020a14 and [[Euclid]], ''[[Euclid's Elements|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 ==
The principle that [[cardinal number]] was to be characterized in terms of [[one-to-one correspondence]] had previously been used by [[Georg Cantor]], whose writings [[Gottlob Frege|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 number]]s in terms of [[ordinal 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 number]]s, as developed in [[axiomatic set theory]].

== References ==
*{{cite journal |last1=Anderson |first1=D. |first2=E. |last2=Zalta |author2-link=Edward Zalta |title=Frege, Boolos, and Logical Objects |journal=Journal of Philosophical Logic |volume=33 |issue= |pages=1–26 |date=2004 |doi=10.1023/B:LOGI.0000019236.64896.fd |s2cid=6620015 |url=https://mally.stanford.edu/Papers/frege-boolos.pdf}}
*{{cite book |author-link=George Boolos |first=George |last=Boolos |chapter=The Standard of Equality of Numbers |chapter-url= |editor-first=G. |editor-last=Boolos |title=Meaning and Method: Essays in Honour of Hilary Putnam |publisher=Cambridge University Press |location= |date=1990 |isbn=978-0-521-36083-8 |pages=261–277 |url={{GBurl|QqBOAAAAIAAJ|pg=PP12}} }}
*{{cite book |first=George |last=Boolos |chapter=§II. "Frege Studies |chapter-url= |title=Logic, Logic, and Logic |publisher=Harvard University Press |date=1998 |isbn=978-0-674-53767-5 |pages=133–342 |url={{GBurl|2BvlvetSrlgC|p=133}} }}
*{{cite book |first=John |last=Burgess |title=Fixing Frege |publisher=Princeton University Press |orig-date=2005 |isbn=978-0-691-18706-8 |url={{GBurl|hf9ZDwAAQBAJ|pg=PP8}} |date=2018 }}
*{{cite book |author-link=Gottlob Frege |first=Gottlob |last=Frege |trans-title=The Foundations of Arithmetic |title-link=The Foundations of Arithmetic |title=Die Grundlagen der Arithmetik: Eine logisch mathematische Untersuchung |publisher=Wilhelm Koebner |location=Breslau |date=1884 }}
*{{cite book |author-link=David Hume |first=David |last=Hume |title=A Treatise of Human Nature |title-link=A Treatise of Human Nature |date=1739–1740 }}
*{{cite book |first=John P. |last=Mayberry |title=The Foundations of Mathematics in the Theory of Sets |publisher=Cambridge University Press |date=2000 |isbn=978-0-521-77034-7 |series=Encyclopedia of Mathematics and its Applications |volume=83 |url={{GBurl|mP1ofko7p6IC|pg=PP1}}}}

=== Citations ===
{{Reflist}}

== External links ==
* [[Stanford Encyclopedia of Philosophy]]: "[http://plato.stanford.edu/entries/frege-logic/ Frege's Logic, Theorem, and Foundations for Arithmetic]" by [[Edward Zalta]].
* [https://web.archive.org/web/20041228032512/http://www.st-andrews.ac.uk/~arche/pages/projects/mathsproject.html "The Logical and Metaphysical Foundations of Classical Mathematics."]
* [https://web.archive.org/web/20070204210030/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.]

{{Hume}}

[[Category:Set theory]]
[[Category:Philosophy of mathematics]]
[[Category:Mathematical principles]]
[[Category:Concepts in logic]]