Editing Hume's principle (section)

== 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).