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Wednesday, September 12, 2012

Hume's Principle in Hume

Hume's Principle (HP), as used nowadays, states that the number of Fs is the same as the number of Gs iff there is a 1-1 correspondence between Fs and Gs. While this sounds pretty obvious, with second-order logic in the background you can use this to derive second-order Peano Arithmetic (PA). (all this is well known, just like the role of HP in a fairly fashionable stream in phil of math called neologicism - check out this or this if you haven't heard of this stuff). Anyway, the person who really used HP to obtain PA was Frege, so just in case you wondered why the principle is called Hume's Principle, I dug up the passage where Hume formulates it (Treatise 1.3.1). It's only moderately interesting, but if you're geeky enough to be still reading this, you might be just geeky enough to be interested in the quote:
We might proceed, after the same manner, in fixing the proportions of quantity or number, and might at one view observe a superiority or inferiority betwixt any numbers, or figures; especially where the difference is very great and remarkable. As to equality or any exact proportion, we can only guess at it from a single consideration; except in very short numbers, or very limited portions of extension; which are comprehended in an instant, and where we perceive an impossibility of falling into any considerable error. In all other cases we must settle the proportions with some liberty, or proceed in a more artificial manner.
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There remain, therefore, algebra and arithmetic as 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 possest 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 an unite answering to every unite of the other, we pronounce them equal; and it is for want of such a standard of equality in extension, that geometry can scarce be esteemed a perfect and infallible science.

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