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Single Idea 8693

[filed under theme 18. Thought / E. Abstraction / 7. Abstracta by Equivalence ]

Full Idea

Hume's Principle has a structure Boolos calls an 'abstraction principle'. Within the scope of two universal quantifiers, a biconditional connects an identity between two things and an equivalence relation. It says we don't care about other differences.

Clarification

Hume's Principle says numbers are identical through one-to-one correspondence

Gist of Idea

An 'abstraction principle' says two things are identical if they are 'equivalent' in some respect

Source

George Boolos (Is Hume's Principle analytic? [1997]), quoted by Michèle Friend - Introducing the Philosophy of Mathematics 3.7

Book Ref

Friend,Michèle: 'Introducing the Philosophy of Mathematics' [Acumen 2007], p.75

A Reaction

This seems to be the traditional principle of abstraction by ignoring some properties, but dressed up in the clothes of formal logic. Frege tries to eliminate psychology, but Boolos implies that what we 'care about' is relevant.

The 31 ideas from George Boolos

10482 The logic of ZF is classical first-order predicate logic with identity [Boolos]
10483 Mathematics and science do not require very high orders of infinity [Boolos]
10484 The iterative conception says sets are formed at stages; some are 'earlier', and must be formed first [Boolos]
10485 Naïve sets are inconsistent: there is no set for things that do not belong to themselves [Boolos]
10488 It is lunacy to think we only see ink-marks, and not word-types [Boolos]
10487 I am a fan of abstract objects, and confident of their existence [Boolos]
10489 We deal with abstract objects all the time: software, poems, mistakes, triangles.. [Boolos]
10490 Mathematics isn't surprising, given that we experience many objects as abstract [Boolos]
10491 Infinite natural numbers is as obvious as infinite sentences in English [Boolos]
10492 A few axioms of set theory 'force themselves on us', but most of them don't [Boolos]
8693 An 'abstraction principle' says two things are identical if they are 'equivalent' in some respect [Boolos]
13547 Limitation of Size is weak (Fs only collect is something the same size does) or strong (fewer Fs than objects) [Boolos, by Potter]
18192 Do the Replacement Axioms exceed the iterative conception of sets? [Boolos, by Maddy]
14249 Boolos reinterprets second-order logic as plural logic [Boolos, by Oliver/Smiley]
13841 Why should compactness be definitive of logic? [Boolos, by Hacking]
10829 A sentence can't be a truth of logic if it asserts the existence of certain sets [Boolos]
10830 Second-order logic metatheory is set-theoretic, and second-order validity has set-theoretic problems [Boolos]
10832 '∀x x=x' only means 'everything is identical to itself' if the range of 'everything' is fixed [Boolos]
10833 Many concepts can only be expressed by second-order logic [Boolos]
10834 Weak completeness: if it is valid, it is provable. Strong: it is provable from a set of sentences [Boolos]
13671 Second-order quantifiers are just like plural quantifiers in ordinary language, with no extra ontology [Boolos, by Shapiro]
10267 We should understand second-order existential quantifiers as plural quantifiers [Boolos, by Shapiro]
7806 Boolos invented plural quantification [Boolos, by Benardete,JA]
10225 Monadic second-order logic might be understood in terms of plural quantifiers [Boolos, by Shapiro]
7785 The use of plurals doesn't commit us to sets; there do not exist individuals and collections [Boolos]
10736 Boolos showed how plural quantifiers can interpret monadic second-order logic [Boolos, by Linnebo]
10780 Any sentence of monadic second-order logic can be translated into plural first-order logic [Boolos, by Linnebo]
10697 Identity is clearly a logical concept, and greatly enhances predicate calculus [Boolos]
10698 Plural forms have no more ontological commitment than to first-order objects [Boolos]
10699 Does a bowl of Cheerios contain all its sets and subsets? [Boolos]
10700 First- and second-order quantifiers are two ways of referring to the same things [Boolos]