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Single Idea 10487

[filed under theme 9. Objects / A. Existence of Objects / 2. Abstract Objects / a. Nature of abstracta ]

Full Idea

I am rather a fan of abstract objects, and confident of their existence. Smaller numbers, sets and functions don't offend my sense of reality.

Gist of Idea

I am a fan of abstract objects, and confident of their existence

Source

George Boolos (Must We Believe in Set Theory? [1997], p.128)

Book Ref

Boolos,George: 'Logic, Logic and Logic' [Harvard 1999], p.128

A Reaction

The great Boolos is rather hard to disagree with, but I disagree. Logicians love abstract objects, indeed they would almost be out of a job without them. It seems to me they smuggle them into our ontology by redefining either 'object' or 'exists'.

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]