15924
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Predicative definitions are acceptable in mathematics if they distinguish objects, rather than creating them? [Zermelo, by Lavine]
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Full Idea:
On Zermelo's view, predicative definitions are not only indispensable to mathematics, but they are unobjectionable since they do not create the objects they define, but merely distinguish them from other objects.
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From:
report of Ernst Zermelo (Investigations in the Foundations of Set Theory I [1908]) by Shaughan Lavine - Understanding the Infinite V.1
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A reaction:
This seems to have an underlying platonism, that there are hitherto undefined 'objects' lying around awaiting the honour of being defined. Hm.
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17608
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We take set theory as given, and retain everything valuable, while avoiding contradictions [Zermelo]
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Full Idea:
Starting from set theory as it is historically given ...we must, on the one hand, restrict these principles sufficiently to exclude as contradiction and, on the other, take them sufficiently wide to retain all that is valuable in this theory.
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From:
Ernst Zermelo (Investigations in the Foundations of Set Theory I [1908], Intro)
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A reaction:
Maddy calls this the one-step-back-from-disaster rule of thumb. Zermelo explicitly mentions the 'Russell antinomy' that blocked Frege's approach to sets.
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17607
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Set theory investigates number, order and function, showing logical foundations for mathematics [Zermelo]
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Full Idea:
Set theory is that branch whose task is to investigate mathematically the fundamental notions 'number', 'order', and 'function', taking them in their pristine, simple form, and to develop thereby the logical foundations of all of arithmetic and analysis.
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From:
Ernst Zermelo (Investigations in the Foundations of Set Theory I [1908], Intro)
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A reaction:
At this point Zermelo seems to be a logicist. Right from the start set theory was meant to be foundational to mathematics, and not just a study of the logic of collections.
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13015
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Zermelo used Foundation to block paradox, but then decided that only Separation was needed [Zermelo, by Maddy]
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Full Idea:
Zermelo used a weak form of the Axiom of Foundation to block Russell's paradox in 1906, but in 1908 felt that the form of his Separation Axiom was enough by itself, and left the earlier axiom off his published list.
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From:
report of Ernst Zermelo (Investigations in the Foundations of Set Theory I [1908]) by Penelope Maddy - Believing the Axioms I §1.2
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A reaction:
Foundation turns out to be fairly controversial. Barwise actually proposes Anti-Foundation as an axiom. Foundation seems to be the rock upon which the iterative view of sets is built. Foundation blocks infinite descending chains of sets, and circularity.
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13027
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Zermelo believed, and Von Neumann seemed to confirm, that numbers are sets [Zermelo, by Maddy]
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Full Idea:
Zermelo was a reductionist, and believed that theorems purportedly about numbers (cardinal or ordinal) are really about sets, and since Von Neumann's definitions of ordinals and cardinals as sets, this has become common doctrine.
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From:
report of Ernst Zermelo (Investigations in the Foundations of Set Theory I [1908]) by Penelope Maddy - Believing the Axioms I §1.8
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A reaction:
Frege has a more sophisticated take on this approach. It may just be an updating of the Greek idea that arithmetic is about treating many things as a unit. A set bestows an identity on a group, and that is all that is needed.
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9627
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Different versions of set theory result in different underlying structures for numbers [Zermelo, by Brown,JR]
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Full Idea:
In Zermelo's set-theoretic definition of number, 2 is a member of 3, but not a member of 4; in Von Neumann's definition every number is a member of every larger number. This means they have two different structures.
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From:
report of Ernst Zermelo (Investigations in the Foundations of Set Theory I [1908]) by James Robert Brown - Philosophy of Mathematics Ch. 4
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A reaction:
This refers back to the dilemma highlighted by Benacerraf, which was supposed to be the motivation for structuralism. My intuition says that the best answer is that they are both wrong. In a pattern, the nodes aren't 'members' of one another.
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16689
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The schools said spirits lack extension, and wonder how many could dance on a needle's point [More,H]
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Full Idea:
Many, not without reason, laugh at those ridiculous fancies of the schools, that rashly take away all extensions from spirits, whether souls or angels, and then dispute how many of them booted and spurred may dance on a needle's point at once.
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From:
Henry More (Immortality of the Soul [1659], III.2.1), quoted by Robert Pasnau - Metaphysical Themes 1274-1671 17.3
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A reaction:
This famous idea originated with William Chillingworth. More's version is the classic one. Pasnau cites Aquinas Summa 1a 52.3 as discussing the actual question (and says this couldn't happen).
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