Combining Philosophers
Ideas for Lycophron, Geoffrey Hellman and Paul Benacerraf
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35 ideas
6. Mathematics / A. Nature of Mathematics / 1. Mathematics
9935
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Mathematical truth is always compromising between ordinary language and sensible epistemology [Benacerraf]
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6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / a. Numbers
13412
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Obtaining numbers by abstraction is impossible - there are too many; only a rule could give them, in order [Benacerraf]
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13413
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We must explain how we know so many numbers, and recognise ones we haven't met before [Benacerraf]
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9912
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There are no such things as numbers [Benacerraf]
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9901
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Numbers can't be sets if there is no agreement on which sets they are [Benacerraf]
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6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / c. Priority of numbers
13411
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If numbers are basically the cardinals (Frege-Russell view) you could know some numbers in isolation [Benacerraf]
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9151
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Benacerraf says numbers are defined by their natural ordering [Benacerraf, by Fine,K]
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6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / f. Cardinal numbers
13891
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To understand finite cardinals, it is necessary and sufficient to understand progressions [Benacerraf, by Wright,C]
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17904
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A set has k members if it one-one corresponds with the numbers less than or equal to k [Benacerraf]
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17906
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To explain numbers you must also explain cardinality, the counting of things [Benacerraf]
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6. Mathematics / A. Nature of Mathematics / 4. Using Numbers / c. Counting procedure
9898
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We can count intransitively (reciting numbers) without understanding transitive counting of items [Benacerraf]
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17903
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Someone can recite numbers but not know how to count things; but not vice versa [Benacerraf]
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6. Mathematics / A. Nature of Mathematics / 4. Using Numbers / g. Applying mathematics
9897
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The application of a system of numbers is counting and measurement [Benacerraf]
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6. Mathematics / B. Foundations for Mathematics / 4. Axioms for Number / a. Axioms for numbers
9900
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For Zermelo 3 belongs to 17, but for Von Neumann it does not [Benacerraf]
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9899
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The successor of x is either x and all its members, or just the unit set of x [Benacerraf]
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6. Mathematics / B. Foundations for Mathematics / 6. Mathematics as Set Theory / b. Mathematics is not set theory
8697
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Disputes about mathematical objects seem irrelevant, and mathematicians cannot resolve them [Benacerraf, by Friend]
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8304
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No particular pair of sets can tell us what 'two' is, just by one-to-one correlation [Benacerraf, by Lowe]
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9906
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If ordinal numbers are 'reducible to' some set-theory, then which is which? [Benacerraf]
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6. Mathematics / B. Foundations for Mathematics / 7. Mathematical Structuralism / a. Structuralism
13415
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An adequate account of a number must relate it to its series [Benacerraf]
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9907
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If any recursive sequence will explain ordinals, then it seems to be the structure which matters [Benacerraf]
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9908
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The job is done by the whole system of numbers, so numbers are not objects [Benacerraf]
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9909
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The number 3 defines the role of being third in a progression [Benacerraf]
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9911
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Number words no more have referents than do the parts of a ruler [Benacerraf]
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8925
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Mathematical objects only have properties relating them to other 'elements' of the same structure [Benacerraf]
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9938
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How can numbers be objects if order is their only property? [Benacerraf, by Putnam]
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8921
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Structuralism is now common, studying relations, with no regard for what the objects might be [Hellman]
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6. Mathematics / B. Foundations for Mathematics / 7. Mathematical Structuralism / c. Nominalist structuralism
8698
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Modal structuralism says mathematics studies possible structures, which may or may not be actualised [Hellman, by Friend]
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9557
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Statements of pure mathematics are elliptical for a sort of modal conditional [Hellman, by Chihara]
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10263
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Modal structuralism can only judge possibility by 'possible' models [Shapiro on Hellman]
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8922
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Maybe mathematical objects only have structural roles, and no intrinsic nature [Hellman]
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6. Mathematics / C. Sources of Mathematics / 1. Mathematical Platonism / b. Against mathematical platonism
9910
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Number-as-objects works wholesale, but fails utterly object by object [Benacerraf]
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17927
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Realists have semantics without epistemology, anti-realists epistemology but bad semantics [Benacerraf, by Colyvan]
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9936
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The platonist view of mathematics doesn't fit our epistemology very well [Benacerraf]
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6. Mathematics / C. Sources of Mathematics / 5. Numbers as Adjectival
9903
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Number words are not predicates, as they function very differently from adjectives [Benacerraf]
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6. Mathematics / C. Sources of Mathematics / 6. Logicism / d. Logicism critique
9904
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The set-theory paradoxes mean that 17 can't be the class of all classes with 17 members [Benacerraf]
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