Combining Texts

All the ideas for 'Mahaprajnaparamitashastra', 'What Numbers Could Not Be' and 'Nominalism and Substitutional Quantifiers'

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38 ideas

5. Theory of Logic / A. Overview of Logic / 7. Second-Order Logic
The nominalist is tied by standard semantics to first-order, denying higher-order abstracta [Marcus (Barcan)]
5. Theory of Logic / F. Referring in Logic / 1. Naming / a. Names
Nominalists see proper names as a main vehicle of reference [Marcus (Barcan)]
Anything which refers tends to be called a 'name', even if it isn't a noun [Marcus (Barcan)]
5. Theory of Logic / G. Quantification / 1. Quantification
Nominalists should quantify existentially at first-order, and substitutionally when higher [Marcus (Barcan)]
5. Theory of Logic / G. Quantification / 2. Domain of Quantification
Quantifiers are needed to refer to infinitely many objects [Marcus (Barcan)]
Substitutional semantics has no domain of objects, but place-markers for substitutions [Marcus (Barcan)]
5. Theory of Logic / G. Quantification / 4. Substitutional Quantification
Maybe a substitutional semantics for quantification lends itself to nominalism [Marcus (Barcan)]
Substitutional language has no ontology, and is just a way of speaking [Marcus (Barcan)]
A true universal sentence might be substitutionally refuted, by an unnamed denumerable object [Marcus (Barcan)]
6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / a. Numbers
There are no such things as numbers [Benacerraf]
Numbers can't be sets if there is no agreement on which sets they are [Benacerraf]
6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / c. Priority of numbers
Benacerraf says numbers are defined by their natural ordering [Benacerraf, by Fine,K]
6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / f. Cardinal numbers
To understand finite cardinals, it is necessary and sufficient to understand progressions [Benacerraf, by Wright,C]
A set has k members if it one-one corresponds with the numbers less than or equal to k [Benacerraf]
To explain numbers you must also explain cardinality, the counting of things [Benacerraf]
6. Mathematics / A. Nature of Mathematics / 4. Using Numbers / c. Counting procedure
We can count intransitively (reciting numbers) without understanding transitive counting of items [Benacerraf]
Someone can recite numbers but not know how to count things; but not vice versa [Benacerraf]
6. Mathematics / A. Nature of Mathematics / 4. Using Numbers / g. Applying mathematics
The application of a system of numbers is counting and measurement [Benacerraf]
6. Mathematics / B. Foundations for Mathematics / 4. Axioms for Number / a. Axioms for numbers
The successor of x is either x and all its members, or just the unit set of x [Benacerraf]
For Zermelo 3 belongs to 17, but for Von Neumann it does not [Benacerraf]
6. Mathematics / B. Foundations for Mathematics / 6. Mathematics as Set Theory / b. Mathematics is not set theory
Disputes about mathematical objects seem irrelevant, and mathematicians cannot resolve them [Benacerraf, by Friend]
No particular pair of sets can tell us what 'two' is, just by one-to-one correlation [Benacerraf, by Lowe]
If ordinal numbers are 'reducible to' some set-theory, then which is which? [Benacerraf]
6. Mathematics / B. Foundations for Mathematics / 7. Mathematical Structuralism / a. Structuralism
The job is done by the whole system of numbers, so numbers are not objects [Benacerraf]
If any recursive sequence will explain ordinals, then it seems to be the structure which matters [Benacerraf]
The number 3 defines the role of being third in a progression [Benacerraf]
Number words no more have referents than do the parts of a ruler [Benacerraf]
Mathematical objects only have properties relating them to other 'elements' of the same structure [Benacerraf]
How can numbers be objects if order is their only property? [Benacerraf, by Putnam]
6. Mathematics / C. Sources of Mathematics / 1. Mathematical Platonism / b. Against mathematical platonism
Number-as-objects works wholesale, but fails utterly object by object [Benacerraf]
6. Mathematics / C. Sources of Mathematics / 5. Numbers as Adjectival
Number words are not predicates, as they function very differently from adjectives [Benacerraf]
6. Mathematics / C. Sources of Mathematics / 6. Logicism / d. Logicism critique
The set-theory paradoxes mean that 17 can't be the class of all classes with 17 members [Benacerraf]
7. Existence / A. Nature of Existence / 3. Being / i. Deflating being
Is being just referent of the verb 'to be'? [Marcus (Barcan)]
8. Modes of Existence / E. Nominalism / 3. Predicate Nominalism
Nominalists say predication is relations between individuals, or deny that it refers [Marcus (Barcan)]
9. Objects / A. Existence of Objects / 3. Objects in Thought
If objects are thoughts, aren't we back to psychologism? [Marcus (Barcan)]
9. Objects / F. Identity among Objects / 2. Defining Identity
Substitutivity won't fix identity, because expressions may be substitutable, but not refer at all [Marcus (Barcan)]
9. Objects / F. Identity among Objects / 6. Identity between Objects
Identity statements make sense only if there are possible individuating conditions [Benacerraf]
23. Ethics / C. Virtue Theory / 3. Virtues / a. Virtues
The six perfections are giving, morality, patience, vigour, meditation, and wisdom [Nagarjuna]