Combining Philosophers

Ideas for Mary Astell, Monroe Beardsley and Stewart Shapiro

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26 ideas

6. Mathematics / B. Foundations for Mathematics / 1. Foundations for Mathematics
10236 There is no grounding for mathematics that is more secure than mathematics [Shapiro]
8764 Categories are the best foundation for mathematics [Shapiro]
6. Mathematics / B. Foundations for Mathematics / 2. Proof in Mathematics
10256 For intuitionists, proof is inherently informal [Shapiro]
6. Mathematics / B. Foundations for Mathematics / 4. Axioms for Number / d. Peano arithmetic
13657 First-order arithmetic can't even represent basic number theory [Shapiro]
10202 Natural numbers just need an initial object, successors, and an induction principle [Shapiro]
6. Mathematics / B. Foundations for Mathematics / 4. Axioms for Number / e. Peano arithmetic 2nd-order
10294 Second-order logic has the expressive power for mathematics, but an unworkable model theory [Shapiro]
6. Mathematics / B. Foundations for Mathematics / 5. Definitions of Number / b. Greek arithmetic
10205 Mathematics originally concerned the continuous (geometry) and the discrete (arithmetic) [Shapiro]
6. Mathematics / B. Foundations for Mathematics / 5. Definitions of Number / f. Zermelo numbers
8762 Two definitions of 3 in terms of sets disagree over whether 1 is a member of 3 [Shapiro]
6. Mathematics / B. Foundations for Mathematics / 6. Mathematics as Set Theory / a. Mathematics is set theory
13656 Some sets of natural numbers are definable in set-theory but not in arithmetic [Shapiro]
6. Mathematics / B. Foundations for Mathematics / 6. Mathematics as Set Theory / b. Mathematics is not set theory
10222 Mathematical foundations may not be sets; categories are a popular rival [Shapiro]
6. Mathematics / B. Foundations for Mathematics / 7. Mathematical Structuralism / a. Structuralism
10218 Baseball positions and chess pieces depend entirely on context [Shapiro]
10224 The even numbers have the natural-number structure, with 6 playing the role of 3 [Shapiro]
10228 Could infinite structures be apprehended by pattern recognition? [Shapiro]
10230 The 4-pattern is the structure common to all collections of four objects [Shapiro]
10249 The main mathematical structures are algebraic, ordered, and topological [Shapiro]
10273 Some structures are exemplified by both abstract and concrete [Shapiro]
10276 Mathematical structures are defined by axioms, or in set theory [Shapiro]
8760 Numbers do not exist independently; the essence of a number is its relations to other numbers [Shapiro]
8761 A 'system' is related objects; a 'pattern' or 'structure' abstracts the pure relations from them [Shapiro]
6. Mathematics / B. Foundations for Mathematics / 7. Mathematical Structuralism / b. Varieties of structuralism
10270 The main versions of structuralism are all definitionally equivalent [Shapiro]
6. Mathematics / B. Foundations for Mathematics / 7. Mathematical Structuralism / c. Nominalist structuralism
10221 Is there is no more to structures than the systems that exemplify them? [Shapiro]
10248 Number statements are generalizations about number sequences, and are bound variables [Shapiro]
6. Mathematics / B. Foundations for Mathematics / 7. Mathematical Structuralism / d. Platonist structuralism
10220 Because one structure exemplifies several systems, a structure is a one-over-many [Shapiro]
10223 There is no 'structure of all structures', just as there is no set of all sets [Shapiro]
8703 Shapiro's structuralism says model theory (comparing structures) is the essence of mathematics [Shapiro, by Friend]
6. Mathematics / B. Foundations for Mathematics / 7. Mathematical Structuralism / e. Structuralism critique
10274 Does someone using small numbers really need to know the infinite structure of arithmetic? [Shapiro]