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'Tropes', 'Philosophy of Mathematics' and 'On Hippocrates and Plato'
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19 ideas
6. Mathematics / B. Foundations for Mathematics / 1. Foundations for Mathematics
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10236
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There is no grounding for mathematics that is more secure than mathematics [Shapiro]
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6. Mathematics / B. Foundations for Mathematics / 2. Proof in Mathematics
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10256
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For intuitionists, proof is inherently informal [Shapiro]
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6. Mathematics / B. Foundations for Mathematics / 4. Axioms for Number / d. Peano arithmetic
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10202
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Natural numbers just need an initial object, successors, and an induction principle [Shapiro]
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6. Mathematics / B. Foundations for Mathematics / 5. Definitions of Number / b. Greek arithmetic
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10205
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Mathematics originally concerned the continuous (geometry) and the discrete (arithmetic) [Shapiro]
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6. Mathematics / B. Foundations for Mathematics / 6. Mathematics as Set Theory / b. Mathematics is not set theory
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10222
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Mathematical foundations may not be sets; categories are a popular rival [Shapiro]
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6. Mathematics / B. Foundations for Mathematics / 7. Mathematical Structuralism / a. Structuralism
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10218
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Baseball positions and chess pieces depend entirely on context [Shapiro]
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10224
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The even numbers have the natural-number structure, with 6 playing the role of 3 [Shapiro]
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10228
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Could infinite structures be apprehended by pattern recognition? [Shapiro]
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10230
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The 4-pattern is the structure common to all collections of four objects [Shapiro]
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10249
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The main mathematical structures are algebraic, ordered, and topological [Shapiro]
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10273
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Some structures are exemplified by both abstract and concrete [Shapiro]
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10276
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Mathematical structures are defined by axioms, or in set theory [Shapiro]
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6. Mathematics / B. Foundations for Mathematics / 7. Mathematical Structuralism / b. Varieties of structuralism
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10270
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The main versions of structuralism are all definitionally equivalent [Shapiro]
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6. Mathematics / B. Foundations for Mathematics / 7. Mathematical Structuralism / c. Nominalist structuralism
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10221
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Is there is no more to structures than the systems that exemplify them? [Shapiro]
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10248
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Number statements are generalizations about number sequences, and are bound variables [Shapiro]
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6. Mathematics / B. Foundations for Mathematics / 7. Mathematical Structuralism / d. Platonist structuralism
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10220
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Because one structure exemplifies several systems, a structure is a one-over-many [Shapiro]
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10223
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There is no 'structure of all structures', just as there is no set of all sets [Shapiro]
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8703
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Shapiro's structuralism says model theory (comparing structures) is the essence of mathematics [Shapiro, by Friend]
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6. Mathematics / B. Foundations for Mathematics / 7. Mathematical Structuralism / e. Structuralism critique
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10274
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Does someone using small numbers really need to know the infinite structure of arithmetic? [Shapiro]
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