Combining Philosophers
Ideas for Anaxarchus, Stewart Shapiro and John Mayberry
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35 ideas
6. Mathematics / B. Foundations for Mathematics / 1. Foundations for Mathematics
17775
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If proof and definition are central, then mathematics needs and possesses foundations [Mayberry]
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17776
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The ultimate principles and concepts of mathematics are presumed, or grasped directly [Mayberry]
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17777
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Foundations need concepts, definition rules, premises, and proof rules [Mayberry]
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17804
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Axiom theories can't give foundations for mathematics - that's using axioms to explain axioms [Mayberry]
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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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8764
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Categories are the best foundation for mathematics [Shapiro]
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6. Mathematics / B. Foundations for Mathematics / 2. Proof in Mathematics
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
17792
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1st-order PA is only interesting because of results which use 2nd-order PA [Mayberry]
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13657
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First-order arithmetic can't even represent basic number theory [Shapiro]
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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 / 4. Axioms for Number / e. Peano arithmetic 2nd-order
10294
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Second-order logic has the expressive power for mathematics, but an unworkable model theory [Shapiro]
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6. Mathematics / B. Foundations for Mathematics / 4. Axioms for Number / g. Incompleteness of Arithmetic
17793
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It is only 2nd-order isomorphism which suggested first-order PA completeness [Mayberry]
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6. Mathematics / B. Foundations for Mathematics / 5. Definitions of Number / b. Greek arithmetic
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 / 5. Definitions of Number / f. Zermelo numbers
8762
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Two definitions of 3 in terms of sets disagree over whether 1 is a member of 3 [Shapiro]
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6. Mathematics / B. Foundations for Mathematics / 6. Mathematics as Set Theory / a. Mathematics is set theory
17794
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Set theory is not just first-order ZF, because that is inadequate for mathematics [Mayberry]
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17802
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We don't translate mathematics into set theory, because it comes embodied in that way [Mayberry]
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17805
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Set theory is not just another axiomatised part of mathematics [Mayberry]
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13656
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Some sets of natural numbers are definable in set-theory but not in arithmetic [Shapiro]
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6. Mathematics / B. Foundations for Mathematics / 6. Mathematics as Set Theory / b. Mathematics is not set theory
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
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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8760
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Numbers do not exist independently; the essence of a number is its relations to other numbers [Shapiro]
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8761
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A 'system' is related objects; a 'pattern' or 'structure' abstracts the pure relations from them [Shapiro]
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6. Mathematics / B. Foundations for Mathematics / 7. Mathematical Structuralism / b. Varieties of structuralism
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
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
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
10274
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Does someone using small numbers really need to know the infinite structure of arithmetic? [Shapiro]
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