Combining Philosophers
Ideas for H.Putnam/P.Oppenheim, Fred Sommers and Stewart Shapiro
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54 ideas
6. Mathematics / A. Nature of Mathematics / 1. Mathematics
10201
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Virtually all of mathematics can be modeled in set theory [Shapiro]
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6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / b. Types of number
13641
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Complex numbers can be defined as reals, which are defined as rationals, then integers, then naturals [Shapiro]
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8763
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The number 3 is presumably identical as a natural, an integer, a rational, a real, and complex [Shapiro]
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6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / d. Natural numbers
13676
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Only higher-order languages can specify that 0,1,2,... are all the natural numbers that there are [Shapiro]
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6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / e. Ordinal numbers
13677
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Natural numbers are the finite ordinals, and integers are equivalence classes of pairs of finite ordinals [Shapiro]
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6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / g. Real numbers
10213
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Real numbers are thought of as either Cauchy sequences or Dedekind cuts [Shapiro]
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18243
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Understanding the real-number structure is knowing usage of the axiomatic language of analysis [Shapiro]
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6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / h. Reals from Cauchy
18249
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Cauchy gave a formal definition of a converging sequence. [Shapiro]
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6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / i. Reals from cuts
18245
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Cuts are made by the smallest upper or largest lower number, some of them not rational [Shapiro]
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6. Mathematics / A. Nature of Mathematics / 5. The Infinite / g. Continuum Hypothesis
13652
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The 'continuum' is the cardinality of the powerset of a denumerably infinite set [Shapiro]
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6. Mathematics / B. Foundations for Mathematics / 1. Foundations for Mathematics
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
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 / 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
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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6. Mathematics / C. Sources of Mathematics / 1. Mathematical Platonism / a. For mathematical platonism
10200
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We distinguish realism 'in ontology' (for objects), and 'in truth-value' (for being either true or false) [Shapiro]
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10210
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If mathematical objects are accepted, then a number of standard principles will follow [Shapiro]
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10215
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Platonists claim we can state the essence of a number without reference to the others [Shapiro]
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10233
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Platonism must accept that the Peano Axioms could all be false [Shapiro]
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6. Mathematics / C. Sources of Mathematics / 2. Intuition of Mathematics
10244
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Intuition is an outright hindrance to five-dimensional geometry [Shapiro]
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6. Mathematics / C. Sources of Mathematics / 4. Mathematical Empiricism / a. Mathematical empiricism
10280
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A stone is a position in some pattern, and can be viewed as an object, or as a location [Shapiro]
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6. Mathematics / C. Sources of Mathematics / 6. Logicism / c. Neo-logicism
13664
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Logicism is distinctive in seeking a universal language, and denying that logic is a series of abstractions [Shapiro]
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6. Mathematics / C. Sources of Mathematics / 6. Logicism / d. Logicism critique
13625
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Mathematics and logic have no border, and logic must involve mathematics and its ontology [Shapiro]
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8744
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Logicism seems to be a non-starter if (as is widely held) logic has no ontology of its own [Shapiro]
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6. Mathematics / C. Sources of Mathematics / 7. Formalism
8749
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Term Formalism says mathematics is just about symbols - but real numbers have no names [Shapiro]
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8750
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Game Formalism is just a matter of rules, like chess - but then why is it useful in science? [Shapiro]
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8752
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Deductivism says mathematics is logical consequences of uninterpreted axioms [Shapiro]
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6. Mathematics / C. Sources of Mathematics / 10. Constructivism / a. Constructivism
10254
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Can the ideal constructor also destroy objects? [Shapiro]
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10255
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Presumably nothing can block a possible dynamic operation? [Shapiro]
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6. Mathematics / C. Sources of Mathematics / 10. Constructivism / b. Intuitionism
8753
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Critics resent the way intuitionism cripples mathematics, but it allows new important distinctions [Shapiro]
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6. Mathematics / C. Sources of Mathematics / 10. Constructivism / c. Conceptualism
8731
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Conceptualist are just realists or idealist or nominalists, depending on their view of concepts [Shapiro]
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6. Mathematics / C. Sources of Mathematics / 10. Constructivism / d. Predicativism
13663
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Some reject formal properties if they are not defined, or defined impredicatively [Shapiro]
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8730
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'Impredicative' definitions refer to the thing being described [Shapiro]
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