Combining Philosophers

Ideas for Rescher,N/Oppenheim,P, Stewart Shapiro and David Galloway

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54 ideas

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