Combining Philosophers

Ideas for Anaxarchus, Stewart Shapiro and John Mayberry

expand these ideas     |    start again     |     choose another area for these philosophers

display all the ideas for this combination of philosophers

15 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
17784 Real numbers can be eliminated, by axiom systems for complete ordered fields [Mayberry]
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 / 4. Using Numbers / b. Quantity
17782 Greek quantities were concrete, and ratio and proportion were their science [Mayberry]
17781 Real numbers were invented, as objects, to simplify and generalise 'quantity' [Mayberry]
6. Mathematics / A. Nature of Mathematics / 5. The Infinite / a. The Infinite
17799 Cantor's infinite is an absolute, of all the sets or all the ordinal numbers [Mayberry]
17797 Cantor extended the finite (rather than 'taming the infinite') [Mayberry]
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]