Combining Philosophers
Ideas for Rescher,N/Oppenheim,P, Stewart Shapiro and Danielle Macbeth
expand these ideas
|
start again
|
choose
another area for these philosophers
display all the ideas for this combination of philosophers
10 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]
|