display all the ideas for this combination of texts
7 ideas
3332 | Greeks saw the science of proportion as the link between geometry and arithmetic [Benardete,JA] |
Full Idea: The Greeks saw the independent science of proportion as the link between geometry and arithmetic. | |
From: José A. Benardete (Metaphysics: the logical approach [1989], Ch.15) |
3330 | Negatives, rationals, irrationals and imaginaries are all postulated to solve baffling equations [Benardete,JA] |
Full Idea: The Negative numbers are postulated (magic word) to solve x=5-8, Rationals postulated to solve 2x=3, Irrationals for x-squared=2, and Imaginaries for x-squared=-1. (…and Zero for x=5-5) …and x/0 remains eternally open. | |
From: José A. Benardete (Metaphysics: the logical approach [1989], Ch.14) |
3337 | Natural numbers are seen in terms of either their ordinality (Peano), or cardinality (set theory) [Benardete,JA] |
Full Idea: One approaches the natural numbers in terms of either their ordinality (Peano), or cardinality (set theory). | |
From: José A. Benardete (Metaphysics: the logical approach [1989], Ch.17) |
13641 | Complex numbers can be defined as reals, which are defined as rationals, then integers, then naturals [Shapiro] |
Full Idea: 'Definitions' of integers as pairs of naturals, rationals as pairs of integers, reals as Cauchy sequences of rationals, and complex numbers as pairs of reals are reductive foundations of various fields. | |
From: Stewart Shapiro (Foundations without Foundationalism [1991], 2.1) | |
A reaction: On p.30 (bottom) Shapiro objects that in the process of reduction the numbers acquire properties they didn't have before. |
13676 | Only higher-order languages can specify that 0,1,2,... are all the natural numbers that there are [Shapiro] |
Full Idea: The main problem of characterizing the natural numbers is to state, somehow, that 0,1,2,.... are all the numbers that there are. We have seen that this can be accomplished with a higher-order language, but not in a first-order language. | |
From: Stewart Shapiro (Foundations without Foundationalism [1991], 9.1.4) |
13677 | Natural numbers are the finite ordinals, and integers are equivalence classes of pairs of finite ordinals [Shapiro] |
Full Idea: By convention, the natural numbers are the finite ordinals, the integers are certain equivalence classes of pairs of finite ordinals, etc. | |
From: Stewart Shapiro (Foundations without Foundationalism [1991], 9.3) |
13652 | The 'continuum' is the cardinality of the powerset of a denumerably infinite set [Shapiro] |
Full Idea: The 'continuum' is the cardinality of the powerset of a denumerably infinite set. | |
From: Stewart Shapiro (Foundations without Foundationalism [1991], 5.1.2) |