display all the ideas for this combination of texts
6 ideas
| 3338 | Numbers have been defined in terms of 'successors' to the concept of 'zero' [Peano, by Blackburn] |
| Full Idea: Dedekind and Peano define the number series as the series of successors to the number zero, according to five postulates. | |
| From: report of Giuseppe Peano (works [1890]) by Simon Blackburn - Oxford Dictionary of Philosophy p.279 |
| 5897 | 0 is a non-successor number, all successors are numbers, successors can't duplicate, if P(n) and P(n+1) then P(all-n) [Peano, by Flew] |
| Full Idea: 1) 0 is a number; 2) The successor of any number is a number; 3) No two numbers have the same successor; 4) 0 is not the successor of any number; 5) If P is true of 0, and if P is true of any number n and of its successor, P is true of every number. | |
| From: report of Giuseppe Peano (works [1890]) by Antony Flew - Pan Dictionary of Philosophy 'Peano' | |
| A reaction: Devised by Dedekind and proposed by Peano, these postulates were intended to avoid references to intuition in specifying the natural numbers. I wonder if they could define 'successor' without reference to 'number'. |
| 17843 | The idea of 'one' is the foundation of number [Aristotle] |
| Full Idea: One is the principle of number qua number. | |
| From: Aristotle (Metaphysics [c.324 BCE], 1052b21) |
| 17850 | Each many is just ones, and is measured by the one [Aristotle] |
| Full Idea: The reason for saying of each number that it is many is just that it is ones and that each number is measured by the one. | |
| From: Aristotle (Metaphysics [c.324 BCE], 1056b16) |
| 17851 | Number is plurality measured by unity [Aristotle] |
| Full Idea: Number is plurality as measured by unity. | |
| From: Aristotle (Metaphysics [c.324 BCE], 1057a04) |
| 9793 | Mathematics studies abstracted relations, commensurability and proportion [Aristotle] |
| Full Idea: Mathematicians abstract perceptible features to study quantity and continuity ...and examine the mutual relations of some and the features of those relations, and commensurabilities of others, and of yet others the proportions. | |
| From: Aristotle (Metaphysics [c.324 BCE], 1061a32) | |
| A reaction: This sounds very much like the intuition of structuralism to me - that the subject is entirely about relations between things, with very little interest in the things themselves. See Aristotle on abstraction (under 'Thought'). |