display all the ideas for this combination of texts
6 ideas
| 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) |
| 10553 | A number is a class of classes of the same cardinality [Frege, by Dummett] |
| Full Idea: For Frege, in 'Grundgesetze', a number is a class of classes of the same cardinality. | |
| From: report of Gottlob Frege (Grundgesetze der Arithmetik 2 (Basic Laws) [1903]) by Michael Dummett - Frege Philosophy of Language (2nd ed) Ch.14 |
| 10020 | Frege's biggest error is in not accounting for the senses of number terms [Hodes on Frege] |
| Full Idea: The inconsistency of Grundgesetze was only a minor flaw. Its fundamental flaw was its inability to account for the way in which the senses of number terms are determined. It leaves the reference-magnetic nature of the standard numberer a mystery. | |
| From: comment on Gottlob Frege (Grundgesetze der Arithmetik 2 (Basic Laws) [1903]) by Harold Hodes - Logicism and Ontological Commits. of Arithmetic p.139 | |
| A reaction: A point also made by Hofweber. As a logician, Frege was only concerned with the inferential role of number terms, and he felt he had captured their logical form, but it is when you come to look at numbers in natural language that he seem in trouble. |
| 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'). |