display all the ideas for this combination of texts
8 ideas
| 14124 | Axiom of Archimedes: a finite multiple of a lesser magnitude can always exceed a greater [Russell] |
| Full Idea: The Axiom of Archimedes asserts that, given any two magnitudes of a kind, some finite multiple of the lesser exceeds the greater. | |
| From: Bertrand Russell (The Principles of Mathematics [1903], §168 n*) |
| 7530 | Russell tried to replace Peano's Postulates with the simple idea of 'class' [Russell, by Monk] |
| Full Idea: What Russell tried to show [at this time] was that Peano's Postulates (based on 'zero', 'number' and 'successor') could in turn be dispensed with, and the whole edifice built upon nothing more than the notion of 'class'. | |
| From: report of Bertrand Russell (The Principles of Mathematics [1903]) by Ray Monk - Bertrand Russell: Spirit of Solitude Ch.4 | |
| A reaction: (See Idea 5897 for Peano) Presumably you can't afford to lose the notion of 'successor' in the account. If you build any theory on the idea of classes, you are still required to explain why a particular is a member of that class, and not another. |
| 18246 | Dedekind failed to distinguish the numbers from other progressions [Shapiro on Russell] |
| Full Idea: Dedekind's demonstrations nowhere - not even where he comes to cardinals - involve any property distinguishing numbers from other progressions. | |
| From: comment on Bertrand Russell (The Principles of Mathematics [1903], p.249) by Stewart Shapiro - Philosophy of Mathematics 5.4 | |
| A reaction: Shapiro notes that his sounds like Frege's Julius Caesar problem, of ensuring that your definition really does capture a number. Russell is objecting to mathematical structuralism. |
| 14147 | Denying mathematical induction gave us the transfinite [Russell] |
| Full Idea: The transfinite was obtained by denying mathematical induction. | |
| From: Bertrand Russell (The Principles of Mathematics [1903], §310) | |
| A reaction: This refers to the work of Dedekind and Cantor. This raises the question (about which thinkers have ceased to care, it seems), of whether it is rational to deny mathematical induction. |
| 14125 | Finite numbers, unlike infinite numbers, obey mathematical induction [Russell] |
| Full Idea: Finite numbers obey the law of mathematical induction: infinite numbers do not. | |
| From: Bertrand Russell (The Principles of Mathematics [1903], §183) |
| 14116 | Numbers were once defined on the basis of 1, but neglected infinities and + [Russell] |
| Full Idea: It used to be common to define numbers by means of 1, with 2 being 1+1 and so on. But this method was only applicable to finite numbers, made a tiresome different between 1 and the other numbers, and left + unexplained. | |
| From: Bertrand Russell (The Principles of Mathematics [1903], §109) | |
| A reaction: Am I alone in hankering after the old approach? The idea of a 'unit' is what connected numbers to the patterns of the world. Russell's approach invites unneeded platonism. + is just 'and', and infinities are fictional extrapolations. Sounds fine to me. |
| 14117 | Numbers are properties of classes [Russell] |
| Full Idea: Numbers are to be regarded as properties of classes. | |
| From: Bertrand Russell (The Principles of Mathematics [1903], §109) | |
| A reaction: If properties are then defined extensionally as classes, you end up with numbers as classes of classes. |
| 9977 | Ordinals can't be defined just by progression; they have intrinsic qualities [Russell] |
| Full Idea: It is impossible that the ordinals should be, as Dedekind suggests, nothing but the terms of such relations as constitute a progression. If they are anything at all, they must be intrinsically something. | |
| From: Bertrand Russell (The Principles of Mathematics [1903], §242) | |
| A reaction: This is the obvious platonist response to the incipient doctrine of structuralism. We have a chicken-and-egg problem. Bricks need intrinsic properties to make a structure. A structure isomorphic to numbers is not thereby the numbers. |