7 ideas
| 6343 | For Russell, both propositions and facts are arrangements of objects, so obviously they correspond [Horwich on Russell] |
| Full Idea: Given Russell's notion of a proposition, as an arrangement of objects and properties, it is hard to see how there could be any difference at all between such a proposition and the fact corresponding to it, since they each involve the same arrangement. | |
| From: comment on Bertrand Russell (On the Nature of Truth and Falsehood [1910]) by Paul Horwich - Truth (2nd edn) Ch.7.35 | |
| A reaction: This seems a little unfair, given that Russell (in 1912) uses the notion now referred to as 'congruence', so that the correspondence is not in the objects and properties, but in how they are 'ordered', which may differ between proposition and fact. |
| 17832 | Zermelo showed that the ZF axioms in 1930 were non-categorical [Zermelo, by Hallett,M] |
| Full Idea: Zermelo's paper sets out to show that the standard set-theoretic axioms (what he calls the 'constitutive axioms', thus the ZF axioms minus the axiom of infinity) have an unending sequence of different models, thus that they are non-categorical. | |
| From: report of Ernst Zermelo (On boundary numbers and domains of sets [1930]) by Michael Hallett - Introduction to Zermelo's 1930 paper p.1209 | |
| A reaction: Hallett says later that Zermelo is working with second-order set theory. The addition of an Axiom of Infinity seems to have aimed at addressing the problem, and the complexities of that were pursued by Gödel. |
| 13028 | Replacement was added when some advanced theorems seemed to need it [Zermelo, by Maddy] |
| Full Idea: Zermelo included Replacement in 1930, after it was noticed that the sequence of power sets was needed, and Replacement gave the ordinal form of the well-ordering theorem, and justification for transfinite recursion. | |
| From: report of Ernst Zermelo (On boundary numbers and domains of sets [1930]) by Penelope Maddy - Believing the Axioms I §1.8 | |
| A reaction: Maddy says that this axiom suits the 'limitation of size' theorists very well, but is not so good for the 'iterative conception'. |
| 17626 | The antinomy of endless advance and of completion is resolved in well-ordered transfinite numbers [Zermelo] |
| Full Idea: Two opposite tendencies of thought, the idea of creative advance and of collection and completion (underlying the Kantian 'antinomies') find their symbolic representation and their symbolic reconciliation in the transfinite numbers based on well-ordering. | |
| From: Ernst Zermelo (On boundary numbers and domains of sets [1930], §5) | |
| A reaction: [a bit compressed] It is this sort of idea, from one of the greatest set-theorists, that leads philosophers to think that the philosophy of mathematics may offer solutions to metaphysical problems. As an outsider, I am sceptical. |
| 7534 | In 1906, Russell decided that propositions did not, after all, exist [Russell, by Monk] |
| Full Idea: With a characteristic readiness to abandon views that he had previously considered definitively correct, Russell declared in 1906 that there were, after all, no such 'things' as propositions. It is judgements that are true or false. | |
| From: report of Bertrand Russell (On the Nature of Truth and Falsehood [1910]) by Ray Monk - Bertrand Russell: Spirit of Solitude Ch.6 | |
| A reaction: Written 1906. Russell developed a 'multiple relation theory of judgement'. But if a judgement is an assessment of truth or falsehood, what is it that is being assessed? |
| 1748 | Archelaus was the first person to say that the universe is boundless [Archelaus, by Diog. Laertius] |
| Full Idea: Archelaus was the first person to say that the universe is boundless. | |
| From: report of Archelaus (fragments/reports [c.450 BCE]) by Diogenes Laertius - Lives of Eminent Philosophers 02.Ar.3 |
| 5989 | Archelaus said life began in a primeval slime [Archelaus, by Schofield] |
| Full Idea: Archelaus wrote that life on Earth began in a primeval slime. | |
| From: report of Archelaus (fragments/reports [c.450 BCE]) by Malcolm Schofield - Archelaus | |
| A reaction: This sounds like a fairly clearcut assertion of the production of life by evolution. Darwin's contribution was to propose the mechanism for achieving it. We should honour the name of Archelaus for this idea. |