9 ideas
| 17879 | Axiomatising set theory makes it all relative [Skolem] |
| Full Idea: Axiomatising set theory leads to a relativity of set-theoretic notions, and this relativity is inseparably bound up with every thoroughgoing axiomatisation. | |
| From: Thoralf Skolem (Remarks on axiomatised set theory [1922], p.296) |
| 13536 | Skolem did not believe in the existence of uncountable sets [Skolem] |
| Full Idea: Skolem did not believe in the existence of uncountable sets. | |
| From: Thoralf Skolem (works [1920], 5.3) | |
| A reaction: Kit Fine refers somewhere to 'unrepentent Skolemites' who still hold this view. |
| 17878 | If a 1st-order proposition is satisfied, it is satisfied in a denumerably infinite domain [Skolem] |
| Full Idea: Löwenheim's theorem reads as follows: If a first-order proposition is satisfied in any domain at all, it is already satisfied in a denumerably infinite domain. | |
| From: Thoralf Skolem (Remarks on axiomatised set theory [1922], p.293) |
| 17880 | Integers and induction are clear as foundations, but set-theory axioms certainly aren't [Skolem] |
| Full Idea: The initial foundations should be immediately clear, natural and not open to question. This is satisfied by the notion of integer and by inductive inference, by it is not satisfied by the axioms of Zermelo, or anything else of that kind. | |
| From: Thoralf Skolem (Remarks on axiomatised set theory [1922], p.299) | |
| A reaction: This is a plea (endorsed by Almog) that the integers themselves should be taken as primitive and foundational. I would say that the idea of successor is more primitive than the integers. |
| 17881 | Mathematician want performable operations, not propositions about objects [Skolem] |
| Full Idea: Most mathematicians want mathematics to deal, ultimately, with performable computing operations, and not to consist of formal propositions about objects called this or that. | |
| From: Thoralf Skolem (Remarks on axiomatised set theory [1922], p.300) |
| 16597 | Quantity is the capacity to be divided [Digby] |
| Full Idea: Quantity …is divisibility, or a capacity to be divided into parts. | |
| From: Kenelm Digby (Two treatises [1644], I.2.8), quoted by Robert Pasnau - Metaphysical Themes 1274-1671 04.1 | |
| A reaction: 'Quantity' is scholastic philosophy is a concept we no longer possess. Without quantity, a thing might potentially exist at a spaceless point. Quantity is what spreads things out. See Pasnau Ch. 4. |
| 579 | Cratylus said you couldn't even step into the same river once [Cratylus, by Aristotle] |
| Full Idea: Cratylus was appalled that Heraclitus said you could not step twice into the same river, because it was already going too far to admit stepping into the same river once. | |
| From: report of Cratylus (fragments/reports [c.425 BCE]) by Aristotle - Metaphysics 1010a | |
| A reaction: Compare Idea 427. |
| 578 | Cratylus decided speech was hopeless, and his only expression was the movement of a finger [Cratylus, by Aristotle] |
| Full Idea: Cratylus thought speech of any kind was radically inappropriate and that expression should be restricted exclusively to the movement of the finger. | |
| From: report of Cratylus (fragments/reports [c.425 BCE]) by Aristotle - Metaphysics 1010a |
| 16731 | Colours arise from the rarity, density and mixture of matter [Digby] |
| Full Idea: The origin of all colours in bodies is plainly deduced out of the various degrees of rarity and density, variously mixed and compounded. | |
| From: Kenelm Digby (Two treatises [1644], I.29.4), quoted by Robert Pasnau - Metaphysical Themes 1274-1671 22.5 | |
| A reaction: We are still struggling with this question, though I think the picture is gradually become clear, once you get the hang of the brain. Easy! See Idea 17396. |