6 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) |
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) |
11052 | Psychological logic can't distinguish justification from causes of a belief [Frege] |
Full Idea: With the psychological conception of logic we lose the distinction between the grounds that justify a conviction and the causes that actually produce it. | |
From: Gottlob Frege (Logic [1897] [1897]) | |
A reaction: Thus Frege kicked the causal theory of justification well into touch long before it had even been properly formulated. That is not to say that there is no psychological aspect to logic, because there is. |
7903 | The six perfections are giving, morality, patience, vigour, meditation, and wisdom [Nagarjuna] |
Full Idea: The six perfections are of giving, morality, patience, vigour, meditation, and wisdom. | |
From: Nagarjuna (Mahaprajnaparamitashastra [c.120], 88) | |
A reaction: What is 'morality', if giving is not part of it? I like patience and vigour being two of the virtues, which immediately implies an Aristotelian mean (which is always what is 'appropriate'). |