10 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) |
14804 | Is chance just unknown laws? But the laws operate the same, whatever chance occurs [Peirce] |
Full Idea: Chance is the name for some law that is unknown to us? If you say 'each die moves under the influence of precise mechanical laws', it seems to me it is not these laws which made the tie turn up sixes, for the laws act the same when other throws come up. | |
From: Charles Sanders Peirce (The Doctrine of Necessity Examined [1892], p.333) |
14805 | Is there any such thing as death among the lower organisms? [Peirce] |
Full Idea: Among some of the lower organisms, it is a moot point with biologists whether there be anything which ought to be called death. | |
From: Charles Sanders Peirce (The Doctrine of Necessity Examined [1892], p.334) | |
A reaction: The point, presumably, is that one phase of an organisms moves into another, and the 'individuals' are not distinct enough for their 'death' to be a significant transition. A nicely mind-expanding thought. |
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 |
14806 | If the world is just mechanical, its whole specification has no more explanation than mere chance [Peirce] |
Full Idea: The mechanical philosopher leaves the whole specification of the world utterly unaccounted for, which is pretty nearly as bad as to baldly attribute it to chance. | |
From: Charles Sanders Peirce (The Doctrine of Necessity Examined [1892], p.337) | |
A reaction: If now complete is even remotely available, then that doesn't seem to matter too much, but if there is one message modern physics teaches philosophy, it is that we should not give up on trying to answer the deeper questions. |
14803 | The more precise the observations, the less reliable appear to be the laws of nature [Peirce] |
Full Idea: Try to verify any law of nature, and you will find that the more precise your observations, the more certain they will be to show irregular departures from the law. | |
From: Charles Sanders Peirce (The Doctrine of Necessity Examined [1892], p.331) | |
A reaction: This nicely encapsulates modern doubts about whether the so-called 'laws' of nature actually capture what is going on in the real world. |
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. |