11 ideas
12427 | All of mathematics is properties of the whole numbers [Kronecker] |
Full Idea: All the results of significant mathematical research must ultimately be expressible in the simple forms of properties of whole numbers. | |
From: Leopold Kronecker (works [1885], Vol 3/274), quoted by Philip Kitcher - The Nature of Mathematical Knowledge 09.5 | |
A reaction: I've always liked Kronecker's line, but I'm beginning to realise that his use of the word 'number' is simply out-of-date. Natural numbers have a special status, but not sufficient to support this claim. |
10091 | God made the integers, all the rest is the work of man [Kronecker] |
Full Idea: God made the integers, all the rest is the work of man. | |
From: Leopold Kronecker (works [1885]), quoted by A.George / D.J.Velleman - Philosophies of Mathematics Intro | |
A reaction: This famous remark was first quoted in Kronecker's obituary. A response to Dedekind, it seems. See Idea 10090. Did he really mean that negative numbers were the work of God? We took a long time to spot them. |
19718 | Indefeasibility does not imply infallibility [Grundmann] |
Full Idea: Infallibility does not follow from indefeasibility. | |
From: Thomas Grundmann (Defeasibility Theory [2011], 'Significance') | |
A reaction: If very little evidence exists then this could clearly be the case. It is especially true of historical and archaeological evidence. |
19717 | Can a defeater itself be defeated? [Grundmann] |
Full Idea: Can the original justification of a belief be regained through a successful defeat of a defeater? | |
From: Thomas Grundmann (Defeasibility Theory [2011], 'Defeater-Defs') | |
A reaction: [Jäger 2005 addresses this] I would have thought the answer is yes. I aspire to coherent justifications, so I don't see justifications as a chain of defeat and counter-defeat, but as collective groups of support and challenge. |
19716 | Simple reliabilism can't cope with defeaters of reliably produced beliefs [Grundmann] |
Full Idea: An unmodified reliabilism does not accommodate defeaters, and surely there can be defeaters against reliably produced beliefs? | |
From: Thomas Grundmann (Defeasibility Theory [2011], 'Defeaters') | |
A reaction: [He cites Bonjour 1980] Reliabilism has plenty of problems anyway, since a generally reliable process can obviously occasionally produce a bad result. 20:20 vision is not perfect vision. Internalist seem to like defeaters. |
19715 | You can 'rebut' previous beliefs, 'undercut' the power of evidence, or 'reason-defeat' the truth [Grundmann] |
Full Idea: There are 'rebutting' defeaters against the truth of a previously justified belief, 'undercutting' defeaters against the power of the evidence, and 'reason-defeating' defeaters against the truth of the reason for the belief. | |
From: Thomas Grundmann (Defeasibility Theory [2011], 'How') | |
A reaction: That is (I think) that you can defeat the background, the likelihood, or the truth. He cites Pollock 1986, and implies that these are standard distinctions about defeaters. |
19713 | Defeasibility theory needs to exclude defeaters which are true but misleading [Grundmann] |
Full Idea: Advocates of the defeasibility theory have tried to exclude true pieces of information that are misleading defeaters. | |
From: Thomas Grundmann (Defeasibility Theory [2011], 'What') | |
A reaction: He gives as an example the genuine news of a claim that the suspect has a twin. |
19714 | Knowledge requires that there are no facts which would defeat its justification [Grundmann] |
Full Idea: The 'defeasibility theory' of knowledge claims that knowledge is only present if there are no facts that - if they were known - would be genuine defeaters of the relevant justification. | |
From: Thomas Grundmann (Defeasibility Theory [2011], 'What') | |
A reaction: Something not right here. A genuine defeater would ensure the proposition was false, so it would simply fail the truth test. So we need a 'defeater' for a truth, which must therefore by definition be misleading. Many qualifications have to be invoked. |
19719 | 'Moderate' foundationalism has basic justification which is defeasible [Grundmann] |
Full Idea: Theories that combine basic justification with the defeasibility of this justification are referred to as 'moderate' foundationalism. | |
From: Thomas Grundmann (Defeasibility Theory [2011], 'Significance') | |
A reaction: I could be more sympathetic to this sort of foundationalism. But it begins to sound more like Neurath's boat (see Quine) than like Descartes' metaphor of building foundations. |
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. |