7 ideas
18085 | Values that approach zero, becoming less than any quantity, are 'infinitesimals' [Cauchy] |
Full Idea: When the successive absolute values of a variable decrease indefinitely in such a way as to become less than any given quantity, that variable becomes what is called an 'infinitesimal'. Such a variable has zero as its limit. | |
From: Augustin-Louis Cauchy (Cours d'Analyse [1821], p.19), quoted by Philip Kitcher - The Nature of Mathematical Knowledge 10.4 | |
A reaction: The creator of the important idea of the limit still talked in terms of infinitesimals. In the next generation the limit took over completely. |
18084 | When successive variable values approach a fixed value, that is its 'limit' [Cauchy] |
Full Idea: When the values successively attributed to the same variable approach indefinitely a fixed value, eventually differing from it by as little as one could wish, that fixed value is called the 'limit' of all the others. | |
From: Augustin-Louis Cauchy (Cours d'Analyse [1821], p.19), quoted by Philip Kitcher - The Nature of Mathematical Knowledge 10.4 | |
A reaction: This seems to be a highly significan proposal, because you can now treat that limit as a number, and adds things to it. It opens the door to Cantor's infinities. Is the 'limit' just a fiction? |
4577 | There is no necessity higher than natural necessity, and that is just regularity [Quine] |
Full Idea: In principle I see no higher or more austere necessity than natural necessity; and in natural necessity, or our attribution of it, I see only Hume's regularities | |
From: Willard Quine (Necessary Truth [1963], p.76) | |
A reaction: Presumably this allows logical necessity as a 'lower' necessity, but denies 'metaphysical' necessity, in line with Hume and other tough empiricists. Personally I adore metaphysical necessities, but they are a bit hard to establish conclusively. |
8830 | A belief can be justified when the person has forgotten the evidence for it [Goldman] |
Full Idea: A characteristic case in which a belief is justified though the cognizer doesn't know that it's justified is where the original evidence for the belief has long since been forgotten. | |
From: Alvin I. Goldman (What is Justified Belief? [1976], II) | |
A reaction: This is a central problem for any very literal version of internalism. The fully rationalist view (to which I incline) will be that the cognizer must make a balanced assessment of whether they once had the evidence. Were my teachers any good? |
8832 | If justified beliefs are well-formed beliefs, then animals and young children have them [Goldman] |
Full Idea: If one shares my view that justified belief is, at least roughly, well-formed belief, surely animals and young children can have justified beliefs. | |
From: Alvin I. Goldman (What is Justified Belief? [1976], III) | |
A reaction: I take this to be a key hallmark of the externalist view of knowledge. Personally I think we should tell the animals that they have got true beliefs, but that they aren't bright enough to aspire to 'knowledge'. Be grateful for what you've got. |
8829 | Justification depends on the reliability of its cause, where reliable processes tend to produce truth [Goldman] |
Full Idea: The justificational status of a belief is a function of the reliability of the processes that cause it, where (provisionally) reliability consists in the tendency of a process to produce beliefs that are true rather than false. | |
From: Alvin I. Goldman (What is Justified Belief? [1976], II) | |
A reaction: Goldman's original first statement of reliabilism, now the favourite version of externalism. The obvious immediate problem is when a normally very reliable process goes wrong. Wise people still get it wrong, or right for the wrong reasons. |
8831 | Introspection is really retrospection; my pain is justified by a brief causal history [Goldman] |
Full Idea: Introspection should be regarded as a form of retrospection. Thus, a justified belief that I am 'now' in pain gets its justificational status from a relevant, though brief, causal history. | |
From: Alvin I. Goldman (What is Justified Belief? [1976], II) | |
A reaction: He cites Hobbes and Ryle as having held this view. See Idea 6668. I am unclear why the history must be 'causal'. I may not know the cause of the pain. I may not believe an event which causes a proposition, or I may form a false belief from it. |