10 ideas
| 18189 | ZFC could contain a contradiction, and it can never prove its own consistency [MacLane] |
| Full Idea: We have at hand no proof that the axioms of ZFC for set theory will never yield a contradiction, while Gödel's second theorem tells us that such a consistency proof cannot be conducted within ZFC. | |
| From: Saunders MacLane (Mathematics: Form and Function [1986], p.406), quoted by Penelope Maddy - Naturalism in Mathematics | |
| A reaction: Maddy quotes this, while defending set theory as the foundation of mathematics, but it clearly isn't the most secure foundation that could be devised. She says the benefits of set theory do not need guaranteed consistency (p.30). |
| 19690 | 'Grasping' a structure seems to be modal, because we must anticipate its behaviour [Grimm] |
| Full Idea: 'Graspng' a structure would seem to bring into play something like a modal sense or ability, not just to register how things are, but also to anticipate how certain elements of the system would behave. | |
| From: Stephen R. Grimm (Understanding [2011], 2) | |
| A reaction: In the case of the chronology of some historical events, talking of 'grasping' or 'understanding' seems wrong because the facts are static and invariant. That seems to support the present idea. But you might 'understand' a pattern if you can reproduce it. |
| 19691 | Unlike knowledge, you can achieve understanding through luck [Grimm] |
| Full Idea: It may be that understanding is compatible with luck, in a way that knowledge is not. | |
| From: Stephen R. Grimm (Understanding [2011], 3) | |
| A reaction: [He cites Kvanvig and Prichard] If so, then we cannot say that knowledge is a lesser type of understanding. If you ask a trusted person how a mechanism works, and they have a wild guess that is luckily right, you would then understand it. |
| 19692 | You may have 'weak' understanding, if by luck you can answer a set of 'why questions' [Grimm] |
| Full Idea: There may be a 'weak' sense of understanding, where all you need to do is to be able to answer 'why questions' successfully, where one might have come by this ability in a lucky way. | |
| From: Stephen R. Grimm (Understanding [2011], 3) | |
| A reaction: We can see this point (in Idea 19691), but the idea that one could come by true complex understanding of something by purely lucky means is a bit absurd. Surely you would get one or two why questions wrong? 100%, just by luck? |
| 411 | If we succeed in speaking the truth, we cannot know we have done it [Xenophanes] |
| Full Idea: No man has seen certain truth, and no man will ever know about the gods and other things I mentioned; for if he succeeds in saying what is fully true, he himself is unaware of it; opinion is fixed by fate on all things. | |
| From: Xenophanes (fragments/reports [c.530 BCE], B34), quoted by Sextus Empiricus - Against the Professors (six books) 7.49.4 |
| 412 | If God had not created honey, men would say figs are sweeter [Xenophanes] |
| Full Idea: If God had not created yellow honey, men would say that figs were sweeter. | |
| From: Xenophanes (fragments/reports [c.530 BCE], B38), quoted by Herodian - On Peculiar Speech 41.5 |
| 1640 | The basic Eleatic belief was that all things are one [Xenophanes, by Plato] |
| Full Idea: The Eleatic tribe, which had its beginnings from Xenophanes and still earlier, proceed on the grounds that all things so-called are one. | |
| From: report of Xenophanes (fragments/reports [c.530 BCE]) by Plato - The Sophist 242d |
| 3055 | Xenophanes said the essence of God was spherical and utterly inhuman [Xenophanes, by Diog. Laertius] |
| Full Idea: Xenophanes taught that the essence of God was of a spherical form, in no respect resembling man. | |
| From: report of Xenophanes (fragments/reports [c.530 BCE]) by Diogenes Laertius - Lives of Eminent Philosophers 09.2.3 |
| 407 | Mortals believe gods are born, and have voices and clothes just like mortals [Xenophanes] |
| Full Idea: Mortals believe the gods to be created by birth, and to have raiment, voice and body like mortals'. | |
| From: Xenophanes (fragments/reports [c.530 BCE], B14), quoted by Clement - Miscellanies 5.109.2 |
| 408 | Ethiopian gods have black hair, and Thracian gods have red hair [Xenophanes] |
| Full Idea: Ethiopians have gods with snub noses and black hair, Thracians have gods with grey eyes and red hair. | |
| From: Xenophanes (fragments/reports [c.530 BCE], B16), quoted by Clement - Miscellanies 7.22.1 |