16 ideas
| 13560 | A wise man is not subservient to anything [Seneca] |
| Full Idea: I do not call any man wise who is subservient to anything. | |
| From: Seneca the Younger (On the Happy Life [c.60], §11) | |
| A reaction: At the very least, a wise man should be subservient to a wiser man. |
| 8679 | We perceive the objects of set theory, just as we perceive with our senses [Gödel] |
| Full Idea: We have something like perception of the objects of set theory, shown by the axioms forcing themselves on us as being true. I don't see why we should have less confidence in this kind of perception (i.e. mathematical intuition) than in sense perception. | |
| From: Kurt Gödel (What is Cantor's Continuum Problem? [1964], p.483), quoted by Michčle Friend - Introducing the Philosophy of Mathematics 2.4 | |
| A reaction: A famous strong expression of realism about the existence of sets. It is remarkable how the ingredients of mathematics spread themselves before the mind like a landscape, inviting journeys - but I think that just shows how minds cope with abstractions. |
| 9942 | Gödel proved the classical relative consistency of the axiom V = L [Gödel, by Putnam] |
| Full Idea: Gödel proved the classical relative consistency of the axiom V = L (which implies the axiom of choice and the generalized continuum hypothesis). This established the full independence of the continuum hypothesis from the other axioms. | |
| From: report of Kurt Gödel (What is Cantor's Continuum Problem? [1964]) by Hilary Putnam - Mathematics without Foundations | |
| A reaction: Gödel initially wanted to make V = L an axiom, but the changed his mind. Maddy has lots to say on the subject. |
| 18062 | Set-theory paradoxes are no worse than sense deception in physics [Gödel] |
| Full Idea: The set-theoretical paradoxes are hardly any more troublesome for mathematics than deceptions of the senses are for physics. | |
| From: Kurt Gödel (What is Cantor's Continuum Problem? [1964], p.271), quoted by Philip Kitcher - The Nature of Mathematical Knowledge 03.4 |
| 10868 | The Continuum Hypothesis is not inconsistent with the axioms of set theory [Gödel, by Clegg] |
| Full Idea: Gödel proved that the Continuum Hypothesis was not inconsistent with the axioms of set theory. | |
| From: report of Kurt Gödel (What is Cantor's Continuum Problem? [1964]) by Brian Clegg - Infinity: Quest to Think the Unthinkable Ch.15 |
| 13517 | If set theory is consistent, we cannot refute or prove the Continuum Hypothesis [Gödel, by Hart,WD] |
| Full Idea: Gödel proved that (if set theory is consistent) we cannot refute the continuum hypothesis, and Cohen proved that (if set theory is consistent) we cannot prove it either. | |
| From: report of Kurt Gödel (What is Cantor's Continuum Problem? [1964]) by William D. Hart - The Evolution of Logic 10 |
| 10271 | Basic mathematics is related to abstract elements of our empirical ideas [Gödel] |
| Full Idea: Evidently the 'given' underlying mathematics is closely related to the abstract elements contained in our empirical ideas. | |
| From: Kurt Gödel (What is Cantor's Continuum Problem? [1964], Suppl) | |
| A reaction: Yes! The great modern mathematical platonist says something with which I can agree. He goes on to hint at a platonic view of the structure of the empirical world, but we'll let that pass. |
| 468 | Musical performance can reveal a range of virtues [Damon of Ath.] |
| Full Idea: In singing and playing the lyre, a boy will be likely to reveal not only courage and moderation, but also justice. | |
| From: Damon (fragments/reports [c.460 BCE], B4), quoted by (who?) - where? |
| 13558 | The supreme good is harmony of spirit [Seneca] |
| Full Idea: The highest good is harmony of spirit. | |
| From: Seneca the Younger (On the Happy Life [c.60], §08) | |
| A reaction: This idea is straight from Plato's Republic. |
| 13559 | I seek virtue, because it is its own reward [Seneca] |
| Full Idea: You ask what I seek from virtue? Virtue herself. For she has nothing better, she is herself her own reward. | |
| From: Seneca the Younger (On the Happy Life [c.60], §09) | |
| A reaction: Presumably this is the source of the popular saying that 'virtue is its own reward'. The trouble is that this doesn't seem a very persuasive thing to say to a sceptic who doubts whether being virtuous is worth the trouble. |
| 13561 | Virtue is always moderate, so excess need not be feared [Seneca] |
| Full Idea: In the case of virtue excess should not be feared, since in virtue resides moderation. | |
| From: Seneca the Younger (On the Happy Life [c.60], §13) | |
| A reaction: This seems to imply that all of the virtues are unified in the one achievement of the virtuous state. It leaves the notion of 'virtue' a bit thin in content, though. |
| 13562 | It is shameful to not even recognise your own slaves [Seneca] |
| Full Idea: Why, to your shame, are you so careless that you do not know your handful of slaves by sight? | |
| From: Seneca the Younger (On the Happy Life [c.60], §17) |
| 13564 | There is far more scope for virtue if you are wealthy; poverty only allows endurance [Seneca] |
| Full Idea: What doubt can there be that the wise man has greater scope for displaying his powers if he is rich than if he is poor, since in the case of poverty only one kind of virtue exists - refusal to be bowed down and crushed. | |
| From: Seneca the Younger (On the Happy Life [c.60], §22) | |
| A reaction: It is against this view that I see Jesus proposing poverty as central to virtue. But then he has the surprising view (to Seneca) that humility is a virtue. What Nietzsche calls the slaves' inversion of values. |
| 13563 | Why does your wife wear in her ears the income of a wealthy house? [Seneca] |
| Full Idea: Why does your wife wear in her ears the income of a wealthy house? | |
| From: Seneca the Younger (On the Happy Life [c.60], §17) |
| 13565 | If wealth was a good, it would make men good [Seneca] |
| Full Idea: Wealth is not a good; for it it was, it would make men good. | |
| From: Seneca the Younger (On the Happy Life [c.60], §24) | |
| A reaction: An immediately attractive argument, but should we assume that anything which is good will enhance our personal goodness? If goodness is a habit, then continual pursuit of wealth is the test case to examine. Seneca is right! |
| 13557 | Unfortunately the majority do not tend to favour what is best [Seneca] |
| Full Idea: Human concerns are not so happily arranged that the majority favours the better things. | |
| From: Seneca the Younger (On the Happy Life [c.60], §02) | |
| A reaction: On the whole Seneca is unimpressed by democracy, as people are rushed into decisions by the crowd, and live to regret them. |