15 ideas
| 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. |
| 3546 | 'Phronesis' should translate as 'practical intelligence', not as prudence [Annas] |
| Full Idea: The best translation of 'phronesis' is probably not 'prudence' (which implies a non-moral motive), or 'practical wisdom' (which makes it sound contemplative), but 'practical intelligence', or just 'intelligence'. | |
| From: Julia Annas (The Morality of Happiness [1993], 2.3) |
| 3547 | Epicureans achieve pleasure through character development [Annas] |
| Full Idea: Since having a virtue does not reduce to performing certain kinds of acts, the Epicurean will achieve pleasure only by aiming at being a certain kind of person. | |
| From: Julia Annas (The Morality of Happiness [1993], 2.4) | |
| A reaction: No Epicurean would want to merely possess virtues, without enacting them. I assume that virtues are sought as guides to finding the finest pleasures (such as friendship). |
| 3543 | Cyrenaics pursue pleasure, but don't equate it with happiness [Annas] |
| Full Idea: Cyrenaics claimed our final good was pleasure, best achieved by seeking maximum intensity of pleasurable experiences, but they explicitly admitted that this was not happiness. | |
| From: Julia Annas (The Morality of Happiness [1993], 1) |
| 3541 | Ancient ethics uses attractive notions, not imperatives [Annas] |
| Full Idea: Instead of modern 'imperative' notions of ethics (involving obligation, duty and rule-following), ancient ethics uses 'attractive' notions like those of goodness and worth | |
| From: Julia Annas (The Morality of Happiness [1993], Intro) |
| 3550 | Principles cover life as a whole, where rules just cover actions [Annas] |
| Full Idea: Principles concern not just types of actions, but one's life as a whole, grasping truths about the nature of justice, and the like; they explain rules, giving the 'why' and not just the 'what'. | |
| From: Julia Annas (The Morality of Happiness [1993], 2.4) |
| 3551 | Virtue theory tries to explain our duties in terms of our character [Annas] |
| Full Idea: An ethics of virtue moves from an initial interest in what we ought to do to an interest in the kinds of people we are and hope to be, because the latter is taken to be the best way of understanding the former. | |
| From: Julia Annas (The Morality of Happiness [1993], 2.5) |
| 3552 | If excessively good actions are admirable but not required, then duty isn't basic [Annas] |
| Full Idea: Supererogatory actions are admirable and valuable, and we praise people for doing them, but they do not generate obligations to perform them, which casts doubt on obligation as the basic notion in ethics. | |
| From: Julia Annas (The Morality of Happiness [1993], 2.6) |
| 3542 | We should do good when necessary, not maximise it [Annas] |
| Full Idea: Why should I want to maximise my acting courageously? I act courageously when it is required. | |
| From: Julia Annas (The Morality of Happiness [1993], 1) |
| 5994 | Is the cosmos open or closed, mechanical or teleological, alive or inanimate, and created or eternal? [Robinson,TM, by PG] |
| Full Idea: The four major disputes in classical cosmology were whether the cosmos is 'open' or 'closed', whether it is explained mechanistically or teleologically, whether it is alive or mere matter, and whether or not it has a beginning. | |
| From: report of T.M. Robinson (Classical Cosmology (frags) [1997]) by PG - Db (ideas) | |
| A reaction: A nice summary. The standard modern view is closed, mechanistic, inanimate and non-eternal. But philosophers can ask deeper questions than physicists, and I say we are entitled to speculate when the evidence runs out. |