9 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. |
5996 | Critolaus redefined Aristotle's moral aim as fulfilment instead of happiness [Critolaus, by White,SA] |
Full Idea: Critolaus reformulated Aristotelian theory by defining happiness as a 'fulfilment' (sumplęrôma) of psychic, physical, and external goods, where virtue vastly outweighs the rest. | |
From: report of Critolaus (fragments/reports [c.170 BCE]) by Stephen A. White - Critolaus | |
A reaction: The sounds more like an attempt at clarification than a real change of Peripatetic doctrine. Occasionally 'fulfilment' is offered as a translation for eudaimonia. Maybe we should just take up Critolaus' suggestion when we are discussing Aristotle. |
16713 | Philosophers are the forefathers of heretics [Tertullian] |
Full Idea: Philosophers are the forefathers of heretics. | |
From: Tertullian (works [c.200]), quoted by Robert Pasnau - Metaphysical Themes 1274-1671 20.2 |
6610 | I believe because it is absurd [Tertullian] |
Full Idea: I believe because it is absurd ('Credo quia absurdum est'). | |
From: Tertullian (works [c.200]), quoted by Robert Fogelin - Walking the Tightrope of Reason n4.2 | |
A reaction: This seems to be a rather desperate remark, in response to what must have been rather good hostile arguments. No one would abandon the support of reason if it was easy to acquire. You can't deny its engaging romantic defiance, though. |