17 ideas
| 15118 | A successful Aristotelian 'definition' is what sciences produces after an investigation [Koslicki] |
| Full Idea: My current use of the Aristotelian term 'definition' is intended to correspond to what is typically accessible to a scientist only at the end of a successful investigation into the nature of a particular phenomenon. | |
| From: Kathrin Koslicki (Essence, Necessity and Explanation [2012], 13.3.1) | |
| A reaction: It is crucial to understand that Aristotle's definitions could be several hundred pages long. It has nothing to do with dictionary definitions. He proposes 'nominal' and 'real' definitions. |
| 15116 | Essences cause necessary features, and definitions describe those necessary features [Koslicki] |
| Full Idea: Since essences cause the other necessary features of a thing, so definitions, as the linguistic correlates of essences, explain, together with other axioms, the propositions describing those necessary features. | |
| From: Kathrin Koslicki (Essence, Necessity and Explanation [2012], 13.3.1) | |
| A reaction: This is nice and clear. Definitions are NOT essences - they are the linguistic correlates of essences, and mirror those essences. The necessary features are not the only things needing explanation. That picture is too passive. |
| 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. |
| 15110 | An essence and what merely follow from it are distinct [Koslicki] |
| Full Idea: We can distinguish (as Aristotle and Fine do) between what belongs to the essence of an object, and what merely follows from the essence of an object. | |
| From: Kathrin Koslicki (Essence, Necessity and Explanation [2012], 13.1) | |
| A reaction: This can help to clarify the confusions that result from treating necessary properties as if they were essential. |
| 15113 | Individuals are perceived, but demonstration and definition require universals [Koslicki] |
| Full Idea: Individual instances of a kind of phenomenon, in Aristotle's view, can only be perceived through sense-perception; but they are not the proper subject-matter of scientific demonstration and definition. | |
| From: Kathrin Koslicki (Essence, Necessity and Explanation [2012], 13.3.1) | |
| A reaction: A footnote (11) explains that this is because they involve syllogisms, which require universals. I take Aristotle, and anyone sensible, to rest on individual essences, but inevitably turn to generic essences when language becomes involved. |
| 15112 | If an object exists, then its essential properties are necessary [Koslicki] |
| Full Idea: If an object has a certain property essentially, then it follows that the object has the property necessarily (if it exists). | |
| From: Kathrin Koslicki (Essence, Necessity and Explanation [2012], 13.2) | |
| A reaction: She is citing Fine, who says that the converse (necessity implying essence) is false. I agree with that. I also willing to challenge the first bit. I suspect an object can retain identity and lose essence. Coma patient; broken clock; aged athlete. |
| 15111 | In demonstration, the explanatory order must mirror the causal order of the phenomena [Koslicki] |
| Full Idea: Demonstration encompasses more than deductive entailment, in that the explanatory order of priority represented in a successful demonstration must mirror precisely the causal order of priority present in the phenomena in question. | |
| From: Kathrin Koslicki (Essence, Necessity and Explanation [2012], 13.1) | |
| A reaction: She is referring to Aristotle's 'Posterior Analytics'. Put so clearly this sounds like an incredibly useful concept in discussing how we present good modern scientific explanations. Reinstating Aristotle is a major priority for philosophy! |
| 15115 | In a demonstration the middle term explains, by being part of the definition [Koslicki] |
| Full Idea: In a proper demonstrative argument, the middle term must be explanatory of the conclusion, in a very specific sense: the middle term must state what properly belongs to the definition of the kind of phenomenon in question. | |
| From: Kathrin Koslicki (Essence, Necessity and Explanation [2012], 13.3.1) | |
| A reaction: So 'All men are mortal, S is a man, so S is mortal'. The middle term is 'man', which gives a generic explanation for why S is mortal. Explanation as categorisation? I don't think this is the whole story of Aristotelian explanation. |
| 15117 | Greek uses the same word for 'cause' and 'explanation' [Koslicki] |
| Full Idea: The Greek does not disambiguate between 'cause' and 'explanation', since the same terms ('aitia' and 'aition') can be translated in both ways. | |
| From: Kathrin Koslicki (Essence, Necessity and Explanation [2012], 13.3.1 n15) | |
| A reaction: This is essential information if we are to understand Aristotle's Four Causes, which are quite baffling if we take 'causes' in the modern way. The are the Four Modes of Explanation. |
| 15114 | Discovering the Aristotelian essence of thunder will tell us why thunder occurs [Koslicki] |
| Full Idea: Both the question 'what is thunder?', and the question 'why does thunder occur?', for Aristotle, are answered simultaneously, once it has been discovered what the essence of thunder it, i.e. what it is to be thunder. | |
| From: Kathrin Koslicki (Essence, Necessity and Explanation [2012], 13.3.1 n10) | |
| A reaction: I take this idea to be pretty much the whole story about essences. |
| 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. |