19 ideas
| 17892 | For clear questions posed by reason, reason can also find clear answers [Gödel] |
| Full Idea: I uphold the belief that for clear questions posed by reason, reason can also find clear answers. | |
| From: Kurt Gödel (works [1930]), quoted by Peter Koellner - On the Question of Absolute Undecidability 1.5 | |
| A reaction: [written in 1961] This contradicts the implication normally taken from his much earlier Incompleteness Theorems. |
| 12223 | It is a fallacy to explain the obscure with the even more obscure [Hale/Wright] |
| Full Idea: The fallacy of 'ad obscurum per obscurius' is to explain the obscure by appeal to what is more obscure. | |
| From: B Hale / C Wright (The Metaontology of Abstraction [2009], §3) | |
| A reaction: Not strictly a fallacy, so much as an example of inadequate explanation, along with circularity and infinite regresses. |
| 9188 | Gödel proved that first-order logic is complete, and second-order logic incomplete [Gödel, by Dummett] |
| Full Idea: Gödel proved the completeness of standard formalizations of first-order logic, including Frege's original one. However, an implication of his famous theorem on the incompleteness of arithmetic is that second-order logic is incomplete. | |
| From: report of Kurt Gödel (works [1930]) by Michael Dummett - The Philosophy of Mathematics 3.1 | |
| A reaction: This must mean that it is impossible to characterise arithmetic fully in terms of first-order logic. In which case we can only characterize the features of abstract reality in general if we employ an incomplete system. We're doomed. |
| 12230 | Singular terms refer if they make certain atomic statements true [Hale/Wright] |
| Full Idea: Anyone should agree that a justification for regarding a singular term as having objectual reference is provided just as soon as one has justification for regarding as true certain atomic statements in which it functions as a singular term. | |
| From: B Hale / C Wright (The Metaontology of Abstraction [2009], §9) | |
| A reaction: The meat of this idea is hidden in the word 'certain'. See Idea 10314 for Hale's explanation. Without that, the proposal strikes me as absurd. |
| 10620 | Originally truth was viewed with total suspicion, and only demonstrability was accepted [Gödel] |
| Full Idea: At that time (c.1930) a concept of objective mathematical truth as opposed to demonstrability was viewed with greatest suspicion and widely rejected as meaningless. | |
| From: Kurt Gödel (works [1930]), quoted by Peter Smith - Intro to Gödel's Theorems 28.2 | |
| A reaction: [quoted from a letter] This is the time of Ramsey's redundancy account, and before Tarski's famous paper of 1933. It is also the high point of Formalism, associated with Hilbert. |
| 17883 | Gödel's Theorems did not refute the claim that all good mathematical questions have answers [Gödel, by Koellner] |
| Full Idea: Gödel was quick to point out that his original incompleteness theorems did not produce instances of absolute undecidability and hence did not undermine Hilbert's conviction that for every precise mathematical question there is a discoverable answer. | |
| From: report of Kurt Gödel (works [1930]) by Peter Koellner - On the Question of Absolute Undecidability Intro | |
| A reaction: The normal simplistic view among philosophes is that Gödel did indeed decisively refute the optimistic claims of Hilbert. Roughly, whether Hilbert is right depends on which axioms of set theory you adopt. |
| 17885 | Gödel eventually hoped for a generalised completeness theorem leaving nothing undecidable [Gödel, by Koellner] |
| Full Idea: Eventually Gödel ...expressed the hope that there might be a generalised completeness theorem according to which there are no absolutely undecidable sentences. | |
| From: report of Kurt Gödel (works [1930]) by Peter Koellner - On the Question of Absolute Undecidability Intro | |
| A reaction: This comes as a bit of a shock to those who associate him with the inherent undecidability of reality. |
| 10614 | The real reason for Incompleteness in arithmetic is inability to define truth in a language [Gödel] |
| Full Idea: The concept of truth of sentences in a language cannot be defined in the language. This is the true reason for the existence of undecidable propositions in the formal systems containing arithmetic. | |
| From: Kurt Gödel (works [1930]), quoted by Peter Smith - Intro to Gödel's Theorems 21.6 | |
| A reaction: [from a letter by Gödel] So they key to Incompleteness is Tarski's observations about truth. Highly significant, as I take it. |
| 12225 | Neo-Fregeanism might be better with truth-makers, rather than quantifier commitment [Hale/Wright] |
| Full Idea: A third way has been offered to 'make sense' of neo-Fregeanism: we should reject Quine's well-known criterion of ontological commitment in favour of one based on 'truth-maker theory'. | |
| From: B Hale / C Wright (The Metaontology of Abstraction [2009], §4 n19) | |
| A reaction: [The cite Ross Cameron for this] They reject this proposal, on the grounds that truth-maker theory is not sufficient to fix the grounding truth-conditions of statements. |
| 12224 | Are neo-Fregeans 'maximalists' - that everything which can exist does exist? [Hale/Wright] |
| Full Idea: It is claimed that neo-Fregeans are committed to 'maximalism' - that whatever can exist does. | |
| From: B Hale / C Wright (The Metaontology of Abstraction [2009], §4) | |
| A reaction: [The cite Eklund] They observe that maximalism denies contingent non-existence (of the Ł20 note I haven't got). There seems to be the related problem of 'hyperinflation', that if abstract objects are generated logically, the process is unstoppable. |
| 12226 | The identity of Pegasus with Pegasus may be true, despite the non-existence [Hale/Wright] |
| Full Idea: Identity is sometimes read so that 'Pegasus is Pegasus' expresses a truth, the non-existence of any winged horse notwithstanding. | |
| From: B Hale / C Wright (The Metaontology of Abstraction [2009], §5) | |
| A reaction: This would give you ontological commitment to truth, without commitment to existence. It undercuts the use of identity statements as the basis of existence claims, which was Frege's strategy. |
| 12229 | Maybe we have abundant properties for semantics, and sparse properties for ontology [Hale/Wright] |
| Full Idea: There is a compatibilist view which says that it is for the abundant properties to play the role of 'bedeutungen' in semantic theory, and the sparse ones to address certain metaphysical concerns. | |
| From: B Hale / C Wright (The Metaontology of Abstraction [2009], §9) | |
| A reaction: Only a philosopher could live with the word 'property' having utterly different extensions in different areas of discourse. They similarly bifurcate words like 'object' and 'exist'. Call properties 'quasi-properties' and I might join in. |
| 18443 | A successful predicate guarantees the existence of a property - the way of being it expresses [Hale/Wright] |
| Full Idea: The good standing of a predicate is already trivially sufficient to ensure the existence of an associated property, a (perhaps complex) way of being which the predicate serves to express. | |
| From: B Hale / C Wright (The Metaontology of Abstraction [2009], §9) | |
| A reaction: 'Way of being' is interesting. Is 'being near Trafalgar Sq' a way of being? I take properties to be 'features', which seems to give a clearer way of demarcating them. They say they are talking about 'abundant' (rather than 'sparse') properties. |
| 12227 | Abstractionism needs existential commitment and uniform truth-conditions [Hale/Wright] |
| Full Idea: Abstractionism needs a face-value, existentially committed reading of the terms occurring on the left-hand sides together with sameness of truth-conditions across the biconditional. | |
| From: B Hale / C Wright (The Metaontology of Abstraction [2009], §5) | |
| A reaction: They employ 'abstractionism' to mean their logical Fregean strategy for defining abstractions, not to mean the older psychological account. Thus the truth-conditions for being 'parallel' and for having the 'same direction' must be consistent. |
| 12228 | Equivalence abstraction refers to objects otherwise beyond our grasp [Hale/Wright] |
| Full Idea: Abstraction principles purport to introduce fundamental means of reference to a range of objects, to which there is accordingly no presumption that we have any prior or independent means of reference. | |
| From: B Hale / C Wright (The Metaontology of Abstraction [2009], §8) | |
| A reaction: There's the rub! They make it sound like a virtue, that we open up yet another heaven of abstract toys to play with. As fictions, they are indeed exciting new fun. As platonic discoveries they strike me as Cloud-Cuckoo Land. |
| 12231 | Reference needs truth as well as sense [Hale/Wright] |
| Full Idea: It takes, over and above the possession of sense, the truth of relevant contexts to ensure reference. | |
| From: B Hale / C Wright (The Metaontology of Abstraction [2009], §9) | |
| A reaction: Reference purely through sense was discredited by Kripke. The present idea challenges Kripke's baptismal realist approach. How do you 'baptise' an abstract object? But isn't reference needed prior to the establishment of truth? |
| 5049 | Intelligent pleasure is the perception of beauty, order and perfection [Leibniz] |
| Full Idea: An intelligent being's pleasure is simply the perception of beauty, order and perfection. | |
| From: Gottfried Leibniz (A Résumé of Metaphysics [1697], §18) | |
| A reaction: Leibniz seems to have inherited this from the Greeks, especially Pythagoras and Plato. Buried in Leibniz's remark I see the Christian fear of physical pleasure. He should have got out more. Must an intelligent being always be intelligent? |
| 5048 | Perfection is simply quantity of reality [Leibniz] |
| Full Idea: Perfection is simply quantity of reality. | |
| From: Gottfried Leibniz (A Résumé of Metaphysics [1697], §11) | |
| A reaction: An interesting claim, but totally beyond my personal comprehension. I presume he inherited 'quantity of reality' from Plato, e.g. as you move up the Line from shadows to Forms you increase the degree of reality. I see 'real' as all-or-nothing. |
| 5050 | Evil serves a greater good, and pain is necessary for higher pleasure [Leibniz] |
| Full Idea: Evils themselves serve a greater good, and the fact that pains are found in minds is necessary if they are to reach greater pleasures. | |
| From: Gottfried Leibniz (A Résumé of Metaphysics [1697], §23) | |
| A reaction: How much pain is needed to qualify for the 'greater pleasures'? Some people receive an awful lot. I am not sure exactly how an evil can 'serve' a greater good. Is he recommending evil? |