9 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. |
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. |
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. |
16640 | Form is the principle that connects a thing's constitution (rather than being operative) [Hill,N] |
Full Idea: Form is the state and condition of a thing, a result of the connection among its material principles; it is a constituting principle, not an operative one. | |
From: Nicholas Hill (Philosophia Epicurea [1610], n 35) | |
A reaction: Pasnau presents this as a denial of form, but it looks to me like someone fishing for what form could be in a more scientific context. Aristotle would have approved of 'principles'. Hill seems to defend the categorical against the dispositional. |
3158 | Theories of intentionality presuppose rationality, so can't explain it [Dennett] |
Full Idea: Intentional theory is vacuous as psychology because it presupposes and does not explain rationality or intelligence. | |
From: Daniel C. Dennett (Brainstorms:Essays on Mind and Psychology [1978], p.15?) | |
A reaction: Virtually every philosophical theory seems to founder because it presupposes something like the thing it is meant to explain. I agree that 'intentionality' is a slightly airy concept that would probably reduce to something better. |
3159 | Beliefs and desires aren't real; they are prediction techniques [Dennett] |
Full Idea: Intentional systems don't really have beliefs and desires, but one can explain and predict their behaviour by ascribing beliefs and desires to them. This strategy is pragmatic, not right or wrong. | |
From: Daniel C. Dennett (Brainstorms:Essays on Mind and Psychology [1978], p.7?) | |
A reaction: If the ascription of beliefs and desires explains behaviour, then that is good grounds for thinking they might be real features of the brain, and even if that is not so, they are real enough as abstractions from brain events, like the 'economic climate'. |