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All the ideas for 'Explanation in Mathematics', 'Dthat' and 'works'

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8 ideas

2. Reason / A. Nature of Reason / 1. On Reason
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.
5. Theory of Logic / A. Overview of Logic / 7. Second-Order Logic
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.
5. Theory of Logic / I. Semantics of Logic / 2. Formal Truth
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.
5. Theory of Logic / K. Features of Logics / 5. Incompleteness
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.
6. Mathematics / B. Foundations for Mathematics / 4. Axioms for Number / g. Incompleteness of Arithmetic
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.
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.
9. Objects / D. Essence of Objects / 7. Essence and Necessity / b. Essence not necessities
Essences are no use in mathematics, if all mathematical truths are necessary [Mancosu]
     Full Idea: Essences and essential properties do not seem to be useful in mathematical contexts, since all mathematical truths are regarded as necessary (though Kit Fine distinguishes between essential and necessary properties).
     From: Paolo Mancosu (Explanation in Mathematics [2008], §6.1)
     A reaction: I take the proviso in brackets to be crucial. This represents a distortion of notion of an essence. There is a world of difference between the central facts about the nature of a square and the peripheral inferences derivable from it.
19. Language / B. Reference / 3. Direct Reference / b. Causal reference
Are causal descriptions part of the causal theory of reference, or are they just metasemantic? [Kaplan, by Schaffer,J]
     Full Idea: Kaplan notes that the causal theory of reference can be understood in two quite different ways, as part of the semantics (involving descriptions of causal processes), or as metasemantics, explaining why a term has the referent it does.
     From: report of David Kaplan (Dthat [1970]) by Jonathan Schaffer - Deflationary Metaontology of Thomasson 1
     A reaction: [Kaplan 'Afterthought' 1989] The theory tends to be labelled as 'direct' rather than as 'causal' these days, but causal chains are still at the heart of the story (even if more diffused socially). Nice question. Kaplan takes the meta- version as orthodox.