Combining Texts

All the ideas for 'Mathematics without Foundations', 'works' and 'Identity, Ostension, and Hypostasis'

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24 ideas

1. Philosophy / E. Nature of Metaphysics / 6. Metaphysics as Conceptual
11103 We aren't stuck with our native conceptual scheme; we can gradually change it [Quine]
2. Reason / A. Nature of Reason / 1. On Reason
17892 For clear questions posed by reason, reason can also find clear answers [Gödel]
4. Formal Logic / F. Set Theory ST / 3. Types of Set / d. Infinite Sets
9944 We understand some statements about all sets [Putnam]
5. Theory of Logic / A. Overview of Logic / 7. Second-Order Logic
9188 Gödel proved that first-order logic is complete, and second-order logic incomplete [Gödel, by Dummett]
5. Theory of Logic / I. Semantics of Logic / 2. Formal Truth
10620 Originally truth was viewed with total suspicion, and only demonstrability was accepted [Gödel]
5. Theory of Logic / K. Features of Logics / 5. Incompleteness
17883 Gödel's Theorems did not refute the claim that all good mathematical questions have answers [Gödel, by Koellner]
6. Mathematics / B. Foundations for Mathematics / 1. Foundations for Mathematics
9937 I do not believe mathematics either has or needs 'foundations' [Putnam]
6. Mathematics / B. Foundations for Mathematics / 4. Axioms for Number / a. Axioms for numbers
9939 It is conceivable that the axioms of arithmetic or propositional logic might be changed [Putnam]
6. Mathematics / B. Foundations for Mathematics / 4. Axioms for Number / g. Incompleteness of Arithmetic
17885 Gödel eventually hoped for a generalised completeness theorem leaving nothing undecidable [Gödel, by Koellner]
10614 The real reason for Incompleteness in arithmetic is inability to define truth in a language [Gödel]
6. Mathematics / C. Sources of Mathematics / 4. Mathematical Empiricism / a. Mathematical empiricism
9940 Maybe mathematics is empirical in that we could try to change it [Putnam]
6. Mathematics / C. Sources of Mathematics / 4. Mathematical Empiricism / b. Indispensability of mathematics
9941 Science requires more than consistency of mathematics [Putnam]
7. Existence / B. Change in Existence / 2. Processes
11092 A river is a process, with stages; if we consider it as one thing, we are considering a process [Quine]
7. Existence / C. Structure of Existence / 7. Abstract/Concrete / a. Abstract/concrete
11093 We don't say 'red' is abstract, unlike a river, just because it has discontinuous shape [Quine]
7. Existence / D. Theories of Reality / 4. Anti-realism
9943 You can't deny a hypothesis a truth-value simply because we may never know it! [Putnam]
7. Existence / D. Theories of Reality / 11. Ontological Commitment / a. Ontological commitment
11101 General terms don't commit us ontologically, but singular terms with substitution do [Quine]
7. Existence / E. Categories / 5. Category Anti-Realism
11096 Discourse generally departmentalizes itself to some degree [Quine]
8. Modes of Existence / E. Nominalism / 4. Concept Nominalism
11099 Understanding 'is square' is knowing when to apply it, not knowing some object [Quine]
8. Modes of Existence / E. Nominalism / 6. Mereological Nominalism
11094 'Red' is a single concrete object in space-time; 'red' and 'drop' are parts of a red drop [Quine]
11097 Red is the largest red thing in the universe [Quine]
9. Objects / F. Identity among Objects / 1. Concept of Identity
17595 To unite a sequence of ostensions to make one object, a prior concept of identity is needed [Quine]
9. Objects / F. Identity among Objects / 7. Indiscernible Objects
11095 We should just identify any items which are indiscernible within a given discourse [Quine]
18. Thought / D. Concepts / 5. Concepts and Language / b. Concepts are linguistic
11104 Concepts are language [Quine]
18. Thought / E. Abstraction / 1. Abstract Thought
11102 Apply '-ness' or 'class of' to abstract general terms, to get second-level abstract singular terms [Quine]