15 ideas
| 4187 | 'There is nothing without a reason why it should be rather than not be' (a generalisation of 'Why?') [Schopenhauer] |
| Full Idea: The Principle may be stated as 'There is nothing without a reason why it should be rather than not be', which is a generalisation of the assumption which justifies the question 'Why?', which is the mother of all science. | |
| From: Arthur Schopenhauer (Abstract of 'The Fourfold Root' [1813], Ch.I) | |
| A reaction: This faith is the core of philosophy, to be maintained against all defeatists like Wittgenstein and Colin McGinn. Reality must be rational, or we wouldn't be here to think about it. (Maybe!) |
| 23445 | Naïve set theory says any formula defines a set, and coextensive sets are identical [Linnebo] |
| Full Idea: Naïve set theory is based on the principles that any formula defines a set, and that coextensive sets are identical. | |
| From: Øystein Linnebo (Philosophy of Mathematics [2017], 4.2) | |
| A reaction: The second principle is a standard axiom of ZFC. The first principle causes the trouble. |
| 23447 | In classical semantics singular terms refer, and quantifiers range over domains [Linnebo] |
| Full Idea: In classical semantics the function of singular terms is to refer, and that of quantifiers, to range over appropriate domains of entities. | |
| From: Øystein Linnebo (Philosophy of Mathematics [2017], 7.1) |
| 23443 | The axioms of group theory are not assertions, but a definition of a structure [Linnebo] |
| Full Idea: Considered in isolation, the axioms of group theory are not assertions but comprise an implicit definition of some abstract structure, | |
| From: Øystein Linnebo (Philosophy of Mathematics [2017], 3.5) | |
| A reaction: The traditional Euclidean approach is that axioms are plausible assertions with which to start. The present idea sums up the modern approach. In the modern version you can work backwards from a structure to a set of axioms. |
| 23444 | To investigate axiomatic theories, mathematics needs its own foundational axioms [Linnebo] |
| Full Idea: Mathematics investigates the deductive consequences of axiomatic theories, but it also needs its own foundational axioms in order to provide models for its various axiomatic theories. | |
| From: Øystein Linnebo (Philosophy of Mathematics [2017], 4.1) | |
| A reaction: This is a problem which faces the deductivist (if-then) approach. The deductive process needs its own grounds. |
| 23446 | You can't prove consistency using a weaker theory, but you can use a consistent theory [Linnebo] |
| Full Idea: If the 2nd Incompleteness Theorem undermines Hilbert's attempt to use a weak theory to prove the consistency of a strong one, it is still possible to prove the consistency of one theory, assuming the consistency of another theory. | |
| From: Øystein Linnebo (Philosophy of Mathematics [2017], 4.6) | |
| A reaction: Note that this concerns consistency, not completeness. |
| 23448 | Mathematics is the study of all possible patterns, and is thus bound to describe the world [Linnebo] |
| Full Idea: Philosophical structuralism holds that mathematics is the study of abstract structures, or 'patterns'. If mathematics is the study of all possible patterns, then it is inevitable that the world is described by mathematics. | |
| From: Øystein Linnebo (Philosophy of Mathematics [2017], 11.1) | |
| A reaction: [He cites the physicist John Barrow (2010) for this] For me this is a major idea, because the concept of a pattern gives a link between the natural physical world and the abstract world of mathematics. No platonism is needed. |
| 23441 | Logical truth is true in all models, so mathematical objects can't be purely logical [Linnebo] |
| Full Idea: Modern logic requires that logical truths be true in all models, including ones devoid of any mathematical objects. It follows immediately that the existence of mathematical objects can never be a matter of logic alone. | |
| From: Øystein Linnebo (Philosophy of Mathematics [2017], 2) | |
| A reaction: Hm. Could there not be a complete set of models for a theory which all included mathematical objects? (I can't answer that). |
| 23442 | Game Formalism has no semantics, and Term Formalism reduces the semantics [Linnebo] |
| Full Idea: Game Formalism seeks to banish all semantics from mathematics, and Term Formalism seeks to reduce any such notions to purely syntactic ones. | |
| From: Øystein Linnebo (Philosophy of Mathematics [2017], 3.3) | |
| A reaction: This approach was stimulated by the need to justify the existence of the imaginary number i. Just say it is a letter! |
| 4192 | All necessity arises from causation, which is conditioned; there is no absolute or unconditioned necessity [Schopenhauer] |
| Full Idea: Necessity has no meaning other than the irresistible sequence of the effect where the cause is given. All necessity is thus conditioned, and absolute or unconditioned necessity is a contradiction in terms. | |
| From: Arthur Schopenhauer (Abstract of 'The Fourfold Root' [1813], Ch.VIII) | |
| A reaction: I.e. there is only natural necessity, and no such thing as metaphysical necessity. But what about logical necessity(e.g. 2+3=5)? I think there may be metaphysical necessity, but we can't know much about it, and we are over-confident in assessing it. |
| 4190 | All understanding is an immediate apprehension of the causal relation [Schopenhauer] |
| Full Idea: All understanding is an immediate apprehension of the causal relation. | |
| From: Arthur Schopenhauer (Abstract of 'The Fourfold Root' [1813], Ch.IV) | |
| A reaction: Based, I take it, on Hume. Presumably he means a posteriori understanding, as it hardly fits an understanding of arithmetic. Understanding needs more than just causation. What aspects of causation? |
| 4191 | What we know in ourselves is not a knower but a will [Schopenhauer] |
| Full Idea: What we know in ourselves is never what knows, but what wills, the will. | |
| From: Arthur Schopenhauer (Abstract of 'The Fourfold Root' [1813], Ch.VII) | |
| A reaction: An interesting slant on Hume's scepticism about personal identity. Hume was hunting for a thing-which-experiences. If he had sought his will, he might have spotted it. |
| 21368 | The knot of the world is the use of 'I' to refer to both willing and knowing [Schopenhauer] |
| Full Idea: The identity of the subject of willing with that of knowing by virtue whereof ...the word 'I' includes and indicates both, is the knot of the world, and hence inexplicable. | |
| From: Arthur Schopenhauer (Abstract of 'The Fourfold Root' [1813], p.211-2), quoted by Christopher Janaway - Schopenhauer 4 'Self' | |
| A reaction: I'm struggling to see this as a deep mystery. If we look objectively at animals and ask 'what is their brain for?' the answer seems obvious. This may be a case of everything looking mysterious after a philosopher has stared at it for a while. |
| 4189 | Time may be defined as the possibility of mutually exclusive conditions of the same thing [Schopenhauer] |
| Full Idea: Time may be defined as the possibility of mutually exclusive conditions of the same thing. | |
| From: Arthur Schopenhauer (Abstract of 'The Fourfold Root' [1813], Ch.IV) | |
| A reaction: An off-beat philosophical view of the question. Sounds more like a consequence of time than its essential nature. |
| 14409 | I am a presentist, and all language and common sense supports my view [Bigelow] |
| Full Idea: I am a presentist: nothing exists which is not present. Everyone believed this until the nineteenth century; it is writing into the grammar of natural languages; it is still assumed in everyday life, even by philosophers who deny it. | |
| From: John Bigelow (Presentism and Properties [1996], p.36), quoted by Trenton Merricks - Truth and Ontology | |
| A reaction: The most likely deniers of presentism seem to be physicists and cosmologists who have overdosed on Einstein. On the whole I vote for presentism, but what justifies truths about the past and future. Traces existing in the present? |