17 ideas
| 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! |
| 3509 | Externalism may be the key idea in philosophical naturalism [Papineau] |
| Full Idea: Some people view an externalist approach to epistemology as the essence of philosophical naturalism. | |
| From: David Papineau (Philosophical Naturalism [1993], Intro) | |
| A reaction: I suspect philosophers avoid psychology and mental events, simply because they are elusive. Externalism is a theory about justification, and independent of naturalism as a metaphysic. |
| 3513 | How does a dualist mind represent, exist outside space, and be transparent to itself? [Papineau] |
| Full Idea: Even dualists must explain how the mind represents things, but then their mind-stuff has so many special powers already (being outside space but in time, being transparent to itself etc.) that one more scarcely seems worth worrying about. | |
| From: David Papineau (Philosophical Naturalism [1993], 3.1 n1) | |
| A reaction: I share the exasperation. It is hard to see how a dualist could even begin to formulate a theory about HOW the mind does so many different things. Could Descartes get a research grant for it? Would we understand God if he tried to explain it to us? |
| 3514 | Functionalism needs causation and intentionality to explain actions [Papineau] |
| Full Idea: The functionalist approach to the mind needs to invoke assumptions about what desires are for and beliefs are about, in order to infer what agents will do. | |
| From: David Papineau (Philosophical Naturalism [1993], 3.2) | |
| A reaction: Isn't the idea that you discover what desires are for and what beliefs are about by examining their function, and what the agent does? Which end should we start? |
| 3510 | Epiphenomenalism is supervenience without physicalism [Papineau] |
| Full Idea: Supervenience is a necessary condition for physicalism, but it is not sufficient. Epiphenomenalism rules out mental variation without physical variation, but says mental properties are quite distinct from physical properties. | |
| From: David Papineau (Philosophical Naturalism [1993], 1.2) | |
| A reaction: I take full epiphenomenalism about mind to be incoherent, and not worth even mentioning (see Idea 7379). Papineau seems to be thinking of so-called property dualism (which may also be incoherent!). |
| 3511 | Supervenience requires all mental events to have physical effects [Papineau] |
| Full Idea: The argument for supervenience rests on the principle that any mental difference must be capable of showing itself in differential physical consequences. | |
| From: David Papineau (Philosophical Naturalism [1993], 1.8) | |
| A reaction: With our current knowledge of the brain, to assume anything less than this sort of correlation would be crazy. |
| 3515 | Knowing what it is like to be something only involves being (physically) that thing [Papineau] |
| Full Idea: Physicalism does not deny that there are conscious experiences, nor that 'it is like something to have them'. The claim is only that this is nothing different from what it is to be a physical system of the relevant kind. | |
| From: David Papineau (Philosophical Naturalism [1993], 4.2) | |
| A reaction: The implication is that no physicalist is an extreme eliminativist about consciousness, which seems to be correct. We all concede that weather exists, but have a reductive view of it. The key question is whether mind is reducible to physics. |
| 3512 | If a mental state is multiply realisable, why does it lead to similar behaviour? [Papineau] |
| Full Idea: If functionalism implies that there is nothing physically in common among the realisations of a given mental state, then there is no possibility of any uniform explanation of why they all give rise to a common physical result. | |
| From: David Papineau (Philosophical Naturalism [1993], 2.2) | |
| A reaction: This is the well known interaction problem for dualism. The standard reply is to accept interaction as a given (with no apparent explanation). A miracle, if you like. |
| 3516 | The Private Language argument only means people may misjudge their experiences [Papineau] |
| Full Idea: I take the moral of the Private Language argument to be that there must be room for error in people's judgements about their experiences, not that those judgements must necessarily be expressed in a language used by a community. | |
| From: David Papineau (Philosophical Naturalism [1993], 4.4 n10) | |
| A reaction: These two readings don't seem to be in conflict, and the argument must have something to say about the communal nature of thought expressed in language. Language imposes introspection on us? |
| 7903 | The six perfections are giving, morality, patience, vigour, meditation, and wisdom [Nagarjuna] |
| Full Idea: The six perfections are of giving, morality, patience, vigour, meditation, and wisdom. | |
| From: Nagarjuna (Mahaprajnaparamitashastra [c.120], 88) | |
| A reaction: What is 'morality', if giving is not part of it? I like patience and vigour being two of the virtues, which immediately implies an Aristotelian mean (which is always what is 'appropriate'). |