20 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! |
| 22121 | The concept of being has only one meaning, whether talking of universals or of God [Duns Scotus, by Dumont] |
| Full Idea: Duns Scotus was the first scholastic to hold that the concept of being and other transcendentals were univocal, not only in application to substance and accidents, but even to God and creatures. | |
| From: report of John Duns Scotus (works [1301]) by Stephen D. Dumont - Duns Scotus p.205 | |
| A reaction: So either it exists or it doesn't. No nonsense about 'subsisting'. Russell flirted with subsistence, but Quine agrees with Duns Scotus (and so do I). |
| 22122 | Being (not sensation or God) is the primary object of the intellect [Duns Scotus, by Dumont] |
| Full Idea: Duns Scotus said the primary object of the created intellect was being, rejecting Aquinas's Aristotelian view that it was limited to the quiddity of the sense particular, and Henry of Ghent's Augustinian view that it was God. | |
| From: report of John Duns Scotus (works [1301]) by Stephen D. Dumont - Duns Scotus p.205 | |
| A reaction: I suppose the 'primary object of the intellect' is the rationalist/empiricism disagreement. So (roughly) Aquinas was an empiricist, Duns Scotus was a rationalist, and Augustine was a transcendentalist? Augustine sounds like Spinoza. |
| 22125 | Duns Scotus was a realist about universals [Duns Scotus, by Dumont] |
| Full Idea: Duns Scotus was a realist on the issue of universals and one of the main adversaries of Ockham's programme of nominalism. | |
| From: report of John Duns Scotus (works [1301]) by Stephen D. Dumont - Duns Scotus p.206 | |
| A reaction: The view of Scotus seems to be the minority view. It is hard to find thinkers who really believe that universals have an independent existence. My interest in Duns Scotus waned when I read this. How does he imagine universals? |
| 22127 | Scotus said a substantial principle of individuation [haecceitas] was needed for an essence [Duns Scotus, by Dumont] |
| Full Idea: Rejecting the standard views that essences are individuated by either actual existence, quantity or matter, Scotus said that the principle of individuation is a further substantial difference added to the species - the so-called haecceitas or 'thisness'. | |
| From: report of John Duns Scotus (works [1301]) by Stephen D. Dumont - Duns Scotus p.206 | |
| A reaction: [Scotus seldom referred to 'haecceitas'] I suppose essences have prior existence, but are too generic, so something must fix an essence as pertaining to this particular object. Is the haecceitas part of the essence, or of the particular? |
| 22126 | Avicenna and Duns Scotus say essences have independent and prior existence [Duns Scotus, by Dumont] |
| Full Idea: Duns Scotus endorsed Avicenna's theory of the common nature, according to which the essences have an independence and priority to their existence as either universal in the mind or singular outside it. | |
| From: report of John Duns Scotus (works [1301]) by Stephen D. Dumont - Duns Scotus p.206 | |
| A reaction: I occasionally meet this weird idea in modern discussions of essence (in Lowe?), and now see its origin. It makes little sense without a divine mind to support the independent essences. Scotus had to add a principle of individuation for essences. |
| 22129 | Certainty comes from the self-evident, from induction, and from self-awareness [Duns Scotus, by Dumont] |
| Full Idea: Duns Scotus grounded certitude in the knowledge of self-evident propositions, induction, and awareness of our own state. | |
| From: report of John Duns Scotus (works [1301]) by Stephen D. Dumont - Duns Scotus p.206 | |
| A reaction: Induction looks like the weak link here. |
| 22130 | Scotus defended direct 'intuitive cognition', against the abstractive view [Duns Scotus, by Dumont] |
| Full Idea: Scotus allocated to the intellect a direct, existential awareness of the intelligible object, called 'intuitive cognition', in contrast to abstractive knowledge, which seized the object independently of its presence to the intellect in actual existence. | |
| From: report of John Duns Scotus (works [1301]) by Stephen D. Dumont - Duns Scotus p.206 | |
| A reaction: Presumably if you see a thing, shut your eyes and then know it, that is 'abstractive'. Scotus says open your eyes for proper knowledge. |
| 22128 | Augustine's 'illumination' theory of knowledge leads to nothing but scepticism [Duns Scotus, by Dumont] |
| Full Idea: Scotus rejected Henry of Ghent's defence of Augustine's of knowledge by 'illumination', as leading to nothing but scepticism. ...After this, illumination never made a serious recovery. | |
| From: report of John Duns Scotus (works [1301]) by Stephen D. Dumont - Duns Scotus p.206 |
| 7628 | Broad rejects the inferential component of the representative theory [Broad, by Maund] |
| Full Idea: Broad, one of the most important modern defenders of the representative theory of perception, explicitly rejects the inferential component of the theory. | |
| From: report of C.D. Broad (Mind and Its Place in Nature [1925]) by Barry Maund - Perception Ch.1 | |
| A reaction: Since the supposed inferences happen much too quickly to be conscious, it is hard to see how we could distinguish an inference from an interpretation mechanism. Personally I interpret things long before the question of truth arises. |
| 22131 | The will retains its power for opposites, even when it is acting [Duns Scotus, by Dumont] |
| Full Idea: Scotus said the will is a power for opposites, in the sense that even when actually willing one thing, it retains a real, active power to will the opposite. He detaches the idea of freedom from time and variability. | |
| From: report of John Duns Scotus (works [1301]) by Stephen D. Dumont - Duns Scotus p.206 | |
| A reaction: In the sense that we can abandon an action when in the middle of it, this seems to be correct. Not just 'I could have done otherwise', but 'I don't have to be doing this'. This shows that the will has wide power, but not that it is 'free'. |
| 22123 | The concept of God is the unique first efficient cause, final cause, and most eminent being [Duns Scotus, by Dumont] |
| Full Idea: Duns Scotus establishes God as first efficient cause, as ultimate final cause, and as most eminent being - his so-called 'triple primacy' - and says there is a unique nature within these primacies. | |
| From: report of John Duns Scotus (works [1301]) by Stephen D. Dumont - Duns Scotus p.206 | |
| A reaction: This is the first stage of Duns Scotus's unusually complex argument for God's existence. Asserting the actual infinity of this unique being concludes his argument. |
| 22124 | We can't infer the infinity of God from creation ex nihilo [Duns Scotus, by Dumont] |
| Full Idea: Duns Scotus rejected the traditional argument that the infinity of God can be inferred from creation ex nihilo. | |
| From: report of John Duns Scotus (works [1301]) by Stephen D. Dumont - Duns Scotus p.206 | |
| A reaction: He accepted the infinity of God, however, but not for this reason. I don't know why he rejected it. I suppose the rejected claim is that something has to be infinite, and if it isn't the Cosmos then that leaves God? |