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! |
| 15565 | Events have inbuilt essences, as necessary conditions for their occurrence [Lewis] |
| Full Idea: Events have their essences built in, in the form of necessary conditions for their occurrence. | |
| From: David Lewis (Events [1986], III) | |
| A reaction: Revealing. He thinks the essence of an event is something which precedes the event. I take it as obvious that if an event has an essence, it will be some features of the event that occur in it and during it. They need to be intrinsic. |
| 15567 | Some events involve no change; they must, because causal histories involve unchanges [Lewis] |
| Full Idea: Not all events involve change. We cannot afford to count the unchanges as nonevents, for the unchanges may be needed to complete causal histories. | |
| From: David Lewis (Events [1986], VI) | |
| A reaction: You end up calling non-changes 'events' if you commit to a simplistic theory that all causal histories consist of events. Why not allow conditions as well as events? Lewis concedes that he may be abusing language. |
| 15566 | Events are classes, and so there is a mereology of their parts [Lewis] |
| Full Idea: If events are classes, as I propose, then they have a mereology in the way that all classes do: the parts of a class are its subclasses. | |
| From: David Lewis (Events [1986], V) | |
| A reaction: Lewis says events are properties, which he regards as classes. It is not clear that events are strictly mereological. Could one happening be two events? Is WWII a simple sum of its parts? [see p.260] |
| 15561 | The events that suit semantics may not be the events that suit causation [Lewis] |
| Full Idea: There is no guarantee that events made for semantics are the same as events that are causes and effects. | |
| From: David Lewis (Events [1986], I) | |
| A reaction: This little cri de couer could be a motto for a huge amount of analytic philosophy, which (for some odd reason) thought that mathematics, logic, set theory and formal semantics were good tools for explaining nature. |
| 15564 | An event is a property of a unique space-time region [Lewis] |
| Full Idea: I propose to identify an event with a property, or in other words with a class, a unique spatio-temporal region corresponding to where that event occurs. | |
| From: David Lewis (Events [1986], II) | |
| A reaction: [I've run together two separate bits, on p.244 and 245] Lewis cites Montague's similar view, that events are properties of times. |
| 15563 | Properties are very abundant (unlike universals), and are used for semantics and higher-order variables [Lewis] |
| Full Idea: Properties are abundant, numbering at least beth-3 for properties of individuals alone; they are suited to serve as semantic values of arbitrarily complex predicates and gerunds, and higher-order variables. (If there are universals, they are sparse). | |
| From: David Lewis (Events [1986], II n2) | |
| A reaction: To me this is an outrageous hijacking of the notion of property which is needed for explaining the natural world. He seems to be talking about predicates. He wants to leave me with his silly universals - well I don't want them, thank you. |
| 1748 | Archelaus was the first person to say that the universe is boundless [Archelaus, by Diog. Laertius] |
| Full Idea: Archelaus was the first person to say that the universe is boundless. | |
| From: report of Archelaus (fragments/reports [c.450 BCE]) by Diogenes Laertius - Lives of Eminent Philosophers 02.Ar.3 |
| 15562 | Causation is a general relation derived from instances of causal dependence [Lewis] |
| Full Idea: Causation is the ancestral of causal dependence: event c causes event e iff either e depends on c, or e depends on an intermediate event which in turn depends on c, or.... | |
| From: David Lewis (Events [1986], I) | |
| A reaction: This is Lewis making sure that we don't postulate some huge bogus thing called 'Causation' which is supposed to be in charge of Nature. Good point. |
| 5989 | Archelaus said life began in a primeval slime [Archelaus, by Schofield] |
| Full Idea: Archelaus wrote that life on Earth began in a primeval slime. | |
| From: report of Archelaus (fragments/reports [c.450 BCE]) by Malcolm Schofield - Archelaus | |
| A reaction: This sounds like a fairly clearcut assertion of the production of life by evolution. Darwin's contribution was to propose the mechanism for achieving it. We should honour the name of Archelaus for this idea. |