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All the ideas for 'Mathematics without Numbers', 'Events as property exemplifications' and 'The Problem of Counterfactual Conditionals'

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10 ideas

6. Mathematics / B. Foundations for Mathematics / 7. Mathematical Structuralism / c. Nominalist structuralism
8698 Modal structuralism says mathematics studies possible structures, which may or may not be actualised [Hellman, by Friend]
     Full Idea: The modal structuralist thinks of mathematical structures as possibilities. The application of mathematics is just the realisation that a possible structure is actualised. As structures are possibilities, realist ontological problems are avoided.
     From: report of Geoffrey Hellman (Mathematics without Numbers [1989]) by Michèle Friend - Introducing the Philosophy of Mathematics 4.3
     A reaction: Friend criticises this and rejects it, but it is appealing. Mathematics should aim to be applicable to any possible world, and not just the actual one. However, does the actual world 'actualise a mathematical structure'?
9557 Statements of pure mathematics are elliptical for a sort of modal conditional [Hellman, by Chihara]
     Full Idea: Hellman represents statements of pure mathematics as elliptical for modal conditionals of a certain sort.
     From: report of Geoffrey Hellman (Mathematics without Numbers [1989]) by Charles Chihara - A Structural Account of Mathematics 5.3
     A reaction: It's a pity there is such difficulty in understanding conditionals (see Graham Priest on the subject). I intuit a grain of truth in this, though I take maths to reflect the structure of the actual world (with possibilities being part of that world).
10263 Modal structuralism can only judge possibility by 'possible' models [Shapiro on Hellman]
     Full Idea: The usual way to show that a sentence is possible is to show that it has a model, but for Hellman presumably a sentence is possible if it might have a model (or if, possibly, it has a model). It is not clear what this move brings us.
     From: comment on Geoffrey Hellman (Mathematics without Numbers [1989]) by Stewart Shapiro - Philosophy of Mathematics 7.3
     A reaction: I can't assess this, but presumably the possibility of the model must be demonstrated in some way. Aren't all models merely possible, because they are based on axioms, which seem to be no more than possibilities?
7. Existence / B. Change in Existence / 4. Events / c. Reduction of events
10369 How fine-grained Kim's events are depends on how finely properties are individuated [Kim, by Schaffer,J]
     Full Idea: How fine-grained Kim's events are depends on how finely properties are individuated.
     From: report of Jaegwon Kim (Events as property exemplifications [1976]) by Jonathan Schaffer - The Metaphysics of Causation 1.2
     A reaction: I don't actually buy the idea that an event could just be an 'exemplification'. Change seems to be required, and processes, or something like them, must be mentioned. Degrees of fine-graining sound good, though, for processes too.
8976 If events are ordered triples of items, such things seem to be sets, and hence abstract [Simons on Kim]
     Full Idea: If Kim's events are just the ordered triple of is that such things are standardly conceived as abstract entities, usually sets, whereas events are concretely located in space and time.
     From: comment on Jaegwon Kim (Events as property exemplifications [1976]) by Peter Simons - Events 2.1
     A reaction: You might reply that the object, and maybe the attribute, are concrete, and the time is natural, but the combination really is an abstraction, even though it is located (like the equator). Where is the set of my books located?
8975 Events cannot be merely ordered triples, but must specify the link between the elements [Kim, by Simons]
     Full Idea: Kim's events cannot just be the ordered triple of , since many such triples do not yield events, such as . Kim has to specify that the object actually has that property at that time.
     From: report of Jaegwon Kim (Events as property exemplifications [1976]) by Peter Simons - Events 2.1
     A reaction: Why should they even be in that particular order? This requirement rather messes up Kim's plan for a very streamlined, Ockhamised ontology. Circles have symmetry at all times. Is 'near Trafalgar Square' a property?
8974 Events are composed of an object with an attribute at a time [Kim, by Simons]
     Full Idea: Kim's events are exemplifications by an object of an attribute at a time...It does not make events basic entities, as the three constituents are more basic, but it gives identity conditions (two events are the same if object, attribute and time the same).
     From: report of Jaegwon Kim (Events as property exemplifications [1976]) by Peter Simons - Events 2.1
     A reaction: [Aristotle is said to be behind this] I am more sympathetic to this view than the claim that events are primitive. If a pebble is ellipsoid for a million years, is that an event? I think the concept of a 'process' is the most fruitful one to investigate.
8977 Since properties like self-identity and being 2+2=4 are timeless, Kim must restrict his properties [Simons on Kim]
     Full Idea: Since some tautologously universal properties such as self-identity or being such that 2+2=4 apply to all things at all times, that is stretching Kim's events too far. Candidate properties need to be realistically restricted, and it is unclear how.
     From: comment on Jaegwon Kim (Events as property exemplifications [1976]) by Peter Simons - Events 2.1
     A reaction: You could deploy Schoemaker's concept of natural properties in terms of the source of causal powers, but the problem would be that you were probably hoping to then use Kim's events to define causation. Answer: treat causation as the primitive.
8980 Kim's theory results in too many events [Simons on Kim]
     Full Idea: The criticism most frequently levelled against Kim's theory is that it results in an unacceptable plurality of finely differentiated events, because of the requirement for identity of the constituent property.
     From: comment on Jaegwon Kim (Events as property exemplifications [1976]) by Peter Simons - Events 4.4
     A reaction: This may mean that the Battle of Waterloo was several trillion events, which seems daft to the historian, but it doesn't to the physicist. A cannon firing is indeed an accumulation of lots of little events.
10. Modality / B. Possibility / 9. Counterfactuals
12191 Counterfactuals are true if logical or natural laws imply the consequence [Goodman, by McFetridge]
     Full Idea: Goodman's central idea was: 'If that match had been scratched, it would have lighted' is true if there are suitable truths from which, with the antecedent, the consequent can be inferred by means of a logical, or more typically natural, law.
     From: report of Nelson Goodman (The Problem of Counterfactual Conditionals [1947]) by Ian McFetridge - Logical Necessity: Some Issues §4
     A reaction: Goodman then discusses the problem of identifying the natural laws, and identifying the suitable truths. I'm inclined to think counterfactuals are vaguer than that; they are plausible if coherent reasons can be offered for the inference.