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All the ideas for 'Philosophy of Mathematics', 'Grounding, Transitivity and Contrastivity' and 'Explanation: the state of play'

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12 ideas

2. Reason / D. Definition / 8. Impredicative Definition
Predicative definitions only refer to entities outside the defined collection [Horsten]
     Full Idea: Definitions are called 'predicative', and are considered sound, if they only refer to entities which exist independently from the defined collection.
     From: Leon Horsten (Philosophy of Mathematics [2007], §2.4)
5. Theory of Logic / J. Model Theory in Logic / 2. Isomorphisms
A theory is 'categorical' if it has just one model up to isomorphism [Horsten]
     Full Idea: If a theory has, up to isomorphism, exactly one model, then it is said to be 'categorical'.
     From: Leon Horsten (Philosophy of Mathematics [2007], §5.2)
6. Mathematics / B. Foundations for Mathematics / 2. Proof in Mathematics
Computer proofs don't provide explanations [Horsten]
     Full Idea: Mathematicians are uncomfortable with computerised proofs because a 'good' proof should do more than convince us that a certain statement is true. It should also explain why the statement in question holds.
     From: Leon Horsten (Philosophy of Mathematics [2007], §5.3)
6. Mathematics / B. Foundations for Mathematics / 6. Mathematics as Set Theory / a. Mathematics is set theory
The concept of 'ordinal number' is set-theoretic, not arithmetical [Horsten]
     Full Idea: The notion of an ordinal number is a set-theoretic, and hence non-arithmetical, concept.
     From: Leon Horsten (Philosophy of Mathematics [2007], §2.3)
7. Existence / C. Structure of Existence / 1. Grounding / a. Nature of grounding
As causation links across time, grounding links the world across levels [Schaffer,J]
     Full Idea: Grounding is something like metaphysical causation. Just as causation links the world across time, grounding links the world across levels. Grounding connects the more fundamental to the less fundamental, and thereby backs a certain form of explanation.
     From: Jonathan Schaffer (Grounding, Transitivity and Contrastivity [2012], Intro)
     A reaction: Obviously you need 'levels' for this, which we should take to be structural levels.
If ground is transitive and irreflexive, it has a strict partial ordering, giving structure [Schaffer,J]
     Full Idea: By treating grounding as transitive (and irreflexive), one generates a strict partial ordering that induces metaphysical structure.
     From: Jonathan Schaffer (Grounding, Transitivity and Contrastivity [2012], Intro)
     A reaction: Schaffer's paper goes on to attach the claim that grounding is transitive, but I didn't find his examples very convincing.
14. Science / D. Explanation / 1. Explanation / b. Aims of explanation
Audience-relative explanation, or metaphysical explanation based on information? [Stanford]
     Full Idea: Rather than an 'interest-relative' notion of explanation (Putnam), it can be informational content which makes an explanation, which is an 'audience-invariant' contraint, which is not pragmatic, but mainly epistemological and also partly metaphysical.
     From: Michael Stanford (Explanation: the state of play [1991], p.172)
     A reaction: [compressed summary of Ruben 1990] Examples given are that Rome burning explains Nero fiddling, even if no one ever says so, and learning that George III had porphyria explains his madness.
Explanation is for curiosity, control, understanding, to make meaningful, or to give authority [Stanford]
     Full Idea: There are a number of reasons why we explain: out of sheer curiosity, to increase our control of a situation, to help understanding by simplifying or making familiar, to confer meaning or significance, and to give scientific authority to some statement.
     From: Michael Stanford (Explanation: the state of play [1991], p.172)
14. Science / D. Explanation / 2. Types of Explanation / b. Contrastive explanations
Explaining 'Adam ate the apple' depends on emphasis, and thus implies a contrast [Schaffer,J]
     Full Idea: Explaining why ADAM ate the apple is a different matter from explaining why he ATE the apple, and from why he ate THE APPLE. ...In my view the best explanations incorporate ....contrastive information.
     From: Jonathan Schaffer (Grounding, Transitivity and Contrastivity [2012], 4.3.1)
     A reaction: But why are the contrasts Eve, or throwing it, or a pear? It occurs to me that this is wrong! The contrast is with anything else which could have gone in subject, verb or object position. It is a matter of categories, not of contrasts.
14. Science / D. Explanation / 2. Types of Explanation / j. Explanations by reduction
We can explain by showing constitution, as well as showing causes [Stanford]
     Full Idea: The powerful engine of my car can be explained by an examination of each of its parts, but it is not caused by them. They do not cause the engine; they constitute it.
     From: Michael Stanford (Explanation: the state of play [1991], p.174)
     A reaction: [example from Ruben 1990:221] This could be challenged, since there is clearly a causal connection between the constitution and the whole. We distinguish engine parts which contribute to the power from those which do not.
26. Natural Theory / A. Speculations on Nature / 1. Nature
I take what is fundamental to be the whole spatiotemporal manifold and its fields [Schaffer,J]
     Full Idea: I myself would prefer to speak of what is fundamental in terms of the whole spatiotemporal manifold and the fields that permeate it, with parts counting as derivative of the whole.
     From: Jonathan Schaffer (Grounding, Transitivity and Contrastivity [2012], 4.1.1)
     A reaction: Not quite the Parmenidean One, since it has parts, but a nice try at updating the great man. Note the reference to 'fields', suggesting that this view is grounded in the physics rather than metaphysics. How many fields has it got?
26. Natural Theory / C. Causation / 1. Causation
Nowadays causation is usually understood in terms of equations and variable ranges [Schaffer,J]
     Full Idea: The leading treatments of causation work within 'structural equation models', with events represented via variables each of which is allotted a range of permitted values, which constitute a 'contrast space'.
     From: Jonathan Schaffer (Grounding, Transitivity and Contrastivity [2012], 4.3.1)
     A reaction: Like Woodward's idea that causation is a graph, this seems to be a matter of plotting or formalising correlations between activities, which is a very Humean approach to causation.