16 ideas
| 14600 | Analysis aims at secure necessary and sufficient conditions [Schaffer,J] |
| Full Idea: An analysis is an attempt at providing finite, non-circular, and intuitively adequate necessary and sufficient conditions. | |
| From: Jonathan Schaffer (Causation and Laws of Nature [2008], 3) | |
| A reaction: Specifying the 'conditions' for something doesn't seem to quite add up to telling you what the thing is. A trivial side-effect might qualify as a sufficient condition for something, if it always happens. |
| 14603 | 'Reification' occurs if we mistake a concept for a thing [Schaffer,J] |
| Full Idea: 'Reification' occurs when a mere concept is mistaken for a thing. We seem generally prone to this sort of error. | |
| From: Jonathan Schaffer (Causation and Laws of Nature [2008], 3.1) | |
| A reaction: Personally I think we should face up to the fact that this is the only way we can think about generalised or abstract entities, and stop thinking of it as an 'error'. We have evolved to think well about objects, so we translate everything that way. |
| 14607 | T adds □p→p for reflexivity, and is ideal for modeling lawhood [Schaffer,J] |
| Full Idea: System T is a normal modal system augmented with the reflexivity-generating axiom □p→p, and is, I think, the best modal logic for modeling lawhood. | |
| From: Jonathan Schaffer (Causation and Laws of Nature [2008], n46) | |
| A reaction: Schaffer shows in the article why transitivity would not be appropriate for lawhood. |
| 17610 | The Axiom of Choice paradoxically allows decomposing a sphere into two identical spheres [Maddy] |
| Full Idea: One feature of the Axiom of Choice that troubled many mathematicians was the so-called Banach-Tarski paradox: using the Axiom, a sphere can be decomposed into finitely many parts and those parts reassembled into two spheres the same size as the original. | |
| From: Penelope Maddy (Defending the Axioms [2011], 1.3) | |
| A reaction: (The key is that the parts are non-measurable). To an outsider it is puzzling that the Axiom has been universally accepted, even though it produces such a result. Someone can explain that, I'm sure. |
| 17620 | Critics of if-thenism say that not all starting points, even consistent ones, are worth studying [Maddy] |
| Full Idea: If-thenism denies that mathematics is in the business of discovering truths about abstracta. ...[their opponents] obviously don't regard any starting point, even a consistent one, as equally worthy of investigation. | |
| From: Penelope Maddy (Defending the Axioms [2011], 3.3) | |
| A reaction: I have some sympathy with if-thenism, in that you can obviously study the implications of any 'if' you like, but deep down I agree with the critics. |
| 17605 | Hilbert's geometry and Dedekind's real numbers were role models for axiomatization [Maddy] |
| Full Idea: At the end of the nineteenth century there was a renewed emphasis on rigor, the central tool of which was axiomatization, along the lines of Hilbert's axioms for geometry and Dedekind's axioms for real numbers. | |
| From: Penelope Maddy (Defending the Axioms [2011], 1.3) |
| 17625 | If two mathematical themes coincide, that suggest a single deep truth [Maddy] |
| Full Idea: The fact that two apparently fruitful mathematical themes turn out to coincide makes it all the more likely that they're tracking a genuine strain of mathematical depth. | |
| From: Penelope Maddy (Defending the Axioms [2011], 5.3ii) |
| 17615 | Every infinite set of reals is either countable or of the same size as the full set of reals [Maddy] |
| Full Idea: One form of the Continuum Hypothesis is the claim that every infinite set of reals is either countable or of the same size as the full set of reals. | |
| From: Penelope Maddy (Defending the Axioms [2011], 2.4 n40) |
| 17618 | Set-theory tracks the contours of mathematical depth and fruitfulness [Maddy] |
| Full Idea: Our set-theoretic methods track the underlying contours of mathematical depth. ...What sets are, most fundamentally, is markers for these contours ...they are maximally effective trackers of certain trains of mathematical fruitfulness. | |
| From: Penelope Maddy (Defending the Axioms [2011], 3.4) | |
| A reaction: This seems to make it more like a map of mathematics than the actual essence of mathematics. |
| 14604 | If a notion is ontologically basic, it should be needed in our best attempt at science [Schaffer,J] |
| Full Idea: Science represents our best systematic understanding of the world, and if a certain notion proves unneeded in our best attempt at that, this provides strong evidence that what this notion concerns is not ontologically basic. | |
| From: Jonathan Schaffer (Causation and Laws of Nature [2008], 3.2) | |
| A reaction: But is the objective of science to find out what is 'ontologically basic'? If scientists can't get a purchase on a question, they have no interest in it. What are electrons made of? |
| 17614 | The connection of arithmetic to perception has been idealised away in modern infinitary mathematics [Maddy] |
| Full Idea: Ordinary perceptual cognition is most likely involved in our grasp of elementary arithmetic, but ...this connection to the physical world has long since been idealized away in the infinitary structures of contemporary pure mathematics. | |
| From: Penelope Maddy (Defending the Axioms [2011], 2.3) | |
| A reaction: Despite this, Maddy's quest is for a 'naturalistic' account of mathematics. She ends up defending 'objectivity' (and invoking Tyler Burge), rather than even modest realism. You can't 'idealise away' the counting of objects. I blame Cantor. |
| 14599 | Three types of reduction: Theoretical (of terms), Definitional (of concepts), Ontological (of reality) [Schaffer,J] |
| Full Idea: Theoretical reduction concerns terms found in a theory; Definitional reduction concerns concepts found in the mind; Ontological reduction is independent of how we conceptualize entities, or theorize about them, and is about reality. | |
| From: Jonathan Schaffer (Causation and Laws of Nature [2008], 1) | |
| A reaction: An Aristotelian definition refers to reality, rather than to our words or concepts. |
| 14605 | Tropes are the same as events [Schaffer,J] |
| Full Idea: Tropes can be identified with events. | |
| From: Jonathan Schaffer (Causation and Laws of Nature [2008], n17) | |
| A reaction: This is presumably on the view of events, associated with Kim, as instantiations of properties. This idea is a new angle on tropes and events which had never occurred to me. |
| 14601 | Individuation aims to count entities, by saying when there is one [Schaffer,J] |
| Full Idea: Individuation principles are attempts to describe how to count entities in a given domain, by saying when there is one. | |
| From: Jonathan Schaffer (Causation and Laws of Nature [2008], 3) | |
| A reaction: At last, someone tells me what they mean by 'individuation'! So it is just saying what your units are prior to counting, followed (presumably) by successful counting. It seems to aim more at kinds than at particulars. |
| 16629 | By comparing qualities and features, reason can gradually infer the nature of substance [Grosseteste] |
| Full Idea: Awakened reason distinguishes color from size and shape from body and then shape and size from the substance of body, and so by drawing distinctions and abstracting, it arrives at a grasp of the substance of body, which supports size, shape and color. | |
| From: Robert Grosseteste (Commentary on 'Posterior Analytics' [1226], I.14), quoted by Robert Pasnau - Metaphysical Themes 1274-1671 07.4 | |
| A reaction: This optimistic view influenced Aquinas, and is called 'incrementalism' by Pasnau. It is the spirit of scientific essentialism, and a nice instance of inference to the best explanation (though 'substance' in itself explains virtually nothing). |
| 14606 | Only ideal conceivability could indicate what is possible [Schaffer,J] |
| Full Idea: The only plausible link from conceivability to possibility is via ideal conceivability. | |
| From: Jonathan Schaffer (Causation and Laws of Nature [2008], n22) | |
| A reaction: [He cites Chalmers 2002] I'm not sure what 'via' could mean here. Since I don't know any other way than attempted conceivability for assessing a possibility, I am a bit baffled by this idea. |