13 ideas
| 15395 | Give up objects necessitating truths, and say their natures cause the truths? [Cameron] |
| Full Idea: We could abandon the view that truthmakers necessitate the truth of that which makes them true, and say that an object makes a truth when its intrinsic nature suffices for that truth. The object would have a different intrinsic nature if the truth failed. | |
| From: Ross P. Cameron (Intrinsic and Extrinsic Properties [2009], 'Truthmakers') | |
| A reaction: [He cites Josh Parsons 1999, 2005 for this] This approach seems closely related to Kit Fine's proposal that necessities arise from the natures of things. It sounds to me as if an object with that intrinsic nature would necessitate that truth. |
| 15394 | Truthmaker requires a commitment to tropes or states of affairs, for contingent truths [Cameron] |
| Full Idea: The most popular view is that an object is a truthmaker if the object couldn't exist and the truth be false. But contingent predications are also held to need truthmakers. Socrates is not necessarily snub-nosed, so a trope or state of affairs is needed. | |
| From: Ross P. Cameron (Intrinsic and Extrinsic Properties [2009], 'Truthmakers') | |
| A reaction: Cameron calls this 'some heavy ontological commitments'. If snub-nosedness is necessitated by the trope of 'being snub-nosed', what is the truthmaker for Socrates having that trope? |
| 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. |
| 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. |
| 15401 | Essentialists say intrinsic properties arise from what the thing is, irrespective of surroundings [Cameron] |
| Full Idea: The essentialist approach would be to say that an intrinsic property is one such that it is no part of what it is to instantiate that property that the bearer stands in some relation to its surroundings. | |
| From: Ross P. Cameron (Intrinsic and Extrinsic Properties [2009], 'Analysis') | |
| A reaction: This is offered as an alternative to the David Lewis account in terms of duplicates across possible worlds. You will have gathered by now, if you have spent days poring over my stuff, that I favour the essentialist approach. |
| 15393 | An object's intrinsic properties are had in virtue of how it is, independently [Cameron] |
| Full Idea: Intrinsic properties are those that an object has solely in virtue of how it is, independently of its surroundings. | |
| From: Ross P. Cameron (Intrinsic and Extrinsic Properties [2009], 'Intro') | |
| A reaction: Better not mention quantum mechanics and fields if you want to talk of objects being independent of their surroundings. Am I 'independent' of gravity, or is gravity 'independent' of me? |
| 15396 | Most criteria for identity over time seem to leave two later objects identical to the earlier one [Cameron] |
| Full Idea: Criteria for identity across times have proven hard to give. Whatever criteria we lay down, it seems that there are possible situations in which two later objects bear the relevant relation to one earlier object, though only one of them can be identical. | |
| From: Ross P. Cameron (Intrinsic and Extrinsic Properties [2009], 'Personal') | |
| A reaction: We only have to think of twins, amoebae that fission, and the Ship of Theseus. We seem to end up inventing a dubious criterion in order to break the tie. |
| 19708 | Rational internal belief is conviction that a proposition enhances a belief system [Foley, by Vahid] |
| Full Idea: In Foley's subjective internalist account it is egocentrically rational for an agent to believe a proposition only if he would think on deep reflection that believing it is conducive to having an accurate and comprehensive belief system. | |
| From: report of Richard Foley (The Theory of Epistemic Rationality [1987], 2.1 B) by Hamid Vahid - Externalism/Internalism | |
| A reaction: I like this idea, because it indicates the link between internalism and coherence about justification. I don't think you can be an externalist coherence theorist for justification. [Reminder: Paul Thagard is the best writer on coherence]. |