20 ideas
| 8074 | There is a five shilling fine for each point of divergence from the thinking of Aristotle [Oxford Univ 1350] |
| Full Idea: Bachelors and Masters of Arts who do not follow Aristotle's philosophy are subject to a fine of five shillings for each point of divergence, as well as for infractions of the rules of the Organon. | |
| From: Oxford Univ 1350 (Oxford University Statutes [1350]), quoted by Keith Devlin - Goodbye Descartes Ch.2 | |
| A reaction: Lovely quotation! We may defend the medieval period as a genuinely philosophical age, but this sort of statement suggests otherwise, and shows what intellectual heroes the few independent thinkers like William of Ockham really were. |
| 15134 | The truthmaker principle requires some specific named thing to make the difference [Williamson] |
| Full Idea: The truthmaker principle seems compelling, because if a proposition is true, something must be different from a world in which it is false. The principle makes this specific, by treating 'something' as a quantifier binding a variable in name position. | |
| From: Timothy Williamson (Truthmakers and Converse Barcan Formula [1999], §2) | |
| A reaction: See Williamson for an examination of the logical implications of this. The point is that the principle seems to require some very specific 'thing', which may be asking too much. For a start, it might be the absence of a thing. |
| 15140 | The converse Barcan formula will not allow contingent truths to have truthmakers [Williamson] |
| Full Idea: The converse Barcan formula does not allow any contingent truths at all to have a truthmaker. Once cannot combine the converse Barcan formula with any truthmaker principle worth having. | |
| From: Timothy Williamson (Truthmakers and Converse Barcan Formula [1999], §3) | |
| A reaction: One might reply, so much the worse for the converse Barcan formula, but Williamson doesn't think that. |
| 15141 | Truthmaker is incompatible with modal semantics of varying domains [Williamson] |
| Full Idea: Friends of the truthmaker principle should reject the Kripke semantics of varying domains. | |
| From: Timothy Williamson (Truthmakers and Converse Barcan Formula [1999], §3) | |
| A reaction: See other ideas from this paper to get a sense of what that is about. |
| 15131 | If metaphysical possibility is not a contingent matter, then S5 seems to suit it best [Williamson] |
| Full Idea: In S5, necessity and possibility are not themselves contingent matters. This is plausible for metaphysical modality, since metaphysical possibility, unlike practical possibility, does not depend on the contingencies of one's situation. | |
| From: Timothy Williamson (Truthmakers and Converse Barcan Formula [1999], §1) | |
| A reaction: This is the clearest statement I have found of why S5 might be preferable for metaphysics. See Nathan Salmon for the rival view. Williamson's point sounds pretty persuasive to me. |
| 15135 | If the domain of propositional quantification is constant, the Barcan formulas hold [Williamson] |
| Full Idea: If the domain of propositional quantification is constant across worlds, the Barcan formula and its converse hold. | |
| From: Timothy Williamson (Truthmakers and Converse Barcan Formula [1999], §2) | |
| A reaction: So the issue is whether we should take metaphysics to be dealing with a constant or varying domains. Williamson seems to favour the former, but my instincts incline towards the latter. |
| 15139 | Converse Barcan: could something fail to meet a condition, if everything meets that condition? [Williamson] |
| Full Idea: The converse Barcan is at least plausible, since its denial says there is something that could fail to meet a condition when everything met that condition; but how could everything meet that condition if that thing did not? | |
| From: Timothy Williamson (Truthmakers and Converse Barcan Formula [1999], §3) | |
| A reaction: Presumably the response involves a discussion of domains, since everything in a given domain might meet a condition, but something in a different domain might fail it. |
| 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. |
| 18492 | Not all quantification is either objectual or substitutional [Williamson] |
| Full Idea: We should not assume that all quantification is either objectual or substitutional. | |
| From: Timothy Williamson (Truthmakers and Converse Barcan Formula [1999], p.262) | |
| A reaction: [see Prior 1971:31-4] He talks of quantifying into sentence position. |
| 15136 | Substitutional quantification is metaphysical neutral, and equivalent to a disjunction of instances [Williamson] |
| Full Idea: If quantification into sentence position is substitutional, then it is metaphysically neutral. A substitutionally interpreted 'existential' quantification is semantically equivalent to the disjunction (possibly infinite) of its substitution instances. | |
| From: Timothy Williamson (Truthmakers and Converse Barcan Formula [1999], §2) | |
| A reaction: Is it not committed to the disjunction, just as the objectual reading commits to objects? Something must make the disjunction true. Or is it too verbal to be about reality? |
| 15138 | Not all quantification is objectual or substitutional [Williamson] |
| Full Idea: We should not assume that all quantification is objectual or substitutional. | |
| From: Timothy Williamson (Truthmakers and Converse Barcan Formula [1999], §2) |
| 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. |
| 15137 | If 'fact' is a noun, can we name the fact that dogs bark 'Mary'? [Williamson] |
| Full Idea: If one uses 'fact' as a noun, the question arises why one cannot name the fact that dogs bark 'Mary'. | |
| From: Timothy Williamson (Truthmakers and Converse Barcan Formula [1999], §2 n10) | |
| A reaction: What an intriguing thought! Must all nouns pass this test? 'The courage of the regiment was called Alfred'? |
| 15142 | Our ability to count objects across possibilities favours the Barcan formulas [Williamson] |
| Full Idea: Consideration of our ability to count objects across possibilities strongly favour both the Barcan formula and its converse. | |
| From: Timothy Williamson (Truthmakers and Converse Barcan Formula [1999], §3) | |
| A reaction: I'm not sure that I can understand counting objects across possibilities. The objects themselves are possibilia, and possibilia seem to include unknowns. The unexpected is highly possible. |
| 15133 | A thing can't be the only necessary existent, because its singleton set would be as well [Williamson] |
| Full Idea: That there is just one necessary existent is surely false, for if x is a necessary, {x} is a distinct necessary existent. | |
| From: Timothy Williamson (Truthmakers and Converse Barcan Formula [1999], §1) | |
| A reaction: You would have to believe that sets actually 'exist' to accept this, but it is a very neat point. |