15 ideas
| 13591 | Quantified modal logic collapses if essence is withdrawn [Quine] |
| Full Idea: The whole of quantified modal logic collapses if essence is withdrawn. | |
| From: Willard Quine (Intensions Revisited [1977], p.121) | |
| A reaction: Quine offers an interesting qualification to this crushing remark in Idea 13590. The point is that objects must retain their identity in modal contexts, as if I say 'John Kennedy might have been Richard Nixon'. What could that mean? |
| 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. |
| 13590 | Essences can make sense in a particular context or enquiry, as the most basic predicates [Quine] |
| Full Idea: The notion of essence makes sense in context. Relative to a particular enquiry, some predicates may play a more basic role than others, or may apply more fixedly; and these may be treated as essential. | |
| From: Willard Quine (Intensions Revisited [1977], p.121) | |
| A reaction: Quine has got a bad press on essentialism, and on modal logic, but I take this point seriously. If you give something a fixed identity by means of essence in some context, you can then go ahead and apply possible world reasoning in that context. |
| 8483 | Necessity is relative to context; it is what is assumed in an inquiry [Quine] |
| Full Idea: The very notion of necessity makes sense to me only relative to context. Typically it is applied to what is assumed in an inquiry, as against what has yet to transpire. | |
| From: Willard Quine (Intensions Revisited [1977], p.121) | |
| A reaction: Lots of things are assumed by an inquiry without an assumption that they must be true. Quine is the greatest opponent of necessity in all of philosophy. Asserting necessities, though, is too much fun to give up. It would ruin philosophy. |
| 13589 | Possible worlds are a way to dramatise essentialism, and yet they presuppose essentialism [Quine] |
| Full Idea: Talk of possible worlds is a graphic way of waging the essentialist philosophy, but it is only that; it is not an explication. Essence is needed to identify an object from one possible world to another. | |
| From: Willard Quine (Intensions Revisited [1977], p.118) | |
| A reaction: He makes the proposal sound circular, but I take a commitment to essences to be prior to talk of possible worlds. Possible worlds are a tool for clarifying modalities, not for clarifying essential identities. |
| 13588 | A rigid designator (for all possible worlds) picks out an object by its essential traits [Quine] |
| Full Idea: A rigid designator differs from others in that it picks out its object by essential traits. It designates the object in all possible worlds in which it exists. | |
| From: Willard Quine (Intensions Revisited [1977], p.118) | |
| A reaction: This states the point more clearly than Kripke ever does, and I presume it is right. Thus when we say that we wish 'our' Hubert Humphrey had won the election, we can allow that his victory elation would change him a bit. Kripke is right. |
| 13592 | Beliefs can be ascribed to machines [Quine] |
| Full Idea: Beliefs have been ascribed to machines, in support of a mechanistic philosophy, and I share this attitude. | |
| From: Willard Quine (Intensions Revisited [1977], p.123) | |
| A reaction: [He cites Raymond Nelson] One suspects that this is Quine's latent behaviourism speaking. It strikes me as a crass misuse of 'belief' to ascribe it to a simple machine like a thermostat. |
| 19518 | Evidentialism says justifications supervene on the available evidence [Conee/Feldman] |
| Full Idea: Fundamentally Evidentialism is a supervenience thesis, according to which facts about whether or not a person is justified in believing a proposition supervene on facts describing the evidence the person has. | |
| From: E Conee / R Feldman (Introduction to 'Evidentialism' [2004], p.1) | |
| A reaction: If facts 'describe', does that make them linguistic? That's not how I use 'facts'. A statement of a fact is not the same as the fact. An ugly fact can be beautifully expressed. I am, however, in favour of evidence. |
| 19519 | Rational decisions are either taken to be based on evidence, or to be explained causally [Conee/Feldman] |
| Full Idea: In decision theory, there is a view according to which the rational basis for all decisions is evidential. This kind of decision theory is typically contrasted with causal decision theory. | |
| From: E Conee / R Feldman (Introduction to 'Evidentialism' [2004], p.3) | |
| A reaction: Your Kantian presumably likes rational reflection on evidence, and your modern reductive scientist prefers causality (which doesn't really sound very rational). |