16 ideas
| 23367 | Even pointing a finger should only be done for a reason [Epictetus] |
| Full Idea: Philosophy says it is not right even to stretch out a finger without some reason. | |
| From: Epictetus (fragments/reports [c.57], 15) | |
| A reaction: The key point here is that philosophy concerns action, an idea on which Epictetus is very keen. He rather despise theory. This idea perfectly sums up the concept of the wholly rational life (which no rational person would actually want to live!). |
| 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. |
| 16422 | The necessity of a proposition concerns reality, not our words or concepts [Stalnaker] |
| Full Idea: The necessity or contingency of a proposition has nothing to do with our concepts or the meanings of our words. The possibilities would have been the same even if we had never conceived of them. | |
| From: Robert C. Stalnaker (Conceptual truth and metaphysical necessity [2003], 1) | |
| A reaction: This sounds in need of qualification, since some of the propositions will be explicitly about words and concepts. Still, I like this idea. |
| 16423 | Conceptual possibilities are metaphysical possibilities we can conceive of [Stalnaker] |
| Full Idea: Conceptual possibilities are just (metaphysical) possibilities that we can conceive of. | |
| From: Robert C. Stalnaker (Conceptual truth and metaphysical necessity [2003], 1) |
| 16421 | Critics say there are just an a priori necessary part, and an a posteriori contingent part [Stalnaker] |
| Full Idea: Critics say there are no irreducible a posteriori truths. They can be factored into a part that is necessary, but knowable a priori through conceptual analysis, and a part knowable only a posteriori, but contingent. 2-D semantics makes this precise. | |
| From: Robert C. Stalnaker (Conceptual truth and metaphysical necessity [2003], 1) | |
| A reaction: [Critics are Sidelle, Jackson and Chalmers] Interesting. If gold is necessarily atomic number 79, or it wouldn't be gold, that sounds like an analytic truth about gold. Discovering the 79 wasn't a discovery of a necessity. Stalnaker rejects this idea. |
| 16429 | A 'centred' world is an ordered triple of world, individual and time [Stalnaker] |
| Full Idea: A 'centred' possible world is an ordered triple consisting of a possible world, an individual in the domain of that world, and a time. | |
| From: Robert C. Stalnaker (Conceptual truth and metaphysical necessity [2003], 2) |
| 16428 | Meanings aren't in the head, but that is because they are abstract [Stalnaker] |
| Full Idea: Meanings ain't in the head. Putnam's famous slogan actually fits Frege's anti-psychologism better than it fits Purnam's and Burge's anti-individualism. The point is that intensions of any kind are abstract objects. | |
| From: Robert C. Stalnaker (Conceptual truth and metaphysical necessity [2003], 2) | |
| A reaction: If intensions are abstract, that leaves (for me) the question of what they are abstracted from. I take it that there are specific brain events that are being abstractly characterised. What do we call those? |
| 16432 | One view says the causal story is built into the description that is the name's content [Stalnaker] |
| Full Idea: In 'causal descriptivism' the causal story is built into the description that is the content of the name (and also incorporates a rigidifying operator to ensure that the descriptions that names abbreviate have wide scope). | |
| From: Robert C. Stalnaker (Conceptual truth and metaphysical necessity [2003], 5) | |
| A reaction: Not very controversial, I would say, since virtually every fact about the world has a 'causal story' built into it. Must we insist on rigidity in order to have wide scope? |
| 16430 | Two-D says that a posteriori is primary and contingent, and the necessity is the secondary intension [Stalnaker] |
| Full Idea: Two-dimensionalism says the necessity of a statement is constituted by the fact that the secondary intensions is a necessary proposition, and their a posteriori character is constituted by the fact that the associated primary intension is contingent. | |
| From: Robert C. Stalnaker (Conceptual truth and metaphysical necessity [2003], 2) | |
| A reaction: This view is found in Sidelle 1989, and then formalised by Jackson and Chalmers. I like metaphysical necessity, but I have some sympathy with the approach. The question must always be 'where does this necessity derive from'? |
| 16431 | In one view, the secondary intension is metasemantic, about how the thinker relates to the content [Stalnaker] |
| Full Idea: On the metasemantic interpretation of the two-dimensional framework, the second dimension is used to represent the metasemantic facts about the relation between a thinker or speaker and the contents of her thoughts or utterances. | |
| From: Robert C. Stalnaker (Conceptual truth and metaphysical necessity [2003], 4) | |
| A reaction: I'm struggling to think what facts there might be about the relation between myself and the contents of my thoughts. I'm more or less constituted by my thoughts. |