27 ideas
| 13011 | New axioms are being sought, to determine the size of the continuum [Maddy] |
| Full Idea: In current set theory, the search is on for new axioms to determine the size of the continuum. | |
| From: Penelope Maddy (Believing the Axioms I [1988], §0) | |
| A reaction: This sounds the wrong way round. Presumably we seek axioms that fix everything else about set theory, and then check to see what continuum results. Otherwise we could just pick our continuum, by picking our axioms. |
| 13013 | The Axiom of Extensionality seems to be analytic [Maddy] |
| Full Idea: Most writers agree that if any sense can be made of the distinction between analytic and synthetic, then the Axiom of Extensionality should be counted as analytic. | |
| From: Penelope Maddy (Believing the Axioms I [1988], §1.1) | |
| A reaction: [Boolos is the source of the idea] In other words Extensionality is not worth discussing, because it simply tells you what the world 'set' means, and there is no room for discussion about that. The set/class called 'humans' varies in size. |
| 13014 | Extensional sets are clearer, simpler, unique and expressive [Maddy] |
| Full Idea: The extensional view of sets is preferable because it is simpler, clearer, and more convenient, because it individuates uniquely, and because it can simulate intensional notions when the need arises. | |
| From: Penelope Maddy (Believing the Axioms I [1988], §1.1) | |
| A reaction: [She cites Fraenkel, Bar-Hillet and Levy for this] The difficulty seems to be whether the extensional notion captures our ordinary intuitive notion of what constitutes a group of things, since that needs flexible size and some sort of unity. |
| 13021 | The Axiom of Infinity states Cantor's breakthrough that launched modern mathematics [Maddy] |
| Full Idea: The Axiom of Infinity is a simple statement of Cantor's great breakthrough. His bold hypothesis that a collection of elements that had lurked in the background of mathematics could be infinite launched modern mathematics. | |
| From: Penelope Maddy (Believing the Axioms I [1988], §1.5) | |
| A reaction: It also embodies one of those many points where mathematics seems to depart from common sense - but then most subjects depart from common sense when they get more sophisticated. Look what happened to art. |
| 13022 | Infinite sets are essential for giving an account of the real numbers [Maddy] |
| Full Idea: If one is interested in analysis then infinite sets are indispensable since even the notion of a real number cannot be developed by means of finite sets alone. | |
| From: Penelope Maddy (Believing the Axioms I [1988], §1.5) | |
| A reaction: [Maddy is citing Fraenkel, Bar-Hillel and Levy] So Cantor's great breakthrough (Idea 13021) actually follows from the earlier acceptance of the real numbers, so that's where the departure from common sense started. |
| 13023 | The Power Set Axiom is needed for, and supported by, accounts of the continuum [Maddy] |
| Full Idea: The Power Set Axiom is indispensable for a set-theoretic account of the continuum, ...and in so far as those attempts are successful, then the power-set principle gains some confirmatory support. | |
| From: Penelope Maddy (Believing the Axioms I [1988], §1.6) | |
| A reaction: The continuum is, of course, notoriously problematic. Have we created an extra problem in our attempts at solving the first one? |
| 13024 | Efforts to prove the Axiom of Choice have failed [Maddy] |
| Full Idea: Jordain made consistent and ill-starred efforts to prove the Axiom of Choice. | |
| From: Penelope Maddy (Believing the Axioms I [1988], §1.7) | |
| A reaction: This would appear to be the fate of most axioms. You would presumably have to use a different system from the one you are engaged with to achieve your proof. |
| 13025 | Modern views say the Choice set exists, even if it can't be constructed [Maddy] |
| Full Idea: Resistance to the Axiom of Choice centred on opposition between existence and construction. Modern set theory thrives on a realistic approach which says the choice set exists, regardless of whether it can be defined, constructed, or given by a rule. | |
| From: Penelope Maddy (Believing the Axioms I [1988], §1.7) | |
| A reaction: This seems to be a key case for the ontology that lies at the heart of theory. Choice seems to be an invaluable tool for proofs, so it won't go away, so admit it to the ontology. Hm. So the tools of thought have existence? |
| 13026 | A large array of theorems depend on the Axiom of Choice [Maddy] |
| Full Idea: Many theorems depend on the Axiom of Choice, including that a countable union of sets is countable, and results in analysis, topology, abstract algebra and mathematical logic. | |
| From: Penelope Maddy (Believing the Axioms I [1988], §1.7) | |
| A reaction: The modern attitude seems to be to admit anything if it leads to interesting results. It makes you wonder about the modern approach of using mathematics and logic as the cutting edges of ontological thinking. |
| 13019 | The Iterative Conception says everything appears at a stage, derived from the preceding appearances [Maddy] |
| Full Idea: The Iterative Conception (Zermelo 1930) says everything appears at some stage. Given two objects a and b, let A and B be the stages at which they first appear. Suppose B is after A. Then the pair set of a and b appears at the immediate stage after B. | |
| From: Penelope Maddy (Believing the Axioms I [1988], §1.3) | |
| A reaction: Presumably this all happens in 'logical time' (a nice phrase I have just invented!). I suppose we might say that the existence of the paired set is 'forced' by the preceding sets. No transcendental inferences in this story? |
| 13018 | Limitation of Size is a vague intuition that over-large sets may generate paradoxes [Maddy] |
| Full Idea: The 'limitation of size' is a vague intuition, based on the idea that being too large may generate the paradoxes. | |
| From: Penelope Maddy (Believing the Axioms I [1988], §1.3) | |
| A reaction: This is an intriguing idea to be found right at the centre of what is supposed to be an incredibly rigorous system. |
| 17954 | Essence is a thing's necessities, but what about its possibilities (which may not be realised)? [Vetter] |
| Full Idea: Essence is, as it were, necessity rooted in things, ...but how about possibility rooted in things? ...Having the potential to Φ, unlike being essentially Φ, does not entail being actually Φ. | |
| From: Barbara Vetter (Essence and Potentiality [2010], §2) | |
| A reaction: To me this invites the question 'what is it about the entity which endows it with its rooted possibilities?' A thing has possibilities because it has a certain nature (at a given time). |
| 17953 | Real definition fits abstracta, but not individual concrete objects like Socrates [Vetter] |
| Full Idea: I can understand the notion of real definition as applying to (some) abstact entities, but I have no idea how to apply it to a concrete object such as Socrates or myself. | |
| From: Barbara Vetter (Essence and Potentiality [2010], §1) | |
| A reaction: She is objecting to Kit Fine's account of essence, which is meant to be clearer than the normal account of essences based on necessities. Aristotle implies that definitions get fuzzy when you reach the level of the individual. |
| 17952 | Modal accounts make essence less mysterious, by basing them on the clearer necessity [Vetter] |
| Full Idea: The modal account was meant, I take it, to make the notion of essence less mysterious by basing it on the supposedly better understood notion of necessity. | |
| From: Barbara Vetter (Essence and Potentiality [2010], §1) |
| 17959 | Metaphysical necessity is even more deeply empirical than Kripke has argued [Vetter] |
| Full Idea: We support the views of metaphysical modality on which metaphysical necessity is an even more deeply empirical matter than Kripke has argued. | |
| From: Barbara Vetter (Essence and Potentiality [2010], p.2) | |
| A reaction: [co-author E. Viebahn] This seems to pinpoint the spirit of scientific essentialism. She cites Bird and Shoemaker. If it is empirical, doesn't that make it a matter of epistemology, and hence further from absolute necessity? |
| 17955 | Possible worlds allow us to talk about degrees of possibility [Vetter] |
| Full Idea: The apparatus of possible worlds affords greater expressive power than mere talk of possibility and necessity. In particular, possible worlds talk allows us to introduce degrees of possibility. | |
| From: Barbara Vetter (Essence and Potentiality [2010], §3) | |
| A reaction: A nice feature, but I'm not sure that either the proportion of possible worlds or the closeness of possible worlds captures what we actually mean by a certain degree of possibility. There is 'accidental closeness', or absence of contingency. See Vetter. |
| 17957 | Maybe possibility is constituted by potentiality [Vetter] |
| Full Idea: We should look at the claim that possibility is constituted by potentiality. | |
| From: Barbara Vetter (Essence and Potentiality [2010], §4) | |
| A reaction: A problem that comes to mind is possibilities arising from coincidence. The whole of reality must be described, to capture all the possibilities for a particular thing. So potentialities of what? Nice thought, though. |
| 17958 | The apparently metaphysically possible may only be epistemically possible [Vetter] |
| Full Idea: Some of what metaphysicians take to be metaphysically possible turns out to be only epistemically possible. | |
| From: Barbara Vetter (Essence and Potentiality [2010], §4) | |
| A reaction: A nice clear expression of the increasingly common view that conceivability may be a limited way to grasp possibility. |
| 17956 | Closeness of worlds should be determined by the intrinsic nature of relevant objects [Vetter] |
| Full Idea: The closeness of possible worlds should be determined by similarity in the intrinsic constitution of whatever object it is whose potentialities are at issue. | |
| From: Barbara Vetter (Essence and Potentiality [2010], §3) | |
| A reaction: Nice thought. This seems to be the essentialist approach to possible worlds, but it makes the natures of the objects more fundamental than the framework of the worlds. She demurs because there are also extrinsic potentialities. |
| 3282 | The general form of moral reasoning is putting yourself in other people's shoes [Nagel] |
| Full Idea: I believe the general form of moral reasoning is to put yourself in other people's shoes. | |
| From: Thomas Nagel (Equality [1977], §9) |
| 3278 | An egalitarian system must give priority to those with the worst prospects in life [Nagel] |
| Full Idea: What makes a system egalitarian is the priority it gives to the claims of those whose overall life prospects put them at the bottom. | |
| From: Thomas Nagel (Equality [1977], §6) |
| 3275 | Equality was once opposed to aristocracy, but now it opposes public utility and individual rights [Nagel] |
| Full Idea: Egalitarianism was once opposed to aristocratic values, but now it is opposed by adherents of two non-aristocratic values: utility (increase benefit, even if unequally) and individual rights (which redistribution violates). | |
| From: Thomas Nagel (Equality [1977], §2) |
| 3281 | The ideal of acceptability to each individual underlies the appeal to equality [Nagel] |
| Full Idea: The ideal of acceptability to each individual underlies the appeal to equality. | |
| From: Thomas Nagel (Equality [1977], §8) |
| 3277 | In judging disputes, should we use one standard, or those of each individual? [Nagel] |
| Full Idea: In assessing equality of claims, it must be decided whether to use a single, objective standard, or whether interests should be ranked by the person's own estimation. Also should they balance momentary or long-term needs? | |
| From: Thomas Nagel (Equality [1977], §6) |
| 3274 | Equality can either be defended as good for society, or as good for individual rights [Nagel] |
| Full Idea: The communitarian defence of equality says it is good for society as a whole, whereas the individualistic defence defends equality as a correct distributive principle. | |
| From: Thomas Nagel (Equality [1977], §2) |
| 3273 | Equality nowadays is seen as political, social, legal and economic [Nagel] |
| Full Idea: Contemporary political debate recognises four types of equality: political, social, legal and economic. | |
| From: Thomas Nagel (Equality [1977], §1) | |
| A reaction: Meaning equality of 1) power and influence, 2) status and respect, 3) rights and justice, 4) wealth. |
| 3276 | A morality of rights is very minimal, leaving a lot of human life without restrictions or duties [Nagel] |
| Full Idea: The morality of rights tends to be a limited, even minimal, morality. It leaves a great deal of human life ungoverned by moral restrictions or requirements. | |
| From: Thomas Nagel (Equality [1977], §5) |