21 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. |
| 11211 | If a sound conclusion comes from two errors that cancel out, the path of the argument must matter [Rumfitt] |
| Full Idea: If a designated conclusion follows from the premisses, but the argument involves two howlers which cancel each other out, then the moral is that the path an argument takes from premisses to conclusion does matter to its logical evaluation. | |
| From: Ian Rumfitt ("Yes" and "No" [2000], II) | |
| A reaction: The drift of this is that our view of logic should be a little closer to the reasoning of ordinary language, and we should rely a little less on purely formal accounts. |
| 11210 | Standardly 'and' and 'but' are held to have the same sense by having the same truth table [Rumfitt] |
| Full Idea: If 'and' and 'but' really are alike in sense, in what might that likeness consist? Some philosophers of classical logic will reply that they share a sense by virtue of sharing a truth table. | |
| From: Ian Rumfitt ("Yes" and "No" [2000]) | |
| A reaction: This is the standard view which Rumfitt sets out to challenge. |
| 11212 | The sense of a connective comes from primitively obvious rules of inference [Rumfitt] |
| Full Idea: A connective will possess the sense that it has by virtue of its competent users' finding certain rules of inference involving it to be primitively obvious. | |
| From: Ian Rumfitt ("Yes" and "No" [2000], III) | |
| A reaction: Rumfitt cites Peacocke as endorsing this view, which characterises the logical connectives by their rules of usage rather than by their pure semantic value. |
| 24016 | Consciousness always transcends itself [Sartre] |
| Full Idea: It is of the essence of consciousness to transcend itself | |
| From: Jean-Paul Sartre (Sketch for a Theory of the Emotions [1939], §III) | |
| A reaction: As usual, I am a bit baffled by these sorts of pronouncement. Sounds like an oxymoron to me. Maybe it is a development of Schopenhauer's thought. |
| 24013 | An emotion and its object form a unity, so emotion is a mode of apprehension [Sartre] |
| Full Idea: Emotion returns to its object every moment, and feeds upon it. …The emotional subject and the object of the emotion are united in an indissoluble synthesis. Emotion is a specific manner of apprehending the world. …[39] It is a transformation of the world. | |
| From: Jean-Paul Sartre (Sketch for a Theory of the Emotions [1939], §III) | |
| A reaction: The last sentence is the essence (or existence?) of Sartre's core theory of the emotions. They are, it seems, a mode of perception, like a colour filter added to a camera. I don't think I agree. I see them as a response to perceptions, not part of them. |
| 24017 | Emotion is one of our modes of understanding our Being-in-the-World [Sartre] |
| Full Idea: Emotion is not an accident, it is a mode of our conscious existence, one of the ways in which consciousness understands (in Heidegger's sense of verstehen) its Being-in-the-World. …It has a meaning. | |
| From: Jean-Paul Sartre (Sketch for a Theory of the Emotions [1939], §III) | |
| A reaction: Calling emotions a 'mode' suggests that this way of understanding is intermittent, which seems wrong. Even performing arithmetical calculations is coloured by emotions, so they go deeper than a 'mode'. |
| 24014 | Emotions are a sort of bodily incantation which brings a magic to the world [Sartre] |
| Full Idea: Joy is the magical behaviour which tries, by incantation, to realise the possession of the desired object as an instantaneous totality. [47] Emotions are all reducible to the constitution of a magic world by using our bodies as instruments of incantation. | |
| From: Jean-Paul Sartre (Sketch for a Theory of the Emotions [1939], §III) | |
| A reaction: I can't pretend to understand this, but I am reminded of the fact that the so-called primary qualities of perception are innately boring, and it is only the secondary qualities (like colour and smell) which make the world interesting. |
| 24015 | Emotions makes us believe in and live in a new world [Sartre] |
| Full Idea: Emotion is a phenomenon of belief. Consciousness does not limit itself to the projection of affective meanings upon the world around it; it lives the new world it has thereby constituted. | |
| From: Jean-Paul Sartre (Sketch for a Theory of the Emotions [1939], §III) | |
| A reaction: There seems to be an implied anti-realism in this, since the emotions prevent us from relating more objectively to the world. The 'magic' seems to be compulsory. |
| 11214 | We learn 'not' along with affirmation, by learning to either affirm or deny a sentence [Rumfitt] |
| Full Idea: The standard view is that affirming not-A is more complex than affirming the atomic sentence A itself, with the latter determining its sense. But we could learn 'not' directly, by learning at once how to either affirm A or reject A. | |
| From: Ian Rumfitt ("Yes" and "No" [2000], IV) | |
| A reaction: [compressed] This seems fairly anti-Fregean in spirit, because it looks at the psychology of how we learn 'not' as a way of clarifying what we mean by it, rather than just looking at its logical behaviour (and thus giving it a secondary role). |
| 20491 | States have a monopoly of legitimate violence [Sartre, by Wolff,J] |
| Full Idea: Max Weber observed that states possess a monopoly of legitimate violence. | |
| From: report of Jean-Paul Sartre (Sketch for a Theory of the Emotions [1939]) by Jonathan Wolff - An Introduction to Political Philosophy (Rev) 2 'State' | |
| A reaction: This sounds rather hair-raising, and often is, but it sounds quite good if we describe it as a denial of legitimate violence to individual citizens. Hobbes would like it, since individual violence breaches some sort of natural contract. Guns in USA. |