31 ideas
| 9065 | S5 collapses iterated modalities (◊□P→□P, and ◊◊P→◊P) [Keefe/Smith] |
| Full Idea: S5 collapses iterated modalities (so ◊□P → □P, and ◊◊P → ◊P). | |
| From: R Keefe / P Smith (Intro: Theories of Vagueness [1997], §5) | |
| A reaction: It is obvious why this might be controversial, and there seems to be a general preference for S4. There may be confusions of epistemic and ontic (and even semantic?) possibilities within a single string of modalities. |
| 12452 | Our dislike of contradiction in logic is a matter of psychology, not mathematics [Brouwer] |
| Full Idea: Not to the mathematician, but to the psychologist, belongs the task of explaining why ...we are averse to so-called contradictory systems in which the negative as well as the positive of certain propositions are valid. | |
| From: Luitzen E.J. Brouwer (Intuitionism and Formalism [1912], p.79) | |
| A reaction: Was the turning point of Graham Priest's life the day he read this sentence? I don't agree. I take the principle of non-contradiction to be a highly generalised observation of how the world works (and Russell agrees with me). |
| 15941 | For intuitionists excluded middle is an outdated historical convention [Brouwer] |
| Full Idea: From the intuitionist standpoint the dogma of the universal validity of the principle of excluded third in mathematics can only be considered as a phenomenon of history of civilization, like the rationality of pi or rotation of the sky about the earth. | |
| From: Luitzen E.J. Brouwer (works [1930]), quoted by Shaughan Lavine - Understanding the Infinite VI.2 | |
| A reaction: [Brouwer 1952:510-11] |
| 18119 | Mathematics is a mental activity which does not use language [Brouwer, by Bostock] |
| Full Idea: Brouwer made the rather extraordinary claim that mathematics is a mental activity which uses no language. | |
| From: report of Luitzen E.J. Brouwer (Mathematics, Science and Language [1928]) by David Bostock - Philosophy of Mathematics 7.1 | |
| A reaction: Since I take language to have far less of a role in thought than is commonly believed, I don't think this idea is absurd. I would say that we don't use language much when we are talking! |
| 18247 | Brouwer saw reals as potential, not actual, and produced by a rule, or a choice [Brouwer, by Shapiro] |
| Full Idea: In his early writing, Brouwer took a real number to be a Cauchy sequence determined by a rule. Later he augmented rule-governed sequences with free-choice sequences, but even then the attitude is that Cauchy sequences are potential, not actual infinities. | |
| From: report of Luitzen E.J. Brouwer (works [1930]) by Stewart Shapiro - Philosophy of Mathematics 6.6 | |
| A reaction: This is the 'constructivist' view of numbers, as espoused by intuitionists like Brouwer. |
| 12451 | Scientific laws largely rest on the results of counting and measuring [Brouwer] |
| Full Idea: A large part of the natural laws introduced by science treat only of the mutual relations between the results of counting and measuring. | |
| From: Luitzen E.J. Brouwer (Intuitionism and Formalism [1912], p.77) | |
| A reaction: His point, I take it, is that the higher reaches of numbers have lost touch with the original point of the system. I now see the whole issue as just depending on conventions about the agreed extension of the word 'number'. |
| 18118 | Brouwer regards the application of mathematics to the world as somehow 'wicked' [Brouwer, by Bostock] |
| Full Idea: Brouwer regards as somehow 'wicked' the idea that mathematics can be applied to a non-mental subject matter, the physical world, and that it might develop in response to the needs which that application reveals. | |
| From: report of Luitzen E.J. Brouwer (Mathematics, Science and Language [1928]) by David Bostock - Philosophy of Mathematics 7.1 | |
| A reaction: The idea is that mathematics only concerns creations of the human mind. It presumably has no more application than, say, noughts-and-crosses. |
| 12454 | Intuitionists only accept denumerable sets [Brouwer] |
| Full Idea: The intuitionist recognises only the existence of denumerable sets. | |
| From: Luitzen E.J. Brouwer (Intuitionism and Formalism [1912], p.80) | |
| A reaction: That takes you up to omega, but not beyond, presumably because it then loses sight of the original intuition of 'bare two-oneness' (Idea 12453). I sympathise, but the word 'number' has shifted its meaning a lot these days. |
| 12453 | Neo-intuitionism abstracts from the reuniting of moments, to intuit bare two-oneness [Brouwer] |
| Full Idea: Neo-intuitionism sees the falling apart of moments, reunited while remaining separated in time, as the fundamental phenomenon of human intellect, passing by abstracting to mathematical thinking, the intuition of bare two-oneness. | |
| From: Luitzen E.J. Brouwer (Intuitionism and Formalism [1912], p.80) | |
| A reaction: [compressed] A famous and somewhat obscure idea. He goes on to say that this creates one and two, and all the finite ordinals. |
| 8728 | Intuitionist mathematics deduces by introspective construction, and rejects unknown truths [Brouwer] |
| Full Idea: Mathematics rigorously treated from the point of view of deducing theorems exclusively by means of introspective construction, is called intuitionistic mathematics. It deviates from classical mathematics, which believes in unknown truths. | |
| From: Luitzen E.J. Brouwer (Consciousness, Philosophy and Mathematics [1948]), quoted by Stewart Shapiro - Thinking About Mathematics 1.2 | |
| A reaction: Clearly intuitionist mathematics is a close cousin of logical positivism and the verification principle. This view would be anathema to Frege, because it is psychological. Personally I believe in the existence of unknown truths, big time! |
| 16062 | A necessary relation between fact-levels seems to be a further irreducible fact [Lynch/Glasgow] |
| Full Idea: It seems unavoidable that the facts about logically necessary relations between levels of facts are themselves logically distinct further facts, irreducible to the microphysical facts. | |
| From: Lynch,MP/Glasgow,JM (The Impossibility of Superdupervenience [2003], C) | |
| A reaction: I'm beginning to think that rejecting every theory of reality that is proposed by carefully exposing some infinite regress hidden in it is a rather lazy way to do philosophy. Almost as bad as rejecting anything if it can't be defined. |
| 16061 | If some facts 'logically supervene' on some others, they just redescribe them, adding nothing [Lynch/Glasgow] |
| Full Idea: Logical supervenience, restricted to individuals, seems to imply strong reduction. It is said that where the B-facts logically supervene on the A-facts, the B-facts simply re-describe what the A-facts describe, and the B-facts come along 'for free'. | |
| From: Lynch,MP/Glasgow,JM (The Impossibility of Superdupervenience [2003], C) | |
| A reaction: This seems to be taking 'logically' to mean 'analytically'. Presumably an entailment is logically supervenient on its premisses, and may therefore be very revealing, even if some people think such things are analytic. |
| 16060 | Nonreductive materialism says upper 'levels' depend on lower, but don't 'reduce' [Lynch/Glasgow] |
| Full Idea: The root intuition behind nonreductive materialism is that reality is composed of ontologically distinct layers or levels. …The upper levels depend on the physical without reducing to it. | |
| From: Lynch,MP/Glasgow,JM (The Impossibility of Superdupervenience [2003], B) | |
| A reaction: A nice clear statement of a view which I take to be false. This relationship is the sort of thing that drives people fishing for an account of it to use the word 'supervenience', which just says two things seem to hang out together. Fluffy materialism. |
| 16064 | The hallmark of physicalism is that each causal power has a base causal power under it [Lynch/Glasgow] |
| Full Idea: Jessica Wilson (1999) says what makes physicalist accounts different from emergentism etc. is that each individual causal power associated with a supervenient property is numerically identical with a causal power associated with its base property. | |
| From: Lynch,MP/Glasgow,JM (The Impossibility of Superdupervenience [2003], n 11) | |
| A reaction: Hence the key thought in so-called (serious, rather than self-evident) 'emergentism' is so-called 'downward causation', which I take to be an idle daydream. |
| 9064 | Objects such as a cloud or Mount Everest seem to have fuzzy boundaries in nature [Keefe/Smith] |
| Full Idea: A common intuition is that a vague object has indeterminate or fuzzy spatio-temporal boundaries, such as a cloud. Mount Everest can only have arbitrary boundaries placed around it, so in nature it must have fuzzy boundaries. | |
| From: R Keefe / P Smith (Intro: Theories of Vagueness [1997], §5) | |
| A reaction: We would have to respond by questioning whether Everest counts precisely as an 'object'. At the microscopic or subatomic level it seems that virtually everything has fuzzy boundaries. Maybe boundaries don't really exist. |
| 9044 | If someone is borderline tall, no further information is likely to resolve the question [Keefe/Smith] |
| Full Idea: If Tek is borderline tall, the unclarity does not seem to be epistemic, because no amount of further information about his exact height (or the heights of others) could help us decide whether he is tall. | |
| From: R Keefe / P Smith (Intro: Theories of Vagueness [1997], §1) | |
| A reaction: One should add also that information about social conventions or conventions about the usage of the word 'tall' will not help either. It seems fairly obvious that God would not know whether Tek is tall, so the epistemic view is certainly counterintuitive. |
| 9048 | The simplest approach, that vagueness is just ignorance, retains classical logic and semantics [Keefe/Smith] |
| Full Idea: The simplest approach to vagueness is to retain classical logic and semantics. Borderline cases are either true or false, but we don't know which, and, despite appearances, vague predicates have well-defined extensions. Vagueness is ignorance. | |
| From: R Keefe / P Smith (Intro: Theories of Vagueness [1997], §1) | |
| A reaction: It seems to me that you must have a rather unhealthy attachment to the logicians' view of the world to take this line. It is the passion of the stamp collector, to want everything in sets, with neatly labelled properties, and inference lines marked out. |
| 9055 | The epistemic view of vagueness must explain why we don't know the predicate boundary [Keefe/Smith] |
| Full Idea: A key question for the epistemic view of vagueness is: why are we ignorant of the facts about where the boundaries of vague predicates lie? | |
| From: R Keefe / P Smith (Intro: Theories of Vagueness [1997], §2) | |
| A reaction: Presumably there is a range of answers, from laziness, to inability to afford the instruments, to limitations on human perception. At the limit, with physical objects, how do we tell whether it is us or the object which is afflicted with vagueness? |
| 9049 | Supervaluationism keeps true-or-false where precision can be produced, but not otherwise [Keefe/Smith] |
| Full Idea: The supervaluationist view of vagueness is that 'tall' comes out true or false on all the ways in which we can make 'tall' precise. There is a gap for borderline cases, but 'tall or not-tall' is still true wherever you draw a boundary. | |
| From: R Keefe / P Smith (Intro: Theories of Vagueness [1997], §1) | |
| A reaction: [Kit Fine is the spokesperson for this; it preserves classical logic, but not semantics] This doesn't seem to solve the problem of vagueness, but it does (sort of) save the principle of excluded middle. |
| 9056 | Vague statements lack truth value if attempts to make them precise fail [Keefe/Smith] |
| Full Idea: The supervaluationist view of vagueness proposes that a sentence is true iff it is true on all precisifications, false iff false on all precisifications, and neither true nor false otherwise. | |
| From: R Keefe / P Smith (Intro: Theories of Vagueness [1997], §3) | |
| A reaction: This seems to be just a footnote to the Russell/Unger view, that logic works if the proposition is precise, but otherwise it is either just the mess of ordinary life, or the predicate doesn't apply at all. |
| 9058 | Some of the principles of classical logic still fail with supervaluationism [Keefe/Smith] |
| Full Idea: Supervaluationist logic (now with a 'definite' operator D) fails to preserve certain classical principles about consequence and rules of inference. For example, reduction ad absurdum, contraposition, the deduction theorem and argument by cases. | |
| From: R Keefe / P Smith (Intro: Theories of Vagueness [1997], §3) | |
| A reaction: The aim of supervaluationism was to try to preserve some classical logic, especially the law of excluded middle, in the face of problems of vagueness. More drastic views, like treating vagueness as irrelevant to logic, or the epistemic view, do better. |
| 9059 | The semantics of supervaluation (e.g. disjunction and quantification) is not classical [Keefe/Smith] |
| Full Idea: The semantics of supervaluational views is not classical. A disjunction can be true without either of its disjuncts being true, and an existential quantification can be true without any of its substitution instances being true. | |
| From: R Keefe / P Smith (Intro: Theories of Vagueness [1997], §3) | |
| A reaction: There is a vaguely plausible story here (either red or orange, but not definitely one nor tother; there exists an x, but which x it is is undecidable), but I think I will vote for this all being very very wrong. |
| 9060 | Supervaluation misunderstands vagueness, treating it as a failure to make things precise [Keefe/Smith] |
| Full Idea: Why should we think vague language is explained away by how things would be if it were made precise? Supervaluationism misrepresents vague expressions, as vague only because we have not bothered to make them precise. | |
| From: R Keefe / P Smith (Intro: Theories of Vagueness [1997], §3) | |
| A reaction: The theory still leaves a gap where vagueness is ineradicable, so the charge doesn't seem quite fair. Logicians always yearn for precision, but common speech enjoys wallowing in a sea of easy-going vagueness, which works fine. |
| 9050 | A third truth-value at borderlines might be 'indeterminate', or a value somewhere between 0 and 1 [Keefe/Smith] |
| Full Idea: One approach to predications in borderline cases is to say that they have a third truth value - 'neutral', 'indeterminate' or 'indefinite', leading to a three-valued logic. Or a degree theory, such as fuzzy logic, with infinite values between 0 and 1. | |
| From: R Keefe / P Smith (Intro: Theories of Vagueness [1997], §1) | |
| A reaction: This looks more like a strategy for computer programmers than for metaphysicians, as it doesn't seem to solve the difficulty of things to which no one can quite assign any value at all. Sometimes you can't be sure if an entity is vague. |
| 9061 | People can't be placed in a precise order according to how 'nice' they are [Keefe/Smith] |
| Full Idea: There is no complete ordering of people by niceness, and two people could be both fairly nice, nice to intermediate degrees, while there is no fact of the matter about who is the nicer. | |
| From: R Keefe / P Smith (Intro: Theories of Vagueness [1997], §4) | |
| A reaction: This is a difficulty if you are trying to decide vague predicates by awarding them degrees of truth. Attempts to place a precise value on 'nice' seem to miss the point, even more than utilitarian attempts to score happiness. |
| 9062 | If truth-values for vagueness range from 0 to 1, there must be someone who is 'completely tall' [Keefe/Smith] |
| Full Idea: Many-valued theories still seem to have a sharp boundary between sentences taking truth-value 1 and those taking value less than 1. So there is a last man in our sorites series who counts as 'completely tall'. | |
| From: R Keefe / P Smith (Intro: Theories of Vagueness [1997], §4) | |
| A reaction: Lovely. Completely nice, totally red, perfectly childlike, an utter mountain, one hundred per cent amused. The enterprise seems to have the same implausibility found in Bayesian approaches to assessing evidence. |
| 9063 | How do we decide if my coat is red to degree 0.322 or 0.321? [Keefe/Smith] |
| Full Idea: What could determine which is the correct function, settling that my coat is red to degree 0.322 rather than 0.321? | |
| From: R Keefe / P Smith (Intro: Theories of Vagueness [1997], §4) | |
| A reaction: It is not just the uncertainty of placing the coat on the scale. The two ends of the scale have all the indeterminacy of being red rather than orange (or, indeed, pink). You are struggling to find a spot on the ruler, when the ruler is placed vaguely. |
| 9045 | Vague predicates involve uncertain properties, uncertain objects, and paradoxes of gradual change [Keefe/Smith] |
| Full Idea: Three interrelated features of vague predicates such as 'tall', 'red', 'heap', 'child' are that they have borderline cases (application is uncertain), they lack well-defined extensions (objects are uncertain), and they're susceptible to sorites paradoxes. | |
| From: R Keefe / P Smith (Intro: Theories of Vagueness [1997], §1) | |
| A reaction: The issue will partly depend on what you think an object is: choose from bundles of properties, total denial, essential substance, or featureless substance with properties. The fungal infection of vagueness could creep in at any point, even the words. |
| 9047 | Many vague predicates are multi-dimensional; 'big' involves height and volume; heaps include arrangement [Keefe/Smith] |
| Full Idea: Many vague predicates are multi-dimensional. 'Big' of people depends on both height and volume; 'nice' does not even have clear dimensions; whether something is a 'heap' depends both the number of grains and their arrangement. | |
| From: R Keefe / P Smith (Intro: Theories of Vagueness [1997], §1) | |
| A reaction: Anyone who was hoping for a nice tidy theory for this problem should abandon hope at this point. Huge numbers of philosophical problems can be simplified by asking 'what exactly do you mean here?' (e.g. tall or bulky?). |
| 9053 | If there is a precise borderline area, that is not a case of vagueness [Keefe/Smith] |
| Full Idea: If a predicate G has a sharply-bounded set of cases falling in between the positive and negative, this shows that merely having borderline cases is not sufficient for vagueness. | |
| From: R Keefe / P Smith (Intro: Theories of Vagueness [1997], §1) | |
| A reaction: Thus you might have 'pass', 'fail' and 'take the test again'. But there seem to be two cases in the border area: will decide later, and decision seems impossible. And the sharp boundaries may be quite arbitrary. |
| 10117 | Intuitonists in mathematics worried about unjustified assertion, as well as contradiction [Brouwer, by George/Velleman] |
| Full Idea: The concern of mathematical intuitionists was that the use of certain forms of inference generates, not contradiction, but unjustified assertions. | |
| From: report of Luitzen E.J. Brouwer (Intuitionism and Formalism [1912]) by A.George / D.J.Velleman - Philosophies of Mathematics Ch.6 | |
| A reaction: This seems to be the real origin of the verificationist idea in the theory of meaning. It is a hugely revolutionary idea - that ideas are not only ruled out of court by contradiction, but that there are other criteria which should also be met. |