42 ideas
| 8729 | Intuitionists deny excluded middle, because it is committed to transcendent truth or objects [Shapiro] |
| Full Idea: Intuitionists in mathematics deny excluded middle, because it is symptomatic of faith in the transcendent existence of mathematical objects and/or the truth of mathematical statements. | |
| From: Stewart Shapiro (Thinking About Mathematics [2000], 1.2) | |
| A reaction: There are other problems with excluded middle, such as vagueness, but on the whole I, as a card-carrying 'realist', am committed to the law of excluded middle. |
| 8763 | The number 3 is presumably identical as a natural, an integer, a rational, a real, and complex [Shapiro] |
| Full Idea: It is surely wise to identify the positions in the natural numbers structure with their counterparts in the integer, rational, real and complex number structures. | |
| From: Stewart Shapiro (Thinking About Mathematics [2000], 10.2) | |
| A reaction: The point is that this might be denied, since 3, 3/1, 3.00.., and -3*i^2 are all arrived at by different methods of construction. Natural 3 has a predecessor, but real 3 doesn't. I agree, intuitively, with Shapiro. Russell (1919) disagreed. |
| 18249 | Cauchy gave a formal definition of a converging sequence. [Shapiro] |
| Full Idea: A sequence a1,a2,... of rational numbers is 'Cauchy' if for each rational number ε>0 there is a natural number N such that for all natural numbers m, n, if m>N and n>N then -ε < am - an < ε. | |
| From: Stewart Shapiro (Thinking About Mathematics [2000], 7.2 n4) | |
| A reaction: The sequence is 'Cauchy' if N exists. |
| 8764 | Categories are the best foundation for mathematics [Shapiro] |
| Full Idea: There is a dedicated contingent who hold that the category of 'categories' is the proper foundation for mathematics. | |
| From: Stewart Shapiro (Thinking About Mathematics [2000], 10.3 n7) | |
| A reaction: He cites Lawvere (1966) and McLarty (1993), the latter presenting the view as a form of structuralism. I would say that the concept of a category will need further explication, and probably reduce to either sets or relations or properties. |
| 8762 | Two definitions of 3 in terms of sets disagree over whether 1 is a member of 3 [Shapiro] |
| Full Idea: Zermelo said that for each number n, its successor is the singleton of n, so 3 is {{{null}}}, and 1 is not a member of 3. Von Neumann said each number n is the set of numbers less than n, so 3 is {null,{null},{null,{null}}}, and 1 is a member of 3. | |
| From: Stewart Shapiro (Thinking About Mathematics [2000], 10.2) | |
| A reaction: See Idea 645 - Zermelo could save Plato from the criticisms of Aristotle! These two accounts are cited by opponents of the set-theoretical account of numbers, because it seems impossible to arbitrate between them. |
| 8760 | Numbers do not exist independently; the essence of a number is its relations to other numbers [Shapiro] |
| Full Idea: The structuralist vigorously rejects any sort of ontological independence among the natural numbers; the essence of a natural number is its relations to other natural numbers. | |
| From: Stewart Shapiro (Thinking About Mathematics [2000], 10.1) | |
| A reaction: This seems to place the emphasis on ordinals (what order?) rather than on cardinality (how many?). I am strongly inclined to think that this is the correct view, though you can't really have relations if there is nothing to relate. |
| 8761 | A 'system' is related objects; a 'pattern' or 'structure' abstracts the pure relations from them [Shapiro] |
| Full Idea: A 'system' is a collection of objects with certain relations among them; a 'pattern' or 'structure' is the abstract form of a system, highlighting the interrelationships and ignoring any features they do not affect how they relate to other objects. | |
| From: Stewart Shapiro (Thinking About Mathematics [2000], 10.1) | |
| A reaction: Note that 'ignoring' features is a psychological account of abstraction, which (thanks to Frege and Geach) is supposed to be taboo - but which I suspect is actually indispensable in any proper account of thought and concepts. |
| 8744 | Logicism seems to be a non-starter if (as is widely held) logic has no ontology of its own [Shapiro] |
| Full Idea: The thesis that principles of arithmetic are derivable from the laws of logic runs against a now common view that logic itself has no ontology. There are no particular logical objects. From this perspective logicism is a non-starter. | |
| From: Stewart Shapiro (Thinking About Mathematics [2000], 5.1) | |
| A reaction: This criticism strikes me as utterly devastating. There are two routes to go: prove that logic does have an ontology of objects (what would they be?), or - better - deny that arithmetic contains any 'objects'. Or give up logicism. |
| 8749 | Term Formalism says mathematics is just about symbols - but real numbers have no names [Shapiro] |
| Full Idea: Term Formalism is the view that mathematics is just about characters or symbols - the systems of numerals and other linguistic forms. ...This will cover integers and rational numbers, but what are real numbers supposed to be, if they lack names? | |
| From: Stewart Shapiro (Thinking About Mathematics [2000], 6.1.1) | |
| A reaction: Real numbers (such as pi and root-2) have infinite decimal expansions, so we can start naming those. We could also start giving names like 'Harry' to other reals, though it might take a while. OK, I give up. |
| 8750 | Game Formalism is just a matter of rules, like chess - but then why is it useful in science? [Shapiro] |
| Full Idea: Game Formalism likens mathematics to chess, where the 'content' of mathematics is exhausted by the rules of operating with its language. ...This, however, leaves the problem of why the mathematical games are so useful to the sciences. | |
| From: Stewart Shapiro (Thinking About Mathematics [2000], 6.1.2) | |
| A reaction: This thought pushes us towards structuralism. It could still be a game, but one we learned from observing nature, which plays its own games. Chess is, after all, modelled on warfare. |
| 8752 | Deductivism says mathematics is logical consequences of uninterpreted axioms [Shapiro] |
| Full Idea: The Deductivist version of formalism (sometimes called 'if-thenism') says that the practice of mathematics consists of determining logical consequences of otherwise uninterpreted axioms. | |
| From: Stewart Shapiro (Thinking About Mathematics [2000], 6.2) | |
| A reaction: [Hilbert is the source] More plausible than Term or Game Formalism (qv). It still leaves the question of why it seems applicable to nature, and why those particular axioms might be chosen. In some sense, though, it is obviously right. |
| 8753 | Critics resent the way intuitionism cripples mathematics, but it allows new important distinctions [Shapiro] |
| Full Idea: Critics commonly complain that the intuitionist restrictions cripple the mathematician. On the other hand, intuitionist mathematics allows for many potentially important distinctions not available in classical mathematics, and is often more subtle. | |
| From: Stewart Shapiro (Thinking About Mathematics [2000], 7.1) | |
| A reaction: The main way in which it cripples is its restriction on talk of infinity ('Cantor's heaven'), which was resented by Hilbert. Since high-level infinities are interesting, it would be odd if we were not allowed to discuss them. |
| 8731 | Conceptualist are just realists or idealist or nominalists, depending on their view of concepts [Shapiro] |
| Full Idea: I classify conceptualists according to what they say about properties or concepts. If someone classified properties as existing independent of language I would classify her as a realist in ontology of mathematics. Or they may be idealists or nominalists. | |
| From: Stewart Shapiro (Thinking About Mathematics [2000], 2.2.1) | |
| A reaction: In other words, Shapiro wants to eliminate 'conceptualist' as a useful label in philosophy of mathematics. He's probably right. All thought involves concepts, but that doesn't produce a conceptualist theory of, say, football. |
| 8730 | 'Impredicative' definitions refer to the thing being described [Shapiro] |
| Full Idea: A definition of a mathematical entity is 'impredicative' if it refers to a collection that contains the defined entity. The definition of 'least upper bound' is impredicative as it refers to upper bounds and characterizes a member of this set. | |
| From: Stewart Shapiro (Thinking About Mathematics [2000], 1.2) | |
| A reaction: The big question is whether mathematics can live with impredicative definitions, or whether they threaten to be viciously circular, and undermine the whole enterprise. |
| 6504 | For physicalists, the only relations are spatial, temporal and causal [Robinson,H] |
| Full Idea: Spatial, temporal and causal relations are the only respectable candidates for relations for a physicalist. | |
| From: Howard Robinson (Perception [1994], V.4) | |
| A reaction: This seems to be true, and is an absolutely crucial principle upon which any respectable physicalist account of the world must be built. It means that physicalists must attempt to explain all mental events in causal terms. |
| 6520 | If reality just has relational properties, what are its substantial ontological features? [Robinson,H] |
| Full Idea: Some thinkers claim the physical world consists just of relational properties - generally of active powers or fields; ..but an ontology of mutual influences is not an ontology at all unless the possessors of the influence have more substantial features. | |
| From: Howard Robinson (Perception [1994], IX.3) | |
| A reaction: I think this idea is one of the keys to wisdom. It is the same problem with functional explanations - you are left asking WHY this thing can have this particular function. Without the buck stopping at essences you are chasing your explanatory tail. |
| 6485 | When a red object is viewed, the air in between does not become red [Robinson,H] |
| Full Idea: When the form of red passes from an object to the eye, the air in between does not become red. | |
| From: Howard Robinson (Perception [1994], 1.2) | |
| A reaction: This strikes me as a crucial and basic fact which must be faced by any philosopher offering a theory of perception. I would have thought it instantly eliminated any sort of direct or naïve realism. The quale of red is created by my brain. |
| 6521 | Representative realists believe that laws of phenomena will apply to the physical world [Robinson,H] |
| Full Idea: One thing which is meant by saying that the phenomenal world represents or resembles the transcendental physical world is that the scientific laws devised to apply to the former, if correct, also apply (at least approximately) to the latter. | |
| From: Howard Robinson (Perception [1994], IX.3) | |
| A reaction: This is not, of course, an argument, or a claim which can be easily substantiated, but it does seem to be a nice statement of a central article of faith for representative realists. The laws of the phenomenal world are the only ones we are going to get. |
| 6509 | Representative realists believe some properties of sense-data are shared by the objects themselves [Robinson,H] |
| Full Idea: A representative realist believes that at least some of the properties that are ostensively demonstrable in virtue of being exemplified in sense-data are of the same kind as some of those exemplified in physical objects. | |
| From: Howard Robinson (Perception [1994], VII.5) | |
| A reaction: It is hard to pin down exactly what is being claimed here. Locke's primary qualities will obviously qualify, but could properties be 'exemplified' in sense-data without them actually being the same as those of the objects? |
| 6522 | Phenomenalism can be theistic (Berkeley), or sceptical (Hume), or analytic (20th century) [Robinson,H] |
| Full Idea: It is useful to identify three kinds of phenomenalism: theistic, sceptical and analytic; the first is represented by Berkeley, the second by Hume, and the third by most twentieth-century phenomenalists. | |
| From: Howard Robinson (Perception [1994], IX.4) | |
| A reaction: In Britain the third group is usually represented by A.J.Ayer. My simple objection to all phenomenalists is that they are intellectual cowards because they won't venture to give an explanation of the phenomena which confront them. |
| 6502 | Can we reduce perception to acquisition of information, which is reduced to causation or disposition? [Robinson,H] |
| Full Idea: Many modern physicalists first analyse perception as no more than the acquisition of beliefs or information through the senses, and then analyse belief and the possession of information in causal or dispositional terms. | |
| From: Howard Robinson (Perception [1994], V.1) | |
| A reaction: (He mentions Armstrong, Dretske and Pitcher). A reduction to dispositions implies behaviourism. This all sounds more like an eliminativist strategy than a reductive one. I would start by saying that perception is only information after interpretation. |
| 6513 | Would someone who recovered their sight recognise felt shapes just by looking? [Robinson,H] |
| Full Idea: Molyneux's Problem is whether someone who was born blind and acquired sight would be able to recognise, on sight, which shapes were which; that is, would they see which shape was the one that felt so-and-so? | |
| From: Howard Robinson (Perception [1994], VIII.7) | |
| A reaction: (Molyneux wrote a letter to John Locke about this). It is a good question, and much discussed in modern times. My estimation is that the person would recognise the shapes. We are partly synaesthetic, and see sharpness as well as feeling it. |
| 6512 | Secondary qualities have one sensory mode, but primary qualities can have more [Robinson,H] |
| Full Idea: Primary qualities and secondary qualities are often distinguished on the grounds that secondaries are restricted to one sensory modality, but primaries can appear in more. | |
| From: Howard Robinson (Perception [1994], VIII.7) | |
| A reaction: This distinction seems to me to be accurate and important. It is not just that the two types are phenomenally different - it is that the best explanation is that the secondaries depend on their one sense, but the primaries are independent. |
| 6497 | We say objects possess no intrinsic secondary qualities because physicists don't need them [Robinson,H] |
| Full Idea: The idea that objects do not possess secondary qualities intrinsically rests on the thought that they do not figure in the physicist's account of the world; ..as they are causally idle, no purpose is served by attributing them to objects. | |
| From: Howard Robinson (Perception [1994], III.1) | |
| A reaction: On the whole I agree with this, but colours (for example) are not causally idle, as they seem to affect the behaviour of insects. They are properties which can only have a causal effect if there is a brain in their vicinity. Physicists ignore brains. |
| 6494 | If objects are not coloured, and neither are sense-contents, we are left saying that nothing is coloured [Robinson,H] |
| Full Idea: If there are good reasons for thinking that physical objects are not literally coloured, and one also refuses to attribute them to sense-contents, then one will have the bizarre theory (which has been recently adopted) that nothing is actually coloured. | |
| From: Howard Robinson (Perception [1994], 1.7) | |
| A reaction: It seems to me that objects are not literally coloured, that the air in between does not become coloured, and that my brain doesn't turn a funny colour, so that only leaves colour as an 'interior' feature of certain brain states. That's how it is. |
| 6499 | Shape can be experienced in different ways, but colour and sound only one way [Robinson,H] |
| Full Idea: Shape can be directly experienced by either touch or sight, which are subjectively different; but colour and sound can be directly experienced only through experiences which are subjectively like sight and hearing. | |
| From: Howard Robinson (Perception [1994], III.1) | |
| A reaction: This seems to be a key argument in support of the distinction between primary and secondary qualities. It seems to me that the distinction may be challenged and questioned, but to deny it completely (as Berkeley and Hume do) is absurd. |
| 6500 | If secondary qualities match senses, would new senses create new qualities? [Robinson,H] |
| Full Idea: As secondary qualities are tailored to match senses, a proliferation of senses would lead to a proliferation of secondary qualities. | |
| From: Howard Robinson (Perception [1994], III.1) | |
| A reaction: One might reply that if we experienced, say, magnetism, we would just be discerning a new fine grained primary quality, not adding something new to the ontological stock of properties in the world. It is a matter of HOW we experience the magnetism. |
| 6484 | Most moderate empiricists adopt Locke's representative theory of perception [Robinson,H] |
| Full Idea: The representative theory of perception is found in Locke, and is adopted by most moderate empiricists. | |
| From: Howard Robinson (Perception [1994], 1.2) | |
| A reaction: This is, I think, my own position. Anything less than fairly robust realism strikes me as being a bit mad (despite Berkeley's endless assertions that he is preaching common sense), and direct realism seems obviously false. |
| 6508 | Sense-data leads to either representative realism or phenomenalism or idealism [Robinson,H] |
| Full Idea: The sense-datum theorist is either a representative realist or a phenomenalist (with which we can classify idealism for present purposes). | |
| From: Howard Robinson (Perception [1994], VII.5) | |
| A reaction: The only alternative to these two positions seems to be some sort of direct realism. I class myself as a representative realist, as this just seems (after a very little thought about colour blindness) to be common sense. I'm open to persuasion. |
| 6480 | Sense-data do not have any intrinsic intentionality [Robinson,H] |
| Full Idea: I understand sense-data as having no intrinsic intentionality; that is, though it may suggest, by habit, things beyond it, in itself it possesses only sensible qualities which do not refer beyond themselves. | |
| From: Howard Robinson (Perception [1994], 1.1) | |
| A reaction: This seems right, as the whole point of proposing sense-data was as something neutral between realism and anti-realism |
| 6482 | For idealists and phenomenalists sense-data are in objects; representative realists say they resemble objects [Robinson,H] |
| Full Idea: For idealists and phenomenalists sense-data are part of physical objects, for objects consist only of actual or actual and possible sense-data; representative realists say they just have an abstract and structural resemblance to objects. | |
| From: Howard Robinson (Perception [1994], 1.1) | |
| A reaction: He puts Berkeley, Hume and Mill in the first group, and Locke in the second. Russell belongs in the second. The very fact that there can be two such different theories about the location of sense-data rather discredits the whole idea. |
| 6505 | Sense-data are rejected because they are a veil between us and reality, leading to scepticism [Robinson,H] |
| Full Idea: Resistance to the sense-datum theory is inspired mainly by the fear that such data constitute a veil of perception which stands between the observer and the external world, threatening scepticism, or even solipsism. | |
| From: Howard Robinson (Perception [1994], VII.1) | |
| A reaction: It is very intellectually dishonest to reject any theory because it leads to scepticism or relativism. This is a common failing among quite good professional philosophers. See Idea 241. |
| 6506 | 'Sense redly' sounds peculiar, but 'senses redly-squarely tablely' sounds far worse [Robinson,H] |
| Full Idea: 'Sense redly' sounds peculiar, but 'senses redly-squarely' or 'red-squarely' or 'senses redly-squarely-tablely' and other variants sound far worse. | |
| From: Howard Robinson (Perception [1994], VII.5) | |
| A reaction: This is a comment on the adverbial theory, which is meant to replace representative theories based on sense-data. The problem is not that it sounds weird; it is that while plain red can be a mode of perception, being a table obviously can't. |
| 6507 | Adverbialism sees the contents of sense-experience as modes, not objects [Robinson,H] |
| Full Idea: The defining claim of adverbialism is that the contents of sense-experience are modes, not objects, of sensory activity. | |
| From: Howard Robinson (Perception [1994], VII.5) | |
| A reaction: This seems quite a good account of simple 'modes' like colour, but not so good when you instantly perceive a house. It never seems wholly satisfactory to sidestep the question of 'what are you perceiving when you perceive red or square?' |
| 6511 | If there are only 'modes' of sensing, then an object can no more be red or square than it can be proud or lazy. [Robinson,H] |
| Full Idea: If only modes of sensing are ostensively available, ..then it is a category mistake to see any resemblance between what is available and properties of bodies; one could as sensibly say that a physical body is proud or lazy as that it is red or square. | |
| From: Howard Robinson (Perception [1994], VII.5) | |
| A reaction: This is an objection to the 'adverbial' theory of perception. It looks to me like a devastating objection, if the theory is meant to cover primary qualities as well as secondary. Red could be a mode of perception, but not square, surely? |
| 8725 | Rationalism tries to apply mathematical methodology to all of knowledge [Shapiro] |
| Full Idea: Rationalism is a long-standing school that can be characterized as an attempt to extend the perceived methodology of mathematics to all of knowledge. | |
| From: Stewart Shapiro (Thinking About Mathematics [2000], 1.1) | |
| A reaction: Sometimes called 'Descartes's Dream', or the 'Enlightenment Project', the dream of proving everything. Within maths, Hilbert's Programme aimed for the same certainty. Idea 22 is the motto for the opposition to this approach. |
| 6515 | An explanation presupposes something that is improbable unless it is explained [Robinson,H] |
| Full Idea: Any search for an explanation presupposes that there is something in need of an explanation - that is, something which is improbable unless explained. | |
| From: Howard Robinson (Perception [1994], IX.3) | |
| A reaction: Elementary enough, but it underlines the human perspective of all explanations. I may need an explanation of baseball, where you don't. |
| 6517 | If all possibilities are equal, order seems (a priori) to need an explanation - or does it? [Robinson,H] |
| Full Idea: The fact that order requires an explanation seems to be an a priori principle; ..we assume all possibilities are equally likely, and so no striking regularities should emerge; the sceptic replies that a highly ordered sequence is as likely as any other. | |
| From: Howard Robinson (Perception [1994], IX.3) | |
| A reaction: An independent notion of 'order' is required. If I write down '14356', and then throw 1 4 3 5 6 on a die, the match is the order; instrinsically 14356 is nothing special. If you threw the die a million times, a run of six sixes seems quite likely. |
| 6481 | If intentional states are intrinsically about other things, what are their own properties? [Robinson,H] |
| Full Idea: Intentional states are mysterious things; if they are intrinsically about other things, what properties, if any, do they possess intrinsically? | |
| From: Howard Robinson (Perception [1994], 1.1) | |
| A reaction: A very nice question, which I suspect to be right at the heart of the tendency towards externalist accounts of the mind. Since you can only talk about the contents of the thoughts, you can't put forward a decent internalist account of what is going on. |
| 6503 | Physicalism cannot allow internal intentional objects, as brain states can't be 'about' anything [Robinson,H] |
| Full Idea: It is generally conceded by reductive physicalists that a state of the brain cannot be intrinsically about anything, for intentionality is not an intrinsic property of anything, so there can be no internal objects for a physicalist. | |
| From: Howard Robinson (Perception [1994], V.4) | |
| A reaction: Perhaps it is best to say that 'aboutness' is not a property of physics. We may say that a brain state 'represents' something, because the something caused the brain state, but representations have to be recognised |
| 20443 | The aesthetic attitude is nothing more than paying close attention [Dickie, by Giovannelli] |
| Full Idea: Once analysed, Dickie claimed, the so-called aesthetic attitude is not special at all, but is rather just a matter of close attention and focus on the subject. | |
| From: report of George Dickie (The Myth of the Aesthetic Attitude [1964]) by Alessandro Giovannelli - Some contemporary developments (aesthetics) 1 | |
| A reaction: Sounds wrong. If a paint specialist gives close attention to a painting, they do not necessarily have an aesthetic view of it. You need to know the aim of the activity, just as when you watch a game. |
| 6519 | Locke's solidity is not matter, because that is impenetrability and hardness combined [Robinson,H] |
| Full Idea: Notoriously, Locke's filler for Descartes's geometrical matter, solidity, will not do, for that quality collapses on examination into a composite of the dispositional-cum-relational propery of impenetrability, and the secondary quality of hardness. | |
| From: Howard Robinson (Perception [1994], IX.3) | |
| A reaction: I would have thought the problem was that 'matter is solidity' turns out on analysis to be a tautology. We have a handful of nearly synonymous words for matter and our experiences of it, but they boil down to some 'given' thing for which we lack words. |