30 ideas
| 18005 | Philosophy aims to become more disciplined about categories [Ryle] |
| Full Idea: Philosophy is the replacement of category-habits by category-disciplines. | |
| From: Gilbert Ryle (The Concept of Mind [1949], Intro p.8), quoted by Ofra Magidor - Category Mistakes 1.2 | |
| A reaction: I rather like this. It fits the view the idea that metaphysics aims to give the structure of reality. If there are not reasonably uniform categories for things, then reality is indescribable. Improving our categories seems a thoroughly laudable aim. |
| 10170 | While true-in-a-model seems relative, true-in-all-models seems not to be [Reck/Price] |
| Full Idea: While truth can be defined in a relative way, as truth in one particular model, a non-relative notion of truth is implied, as truth in all models. | |
| From: E Reck / M Price (Structures and Structuralism in Phil of Maths [2000], §4) | |
| A reaction: [The article is actually discussing arithmetic] This idea strikes me as extremely important. True-in-all-models is usually taken to be tautological, but it does seem to give a more universal notion of truth. See semantic truth, Tarski, Davidson etc etc. |
| 10166 | ZFC set theory has only 'pure' sets, without 'urelements' [Reck/Price] |
| Full Idea: In standard ZFC ('Zermelo-Fraenkel with Choice') set theory we deal merely with pure sets, not with additional urelements. | |
| From: E Reck / M Price (Structures and Structuralism in Phil of Maths [2000], §2) | |
| A reaction: The 'urelements' would the actual objects that are members of the sets, be they physical or abstract. This idea is crucial to understanding philosophy of mathematics, and especially logicism. Must the sets exist, just as the urelements do? |
| 10175 | Three types of variable in second-order logic, for objects, functions, and predicates/sets [Reck/Price] |
| Full Idea: In second-order logic there are three kinds of variables, for objects, for functions, and for predicates or sets. | |
| From: E Reck / M Price (Structures and Structuralism in Phil of Maths [2000], §5) | |
| A reaction: It is interesting that a predicate seems to be the same as a set, which begs rather a lot of questions. For those who dislike second-order logic, there seems nothing instrinsically wicked in having variables ranging over innumerable multi-order types. |
| 10165 | 'Analysis' is the theory of the real numbers [Reck/Price] |
| Full Idea: 'Analysis' is the theory of the real numbers. | |
| From: E Reck / M Price (Structures and Structuralism in Phil of Maths [2000], §2) | |
| A reaction: 'Analysis' began with the infinitesimal calculus, which later built on the concept of 'limit'. A continuum of numbers seems to be required to make that work. |
| 10174 | Mereological arithmetic needs infinite objects, and function definitions [Reck/Price] |
| Full Idea: The difficulties for a nominalistic mereological approach to arithmetic is that an infinity of physical objects are needed (space-time points? strokes?), and it must define functions, such as 'successor'. | |
| From: E Reck / M Price (Structures and Structuralism in Phil of Maths [2000], §4) | |
| A reaction: Many ontologically austere accounts of arithmetic are faced with the problem of infinity. The obvious non-platonist response seems to be a modal or if-then approach. To postulate infinite abstract or physical entities so that we can add 3 and 2 is mad. |
| 10164 | Peano Arithmetic can have three second-order axioms, plus '1' and 'successor' [Reck/Price] |
| Full Idea: A common formulation of Peano Arithmetic uses 2nd-order logic, the constant '1', and a one-place function 's' ('successor'). Three axioms then give '1 is not a successor', 'different numbers have different successors', and induction. | |
| From: E Reck / M Price (Structures and Structuralism in Phil of Maths [2000], §2) | |
| A reaction: This is 'second-order' Peano Arithmetic, though it is at least as common to formulate in first-order terms (only quantifying over objects, not over properties - as is done here in the induction axiom). I like the use of '1' as basic instead of '0'! |
| 10172 | Set-theory gives a unified and an explicit basis for mathematics [Reck/Price] |
| Full Idea: The merits of basing an account of mathematics on set theory are that it allows for a comprehensive unified treatment of many otherwise separate branches of mathematics, and that all assumption, including existence, are explicit in the axioms. | |
| From: E Reck / M Price (Structures and Structuralism in Phil of Maths [2000], §4) | |
| A reaction: I am forming the impression that set-theory provides one rather good model (maybe the best available) for mathematics, but that doesn't mean that mathematics is set-theory. The best map of a landscape isn't a landscape. |
| 10167 | Structuralism emerged from abstract algebra, axioms, and set theory and its structures [Reck/Price] |
| Full Idea: Structuralism has emerged from the development of abstract algebra (such as group theory), the creation of axiom systems, the introduction of set theory, and Bourbaki's encyclopaedic survey of set theoretic structures. | |
| From: E Reck / M Price (Structures and Structuralism in Phil of Maths [2000], §2) | |
| A reaction: In other words, mathematics has gradually risen from one level of abstraction to the next, so that mathematical entities like points and numbers receive less and less attention, with relationships becoming more prominent. |
| 10169 | Relativist Structuralism just stipulates one successful model as its arithmetic [Reck/Price] |
| Full Idea: Relativist Structuralism simply picks one particular model of axiomatised arithmetic (i.e. one particular interpretation that satisfies the axioms), and then stipulates what the elements, functions and quantifiers refer to. | |
| From: E Reck / M Price (Structures and Structuralism in Phil of Maths [2000], §4) | |
| A reaction: The point is that a successful model can be offered, and it doesn't matter which one, like having any sort of aeroplane, as long as it flies. I don't find this approach congenial, though having a model is good. What is the essence of flight? |
| 10179 | There are 'particular' structures, and 'universal' structures (what the former have in common) [Reck/Price] |
| Full Idea: The term 'structure' has two uses in the literature, what can be called 'particular structures' (which are particular relational systems), but also what can be called 'universal structures' - what particular systems share, or what they instantiate. | |
| From: E Reck / M Price (Structures and Structuralism in Phil of Maths [2000], §6) | |
| A reaction: This is a very helpful distinction, because it clarifies why (rather to my surprise) some structuralists turn out to be platonists in a new guise. Personal my interest in structuralism has been anti-platonist from the start. |
| 10181 | Pattern Structuralism studies what isomorphic arithmetic models have in common [Reck/Price] |
| Full Idea: According to 'pattern' structuralism, what we study are not the various particular isomorphic models of arithmetic, but something in addition to them: a corresponding pattern. | |
| From: E Reck / M Price (Structures and Structuralism in Phil of Maths [2000], §7) | |
| A reaction: Put like that, we have to feel a temptation to wield Ockham's Razor. It's bad enough trying to give the structure of all the isomorphic models, without seeking an even more abstract account of underlying patterns. But patterns connect to minds.. |
| 10182 | There are Formalist, Relativist, Universalist and Pattern structuralism [Reck/Price] |
| Full Idea: There are four main variants of structuralism in the philosophy of mathematics - formalist structuralism, relativist structuralism, universalist structuralism (with modal variants), and pattern structuralism. | |
| From: E Reck / M Price (Structures and Structuralism in Phil of Maths [2000], §9) | |
| A reaction: I'm not sure where Chihara's later book fits into this, though it is at the nominalist end of the spectrum. Shapiro and Resnik do patterns (the latter more loosely); Hellman does modal universalism; Quine does the relativist version. Dedekind? |
| 10168 | Formalist Structuralism says the ontology is vacuous, or formal, or inference relations [Reck/Price] |
| Full Idea: Formalist Structuralism endorses structural methodology in mathematics, but rejects semantic and metaphysical problems as either meaningless, or purely formal, or as inference relations. | |
| From: E Reck / M Price (Structures and Structuralism in Phil of Maths [2000], §3) | |
| A reaction: [very compressed] I find the third option fairly congenial, certainly in preference to rather platonist accounts of structuralism. One still needs to distinguish the mathematical from the non-mathematical in the inference relations. |
| 10178 | Maybe we should talk of an infinity of 'possible' objects, to avoid arithmetic being vacuous [Reck/Price] |
| Full Idea: It is tempting to take a modal turn, and quantify over all possible objects, because if there are only a finite number of actual objects, then there are no models (of the right sort) for Peano Arithmetic, and arithmetic is vacuously true. | |
| From: E Reck / M Price (Structures and Structuralism in Phil of Maths [2000], §5) | |
| A reaction: [compressed; Geoffrey Hellman is the chief champion of this view] The article asks whether we are not still left with the puzzle of whether infinitely many objects are possible, instead of existent. |
| 10176 | Universalist Structuralism is based on generalised if-then claims, not one particular model [Reck/Price] |
| Full Idea: Universalist Structuralism is a semantic thesis, that an arithmetical statement asserts a universal if-then statement. We build an if-then statement (using quantifiers) into the structure, and we generalise away from any one particular model. | |
| From: E Reck / M Price (Structures and Structuralism in Phil of Maths [2000], §5) | |
| A reaction: There remains the question of what is distinctively mathematical about the highly generalised network of inferences that is being described. Presumable the axioms capture that, but why those particular axioms? Russell is cited as an originator. |
| 10177 | Universalist Structuralism eliminates the base element, as a variable, which is then quantified out [Reck/Price] |
| Full Idea: Universalist Structuralism is eliminativist about abstract objects, in a distinctive form. Instead of treating the base element (say '1') as an ambiguous referring expression (the Relativist approach), it is a variable which is quantified out. | |
| From: E Reck / M Price (Structures and Structuralism in Phil of Maths [2000], §5) | |
| A reaction: I am a temperamental eliminativist on this front (and most others) so this is tempting. I am also in love with the concept of a 'variable', which I take to be utterly fundamental to all conceptual thought, even in animals, and not just a trick of algebra. |
| 10171 | The existence of an infinite set is assumed by Relativist Structuralism [Reck/Price] |
| Full Idea: Relativist Structuralism must first assume the existence of an infinite set, otherwise there would be no model to pick, and arithmetical terms would have no reference. | |
| From: E Reck / M Price (Structures and Structuralism in Phil of Maths [2000], §4) | |
| A reaction: See Idea 10169 for Relativist Structuralism. They point out that ZFC has an Axiom of Infinity. |
| 8628 | I hold that algebra and number are developments of logic [Jevons] |
| Full Idea: I hold that algebra is a highly developed logic, and number but logical discrimination. | |
| From: William S. Jevons (The Principles of Science [1879], p.156), quoted by Gottlob Frege - Grundlagen der Arithmetik (Foundations) §15 | |
| A reaction: Thus Frege shows that logicism was an idea that was in the air before he started writing. Riemann's geometry and Boole's logic presumably had some influence here. |
| 14297 | A dispositional property is not a state, but a liability to be in some state, given a condition [Ryle] |
| Full Idea: To possess a dispositional property is not to be in a particular state;..it is to be bound or liable to be in a particular state, or undergo a particular change, when a particular condition is realized. | |
| From: Gilbert Ryle (The Concept of Mind [1949], II (7)) | |
| A reaction: Whether this view is correct is the central question about dispositions. Ryle's view is tied in with Humean regularities and behaviourism about mind. The powers view, which I favour, says a disposition is a drawn bow, an actual state of power. |
| 14300 | No physical scientist now believes in an occult force-exerting agency [Ryle] |
| Full Idea: The old error treating the term 'Force' as denoting an occult force-exerting agency has been given up in the physical sciences. | |
| From: Gilbert Ryle (The Concept of Mind [1949], V (1)) | |
| A reaction: Since 1949 they seem to have made a revival, once they are divested of their religious connotations. The word 'agency' is the misleading bit. Even Leibniz's monads weren't actual agents - he always said that was 'an analogy'. |
| 10173 | A nominalist might avoid abstract objects by just appealing to mereological sums [Reck/Price] |
| Full Idea: One way for a nominalist to reject appeal to all abstract objects, including sets, is to only appeal to nominalistically acceptable objects, including mereological sums. | |
| From: E Reck / M Price (Structures and Structuralism in Phil of Maths [2000], §4) | |
| A reaction: I'm suddenly thinking that this looks very interesting and might be the way to go. The issue seems to be whether mereological sums should be seen as constrained by nature, or whether they are unrestricted. See Mereology in Ontology...|Intrinsic Identity. |
| 2622 | Can one movement have a mental and physical cause? [Ryle] |
| Full Idea: The dogma of the Ghost in the Machine maintains that there exist both minds and bodies; that there are mechanical causes of corporeal movements, and mental causes of corporeal movements. | |
| From: Gilbert Ryle (The Concept of Mind [1949], I (3)) | |
| A reaction: This nicely identifies the problem of double causation, which can be found in Spinoza (Idea 4862). The dualists have certainly got a problem here, but they can deny a conflict. The initiation of a hand movement is not mechanical at all. |
| 1354 | We cannot introspect states of anger or panic [Ryle] |
| Full Idea: No one could introspectively scrutinize the state of panic or fury. | |
| From: Gilbert Ryle (The Concept of Mind [1949], Ch.6) | |
| A reaction: It depends what you mean by 'scrutinize'. No human being ever loses their temper or panics without a background thought of "Oh dear, I'm losing it - it would probably be better if I didn't" (or, as Aristotle might say, "I'm angry, and so I should be"). |
| 1353 | Reporting on myself has the same problems as reporting on you [Ryle] |
| Full Idea: My reports on myself are subject to the same kinds of defects as are my reports on you. | |
| From: Gilbert Ryle (The Concept of Mind [1949], Ch.6) | |
| A reaction: This may be true of memories or of motives, but it hardly seems to apply to being in pain, where you might be totally lying, where the worst I could do to myself is exaggerate. "You're fine; how am I?" |
| 2624 | I cannot prepare myself for the next thought I am going to think [Ryle] |
| Full Idea: One thing that I cannot prepare myself for is the next thought that I am going to think. | |
| From: Gilbert Ryle (The Concept of Mind [1949], VI (7)) |
| 2620 | Dualism is a category mistake [Ryle] |
| Full Idea: The official theory of mind (as private, non-spatial, outside physical laws) I call 'the dogma of the Ghost in the Machine'. I hope to prove it entirely false, and show that it is one big mistake, namely a 'category mistake'. | |
| From: Gilbert Ryle (The Concept of Mind [1949], I (2)) | |
| A reaction: This is the essence of Ryle's eliminitavist behaviourism. Personally I agree that the idea of a separate 'ghost' running the machine is utterly implausible, but it isn't a 'category mistake'. The mind clearly exists, but the confusion is about what it is. |
| 2388 | Behaviour depends on desires as well as beliefs [Chalmers on Ryle] |
| Full Idea: Another problem for Ryle (from Chisholm and Geach) is that no mental state could be defined by a single range of behavioural dispositions, independent of any other mental states. (Behaviour depends upon desires as well as beliefs). | |
| From: comment on Gilbert Ryle (The Concept of Mind [1949]) by David J.Chalmers - The Conscious Mind 1.1.2 | |
| A reaction: The defence of behaviourism is to concede this point, but suggest that behavioural dispositions come in large groups of interdependent sets, some relating to beliefs, others relating to desires, and each group leads to a behaviour. |
| 3354 | You can't explain mind as dispositions, if they aren't real [Benardete,JA on Ryle] |
| Full Idea: Ryle is tough-minded to the point of incoherence when he combines a dispositional account of the mind with an anti-realist account of dispositions. | |
| From: comment on Gilbert Ryle (The Concept of Mind [1949]) by José A. Benardete - Metaphysics: the logical approach Ch.22 | |
| A reaction: A nice point, but it strikes me that Ryle was, by temperament at least, an eliminativist about the mind, so the objection would not bother him. Maybe a disposition and a property are the same thing? |
| 2387 | How can behaviour be the cause of behaviour? [Chalmers on Ryle] |
| Full Idea: A problem for Ryle is that mental states may cause behaviour, but if mental states are themselves behavioural or behavioural dispositions, as opposed to internal states, then it is hard to see how they could do the job. | |
| From: comment on Gilbert Ryle (The Concept of Mind [1949]) by David J.Chalmers - The Conscious Mind 1.1.2 | |
| A reaction: I strongly approve of this, as an objection to any form of behaviourism or functionalism. If you identify something by its related behaviour, or its apparent function, this leaves the question 'WHY does it behave or function in this way?' |