43 ideas
| 4697 | There has been a distinct 'Social Turn' in recent philosophy, like the earlier 'Linguistic Turn' [O'Grady] |
| Full Idea: The Social Turn is as defining a characteristic of contemporary philosophy as the Linguistic Turn has been of the earlier twentieth century period. | |
| From: Paul O'Grady (Relativism [2002], Ch.1) | |
| A reaction: A helpful observation. It ties in with externalism about concepts (Twin Earth), impossibility of Private Language, and externalism about knowledge. |
| 4731 | Good reasoning will avoid contradiction, enhance coherence, not ignore evidence, and maximise evidence [O'Grady] |
| Full Idea: The four basic principles of rationality are 1) avoid contradiction, 2) enhance coherence, 3) avoid ignoring evidence, and 4) maximise evidence. | |
| From: Paul O'Grady (Relativism [2002], Ch.5) | |
| A reaction: I like this, and can't think of any additions. 'Coherence' is the vaguest of the conditions. Maximising evidence is still the driving force of science, even if it does sound quaintly positivist. |
| 4735 | Just as maps must simplify their subject matter, so thought has to be reductionist about reality [O'Grady] |
| Full Idea: A map that is identical in all respects with that which is mapped is just useless. So reductionism is not just a good thing - it is essential to thought. | |
| From: Paul O'Grady (Relativism [2002], Ch.6) | |
| A reaction: A useful warning, when thinking about truth. It is folly to want your thoughts to exactly correspond to reality. I want to understand the world, but not if it requires being the world. |
| 4703 | The epistemic theory of truth presents it as 'that which is licensed by our best theory of reality' [O'Grady] |
| Full Idea: The epistemic theory of truth presents it as 'that which is licensed by our best theory of reality'. | |
| From: Paul O'Grady (Relativism [2002], Ch.2) | |
| A reaction: Dangerous nonsense. This leaves truth shifting as our theories change, it leads to different truths in different cultures, and no palpable falsehood in ignorant cultures. Don't give it house-room. |
| 4701 | To say a relative truth is inexpressible in other frameworks is 'weak', while saying it is false is 'strong' [O'Grady] |
| Full Idea: Weak alethic relativism holds that while a statement may be true in one framework, it is inexpressible in another. Strong alethic relativism is where a sentence is true relative to one framework, but false relative to another. | |
| From: Paul O'Grady (Relativism [2002], Ch.2) | |
| A reaction: The weak version will be Kuhn's 'incommensurability' of scientific theories, while the strong version will be full Protagorean relativism, saying all beliefs are true. |
| 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? |
| 4705 | Logical relativism appears if we allow more than one legitimate logical system [O'Grady] |
| Full Idea: Logical relativism emerges if one defends the existence of two or more rival systems that one may legitimately choose between, or move back and forth between. | |
| From: Paul O'Grady (Relativism [2002], Ch.2) | |
| A reaction: All my instincts rebel against this possibility. All of Aristotle's and Kant's philosophy would be rendered meaningless. Obviously you can create artificial logics (like games), but I believe there is a truth logic. (Pathetic, isn't it?) |
| 4700 | A third value for truth might be "indeterminate", or a point on a scale between 'true' and 'false' [O'Grady] |
| Full Idea: Suggestions for a third value for truth are "indeterminate", or a scale running from "true", through "mostly true", "mainly true", "half true", "mainly false", "mostly false", to "false", or maybe even "0.56 true". | |
| From: Paul O'Grady (Relativism [2002], Ch.2) | |
| A reaction: Anything on a sliding scale sounds wrong, as it seems to be paracitic on an underlying fixed idea of 'true'. "Indeterminate", though, seems just right for the truth of predictions ('sea-fight tomorrow'). |
| 4704 | Wittgenstein reduced Russell's five primitive logical symbols to a mere one [O'Grady] |
| Full Idea: While Russell and Whitehead used five primitive logical symbols in their system, Wittgenstein suggested in his 'Tractatus' that this be reduced to one. | |
| From: Paul O'Grady (Relativism [2002], Ch.2) | |
| A reaction: This certainly captures why Russell was so impressed by him. In retrospect what looked like progress presumably now looks like the beginning of the collapse of the enterprise. |
| 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. |
| 4711 | Anti-realists say our theories (such as wave-particle duality) give reality incompatible properties [O'Grady] |
| Full Idea: The anti-realist says we have theories about the world that are incompatible with each other, and irreducible to each other. They often cite wave-particle duality, which postulate incompatible properties to reality. | |
| From: Paul O'Grady (Relativism [2002], Ch.3) | |
| A reaction: Most physicists, of course, hate this duality, precisely because they can't conceive how the two properties could be real. I say realism comes first, and the theories must try to accommodate that assumption. |
| 4698 | What counts as a fact partly depends on the availability of human concepts to describe them [O'Grady] |
| Full Idea: What counts as a fact partly depends on human input, such as the availability of concepts to describe such facts. | |
| From: Paul O'Grady (Relativism [2002], Ch.1) | |
| A reaction: The point must be taken. I am happy to generalise about 'The Facts', meaning 'whatever is the case', but the individuation of specific facts is bound to hit the current problem. |
| 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. |
| 4715 | We may say that objects have intrinsic identity conditions, but still allow multiple accounts of them [O'Grady] |
| Full Idea: Those defending the claim that objects exist with identity conditions not imposed by us, do not have to say that there is just one account of those objects possible. | |
| From: Paul O'Grady (Relativism [2002], Ch.3) | |
| A reaction: This seems right, but the test question is whether the mind of God contains a single unified theory/account. Are multiple accounts the result of human inadequacy? Yes, I surmise. |
| 16766 | One thing needs a single thing to unite it; if there were two forms, something must unite them [Aquinas] |
| Full Idea: One thing simpliciter is produced out of many actually existing things only if there is something uniting and tying them to each other. If Socrates were animal and rational by different forms, then to be united they would need something to make them one. | |
| From: Thomas Aquinas (Quaestiones de anima [1269], 11c), quoted by Robert Pasnau - Metaphysical Themes 1274-1671 25.2 | |
| A reaction: This is the reply to the idea that a single thing is just an interesting of many sortal essences. It presumes, of course, that a thing like a horse has something called 'unity'. |
| 4719 | Maybe developments in logic and geometry have shown that the a priori may be relative [O'Grady] |
| Full Idea: A weaker form of relativism holds that developments in logic, in maths and in geometry have shown how a relativised notion of the a priori is possible. | |
| From: Paul O'Grady (Relativism [2002], Ch.4) | |
| A reaction: This is non-Euclidean geometry, and multiple formalisations of logic. Personally I don't believe it. You can expand these subjects, and pursue whimsical speculations, but I have faith in their stable natural core. Neo-Platonism. |
| 4720 | Sense-data are only safe from scepticism if they are primitive and unconceptualised [O'Grady] |
| Full Idea: The reason sense-data were immune from doubt was because they were so primitive; they were unstructured and below the level of conceptualisation. Once they were given structure and conceptualised, they were no longer safe from sceptical challenge. | |
| From: Paul O'Grady (Relativism [2002], Ch.4) | |
| A reaction: The question of whether sense-data are conceptualised doesn't have to be all-or-nothing. As concepts creep in, so does scepticism, but so what? Sensible philosophers live with scepticism, like a mad aunt in the attic. |
| 4722 | Modern epistemology centres on debates about foundations, and about external justification [O'Grady] |
| Full Idea: The two dichotomies which set the agenda in contemporary epistemology are the foundationalist-coherentist debate, and the internalist-externalist debate. | |
| From: Paul O'Grady (Relativism [2002], Ch.4) | |
| A reaction: Helpful. Roughly, foundationalists are often externalists (if they are empiricists), and coherentists are often internalists (esp. if they are rationalists). An eccentric combination would make a good PhD. |
| 4724 | Internalists say the reasons for belief must be available to the subject, and externalists deny this [O'Grady] |
| Full Idea: Internalism about justification says that the reasons one has for a belief must be in some sense available to the knowing subject, ..while externalism holds that it is possible for a person to have a justified belief without having access to the reason. | |
| From: Paul O'Grady (Relativism [2002], Ch.4) | |
| A reaction: It strikes me that internalists are talking about the believer being justified, and externalists talk about the belief being justified. I'm with the internalists. If this means cats don't know much, so much the worse for cats. |
| 4723 | Coherence involves support from explanation and evidence, and also probability and confirmation [O'Grady] |
| Full Idea: Coherentist justification is more than absence of contradictions, and will involve issues like explanatory support and evidential support, and perhaps issues about probability and confirmation too. | |
| From: Paul O'Grady (Relativism [2002], Ch.4) | |
| A reaction: Something like this is obviously essential. Is the notion of 'relevance' also needed (e.g. to avoid the raven paradox of induction)? Coherence of justification will combine with correspondence for truth. |
| 4709 | Ontological relativists are anti-realists, who deny that our theories carve nature at the joints [O'Grady] |
| Full Idea: Ontological relativists are anti-realists in the strong sense; they hold as meaningless the view that our theories carve nature at the joints. | |
| From: Paul O'Grady (Relativism [2002], Ch.3) | |
| A reaction: This pinpoints my disagreement with such relativism, as it seems obvious to me that nature has 'joints', and that we would agree with any sensible alien about lots of things. |
| 4725 | Contextualism says that knowledge is relative to its context; 'empty' depends on your interests [O'Grady] |
| Full Idea: Contextualist about knowledge say that "to know" means different things in different context. For example, a warehouse may be empty for a furniture owner, but not for a bacteriologist or a physicist. | |
| From: Paul O'Grady (Relativism [2002], Ch.4) | |
| A reaction: There is obviously some truth in this, but we might say that 'empty' is a secondary quality, or that 'empty for furniture' is not relative. We needn't accept relativism here. |
| 4732 | One may understand a realm of ideas, but be unable to judge their rationality or truth [O'Grady] |
| Full Idea: It is possible to conceive of one understanding the meaning of a realm of ideas, but holding that one cannot judge as to the truth or rationality of the claims made in it. | |
| From: Paul O'Grady (Relativism [2002], Ch.5) | |
| A reaction: I think Davidson gives good grounds for challenging this, by doubt whether one 'conceptual scheme' can know another without grasping its rationality and truth-conditions. |
| 4710 | Verificationism was attacked by the deniers of the analytic-synthetic distinction, needed for 'facts' [O'Grady] |
| Full Idea: Verificationism came under attack from empiricists who were friendly to the banishment of traditional metaphysics, but unfriendly to the analytic-synthetic distinction, on which the idea of a 'factual statement' depended. | |
| From: Paul O'Grady (Relativism [2002], Ch.3) | |
| A reaction: I don't accept this move because I don't consider the 'facts' to be language-dependent. They are pre-linguistic, they outrun that capacity of our language, and they are available to animals. |
| 4717 | If we abandon the analytic-synthetic distinction, scepticism about meaning may be inevitable [O'Grady] |
| Full Idea: There may be no way to avoid scepticism about meaning if you abandon the analytic-synthetic distinction in the way Quine does. | |
| From: Paul O'Grady (Relativism [2002], Ch.3) | |
| A reaction: My suspicion was always that Quine's proposal began the slippery road to hell. It appears to be pragmatists who are most drawn to Quine's idea. The proposal that all my analytic propositions could be treated as synthetic totally baffles me. |
| 4706 | Early Quine says all beliefs could be otherwise, but later he said we would assume mistranslation [O'Grady] |
| Full Idea: In his earlier work, Quine defended the view that no belief (including logic) is in principle unrevisable, but in his later work (1970) he took the conservative view that we would always impute mistranslation rather than deviancy. | |
| From: Paul O'Grady (Relativism [2002], Ch.2) | |
| A reaction: I take it he was influenced by Davidson's 'principle of charity'. He says that if someone asserts 'p and not-p', we would assume a misunderstanding of 'and' or 'not'. |
| 4734 | Cryptographers can recognise that something is a language, without translating it [O'Grady] |
| Full Idea: It makes sense to think that one could recognise that something is a language without necessarily being able to translate it; cryptographers do this all the time. | |
| From: Paul O'Grady (Relativism [2002], Ch.5) | |
| A reaction: Maybe, but cryptographers usually have a lot of context to work with. If we met extraterrestrials if might not be so clear. One can only spot patterns, and crystals have those. |
| 4727 | The chief problem for fideists is other fideists who hold contrary ideas [O'Grady] |
| Full Idea: The chief problem for fideists is other fideists who hold contrary ideas. | |
| From: Paul O'Grady (Relativism [2002], Ch.4) | |
| A reaction: The other problem is trying to find grounds for sticking to the object of one's faith, rather than changing from time to time. |