40 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. |
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'). |
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. |
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. |
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. |
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. |
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. |
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. |
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. |
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. |
6005 | Animals are dangerous and nourishing, and can't form contracts of justice [Hermarchus, by Sedley] |
Full Idea: Hermarchus said that animal killing is justified by considerations of human safety and nourishment and by animals' inability to form contractual relations of justice with us. | |
From: report of Hermarchus (fragments/reports [c.270 BCE]) by David A. Sedley - Hermarchus | |
A reaction: Could the last argument be used to justify torturing animals? Or could we eat a human who was too brain-damaged to form contracts? |
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. |