36 ideas
| 18986 | Truth is just a name for verification-processes [James] |
| Full Idea: Truth for us is simply a collective name for verification-processes, just as 'health' is a name for other processes in life. | |
| From: William James (Pragmatism - eight lectures [1907], Lec 6) | |
| A reaction: So the slogan is 'truth is success in belief'? Suicide and racist genocide can be 'successful'. I would have thought that truth was the end of a process, rather than the process itself. |
| 18983 | In many cases there is no obvious way in which ideas can agree with their object [James] |
| Full Idea: When you speak of the 'time-keeping function' of a clock, it is hard to see exactly what your ideas can copy. ...Where our ideas cannot copy definitely their object, what does agreement with that object mean? | |
| From: William James (Pragmatism - eight lectures [1907], Lec 6) | |
| A reaction: This is a very good criticism of the correspondence theory of truth. It looks a lovely theory when you can map components of a sentence (like 'the pen is in the drawer') onto components of reality - but it has to cover the hard cases. |
| 18972 | Ideas are true in so far as they co-ordinate our experiences [James] |
| Full Idea: Pragmatists say that ideas (which themselves are but parts of our experience) become true just in so far as they help us to get into satisfactory relation with other parts of our experience. | |
| From: William James (Pragmatism - eight lectures [1907], Lec 2) | |
| A reaction: I'm struck by the close similarity (at least in James) of the pragmatic view of truth and the coherence theory of truth (associated later with Blanshard). Perhaps the coherence theory is one version of the pragmatic account |
| 18973 | New opinions count as 'true' if they are assimilated to an individual's current beliefs [James] |
| Full Idea: A new opinion counts as 'true' just in proportion as it gratifies the individual's desire to assimilate the novel in his experience to his beliefs in stock. | |
| From: William James (Pragmatism - eight lectures [1907], Lec 2) | |
| A reaction: Note the tell-tale locution 'counts as' true, rather than 'is' true. The obvious problem is that someone with a big stock of foolish beliefs will 'count as' true some bad interpretation which is gratifyingly assimilated to their current confusions. |
| 18984 | True ideas are those we can assimilate, validate, corroborate and verify (and false otherwise) [James] |
| Full Idea: True ideas are those that we can assimilate, validate, corroborate and verify. False ideas are those that we cannot. | |
| From: William James (Pragmatism - eight lectures [1907], Lec 6) | |
| A reaction: The immediate question is why you should label something as 'false' simply on the grounds that you can't corroborate it. Proving the falsity is a stronger position than the ignorance James seems happy with. 'Assimilate' implies coherence. |
| 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. |
| 18987 | A 'thing' is simply carved out of reality for human purposes [James] |
| Full Idea: What shall we call a 'thing' anyhow? It seems quite arbitrary, for we carve out everything, just as we carve out constellations, to suit our human purposes. | |
| From: William James (Pragmatism - eight lectures [1907], Lec 7) | |
| A reaction: James wrote just before the discovery of galaxies, which are much more obviously 'things' than constellations like the Plough are! This idea suggests a connection between pragmatism and the nihilist view of objects of Van Inwagen and co. |
| 18981 | 'Substance' is just a word for groupings and structures in experience [James] |
| Full Idea: 'Substance' appears now only as another name for the fact that phenomena as they come are actually grouped and given in coherent forms. | |
| From: William James (Pragmatism - eight lectures [1907], Lec 4) | |
| A reaction: This is the strongly empirical strain in James's empiricism. This sounds like a David Lewis comment on the Humean mosaic of experience. We Aristotelians at least believe that the groups run much deeper than the surface of experience. |
| 18974 | Truth is a species of good, being whatever proves itself good in the way of belief [James] |
| Full Idea: Truth is one species of good, and not, as is usually supposed, a category distinct from good, and co-ordinate with it. The true is whatever proves itself to be good in the way of belief, and good, too, for definite, assignable reasons. | |
| From: William James (Pragmatism - eight lectures [1907], Lec 2) | |
| A reaction: The trouble is that false optimism can often often be what is 'good in the way of belief'. That said, I think quite a good way to specify 'truth' is 'success in belief', but I mean intrinsically successful, not pragmatically successful. |
| 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. |
| 18989 | Pragmatism accepts any hypothesis which has useful consequences [James] |
| Full Idea: On pragmatic principles we cannot reject any hypothesis if consequences useful to life flow from it. | |
| From: William James (Pragmatism - eight lectures [1907], Lec 8) | |
| A reaction: Most governments seem to find lies more useful than the truth. Maybe most children are better off not knowing the truth about their parents. It might be disastrous to know the truth about what other people are thinking. Is 'useful but false' meaningful? |
| 18971 | Theories are practical tools for progress, not answers to enigmas [James] |
| Full Idea: Theories are instruments, not answers to enigmas, in which we can rest. We don't lie back upon them, we move forward, and, on occasion, make nature over again by their aid. Pragmatism unstiffens all our theories, limbers them up and sets each one to work. | |
| From: William James (Pragmatism - eight lectures [1907], Lec 2) | |
| A reaction: This follows his criticism of the quest for 'solving names' - big words that give bogus solutions to problems. James's view is not the same as 'instrumentalism', though he would probably sympathise with that view. The defines theories badly. |
| 18985 | True thoughts are just valuable instruments of action [James] |
| Full Idea: The possession of true thoughts means everywhere the possession of invaluable instruments of action. | |
| From: William James (Pragmatism - eight lectures [1907], Lec 6) | |
| A reaction: It looks to me like we should distinguish 'active' and 'passive' instrumentalism. The passive version says there is no more to theories and truth than what instruments record. James's active version says truth is an instrument for doing things well. |
| 18982 | Pragmatism says all theories are instrumental - that is, mental modes of adaptation to reality [James] |
| Full Idea: The pragmatist view is that all our theories are instrumental, are mental modes of adaptation to reality, rather than revelations or gnostic answers to some divinely instituted world enigma. | |
| From: William James (Pragmatism - eight lectures [1907], Lec 5) | |
| A reaction: This treats instrumentalism as the pragmatic idea of theories as what works (and nothing more), with, presumably, no interest in grasping something called 'reality'. Presumably instrumentalism might have other motivations - such as fun. |
| 18975 | We return to experience with concepts, where they show us differences [James] |
| Full Idea: Concepts for the pragmatist are things to come back into experience with, things to make us look for differences. | |
| From: William James (Pragmatism - eight lectures [1907], Lec 3) | |
| A reaction: That's good. I like both halves of this. Experience gives us the concepts, but then we 'come back' into experience equipped with them. Presumably animals can look for differences, but concepts enhance that hugely. Know the names of the flowers. |
| 18980 | If there is a 'greatest knower', it doesn't follow that they know absolutely everything [James] |
| Full Idea: The greatest knower of them all may yet not know the whole of everything, or even know what he does know at one single stroke: - he may be liable to forget. | |
| From: William James (Pragmatism - eight lectures [1907], Lec 4) | |
| A reaction: And that's before you get to the problem of how the greatest knower could possibly know whether or not they knew absolutely everything, or whether there might be some fact which was irremediably hidden from them. |
| 18978 | It is hard to grasp a cosmic mind which produces such a mixture of goods and evils [James] |
| Full Idea: We can with difficulty comprehend the character of a cosmic mind whose purposes are fully revealed by the strange mixture of good and evils that we find in this actual world's particulars. | |
| From: William James (Pragmatism - eight lectures [1907], Lec 3) | |
| A reaction: And, of course, what counts as 'goods' or 'evils' seems to have a highly relative aspect to it. To claim that really it is all good is massive hope based on flimsy evidence. |
| 8141 | He that does evil has not seen God [John] |
| Full Idea: He that doeth evil hath not seen God. | |
| From: St John (25: Third Epistle of John [c.90], 11) | |
| A reaction: This gives God a role striking similar to Plato's Form of the Good. Plato thought the Good was prior to the gods, but he gives the good a quasi-religious role. I say we would only be inspired by the sight of God if we already had a moral sense. |
| 18991 | If the God hypothesis works well, then it is true [James] |
| Full Idea: On pragmatistic principles, if the hypothesis of God works satisfactorily in the widest sense of the word, it is true. | |
| From: William James (Pragmatism - eight lectures [1907], Lec 8) | |
| A reaction: The truth of God's existence certainly is a challenging test case for the pragmatic theory of truth, and James really bites the bullet here. Pragmatism may ultimately founder on the impossibility of specifying what 'works satisfactorily' means. |
| 18977 | The wonderful design of a woodpecker looks diabolical to its victims [James] |
| Full Idea: To the grub under the bark the exquisite fitness of the woodpecker's organism to extract him would certainly argue a diabolical designer. | |
| From: William James (Pragmatism - eight lectures [1907], Lec 3) | |
| A reaction: What an elegant sentence! The huge problem for religious people who accept (probably reluctantly) evolution by natural selection is the moral nature of the divine being who could use such a ruthless method of design. |
| 18979 | Things with parts always have some structure, so they always appear to be designed [James] |
| Full Idea: The parts of things must always make some definite resultant, be it chaotic or harmonious. When we look at what has actually come, the conditions must always appear perfectly designed to ensure it. | |
| From: William James (Pragmatism - eight lectures [1907], Lec 3) | |
| A reaction: In so far as the design argument is an analogy with human affairs, we can't deny that high levels of order suggest an organising mind, and mere chaos suggests a coincidence of unco-ordinated forces. |
| 18976 | Private experience is the main evidence for God [James] |
| Full Idea: I myself believe that the evidence for God lies primarily in inner personal experience. | |
| From: William James (Pragmatism - eight lectures [1907], Lec 3) | |
| A reaction: There is not much you can say to someone who claims incontrovertible evidence which is utterly private to themselves. Does total absence of private religious experience count as evidence on the subject? |
| 18990 | Nirvana means safety from sense experience, and hindus and buddhists are just afraid of life [James] |
| Full Idea: Nirvana means safety from the everlasting round of adventures of which the world of sense consists. The hindoo and the buddhist for this is essentially their attitude, are simply afraid, afraid of more experience, afraid of life. | |
| From: William James (Pragmatism - eight lectures [1907], Lec 8) | |
| A reaction: Wonderfully American! From what I have seen of eastern thought, including Taoism, I agree with James, in general. There is a rejection of knowledge and of human life which I find shocking. I suspect it is a defence mechanism for downtrodden people. |