47 ideas
| 23367 | Even pointing a finger should only be done for a reason [Epictetus] |
| Full Idea: Philosophy says it is not right even to stretch out a finger without some reason. | |
| From: Epictetus (fragments/reports [c.57], 15) | |
| A reaction: The key point here is that philosophy concerns action, an idea on which Epictetus is very keen. He rather despise theory. This idea perfectly sums up the concept of the wholly rational life (which no rational person would actually want to live!). |
| 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. |
| 18194 | 'Forcing' can produce new models of ZFC from old models [Maddy] |
| Full Idea: Cohen's method of 'forcing' produces a new model of ZFC from an old model by appending a carefully chosen 'generic' set. | |
| From: Penelope Maddy (Naturalism in Mathematics [1997], I.4) |
| 18195 | A Large Cardinal Axiom would assert ever-increasing stages in the hierarchy [Maddy] |
| Full Idea: A possible axiom is the Large Cardinal Axiom, which asserts that there are more and more stages in the cumulative hierarchy. Infinity can be seen as the first of these stages, and Replacement pushes further in this direction. | |
| From: Penelope Maddy (Naturalism in Mathematics [1997], I.5) |
| 18191 | Axiom of Infinity: completed infinite collections can be treated mathematically [Maddy] |
| Full Idea: The axiom of infinity: that there are infinite sets is to claim that completed infinite collections can be treated mathematically. In its standard contemporary form, the axioms assert the existence of the set of all finite ordinals. | |
| From: Penelope Maddy (Naturalism in Mathematics [1997], I.3) |
| 18193 | The Axiom of Foundation says every set exists at a level in the set hierarchy [Maddy] |
| Full Idea: In the presence of other axioms, the Axiom of Foundation is equivalent to the claim that every set is a member of some Vα. | |
| From: Penelope Maddy (Naturalism in Mathematics [1997], I.3) |
| 18169 | Axiom of Reducibility: propositional functions are extensionally predicative [Maddy] |
| Full Idea: The Axiom of Reducibility states that every propositional function is extensionally equivalent to some predicative proposition function. | |
| From: Penelope Maddy (Naturalism in Mathematics [1997], I.1) |
| 18168 | 'Propositional functions' are propositions with a variable as subject or predicate [Maddy] |
| Full Idea: A 'propositional function' is generated when one of the terms of the proposition is replaced by a variable, as in 'x is wise' or 'Socrates'. | |
| From: Penelope Maddy (Naturalism in Mathematics [1997], I.1) | |
| A reaction: This implies that you can only have a propositional function if it is derived from a complete proposition. Note that the variable can be in either subject or in predicate position. It extends Frege's account of a concept as 'x is F'. |
| 18190 | Completed infinities resulted from giving foundations to calculus [Maddy] |
| Full Idea: The line of development that finally led to a coherent foundation for the calculus also led to the explicit introduction of completed infinities: each real number is identified with an infinite collection of rationals. | |
| From: Penelope Maddy (Naturalism in Mathematics [1997], I.3) | |
| A reaction: Effectively, completed infinities just are the real numbers. |
| 18171 | Cantor and Dedekind brought completed infinities into mathematics [Maddy] |
| Full Idea: Both Cantor's real number (Cauchy sequences of rationals) and Dedekind's cuts involved regarding infinite items (sequences or sets) as completed and subject to further manipulation, bringing the completed infinite into mathematics unambiguously. | |
| From: Penelope Maddy (Naturalism in Mathematics [1997], I.1 n39) | |
| A reaction: So it is the arrival of the real numbers which is the culprit for lumbering us with weird completed infinites, which can then be the subject of addition, multiplication and exponentiation. Maybe this was a silly mistake? |
| 18172 | Infinity has degrees, and large cardinals are the heart of set theory [Maddy] |
| Full Idea: The stunning discovery that infinity comes in different degrees led to the theory of infinite cardinal numbers, the heart of contemporary set theory. | |
| From: Penelope Maddy (Naturalism in Mathematics [1997], I.1) | |
| A reaction: It occurs to me that these huge cardinals only exist in set theory. If you took away that prop, they would vanish in a puff. |
| 18175 | For any cardinal there is always a larger one (so there is no set of all sets) [Maddy] |
| Full Idea: By the mid 1890s Cantor was aware that there could be no set of all sets, as its cardinal number would have to be the largest cardinal number, while his own theorem shows that for any cardinal there is a larger. | |
| From: Penelope Maddy (Naturalism in Mathematics [1997], I.1) | |
| A reaction: There is always a larger cardinal because of the power set axiom. Some people regard that with suspicion. |
| 18196 | An 'inaccessible' cardinal cannot be reached by union sets or power sets [Maddy] |
| Full Idea: An 'inaccessible' cardinal is one that cannot be reached by taking unions of small collections of smaller sets or by taking power sets. | |
| From: Penelope Maddy (Naturalism in Mathematics [1997], I.5) | |
| A reaction: They were introduced by Hausdorff in 1908. |
| 18187 | Theorems about limits could only be proved once the real numbers were understood [Maddy] |
| Full Idea: Even the fundamental theorems about limits could not [at first] be proved because the reals themselves were not well understood. | |
| From: Penelope Maddy (Naturalism in Mathematics [1997], I.2) | |
| A reaction: This refers to the period of about 1850 (Weierstrass) to 1880 (Dedekind and Cantor). |
| 18182 | The extension of concepts is not important to me [Maddy] |
| Full Idea: I attach no decisive importance even to bringing in the extension of the concepts at all. | |
| From: Penelope Maddy (Naturalism in Mathematics [1997], §107) | |
| A reaction: He almost seems to equate the concept with its extension, but that seems to raise all sorts of questions, about indeterminate and fluctuating extensions. |
| 18177 | In the ZFC hierarchy it is impossible to form Frege's set of all three-element sets [Maddy] |
| Full Idea: In the ZFC cumulative hierarchy, Frege's candidates for numbers do not exist. For example, new three-element sets are formed at every stage, so there is no stage at which the set of all three-element sets could he formed. | |
| From: Penelope Maddy (Naturalism in Mathematics [1997], I.2) | |
| A reaction: Ah. This is a very important fact indeed if you are trying to understand contemporary discussions in philosophy of mathematics. |
| 18164 | Frege solves the Caesar problem by explicitly defining each number [Maddy] |
| Full Idea: To solve the Julius Caesar problem, Frege requires explicit definitions of the numbers, and he proposes his well-known solution: the number of Fs = the extension of the concept 'equinumerous with F' (based on one-one correspondence). | |
| From: Penelope Maddy (Naturalism in Mathematics [1997], I.1) | |
| A reaction: Why do there have to be Fs before there can be the corresponding number? If there were no F for 523, would that mean that '523' didn't exist (even if 522 and 524 did exist)? |
| 18184 | Making set theory foundational to mathematics leads to very fruitful axioms [Maddy] |
| Full Idea: The set theory axioms developed in producing foundations for mathematics also have strong consequences for existing fields, and produce a theory that is immensely fruitful in its own right. | |
| From: Penelope Maddy (Naturalism in Mathematics [1997], I.2) | |
| A reaction: [compressed] Second of Maddy's three benefits of set theory. This benefit is more questionable than the first, because the axioms may be invented because of their nice fruit, instead of their accurate account of foundations. |
| 18185 | Unified set theory gives a final court of appeal for mathematics [Maddy] |
| Full Idea: The single unified area of set theory provides a court of final appeal for questions of mathematical existence and proof. | |
| From: Penelope Maddy (Naturalism in Mathematics [1997], I.2) | |
| A reaction: Maddy's third benefit of set theory. 'Existence' means being modellable in sets, and 'proof' means being derivable from the axioms. The slightly ad hoc character of the axioms makes this a weaker defence. |
| 18183 | Set theory brings mathematics into one arena, where interrelations become clearer [Maddy] |
| Full Idea: Set theoretic foundations bring all mathematical objects and structures into one arena, allowing relations and interactions between them to be clearly displayed and investigated. | |
| From: Penelope Maddy (Naturalism in Mathematics [1997], I.2) | |
| A reaction: The first of three benefits of set theory which Maddy lists. The advantages of the one arena seem to be indisputable. |
| 18186 | Identifying geometric points with real numbers revealed the power of set theory [Maddy] |
| Full Idea: The identification of geometric points with real numbers was among the first and most dramatic examples of the power of set theoretic foundations. | |
| From: Penelope Maddy (Naturalism in Mathematics [1997], I.2) | |
| A reaction: Hence the clear definition of the reals by Dedekind and Cantor was the real trigger for launching set theory. |
| 18188 | The line of rationals has gaps, but set theory provided an ordered continuum [Maddy] |
| Full Idea: The structure of a geometric line by rational points left gaps, which were inconsistent with a continuous line. Set theory provided an ordering that contained no gaps. These reals are constructed from rationals, which come from integers and naturals. | |
| From: Penelope Maddy (Naturalism in Mathematics [1997], I.2) | |
| A reaction: This completes the reduction of geometry to arithmetic and algebra, which was launch 250 years earlier by Descartes. |
| 18163 | Mathematics rests on the logic of proofs, and on the set theoretic axioms [Maddy] |
| Full Idea: Our much loved mathematical knowledge rests on two supports: inexorable deductive logic (the stuff of proof), and the set theoretic axioms. | |
| From: Penelope Maddy (Naturalism in Mathematics [1997], I Intro) |
| 18207 | Maybe applications of continuum mathematics are all idealisations [Maddy] |
| Full Idea: It could turn out that all applications of continuum mathematics in natural sciences are actually instances of idealisation. | |
| From: Penelope Maddy (Naturalism in Mathematics [1997], II.6) |
| 18204 | Scientists posit as few entities as possible, but set theorist posit as many as possible [Maddy] |
| Full Idea: Crudely, the scientist posits only those entities without which she cannot account for observations, while the set theorist posits as many entities as she can, short of inconsistency. | |
| From: Penelope Maddy (Naturalism in Mathematics [1997], II.5) |
| 18167 | We can get arithmetic directly from HP; Law V was used to get HP from the definition of number [Maddy] |
| Full Idea: Recent commentators have noted that Frege's versions of the basic propositions of arithmetic can be derived from Hume's Principle alone, that the fatal Law V is only needed to derive Hume's Principle itself from the definition of number. | |
| From: Penelope Maddy (Naturalism in Mathematics [1997], I.1) | |
| A reaction: Crispin Wright is the famous exponent of this modern view. Apparently Charles Parsons (1965) first floated the idea. |
| 18205 | The theoretical indispensability of atoms did not at first convince scientists that they were real [Maddy] |
| Full Idea: The case of atoms makes it clear that the indispensable appearance of an entity in our best scientific theory is not generally enough to convince scientists that it is real. | |
| From: Penelope Maddy (Naturalism in Mathematics [1997], II.6) | |
| A reaction: She refers to the period between Dalton and Einstein, when theories were full of atoms, but there was strong reluctance to actually say that they existed, until the direct evidence was incontrovertable. Nice point. |
| 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. |
| 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. |
| 18206 | Science idealises the earth's surface, the oceans, continuities, and liquids [Maddy] |
| Full Idea: In science we treat the earth's surface as flat, we assume the ocean to be infinitely deep, we use continuous functions for what we know to be quantised, and we take liquids to be continuous despite atomic theory. | |
| From: Penelope Maddy (Naturalism in Mathematics [1997], II.6) | |
| A reaction: If fussy people like scientists do this all the time, how much more so must the confused multitude be doing the same thing all day? |
| 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. |
| 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. |