42 ideas
| 7623 | For ancient Greeks being wise was an ethical value [Putnam] |
| Full Idea: An ancient Greek would have said that being wise is an ethical value. | |
| From: Hilary Putnam (Reason, Truth and History [1981], Ch.6) | |
| A reaction: This is instantly appealing, but since the Enlightenment we are under an obligation to attempt to justify absolutely everything, including the value of wisdom. I'm thinking that it only has value if it leads to eudaimonia. |
| 4714 | Putnam's epistemic notion of truth replaces the realism of correspondence with ontological relativism [Putnam, by O'Grady] |
| Full Idea: Putnam replaces a correspondence theory of truth with an epistemic notion of truth - truth is idealized rational acceptability. The correspondence theory is committed to realism, but his allows ontological relativism. | |
| From: report of Hilary Putnam (Reason, Truth and History [1981]) by Paul O'Grady - Relativism Ch.3 | |
| A reaction: This seems to be part of a slide by Putnam away from realism towards pragmatism. As a robust and defiant realist, this always strikes me as the road to hell. |
| 7617 | Before Kant, all philosophers had a correspondence theory of truth [Putnam] |
| Full Idea: Before Kant it is impossible to find any philosopher who did not have a correspondence theory of truth. | |
| From: Hilary Putnam (Reason, Truth and History [1981], Ch.3) | |
| A reaction: I don't believe this is true of Descartes. See ideas 2266 and 4298. Truth is 'clear and distinct' conceptions, but if you enlarge (and maybe socialise) 'clear' you get coherent. Descartes firmly avoids correspondence, because he can't trust 'facts'. |
| 4716 | The correspondence theory is wrong, because there is no one correspondence between reality and fact [Putnam, by O'Grady] |
| Full Idea: Putnam argues that theory does not correspond to reality, because there are myriad correspondences possible, and we cannot single out "the" relation of correspondence. | |
| From: report of Hilary Putnam (Reason, Truth and History [1981]) by Paul O'Grady - Relativism Ch.3 | |
| A reaction: This obviously depends on views about reference and meaning. I don't see the problem in simple cases, which is all the correspondence theory needs. Complex cases, like chemistry, may well have ambiguities, but so what? |
| 7616 | Truth is an idealisation of rational acceptability [Putnam] |
| Full Idea: Truth is an idealisation of rational acceptability; we speak as if there were such things as epistemically ideal conditions, and we call a statement 'true' if it would be justified under such conditions. | |
| From: Hilary Putnam (Reason, Truth and History [1981], Ch.3) | |
| A reaction: The second part makes human beings sound stupid (which they are not), but the first part is right, and incredibly important. Peirce is behind Putnam's thought. Truth is the target of belief. It isn't a nonsense just because we can't be infallible. |
| 10170 | While true-in-a-model seems relative, true-in-all-models seems not to be [Reck/Price] |
| Full Idea: While truth can be defined in a relative way, as truth in one particular model, a non-relative notion of truth is implied, as truth in all models. | |
| From: E Reck / M Price (Structures and Structuralism in Phil of Maths [2000], §4) | |
| A reaction: [The article is actually discussing arithmetic] This idea strikes me as extremely important. True-in-all-models is usually taken to be tautological, but it does seem to give a more universal notion of truth. See semantic truth, Tarski, Davidson etc etc. |
| 10166 | ZFC set theory has only 'pure' sets, without 'urelements' [Reck/Price] |
| Full Idea: In standard ZFC ('Zermelo-Fraenkel with Choice') set theory we deal merely with pure sets, not with additional urelements. | |
| From: E Reck / M Price (Structures and Structuralism in Phil of Maths [2000], §2) | |
| A reaction: The 'urelements' would the actual objects that are members of the sets, be they physical or abstract. This idea is crucial to understanding philosophy of mathematics, and especially logicism. Must the sets exist, just as the urelements do? |
| 10175 | Three types of variable in second-order logic, for objects, functions, and predicates/sets [Reck/Price] |
| Full Idea: In second-order logic there are three kinds of variables, for objects, for functions, and for predicates or sets. | |
| From: E Reck / M Price (Structures and Structuralism in Phil of Maths [2000], §5) | |
| A reaction: It is interesting that a predicate seems to be the same as a set, which begs rather a lot of questions. For those who dislike second-order logic, there seems nothing instrinsically wicked in having variables ranging over innumerable multi-order types. |
| 14203 | Intension is not meaning, as 'cube' and 'square-faced polyhedron' are intensionally the same [Putnam] |
| Full Idea: Intension cannot be identified with meaning. ..'Cube' and 'regular polyhedron with six square faces' are logically equivalent predicates. The intension is the same (the function giving the cubes in any possible world) but there is a difference of meaning. | |
| From: Hilary Putnam (Reason, Truth and History [1981], Ch.2) |
| 14207 | If cats equal cherries, model theory allows reinterpretation of the whole language preserving truth [Putnam] |
| Full Idea: If the number of cats happens to equal the cherries, then it follows from the theory of models that there is a reinterpretation of the entire language that leaves all sentences unchanged in truth value while permuting the extensions of 'cat' and 'cherry'. | |
| From: Hilary Putnam (Reason, Truth and History [1981], Ch.2) | |
| A reaction: This horrifying result seems to come simply from the fact that there is an isomorphism between two models, which in turn seems to rest largely on the cardinality of the models. There seems to be something wrong with model theory here (?). |
| 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. |
| 14214 | If we try to cure the abundance of theories with causal links, this is 'just more theory' [Putnam, by Lewis] |
| Full Idea: If we try to base determinate reference on natural causal connection, Putnam says this is just more theory, as subject as any theory to overabundant, conflicting intended interpretations. | |
| From: report of Hilary Putnam (Reason, Truth and History [1981]) by David Lewis - Putnam's Paradox 'Why Are' | |
| A reaction: This is the 1981 Putnam, moving away from the realism that was implicit in the original causal theory of reference developed by himself and Kripke. His 'just more theory' is the slogan of Putnam's later anti-realism. |
| 14205 | The sentence 'A cat is on a mat' remains always true when 'cat' means cherry and 'mat' means tree [Putnam] |
| Full Idea: The sentence 'A cat is on a mat' can be reinterpreted so that in the actual world 'cat' refers to cherries and 'mat' refers to trees, without affecting the truth-value of the sentence in any possible world. | |
| From: Hilary Putnam (Reason, Truth and History [1981], Ch.2) | |
| A reaction: This simple suggestion is the basis of a notorious argument in favour of anti-realism. See D.Lewis's 'Putnam's Paradox'. It tracks back to Skolem's doubts about whether infinitary mathematics is possible. Putnam's conclusion sounds daft. |
| 7610 | A fact is simply what it is rational to accept [Putnam] |
| Full Idea: I propose that the only criterion for what is a fact is what it is rational to accept. | |
| From: Hilary Putnam (Reason, Truth and History [1981], Pref) | |
| A reaction: An epistemological-ontological confusion here. The concept of a fact is of something which is the case quite independently of our criteria for believing it. There are facts which are unknowable for humans. It is, of course, rational to accept facts. |
| 7618 | Very nominalistic philosophers deny properties, though scientists accept them [Putnam] |
| Full Idea: Some philosophers are so nominalistic that they would deny the existence of such entities as 'properties' altogether; but science itself does not hesitate to talk freely of properties. | |
| From: Hilary Putnam (Reason, Truth and History [1981], Ch.3) | |
| A reaction: Maybe scientists aren't very good at ontology? They talk about forces and energy, but don't seem to know what they are. I am inclined to think that we must include properties in the working ontology of humans, but not into strict physics. |
| 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. |
| 4718 | If necessity is always relative to a description in a language, then there is only 'de dicto' necessity [Putnam, by O'Grady] |
| Full Idea: Putnam endorses the view that necessity is relative to a description, so there is only necessity 'de dicto': relative to language, not to reality. | |
| From: report of Hilary Putnam (Reason, Truth and History [1981]) by Paul O'Grady - Relativism Ch.3 | |
| A reaction: Even a realist must take this proposal seriously. The facts may contain de re necessities, but we could be very sceptical about our capacity to know them. Personally I enjoy speculating about de re necessities. They can't stop you. |
| 7620 | Some kind of objective 'rightness' is a presupposition of thought itself [Putnam] |
| Full Idea: What the relativist fails to see is that it is a presupposition of thought itself that some kind of objective 'rightness' exists. | |
| From: Hilary Putnam (Reason, Truth and History [1981], Ch.5) | |
| A reaction: This may be the key objection to relativism. If you have a frame of reference, is it a good one? If you have a new perspective, is it better than your old one? Is the culture you live in confused or clear-thinking? Jokes and metaphors rely on truth. |
| 14204 | Naïve operationalism would have meanings change every time the tests change [Putnam] |
| Full Idea: On a naïve operationalist account every time a new way of testing whether a substance is really gold is discovered, the meaning and reference of 'gold' undergoes a change. | |
| From: Hilary Putnam (Reason, Truth and History [1981], Ch.2) |
| 7611 | Rationality is one part of our conception of human flourishing [Putnam] |
| Full Idea: Our notion of rationality is, at bottom, just one part of our conception of human flourishing, our idea of the good. | |
| From: Hilary Putnam (Reason, Truth and History [1981], Pref) | |
| A reaction: This looks like the beginnings of virtue epistemology, since rationality will have criteria, which would seem to be virtues. I find this idea appealing, both as a view of rationality, and as a view of the human good. |
| 14200 | 'Water' on Twin Earth doesn't refer to water, but no mental difference can account for this [Putnam] |
| Full Idea: The word 'water' used on Twin Earth refers not to water but to this other liquid (XYZ). Yet there is no relevant difference in the mental state of Twin Earth speakers and speakers on Earth (in 1750) to account for this difference of reference. | |
| From: Hilary Putnam (Reason, Truth and History [1981], Ch.2) | |
| A reaction: In this world, if you and I separately meet twins, and I think about this twin while you think about that one, our mental states are different even if they are indistinguishable. I know I'm thinking about my twin, not yours. Indexicals. |
| 7612 | Reference is social not individual, because we defer to experts when referring to elm trees [Putnam] |
| Full Idea: My concept of an elm tree is exactly the same as my concept of a beech tree (I blush to confess), which shows that the determination of reference is social and not individual - both you and I defer to experts who can tell elms from beeches. | |
| From: Hilary Putnam (Reason, Truth and History [1981], Ch.1) | |
| A reaction: If I said 'that tree looks nice' I wouldn't be deferring to experts. Nor if I said 'that tree, which I take to be an elm, looks nice'. If I am an expert I don't defer to experts. |
| 7613 | Concepts are (at least in part) abilities and not occurrences [Putnam] |
| Full Idea: Concepts are (at least in part) abilities and not occurrences. | |
| From: Hilary Putnam (Reason, Truth and History [1981], Ch.1) | |
| A reaction: This seems to be building on the idea that meaning is use, and also arises from a background of pragmatism. Perhaps a concept is an acquaintance with a node in platonic space? Lots of abilities aren't concepts, so what distinguishes the concepts? |
| 14202 | Neither individual nor community mental states fix reference [Putnam] |
| Full Idea: Mental state (in either the individualistic or the collective sense) does not fix reference. | |
| From: Hilary Putnam (Reason, Truth and History [1981], Ch.2) | |
| A reaction: The idea that communities fix reference seems to me plausible. See Tyler Burge on this. |
| 14201 | Maybe the total mental state of a language community fixes the reference of a term [Putnam] |
| Full Idea: One might concede that the reference of a person's term isn't fixed by his individual mental state, but insist that the total mental state of all the members of the language community fixes the reference of the term. | |
| From: Hilary Putnam (Reason, Truth and History [1981], Ch.2) | |
| A reaction: I like this reading of the problem, though Putnam himself prefers to say that things fix the reference. I take reference to be a human action, not a natural causal relation. Animals connecting thought to object may not count as reference at all. |
| 14206 | There are infinitely many interpretations of a sentence which can all seem to be 'correct' [Putnam] |
| Full Idea: There are always infinitely many different interpretations of the predicates of a language which assign 'correct' truth-values to the sentences in all possible worlds, no matter how those 'correct' truth-values are singled out. | |
| From: Hilary Putnam (Reason, Truth and History [1981], Ch.2) | |
| A reaction: Putnam says that he is using this argument from model theory to endorse the scepticism about 'gavagai' that Quine expressed in 1960. It is based on the ideas of Skolem, who was a renegade philosopher of mathematics. See Tim Button. |
| 7624 | The word 'inconsiderate' nicely shows the blurring of facts and values [Putnam] |
| Full Idea: The use of the word 'inconsiderate' seems to me a very fine example of the way in which the fact/value distinction is hopelessly fuzzy in the real world and in the real language. | |
| From: Hilary Putnam (Reason, Truth and History [1981]) | |
| A reaction: Interesting, but not much of an argument. What would Nietzsche say? Was Agamemnon morally deficient because we might think him 'inconsiderate'? |
| 22439 | There are only duties if there are rights, so truth is only for those with a right to it [Constant] |
| Full Idea: A duty is that on the part of one being which corresponds to the rights of another. Where there are no rights there are no duties. To tell the truth is therefore a duty, but only to the one who has the right to the truth. | |
| From: Benjamin Constant (On Political Reactions [1797], p.123), quoted by Immanuel Kant - On a supposed right to lie p.28 | |
| A reaction: We can't claim a right to have all questions answered truthfully (because there is a right to privacy), but we might claim a right not to be lied to (as long as we accept a refusal to answer). Kant rejected this idea. |
| 22440 | Unconditional truth-telling makes a society impossible [Constant] |
| Full Idea: The moral principle 'it is a duty to tell the truth' would, if taken unconditionally and singly, make any society impossible. | |
| From: Benjamin Constant (On Political Reactions [1797], p.124), quoted by Immanuel Kant - On a supposed right to lie p.28 | |
| A reaction: He gives the well known example of the murderer at the door asking if your friend is inside. Compare everyone becoming perfectly telepathic. Our society would collapse, but a new society would learn to live with it. |