40 ideas
6267 | A culture needs to admit that knowledge is more extensive than just 'science' [Putnam] |
Full Idea: The acknowledgement that the sphere of knowledge is wider than the sphere of 'science' seems to me to be a cultural necessity if we are to arrive at a sane and human view of ourselves or of science. | |
From: Hilary Putnam (Meaning and the Moral Sciences [1978], Intro) | |
A reaction: A very nice remark, with which I intuitively agree, but then you are left with the problem of explaining how something can qualify as knowledge when it can't pass the stringent tests of science. How wide to we spread, and why? |
6272 | 'True' and 'refers' cannot be made scientically precise, but are fundamental to science [Putnam] |
Full Idea: Some non-scientific knowledge is presupposed by science; for example, I have been arguing that 'refers' and 'true' cannot be made scientifically precise; yet truth is a fundamental term in logic - a precise science. | |
From: Hilary Putnam (Meaning and the Moral Sciences [1978], Lec VI) | |
A reaction: We might ask whether we 'know' what 'true' and 'refers' mean, as opposed to being able to use them. If their usage doesn't count as knowledge, then we could still end up with all actual knowledge being somehow 'scientific'. |
6276 | 'The rug is green' might be warrantedly assertible even though the rug is not green [Putnam] |
Full Idea: 'The rug is green' might be warrantedly assertible even though the rug is not green. | |
From: Hilary Putnam (Meaning and the Moral Sciences [1978], Pt Three) | |
A reaction: The word 'warranted' seems to be ambiguous in modern philosophy. See Idea 6150. There seem to be internalist and externalist versions. It seems clear to say that a belief could be well-justified and yet false. |
6266 | We need the correspondence theory of truth to understand language and science [Putnam] |
Full Idea: A correspondence theory of truth is needed to understand how language works, and how science works. | |
From: Hilary Putnam (Meaning and the Moral Sciences [1978], Intro) | |
A reaction: Putnam retreated from this position to a more pragmatic one later on, but all my sympathies are with the present view, despite being repeatedly told by modern philosophers that I am wrong. See McGinn (Idea 6085) and Searle (Idea 3508). |
6277 | Correspondence between concepts and unconceptualised reality is impossible [Putnam] |
Full Idea: The great nineteenth century argument against the correspondence theory of truth was that one cannot think of truth as correspondence to facts (or 'reality') because one would need to compare concepts directly with unconceptualised reality. | |
From: Hilary Putnam (Meaning and the Moral Sciences [1978], Pt Three) | |
A reaction: Presumably the criticism was offered by idealists, who preferred a coherence theory. The defence is to say that there is a confusion here between a concept and the contents of a concept. The contents of a concept are designed to be facts. |
6264 | In Tarski's definition, you understand 'true' if you accept the notions of the object language [Putnam] |
Full Idea: Anyone who accepts the notions of whatever object language is in question - and this can be chosen arbitrarily - can also understand 'true' as defined by Tarski for that object language. | |
From: Hilary Putnam (Meaning and the Moral Sciences [1978], Intro) | |
A reaction: Thus if we say "'snow is white' is true iff snow is white", then if you 'accept the notion' that snow is white in English, you understand what 'true' means. This seems to leave you with the meaning of 'snow is white' being its truth conditions. |
6265 | Tarski has given a correct account of the formal logic of 'true', but there is more to the concept [Putnam] |
Full Idea: What Tarski has done is to give us a perfectly correct account of the formal logic of the concept 'true', but the formal logic of the concept is not all there is to the notion of truth. | |
From: Hilary Putnam (Meaning and the Moral Sciences [1978], Intro) | |
A reaction: I find this refreshing. A lot of modern philosophers seem to think that truth is no longer an interesting philosophical topic, because deflationary accounts have sidelined it, but I take the concept to be at the heart of metaphysics. |
6269 | Only Tarski has found a way to define 'true' [Putnam] |
Full Idea: There is only one way anyone knows how to define 'true' and that is Tarski's way. | |
From: Hilary Putnam (Meaning and the Moral Sciences [1978], Lec II.5) | |
A reaction: However, Davidson wrote a paper called 'On the Folly of Trying to Define Truth', which seems to reject even Tarski. Also bear in mind Putnam's earlier remark (Idea 6265) that there is more to truth than Tarski's definition. Just take 'true' as primitive. |
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. |
6280 | Realism is a theory, which explains the convergence of science and the success of language [Putnam] |
Full Idea: Realism is an empirical theory; it explains the convergence of scientific theories, where earlier theories are often limiting cases of later theories (which is why theoretical terms preserve their reference); and it explains the success of language. | |
From: Hilary Putnam (Meaning and the Moral Sciences [1978], Pt Four) | |
A reaction: I agree. Personally, I think of Plato's Theory of Forms and all religions as empirical theories. The response from anti-realists is generally to undermine confidence in the evidence which these 'empirical theories' are said to explain. |
8508 | A 'trope' is an abstract particular, the occurrence of an essence [Williams,DC] |
Full Idea: I shall divert the word 'trope' to stand for the abstract particular which is, so to speak, the occurrence of an essence. | |
From: Donald C. Williams (On the Elements of Being: I [1953], p.115) | |
A reaction: Thus tropes entered philosophical discussion. Presumably the precedent for an 'abstract particular' would be a particular occurrence of the number 7. |
8509 | A world is completely constituted by its tropes and their connections [Williams,DC] |
Full Idea: Any possible world, and hence, of course, this one, is completely constituted by its tropes and connections of location and similarity. | |
From: Donald C. Williams (On the Elements of Being: I [1953], p.116) | |
A reaction: Note that Williams regularly referred to possible worlds in 1953. This is a full-blooded trope theory, which asserts that objects are bundles of tropes, so that both particulars and universals are ontologically taken care of. |
8510 | 'Socrates is wise' means a concurrence sum contains a member of a similarity set [Williams,DC] |
Full Idea: 'Socrates is wise' means that the concurrence sum (Socrates) includes a trope which is a member of the similarity set (Wisdom). | |
From: Donald C. Williams (On the Elements of Being: I [1953], p.119) | |
A reaction: Resemblance has to be taken as a basic (and presumably unanalysable) concept, which invites Russell's objection (Idea 4441). |
6284 | If a tautology is immune from revision, why would that make it true? [Putnam] |
Full Idea: If we held, say, 'All unmarried men are unmarried' as absolutely immune from revision, why would this make it true? | |
From: Hilary Putnam (Meaning and the Moral Sciences [1978], Pt Four) | |
A reaction: A very nice question. Like most American philosophers, Putnam accepts Quine's attack on the unrevisability of analytic truths. His point here is that defenders of analytic truths are probably desperate to preserve basic truths, but it won't work. |
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. |
6273 | Knowledge depends on believing others, which must be innate, as inferences are not strong enough [Putnam] |
Full Idea: Our ability to picture how people are likely to respond may well be innate; indeed, our disposition to believe what other people tell us (which is fundamental to knowledge) could hardly be an inference, as that isn’t good enough for knowledge. | |
From: Hilary Putnam (Meaning and the Moral Sciences [1978], Lec VI) | |
A reaction: An interesting claim. There could be an intermediate situation, which is a hard-wired non-conscious inference. When dismantled, the 'innate' brain circuits for assessing testimony could turn out to work on logic and evidence. |
6274 | Empathy may not give knowledge, but it can give plausibility or right opinion [Putnam] |
Full Idea: Empathy with others may give less than 'Knowledge', but it gives more than mere logical or physical possibility; it gives plausibility, or (to revive Platonic terminology) it provides 'right opinion'. | |
From: Hilary Putnam (Meaning and the Moral Sciences [1978], Lec VI) | |
A reaction: See Ideas 174 and 2140 for Plato. Putnam is exploring areas of knowledge outside the limits of strict science. Behind this claim seems to lie the Principle of Charity (3971), but a gang of systematic liars (e.g. evil students) would be a problem case. |
17084 | You can't decide which explanations are good if you don't attend to the interest-relative aspects [Putnam] |
Full Idea: Explanation is an interest-relative notion …explanation has to be partly a pragmatic concept. To regard the 'pragmatics' of explanation as no part of the concept is to abdicate the job of figuring out what makes an explanation good. | |
From: Hilary Putnam (Meaning and the Moral Sciences [1978], p. 41-2), quoted by David-Hillel Ruben - Explaining Explanation Ch 1 | |
A reaction: I suppose this is just obvious, depending on how far you want to take the 'interest-relative' bit. If a fool is fobbed off with a trivial explanation, there must be some non-relative criterion for assessing that. |
6282 | Theory of meaning presupposes theory of understanding and reference [Putnam] |
Full Idea: Theory of meaning presupposes theory of understanding and reference. | |
From: Hilary Putnam (Meaning and the Moral Sciences [1978], Pt Four) | |
A reaction: How can you have a theory of understanding without a meaning that requires to be understood? Personally I think about the minds of small animals when pondering this, and that seems to put reference and truth at the front of the queue. |
6281 | Truth conditions can't explain understanding a sentence, because that in turn needs explanation [Putnam] |
Full Idea: You can't treat understanding a sentence as knowing its truth conditions, because it then becomes unintelligible what that knowledge in turn consists in. | |
From: Hilary Putnam (Meaning and the Moral Sciences [1978], Pt Four) | |
A reaction: The implication, I take it, is circularity; how can you specify truth conditions if you don't understand sentences? Putnam here agrees with Dummett that verification must be involved. Something has to be taken as axiomatic in all this. |
6278 | We should reject the view that truth is prior to meaning [Putnam] |
Full Idea: I am suggesting that we reject the view that truth (based on the semantic theory) is prior to meaning. | |
From: Hilary Putnam (Meaning and the Moral Sciences [1978], Pt Three) | |
A reaction: It is a nice question which of truth or meaning has logical priority. One might start by speculating about how and why animals think. A moth attracted to flame is probably working on truth without much that could be called 'meaning'. |
6271 | How reference is specified is not what reference is [Putnam] |
Full Idea: A theory of how reference is specified isn't a theory of what reference is. | |
From: Hilary Putnam (Meaning and the Moral Sciences [1978], Lec V) | |
A reaction: A simple and important point. We may achieve reference by naming, describing, grunting or pointing, but the question is, what have we achieved when we get there? |
6268 | The claim that scientific terms are incommensurable can be blocked if scientific terms are not descriptions [Putnam] |
Full Idea: The line of reasoning of Kuhn and Feyerabend can be blocked by arguing (as I have in various places, and as Saul Kripke has) that scientific terms are not synonymous with descriptions. | |
From: Hilary Putnam (Meaning and the Moral Sciences [1978], Lec II.2) | |
A reaction: A nice clear statement of the motivation for creating the causal theory of reference. See Idea 6162. We could still go back and ask whether we could block scientific relativism by rethinking how descriptions work, instead of abandoning them. |
6279 | A private language could work with reference and beliefs, and wouldn't need meaning [Putnam] |
Full Idea: A language made up and used by a being who belonged to no community would have no need for such a concept as the 'meaning' of a term. To state the reference of each term and what the language speaker believes is to tell the whole story. | |
From: Hilary Putnam (Meaning and the Moral Sciences [1978], Pt Three) | |
A reaction: A subtle response to Wittgenstein's claim (e.g. Ideas 4152,4158), but I am not sure what Putnam means. I would have thought that beliefs had to be embodied in propositions. They may not need 'meaning' quite as urgently as sentences, but still… |
6270 | The correct translation is the one that explains the speaker's behaviour [Putnam] |
Full Idea: What it is to be a correct translation is to be the translation that best explains the behaviour of the speaker. | |
From: Hilary Putnam (Meaning and the Moral Sciences [1978], Lec III) | |
A reaction: This seems fairly close to Quine, but rather puzzlingly uses the word 'correct'. If our criteria of translation are purely behavioural, there is no way we can be correct after one word ('gavagai'), so at what point does it become 'correct'? |
6283 | Language maps the world in many ways (because it maps onto other languages in many ways) [Putnam] |
Full Idea: We could say that the language has more than one correct way of being mapped onto the world (it must, since it has more than one way of being correctly mapped onto a language which is itself correctly mapped onto the world). | |
From: Hilary Putnam (Meaning and the Moral Sciences [1978], Pt Four) | |
A reaction: This spells out nicely the significance of Quine's 'indeterminacy of translation'. Others have pointed out that the fact that language maps onto world in many ways need not be anti-realist; the world is endless, and language is limited. |
6275 | You can't say 'most speaker's beliefs are true'; in some areas this is not so, and you can't count beliefs [Putnam] |
Full Idea: The maxim that 'most of a speaker's beliefs are true' as an a priori principle governing radical translation seems to me to go too far; first, I don't know how to count beliefs; second, most people's beliefs on some topics (philosophy) are probably false. | |
From: Hilary Putnam (Meaning and the Moral Sciences [1978], Pt Three) | |
A reaction: Putnam prefers a pragmatic view, where charity is applicable if behaviour is involved. Philosophy is too purely theoretical. The extent to which Charity should apply in philosophy seminars is a nice question, which all students should test in practice. |