193 ideas
| 7623 | For ancient Greeks being wise was an ethical value [Putnam] |
| 2352 | The job of the philosopher is to distinguish facts about the world from conventions [Putnam] |
| 6782 | Realism is the only philosophy of science that doesn't make the success of science a miracle [Putnam] |
| 6267 | A culture needs to admit that knowledge is more extensive than just 'science' [Putnam] |
| 6272 | 'True' and 'refers' cannot be made scientically precise, but are fundamental to science [Putnam] |
| 6276 | 'The rug is green' might be warrantedly assertible even though the rug is not green [Putnam] |
| 4714 | Putnam's epistemic notion of truth replaces the realism of correspondence with ontological relativism [Putnam, by O'Grady] |
| 6266 | We need the correspondence theory of truth to understand language and science [Putnam] |
| 7617 | Before Kant, all philosophers had a correspondence theory of truth [Putnam] |
| 6277 | Correspondence between concepts and unconceptualised reality is impossible [Putnam] |
| 4716 | The correspondence theory is wrong, because there is no one correspondence between reality and fact [Putnam, by O'Grady] |
| 8828 | Truth is rational acceptability [Putnam] |
| 7616 | Truth is an idealisation of rational acceptability [Putnam] |
| 18951 | For scientific purposes there is a precise concept of 'true-in-L', using set theory [Putnam] |
| 6264 | In Tarski's definition, you understand 'true' if you accept the notions of the object language [Putnam] |
| 6265 | Tarski has given a correct account of the formal logic of 'true', but there is more to the concept [Putnam] |
| 6269 | Only Tarski has found a way to define 'true' [Putnam] |
| 2345 | Semantic notions do not occur in Tarski's definitions, but assessing their correctness involves translation [Putnam] |
| 2347 | Asserting the truth of an indexical statement is not the same as uttering the statement [Putnam] |
| 18953 | Modern notation frees us from Aristotle's restriction of only using two class-names in premises [Putnam] |
| 18949 | The universal syllogism is now expressed as the transitivity of subclasses [Putnam] |
| 18952 | '⊃' ('if...then') is used with the definition 'Px ⊃ Qx' is short for '¬(Px & ¬Qx)' [Putnam] |
| 15901 | Trying to represent curves, we study arbitrary functions, leading to the ordinals, which produces set theory [Cantor, by Lavine] |
| 15946 | Cantor developed sets from a progression into infinity by addition, multiplication and exponentiation [Cantor, by Lavine] |
| 9616 | A set is a collection into a whole of distinct objects of our intuition or thought [Cantor] |
| 13444 | Cantor's Theorem: for any set x, its power set P(x) has more members than x [Cantor, by Hart,WD] |
| 18098 | Cantor proved that all sets have more subsets than they have members [Cantor, by Bostock] |
| 18958 | In type theory, 'x ∈ y' is well defined only if x and y are of the appropriate type [Putnam] |
| 15505 | If a set is 'a many thought of as one', beginners should protest against singleton sets [Cantor, by Lewis] |
| 10701 | Cantor showed that supposed contradictions in infinity were just a lack of clarity [Cantor, by Potter] |
| 9944 | We understand some statements about all sets [Putnam] |
| 10865 | The continuum is the powerset of the integers, which moves up a level [Cantor, by Clegg] |
| 17831 | Cantor gives informal versions of ZF axioms as ways of getting from one set to another [Cantor, by Lake] |
| 13016 | The Axiom of Union dates from 1899, and seems fairly obvious [Cantor, by Maddy] |
| 13655 | The Löwenheim-Skolem theorems show that whether all sets are constructible is indeterminate [Putnam, by Shapiro] |
| 9915 | V = L just says all sets are constructible [Putnam] |
| 14199 | Cantor's sets were just collections, but Dedekind's were containers [Cantor, by Oliver/Smiley] |
| 18954 | Before the late 19th century logic was trivialised by not dealing with relations [Putnam] |
| 18956 | Asserting first-order validity implicitly involves second-order reference to classes [Putnam] |
| 18962 | Unfashionably, I think logic has an empirical foundation [Putnam] |
| 10066 | Putnam coined the term 'if-thenism' [Putnam, by Musgrave] |
| 18961 | We can identify functions with certain sets - or identify sets with certain functions [Putnam] |
| 17505 | Using proper names properly doesn't involve necessary and sufficient conditions [Putnam] |
| 18955 | Having a valid form doesn't ensure truth, as it may be meaningless [Putnam] |
| 14203 | Intension is not meaning, as 'cube' and 'square-faced polyhedron' are intensionally the same [Putnam] |
| 14207 | If cats equal cherries, model theory allows reinterpretation of the whole language preserving truth [Putnam] |
| 9913 | The Löwenheim-Skolem Theorem is close to an antinomy in philosophy of language [Putnam] |
| 10082 | There are infinite sets that are not enumerable [Cantor, by Smith,P] |
| 13483 | Cantor's Paradox: the power set of the universe must be bigger than the universe, yet a subset of it [Cantor, by Hart,WD] |
| 8710 | The powerset of all the cardinal numbers is required to be greater than itself [Cantor, by Friend] |
| 15910 | Cantor named the third realm between the finite and the Absolute the 'transfinite' [Cantor, by Lavine] |
| 15905 | Cantor proved the points on a plane are in one-to-one correspondence to the points on a line [Cantor, by Lavine] |
| 9983 | Cantor took the ordinal numbers to be primary [Cantor, by Tait] |
| 17798 | Cantor presented the totality of natural numbers as finite, not infinite [Cantor, by Mayberry] |
| 9971 | Cantor introduced the distinction between cardinals and ordinals [Cantor, by Tait] |
| 9892 | Cantor showed that ordinals are more basic than cardinals [Cantor, by Dummett] |
| 15911 | Ordinals are generated by endless succession, followed by a limit ordinal [Cantor, by Lavine] |
| 14136 | A cardinal is an abstraction, from the nature of a set's elements, and from their order [Cantor] |
| 15906 | Cantor tried to prove points on a line matched naturals or reals - but nothing in between [Cantor, by Lavine] |
| 11015 | Cantor's diagonal argument proved you can't list all decimal numbers between 0 and 1 [Cantor, by Read] |
| 15903 | A real is associated with an infinite set of infinite Cauchy sequences of rationals [Cantor, by Lavine] |
| 18251 | Irrational numbers are the limits of Cauchy sequences of rational numbers [Cantor, by Lavine] |
| 15902 | Irrationals and the Dedekind Cut implied infinite classes, but they seemed to have logical difficulties [Cantor, by Lavine] |
| 15908 | It was Cantor's diagonal argument which revealed infinities greater than that of the real numbers [Cantor, by Lavine] |
| 13464 | Cantor proposes that there won't be a potential infinity if there is no actual infinity [Cantor, by Hart,WD] |
| 10112 | The naturals won't map onto the reals, so there are different sizes of infinity [Cantor, by George/Velleman] |
| 15896 | Cantor needed Power Set for the reals, but then couldn't count the new collections [Cantor, by Lavine] |
| 18959 | Sets larger than the continuum should be studied in an 'if-then' spirit [Putnam] |
| 17889 | CH: An infinite set of reals corresponds 1-1 either to the naturals or to the reals [Cantor, by Koellner] |
| 13447 | Cantor: there is no size between naturals and reals, or between a set and its power set [Cantor, by Hart,WD] |
| 10883 | Cantor's Continuum Hypothesis says there is a gap between the natural and the real numbers [Cantor, by Horsten] |
| 8733 | The Continuum Hypothesis says there are no sets between the natural numbers and reals [Cantor, by Shapiro] |
| 13528 | Continuum Hypothesis: there are no sets between N and P(N) [Cantor, by Wolf,RS] |
| 9555 | Continuum Hypothesis: no cardinal greater than aleph-null but less than cardinality of the continuum [Cantor, by Chihara] |
| 15893 | Cantor's theory concerns collections which can be counted, using the ordinals [Cantor, by Lavine] |
| 18174 | Cantor extended ordinals into the transfinite, and they can thus measure infinite cardinalities [Cantor, by Maddy] |
| 18200 | Very large sets should be studied in an 'if-then' spirit [Putnam] |
| 18173 | Cardinality strictly concerns one-one correspondence, to test infinite sameness of size [Cantor, by Maddy] |
| 9937 | I do not believe mathematics either has or needs 'foundations' [Putnam] |
| 9939 | It is conceivable that the axioms of arithmetic or propositional logic might be changed [Putnam] |
| 9992 | The 'extension of a concept' in general may be quantitatively completely indeterminate [Cantor] |
| 10232 | Property extensions outstrip objects, so shortage of objects caused the Caesar problem [Cantor, by Shapiro] |
| 18176 | Pure mathematics is pure set theory [Cantor] |
| 3663 | How can you contemplate Platonic entities without causal transactions with them? [Putnam] |
| 8631 | Cantor says that maths originates only by abstraction from objects [Cantor, by Frege] |
| 9940 | Maybe mathematics is empirical in that we could try to change it [Putnam] |
| 9914 | It is unfashionable, but most mathematical intuitions come from nature [Putnam] |
| 9941 | Science requires more than consistency of mathematics [Putnam] |
| 18199 | Indispensability strongly supports predicative sets, and somewhat supports impredicative sets [Putnam] |
| 8857 | We must quantify over numbers for science; but that commits us to their existence [Putnam] |
| 6280 | Realism is a theory, which explains the convergence of science and the success of language [Putnam] |
| 17644 | Metaphysical realism is committed to there being one ultimate true theory [Putnam] |
| 2349 | Realists believe truth is correspondence, independent of humans, is bivalent, and is unique [Putnam] |
| 9943 | You can't deny a hypothesis a truth-value simply because we may never know it! [Putnam] |
| 22181 | Putnam says anti-realism is a bad explanation of accurate predictions [Putnam, by Okasha] |
| 14214 | If we try to cure the abundance of theories with causal links, this is 'just more theory' [Putnam, by Lewis] |
| 17648 | It is an illusion to think there could be one good scientific theory of reality [Putnam] |
| 14205 | The sentence 'A cat is on a mat' remains always true when 'cat' means cherry and 'mat' means tree [Putnam] |
| 7610 | A fact is simply what it is rational to accept [Putnam] |
| 7618 | Very nominalistic philosophers deny properties, though scientists accept them [Putnam] |
| 18957 | Nominalism only makes sense if it is materialist [Putnam] |
| 2351 | Aristotle says an object (e.g. a lamp) has identity if its parts stay together when it is moved [Putnam] |
| 18950 | Physics is full of non-physical entities, such as space-vectors [Putnam] |
| 17643 | Shape is essential relative to 'statue', but not essential relative to 'clay' [Putnam] |
| 11908 | Putnam bases essences on 'same kind', but same kinds may not share properties [Mackie,P on Putnam] |
| 18890 | Putnam smuggles essentialism about liquids into his proof that water must be H2O [Salmon,N on Putnam] |
| 4718 | If necessity is always relative to a description in a language, then there is only 'de dicto' necessity [Putnam, by O'Grady] |
| 10269 | Mathematics eliminates possibility, as being simultaneous actuality in sets [Putnam] |
| 9169 | A statement can be metaphysically necessary and epistemologically contingent [Putnam] |
| 5819 | Conceivability is no proof of possibility [Putnam] |
| 19542 | It is nonsense that understanding does not involve knowledge; to understand, you must know [Dougherty/Rysiew] |
| 19543 | To grasp understanding, we should be more explicit about what needs to be known [Dougherty/Rysiew] |
| 19541 | Rather than knowledge, our epistemic aim may be mere true belief, or else understanding and wisdom [Dougherty/Rysiew] |
| 6284 | If a tautology is immune from revision, why would that make it true? [Putnam] |
| 17642 | The old view that sense data are independent of mind is quite dotty [Putnam] |
| 19540 | Don't confuse justified belief with justified believers [Dougherty/Rysiew] |
| 19539 | If knowledge is unanalysable, that makes justification more important [Dougherty/Rysiew] |
| 6273 | Knowledge depends on believing others, which must be innate, as inferences are not strong enough [Putnam] |
| 6274 | Empathy may not give knowledge, but it can give plausibility or right opinion [Putnam] |
| 7620 | Some kind of objective 'rightness' is a presupposition of thought itself [Putnam] |
| 18960 | Most predictions are uninteresting, and are only sought in order to confirm a theory [Putnam] |
| 17508 | Science aims at truth, not at 'simplicity' [Putnam] |
| 14204 | Naïve operationalism would have meanings change every time the tests change [Putnam] |
| 17084 | You can't decide which explanations are good if you don't attend to the interest-relative aspects [Putnam] |
| 7705 | The Twin Earth theory suggests that intentionality is independent of qualia [Jacquette on Putnam] |
| 2590 | Dispositions need mental terms to define them [Putnam] |
| 3460 | Superactors and superspartans count against behaviourism [Putnam, by Searle] |
| 2591 | Total paralysis would mean that there were mental states but no behaviour at all [Putnam] |
| 2588 | Is pain a functional state of a complete organism? [Putnam] |
| 2589 | Functionalism is compatible with dualism, as pure mind could perform the functions [Putnam] |
| 2592 | Functional states correlate with AND explain pain behaviour [Putnam] |
| 5495 | Instances of pain are physical tokens, but the nature of pain is more abstract [Putnam, by Lycan] |
| 2331 | Functionalism says robots and people are the same at one level of abstraction [Putnam] |
| 2071 | If concepts have external meaning, computational states won't explain psychology [Putnam] |
| 2332 | Functionalism can't explain reference and truth, which are needed for logic [Putnam] |
| 2348 | Is there just one computational state for each specific belief? [Putnam] |
| 2587 | Temperature is mean molecular kinetic energy, but they are two different concepts [Putnam] |
| 2344 | If we are going to eliminate folk psychology, we must also eliminate folk logic [Putnam] |
| 6376 | Neuroscience does not support multiple realisability, and tends to support identity [Polger on Putnam] |
| 2330 | If humans and molluscs both feel pain, it can't be a single biological state [Putnam, by Kim] |
| 2074 | Can we give a scientific, computational account of folk psychology? [Putnam] |
| 7611 | Rationality is one part of our conception of human flourishing [Putnam] |
| 2605 | If everything uses mentalese, ALL concepts must be innate! [Putnam] |
| 2606 | No machine language can express generalisations [Putnam] |
| 4099 | If Twins talking about 'water' and 'XYZ' have different thoughts but identical heads, then thoughts aren't in the head [Putnam, by Crane] |
| 12026 | We say ice and steam are different forms of water, but not that they are different forms of H2O [Forbes,G on Putnam] |
| 3208 | Does 'water' mean a particular substance that was 'dubbed'? [Putnam, by Rey] |
| 14200 | 'Water' on Twin Earth doesn't refer to water, but no mental difference can account for this [Putnam] |
| 2343 | Reference may be different while mental representation is the same [Putnam] |
| 9168 | I can't distinguish elm trees, but I mean by 'elm' the same set of trees as everybody else [Putnam] |
| 5820 | 'Water' has an unnoticed indexical component, referring to stuff around here [Putnam] |
| 7612 | Reference is social not individual, because we defer to experts when referring to elm trees [Putnam] |
| 8715 | Infinities expand the bounds of the conceivable; we explore concepts to explore conceivability [Cantor, by Friend] |
| 7613 | Concepts are (at least in part) abilities and not occurrences [Putnam] |
| 13454 | Cantor says (vaguely) that we abstract numbers from equal sized sets [Hart,WD on Cantor] |
| 9145 | We form the image of a cardinal number by a double abstraction, from the elements and from their order [Cantor] |
| 6282 | Theory of meaning presupposes theory of understanding and reference [Putnam] |
| 2346 | Meaning and translation (which are needed to define truth) both presuppose the notion of reference [Putnam] |
| 6281 | Truth conditions can't explain understanding a sentence, because that in turn needs explanation [Putnam] |
| 6278 | We should reject the view that truth is prior to meaning [Putnam] |
| 2354 | "Meaning is use" is not a definition of meaning [Putnam] |
| 2336 | Holism seems to make fixed definition more or less impossible [Putnam] |
| 2334 | Meaning holism tried to show that you can't get fixed meanings built out of observation terms [Putnam] |
| 2335 | Understanding a sentence involves background knowledge and can't be done in isolation [Putnam] |
| 6271 | How reference is specified is not what reference is [Putnam] |
| 2340 | We should separate how the reference of 'gold' is fixed from its conceptual content [Putnam] |
| 2341 | Like names, natural kind terms have their meaning fixed by extension and reference [Putnam] |
| 17506 | I now think reference by the tests of experts is a special case of being causally connected [Putnam] |
| 2338 | Reference (say to 'elms') is a social phenomenon which we can leave to experts [Putnam] |
| 14202 | Neither individual nor community mental states fix reference [Putnam] |
| 9170 | We need to recognise the contribution of society and of the world in determining reference [Putnam] |
| 14201 | Maybe the total mental state of a language community fixes the reference of a term [Putnam] |
| 2339 | Aristotle implies that we have the complete concepts of a language in our heads, but we don't [Putnam] |
| 3893 | Often reference determines sense, and not (as Frege thought) vice versa [Putnam, by Scruton] |
| 6268 | The claim that scientific terms are incommensurable can be blocked if scientific terms are not descriptions [Putnam] |
| 19538 | Entailment is modelled in formal semantics as set inclusion (where 'mammals' contains 'cats') [Dougherty/Rysiew] |
| 5817 | Language is more like a cooperative steamship than an individual hammer [Putnam] |
| 6279 | A private language could work with reference and beliefs, and wouldn't need meaning [Putnam] |
| 6270 | The correct translation is the one that explains the speaker's behaviour [Putnam] |
| 6283 | Language maps the world in many ways (because it maps onto other languages in many ways) [Putnam] |
| 14206 | There are infinitely many interpretations of a sentence which can all seem to be 'correct' [Putnam] |
| 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] |
| 7624 | The word 'inconsiderate' nicely shows the blurring of facts and values [Putnam] |
| 11191 | The hidden structure of a natural kind determines membership in all possible worlds [Putnam] |
| 17507 | Natural kind stereotypes are 'strong' (obvious, like tiger) or 'weak' (obscure, like molybdenum) [Putnam] |
| 11904 | Express natural kinds as a posteriori predicate connections, not as singular terms [Putnam, by Mackie,P] |
| 2342 | "Water" is a natural kind term, but "H2O" is a description [Putnam] |
| 17645 | An alien might think oxygen was the main cause of a forest fire [Putnam] |
| 11192 | If causes are the essence of diseases, then disease is an example of a relational essence [Putnam, by Williams,NE] |
| 11190 | Archimedes meant by 'gold' the hidden structure or essence of the stuff [Putnam] |
| 5818 | If water is H2O in the actual world, there is no possible world where it isn't H2O [Putnam] |
| 10863 | Cantor proved that three dimensions have the same number of points as one dimension [Cantor, by Clegg] |
| 13465 | Only God is absolutely infinite [Cantor, by Hart,WD] |