96 ideas
| 11006 | Russell started a whole movement in philosophy by providing an analysis of descriptions [Read on Russell] |
| Full Idea: Russell started a whole movement in philosophy by providing an analysis of descriptions. | |
| From: comment on Bertrand Russell (On Denoting [1905]) by Stephen Read - Thinking About Logic Ch.5 |
| 6564 | To affirm 'p and not-p' is to have mislearned 'and' or 'not' [Quine] |
| Full Idea: To affirm a compound of the form 'p and not-p' is just to have mislearned one or both of these particles. | |
| From: Willard Quine (From Stimulus to Science [1995], p.23), quoted by Robert Fogelin - Walking the Tightrope of Reason Ch.1 | |
| A reaction: Quoted by Fogelin. This summarises the view of logic developed by the young Wittgenstein, that logical terms are 'operators', rather than referring terms. Of course the speaker may have a compartmentalised mind, or not understand 'p' properly. |
| 18137 | Impredicative definitions are wrong, because they change the set that is being defined? [Bostock] |
| Full Idea: Poincaré suggested that what is wrong with an impredicative definition is that it allows the set defined to alter its composition as more sets are added to the theory. | |
| From: David Bostock (Philosophy of Mathematics [2009], 8.3) |
| 18122 | Classical interdefinitions of logical constants and quantifiers is impossible in intuitionism [Bostock] |
| Full Idea: None of the classical ways of defining one logical constant in terms of others is available in intuitionist logic (and this includes the two quantifiers). | |
| From: David Bostock (Philosophy of Mathematics [2009], 7.2) |
| 18114 | There is no single agreed structure for set theory [Bostock] |
| Full Idea: There is so far no agreed set of axioms for set theory which is categorical, i.e. which does pick just one structure. | |
| From: David Bostock (Philosophy of Mathematics [2009], 6.4) | |
| A reaction: This contrasts with Peano Arithmetic, which is categorical in its second-order version. |
| 18107 | A 'proper class' cannot be a member of anything [Bostock] |
| Full Idea: A 'proper class' cannot be a member of anything, neither of a set nor of another proper class. | |
| From: David Bostock (Philosophy of Mathematics [2009], 5.4) |
| 18115 | We could add axioms to make sets either as small or as large as possible [Bostock] |
| Full Idea: We could add the axiom that all sets are constructible (V = L), making the universe of sets as small as possible, or add the axiom that there is a supercompact cardinal (SC), making the universe as large as we no know how to. | |
| From: David Bostock (Philosophy of Mathematics [2009], 6.4) | |
| A reaction: Bostock says most mathematicians reject the first option, and are undecided about the second option. |
| 18139 | The Axiom of Choice relies on reference to sets that we are unable to describe [Bostock] |
| Full Idea: The usual accounts of ZF are not restricted to subsets that we can describe, and that is what justifies the axiom of choice. | |
| From: David Bostock (Philosophy of Mathematics [2009], 8.4 n36) | |
| A reaction: This contrasts interestingly with predicativism, which says we can only discuss things which we can describe or define. Something like verificationism hovers in the background. |
| 18105 | Replacement enforces a 'limitation of size' test for the existence of sets [Bostock] |
| Full Idea: The Axiom of Replacement (or the Axiom of Subsets, 'Aussonderung', Fraenkel 1922) in effect enforces the idea that 'limitation of size' is a crucial factor when deciding whether a proposed set or does not not exist. | |
| From: David Bostock (Philosophy of Mathematics [2009], 5.4) |
| 18108 | First-order logic is not decidable: there is no test of whether any formula is valid [Bostock] |
| Full Idea: First-order logic is not decidable. That is, there is no test which can be applied to any arbitrary formula of that logic and which will tell one whether the formula is or is not valid (as proved by Church in 1936). | |
| From: David Bostock (Philosophy of Mathematics [2009], 5.5) |
| 18109 | The completeness of first-order logic implies its compactness [Bostock] |
| Full Idea: From the fact that the usual rules for first-level logic are complete (as proved by Gödel 1930), it follows that this logic is 'compact'. | |
| From: David Bostock (Philosophy of Mathematics [2009], 5.5) | |
| A reaction: The point is that the completeness requires finite proofs. |
| 18944 | Russell's theories aim to preserve excluded middle (saying all sentences are T or F) [Sawyer on Russell] |
| Full Idea: Russell's account of names and definite descriptions was concerned to preserve the law of excluded middle, according to which every sentence is either true or false (but it is not obvious that the law ought to be preserved). | |
| From: comment on Bertrand Russell (On Denoting [1905]) by Sarah Sawyer - Empty Names 3 | |
| A reaction: That is the strongest form of excluded middle, but things work better if every sentence is either 'true' or 'not true', leaving it open whether 'not true' actually means 'false'. |
| 7758 | 'Elizabeth = Queen of England' is really a predication, not an identity-statement [Russell, by Lycan] |
| Full Idea: On Russell's view 'Elizabeth II = Queen of England' is only superficially an identity-statement; really it is a predication, and attributes a complex relational property to Elizabeth. | |
| From: report of Bertrand Russell (On Denoting [1905]) by William Lycan - Philosophy of Language Ch.1 | |
| A reaction: The original example is 'Scott = author of Waverley'. Why can't such statements be identities, in which the reference of one half of the identity is not yet known? 'The murderer is violent' and 'Smith is violent' suggests 'Smith is the murderer'. |
| 5772 | The idea of a variable is fundamental [Russell] |
| Full Idea: I take the notion of the variable as fundamental. | |
| From: Bertrand Russell (On Denoting [1905], p.42) | |
| A reaction: A key idea of twentieth century philosophy, derived from Frege and handed on to Quine. A universal term, such as 'horse', is a variable, for which any particular horse can be its value. You can calculate using x, and generalise about horses. |
| 18941 | Names don't have a sense, but are disguised definite descriptions [Russell, by Sawyer] |
| Full Idea: Russell proposed that names do not express a Fregean sense, ...but are disguised definite descriptions, of the form 'the F'. | |
| From: report of Bertrand Russell (On Denoting [1905]) by Sarah Sawyer - Empty Names 3 | |
| A reaction: Of course, Russell then has a famous theory about definite descriptions, which turns them into quantifications. |
| 4945 | Russell says names are not denotations, but definite descriptions in disguise [Russell, by Kripke] |
| Full Idea: Russell (and Frege) thought that Mill was wrong about names: really a proper name, properly used, simply was a definite description abbreviated or disguised. | |
| From: report of Bertrand Russell (On Denoting [1905]) by Saul A. Kripke - Naming and Necessity lectures Lecture 1 | |
| A reaction: It is tempting to oversimplify this issue, one way or the other, but essentially one has to agree with Kripke that naming does not inherently involve description, but is a 'baptism', without initial content. Connotations and descriptions accrue to a name. |
| 18942 | Russell says a name contributes a complex of properties, rather than an object [Russell, by Sawyer] |
| Full Idea: Russell's view of names, understood as a definite description, which is understood as a quantificational phrase, is not to contribute an object to propositions, but to contribute a complex of properties. | |
| From: report of Bertrand Russell (On Denoting [1905]) by Sarah Sawyer - Empty Names 3 | |
| A reaction: This seems to contradict the role of constants in first-logic, which are the paradigm names, picking out an object in the domain. Kripke says names and definite descriptions have different modal profiles. |
| 7745 | Are names descriptions, if the description is unknown, false, not special, or contains names? [McCullogh on Russell] |
| Full Idea: Russell's proposal that a natural name is an abbreviated description invites four objections: not all speakers can produce descriptions; the description could be false; no one description seems special; and descriptions usually contain names. | |
| From: comment on Bertrand Russell (On Denoting [1905]) by Gregory McCullogh - The Game of the Name 8.74 | |
| A reaction: The best reply on behalf of Russell is probably to concede all of these points, but deny that any of them are fatal. Most replies will probably say that they are possible true descriptions, rather than actual limited, confused or false ones. |
| 10449 | Logically proper names introduce objects; definite descriptions introduce quantifications [Russell, by Bach] |
| Full Idea: For Russell, a logically proper name introduces its referent into the proposition, whereas a description introduces a certain quantificational structure, not its denotation. | |
| From: report of Bertrand Russell (On Denoting [1905]) by Kent Bach - What Does It Take to Refer? 22.2 L0 | |
| A reaction: I have very strong resistance to the idea that the actual referent could ever become part of a proposition. I am not, and never have been, part of a proposition! Russell depended on narrow 'acquaintance', which meant that few things qualified. |
| 15159 | The meaning of a logically proper name is its referent, but most names are not logically proper [Russell, by Soames] |
| Full Idea: Russell defined a logically proper name to be one the meaning of which is its referent. However, his internalist epistemology led him to deny that the words we ordinarily call names are logically proper. | |
| From: report of Bertrand Russell (On Denoting [1905]) by Scott Soames - Philosophy of Language 1.25 |
| 2612 | Russell rewrote singular term names as predicates [Russell, by Ayer] |
| Full Idea: Russell's theory used quantification to eliminate singular terms, which could be meaningful without denoting anything. He reparsed such sentences so they appeared as predicates instead of names. | |
| From: report of Bertrand Russell (On Denoting [1905]) by A.J. Ayer - The Central Questions of Philosophy IX.A.2 |
| 7757 | "Nobody" is not a singular term, but a quantifier [Russell, by Lycan] |
| Full Idea: Though someone just beginning to learn English might take it as one, "nobody" is not a singular term, but a quantifier. | |
| From: report of Bertrand Russell (On Denoting [1905]) by William Lycan - Philosophy of Language Ch.1 | |
| A reaction: If someone replies to "nobody's there" with "show him to me!", presumably it IS a singular term - just one that doesn't work very well. If you want to get on in life, treat it as a quantifier; if you just want to have fun... |
| 18943 | Russell implies that all sentences containing empty names are false [Sawyer on Russell] |
| Full Idea: Russell's account implies that all sentences composed of an empty name and a predicate are false, including 'Pegasus was a mythical creature'. | |
| From: comment on Bertrand Russell (On Denoting [1905]) by Sarah Sawyer - Empty Names 4 | |
| A reaction: Russell insists that such sentences contain a concealed existence claim, which they clearly don't. |
| 6411 | Critics say definite descriptions can refer, and may not embody both uniqueness and existence claims [Grayling on Russell] |
| Full Idea: The main objections to Russell's theory of descriptions are to say that definite descriptions sometime are referring expressions, and disputing the claim that definite descriptions embody both uniqueness and existence claims. | |
| From: comment on Bertrand Russell (On Denoting [1905]) by A.C. Grayling - Russell Ch.2 | |
| A reaction: The first one seems particularly correct, as you can successfully refer with a false description. See Colin McGinn (Idea 6067) for criticism of the existence claim made by the so-called 'existential' quantifier. |
| 10433 | Definite descriptions fail to refer in three situations, so they aren't essentially referring [Russell, by Sainsbury] |
| Full Idea: Russell's reasons for saying that definite descriptions are not referring expressions are: some definite descriptions have no referent, and they cannot be referring when used in negative existential truths, or in informative identity sentences. | |
| From: report of Bertrand Russell (On Denoting [1905]) by Mark Sainsbury - The Essence of Reference 18.5 | |
| A reaction: The idea is that by 'parity of form', if they aren't referring in these situations, they aren't really referring in others. Sainsbury notes that if there are two different forms of definite description (referential and attributive) these arguments fail. |
| 1608 | The theory of descriptions eliminates the name of the entity whose existence was presupposed [Russell, by Quine] |
| Full Idea: When a statement of being or non-being is analysed by Russell's theory of descriptions it ceases to contain any expression which even purports to name the alleged entity, so the being of such an entity is no longer presupposed. | |
| From: report of Bertrand Russell (On Denoting [1905]) by Willard Quine - On What There Is p.6 |
| 7754 | Russell's theory explains non-existents, negative existentials, identity problems, and substitutivity [Russell, by Lycan] |
| Full Idea: Russell showed that his theory of definite descriptions affords solutions to each of four vexing logical problems: the Problems of Apparent Reference to Non-existents and Negative existentials, Frege's Puzzle about Identity, and Substitutivity. | |
| From: report of Bertrand Russell (On Denoting [1905]) by William Lycan - Philosophy of Language 2.Over | |
| A reaction: You must seek elsewhere for the explanations of the four problems, but this gives some indication of why Russell's theory was famous, and was felt to be a breakthrough in explaining logical forms. |
| 21529 | Russell showed how to define 'the', and thereby reduce the ontology of logic [Russell, by Lackey] |
| Full Idea: With the devices of the Theory of Descriptions at hand, it was no longer necessary to take 'the' as indefinable, and it was possible to diminish greatly the number of entities to which a logical system is ontologically committed. | |
| From: report of Bertrand Russell (On Denoting [1905]) by Douglas Lackey - Intros to Russell's 'Essays in Analysis' p.13 | |
| A reaction: Illuminating, because it shows that ontology is what drove Russell at this time, and really they were all searching for Quine's 'desert landscapes', which minimalise commitment. |
| 6333 | The theory of definite descriptions reduces the definite article 'the' to the concepts of predicate logic [Russell, by Horwich] |
| Full Idea: Russell's theory of definite descriptions reduces the definite article 'the' to the notions of predicate logic - specifically, 'some', 'every', and 'same as'. | |
| From: report of Bertrand Russell (On Denoting [1905]) by Paul Horwich - Truth (2nd edn) Ch.2.7 | |
| A reaction: This helpfully clarifies Russell's project - to find the logical form of every sentence, expressed in terms which are strictly defined and consistent. This huge project now looks rather too optimistic. Artificial Intelligence would love to complete it. |
| 6412 | Russell implies that 'the baby is crying' is only true if the baby is unique [Grayling on Russell] |
| Full Idea: Russell's analysis of 'the baby is crying' seems to imply that this can only be true if there is just one baby in the world; ..to dispose of the objection, it seems necessary to appeal implicitly or explicitly to a 'domain of discourse'. | |
| From: comment on Bertrand Russell (On Denoting [1905]) by A.C. Grayling - Russell Ch.2 | |
| A reaction: This objection leads to ordinary language philosophy, and the 'pragmatics' of language. It is standard in modern predicate logic to specify the domain over which an expression is quantified. |
| 7743 | Russell explained descriptions with quantifiers, where Frege treated them as names [Russell, by McCullogh] |
| Full Idea: Russell proposed that descriptions be treated along with the quantifiers, which departs from Frege, who treated descriptions as proper names. ...the problem was that names invoke objects, and there is no object in failed descriptions. | |
| From: report of Bertrand Russell (On Denoting [1905]) by Gregory McCullogh - The Game of the Name 2.16 | |
| A reaction: Maybe we just allow intentional objects (such as unicorns) into our ontology? Producing a parsimonious ontology seems to be the main motivation of most philosophy of language. Or maybe names are just not committed to actual existence? |
| 7310 | Russell avoids non-existent objects by denying that definite descriptions are proper names [Russell, by Miller,A] |
| Full Idea: Russell attempted to avoid Meinong's strategy (of saying 'The present King of France' refers to a 'non-existent object') by denying that definite descriptions are proper names. | |
| From: report of Bertrand Russell (On Denoting [1905]) by Alexander Miller - Philosophy of Language 2.7 | |
| A reaction: Russell claimed that there was a covert existence claim built into a definite description. What about descriptions in known counterfactual situations ('Queen of the Fairies')? |
| 12006 | Denying definite description sentences are subject-predicate in form blocks two big problems [Russell, by Forbes,G] |
| Full Idea: Since Russell did not want to introduce non-existent objects, or declare many sentences meaningless, he prevented the problem from getting started, by denying that 'the present King of France is bald' is really a subject-predicate sentence. | |
| From: report of Bertrand Russell (On Denoting [1905]) by Graeme Forbes - The Metaphysics of Modality 4.1 |
| 4569 | Russell says apparent referring expressions are really assertions about properties [Russell, by Cooper,DE] |
| Full Idea: Russell's theory says that sentences which apparently serve to refer to particulars are really assertions about properties. | |
| From: report of Bertrand Russell (On Denoting [1905]) by David E. Cooper - Philosophy and the Nature of Language §4.1 | |
| A reaction: Right. Which is why particulars get marginalised in Russell, and universals take centre stage. I can't help suspecting that talk of de re/de dicto reference handles this problem better. |
| 11009 | Russell's theory must be wrong if it says all statements about non-existents are false [Read on Russell] |
| Full Idea: Russell's theory makes an exciting distinction between logical and grammatical form, but any theory which says that every positive statement, without distinction, about objects which don't exist is false, has to be wrong. | |
| From: comment on Bertrand Russell (On Denoting [1905]) by Stephen Read - Thinking About Logic Ch.5 |
| 21549 | The theory of descriptions lacks conventions for the scope of quantifiers [Lackey on Russell] |
| Full Idea: Some logicians charge that the theory of descriptions as it stands is formally inadequate because it lacks explicit conventions for the scope of quantifiers, and that when these conventions are added the theory becomes unduly complex. | |
| From: comment on Bertrand Russell (On Denoting [1905]) by Douglas Lackey - Intros to Russell's 'Essays in Analysis' p.97 | |
| A reaction: [Especially in modal contexts, apparently] I suppose if the main point is to spell out the existence commitments of the description, then that has to include quantification, for full generality. |
| 12796 | Non-count descriptions don't threaten Russell's theory, which is only about singulars [Laycock on Russell] |
| Full Idea: It is sometimes claimed that the behaviour of definite non-count descriptions shows Russell's Theory of Descriptions itself to be false. ....but it isn't a general theory of descriptions, but precisely a theory of singular descriptions. | |
| From: comment on Bertrand Russell (On Denoting [1905]) by Henry Laycock - Words without Objects 3.1 |
| 7532 | Denoting is crucial in Russell's account of mathematics, for identifying classes [Russell, by Monk] |
| Full Idea: Denoting phrases are central to mathematics, especially in Russell's 'logicist' theory, in which they are crucial to identifying classes ('the class of all mortal beings', 'the class of natural numbers'). | |
| From: report of Bertrand Russell (On Denoting [1905]) by Ray Monk - Bertrand Russell: Spirit of Solitude Ch.6 | |
| A reaction: This explains the motivation for Russell's theory of definite descriptions, since he thinks reference is achieved by description. Russell nearly achieved an extremely complete philosophical system. |
| 11988 | Russell's analysis means molecular sentences are ambiguous over the scope of the description [Kaplan on Russell] |
| Full Idea: Russell's analysis of sentences containing definite descriptions has as an immediate consequence the doctrine that molecular sentences containing definite descriptions are syntactically ambiguous as regards the scope of the definite description. | |
| From: comment on Bertrand Russell (On Denoting [1905]) by David Kaplan - How to Russell a Frege-Church I | |
| A reaction: Presumably this is a virtue of Russell's account, and an advert for analytic philosophy, because it reveals an ambiguity which was there all the time. |
| 6061 | Existence is entirely expressed by the existential quantifier [Russell, by McGinn] |
| Full Idea: Nowadays Russell's position is routinely put by saying that existence is what is expressed by the existential quantifier and only by that. | |
| From: report of Bertrand Russell (On Denoting [1905]) by Colin McGinn - Logical Properties Ch.2 | |
| A reaction: We must keep separate how you express existence, and what it is. Quantifiers seem only to be a style of expressing existence; they don't offer any insight into what existence actually is, or what we mean by 'exist'. McGinn dislikes quantifiers. |
| 18123 | Substitutional quantification is just standard if all objects in the domain have a name [Bostock] |
| Full Idea: Substitutional quantification and quantification understood in the usual 'ontological' way will coincide when every object in the (ontological) domain has a name. | |
| From: David Bostock (Philosophy of Mathematics [2009], 7.3 n23) |
| 18120 | The Deduction Theorem is what licenses a system of natural deduction [Bostock] |
| Full Idea: The Deduction Theorem is what licenses a system of 'natural deduction' in the first place. | |
| From: David Bostock (Philosophy of Mathematics [2009], 7.2) |
| 18125 | Berry's Paradox considers the meaning of 'The least number not named by this name' [Bostock] |
| Full Idea: Berry's Paradox can be put in this form, by considering the alleged name 'The least number not named by this name'. | |
| From: David Bostock (Philosophy of Mathematics [2009], 8.1) |
| 18101 | Each addition changes the ordinality but not the cardinality, prior to aleph-1 [Bostock] |
| Full Idea: If you add to the ordinals you produce many different ordinals, each measuring the length of the sequence of ordinals less than it. They each have cardinality aleph-0. The cardinality eventually increases, but we can't say where this break comes. | |
| From: David Bostock (Philosophy of Mathematics [2009], 4.5) |
| 18100 | ω + 1 is a new ordinal, but its cardinality is unchanged [Bostock] |
| Full Idea: If we add ω onto the end of 0,1,2,3,4..., it then has a different length, of ω+1. It has a different ordinal (since it can't be matched with its first part), but the same cardinal (since adding 1 makes no difference). | |
| From: David Bostock (Philosophy of Mathematics [2009], 4.5) | |
| A reaction: [compressed] The ordinals and cardinals coincide up to ω, but this is the point at which they come apart. |
| 18102 | A cardinal is the earliest ordinal that has that number of predecessors [Bostock] |
| Full Idea: It is the usual procedure these days to identify a cardinal number with the earliest ordinal number that has that number of predecessors. | |
| From: David Bostock (Philosophy of Mathematics [2009], 4.5) | |
| A reaction: This sounds circular, since you need to know the cardinal in order to decide which ordinal is the one you want, but, hey, what do I know? |
| 18106 | Aleph-1 is the first ordinal that exceeds aleph-0 [Bostock] |
| Full Idea: The cardinal aleph-1 is identified with the first ordinal to have more than aleph-0 members, and so on. | |
| From: David Bostock (Philosophy of Mathematics [2009], 5.4) | |
| A reaction: That is, the succeeding infinite ordinals all have the same cardinal number of members (aleph-0), until the new total is triggered (at the number of the reals). This is Continuum Hypothesis territory. |
| 18095 | Instead of by cuts or series convergence, real numbers could be defined by axioms [Bostock] |
| Full Idea: In addition to cuts, or converging series, Cantor suggests we can simply lay down a set of axioms for the real numbers, and this can be done without any explicit mention of the rational numbers [note: the axioms are those for a complete ordered field]. | |
| From: David Bostock (Philosophy of Mathematics [2009], 4.4) | |
| A reaction: It is interesting when axioms are best, and when not. Set theory depends entirely on axioms. Horsten and Halbach are now exploring treating truth as axiomatic. You don't give the 'nature' of the thing - just rules for its operation. |
| 18099 | The number of reals is the number of subsets of the natural numbers [Bostock] |
| Full Idea: It is not difficult to show that the number of the real numbers is the same as the number of all the subsets of the natural numbers. | |
| From: David Bostock (Philosophy of Mathematics [2009], 4.5) | |
| A reaction: The Continuum Hypothesis is that this is the next infinite number after the number of natural numbers. Why can't there be a number which is 'most' of the subsets of the natural numbers? |
| 18093 | For Eudoxus cuts in rationals are unique, but not every cut makes a real number [Bostock] |
| Full Idea: As Eudoxus claimed, two distinct real numbers cannot both make the same cut in the rationals, for any two real numbers must be separated by a rational number. He did not say, though, that for every such cut there is a real number that makes it. | |
| From: David Bostock (Philosophy of Mathematics [2009], 4.4) | |
| A reaction: This is in Bostock's discussion of Dedekind's cuts. It seems that every cut is guaranteed to produce a real. Fine challenges the later assumption. |
| 18110 | Infinitesimals are not actually contradictory, because they can be non-standard real numbers [Bostock] |
| Full Idea: Non-standard natural numbers will yield non-standard rational and real numbers. These will include reciprocals which will be closer to 0 than any standard real number. These are like 'infinitesimals', so that notion is not actually a contradiction. | |
| From: David Bostock (Philosophy of Mathematics [2009], 5.5) |
| 18156 | Modern axioms of geometry do not need the real numbers [Bostock] |
| Full Idea: A modern axiomatisation of geometry, such as Hilbert's (1899), does not need to claim the existence of real numbers anywhere in its axioms. | |
| From: David Bostock (Philosophy of Mathematics [2009], 9.B.5.ii) | |
| A reaction: This is despite the fact that geometry is reduced to algebra, and the real numbers are the equivalent of continuous lines. Bostock votes for a Greek theory of proportion in this role. |
| 18097 | The Peano Axioms describe a unique structure [Bostock] |
| Full Idea: The Peano Axioms are categorical, meaning that they describe a unique structure. | |
| From: David Bostock (Philosophy of Mathematics [2009], 4.4 n20) | |
| A reaction: So if you think there is nothing more to the natural numbers than their structure, then the Peano Axioms give the essence of arithmetic. If you think that 'objects' must exist to generate a structure, there must be more to the numbers. |
| 18148 | Hume's Principle is a definition with existential claims, and won't explain numbers [Bostock] |
| Full Idea: Hume's Principle will not do as an implicit definition because it makes a positive claim about the size of the universe (which no mere definition can do), and because it does not by itself explain what the numbers are. | |
| From: David Bostock (Philosophy of Mathematics [2009], 9.A.2) |
| 18145 | Many things will satisfy Hume's Principle, so there are many interpretations of it [Bostock] |
| Full Idea: Hume's Principle gives a criterion of identity for numbers, but it is obvious that many other things satisfy that criterion. The simplest example is probably the numerals (in any notation, decimal, binary etc.), giving many different interpretations. | |
| From: David Bostock (Philosophy of Mathematics [2009], 9.A.2) |
| 18149 | There are many criteria for the identity of numbers [Bostock] |
| Full Idea: There is not just one way of giving a criterion of identity for numbers. | |
| From: David Bostock (Philosophy of Mathematics [2009], 9.A.2) |
| 18143 | Frege makes numbers sets to solve the Caesar problem, but maybe Caesar is a set! [Bostock] |
| Full Idea: The Julius Caesar problem was one reason that led Frege to give an explicit definition of numbers as special sets. He does not appear to notice that the same problem affects his Axiom V for introducing sets (whether Caesar is or is not a set). | |
| From: David Bostock (Philosophy of Mathematics [2009], 9.A.2) | |
| A reaction: The Julius Caesar problem is a sceptical acid that eats into everything in philosophy of mathematics. You give all sorts of wonderful accounts of numbers, but at what point do you know that you now have a number, and not something else? |
| 18116 | Numbers can't be positions, if nothing decides what position a given number has [Bostock] |
| Full Idea: There is no ground for saying that a number IS a position, if the truth is that there is nothing to determine which number is which position. | |
| From: David Bostock (Philosophy of Mathematics [2009], 6.4) | |
| A reaction: If numbers lose touch with the empirical ability to count physical objects, they drift off into a mad world where they crumble away. |
| 18117 | Structuralism falsely assumes relations to other numbers are numbers' only properties [Bostock] |
| Full Idea: Structuralism begins from a false premise, namely that numbers have no properties other than their relations to other numbers. | |
| From: David Bostock (Philosophy of Mathematics [2009], 6.5) | |
| A reaction: Well said. Describing anything purely relationally strikes me as doomed, because you have to say why those things relate in those ways. |
| 18141 | Nominalism about mathematics is either reductionist, or fictionalist [Bostock] |
| Full Idea: Nominalism has two main versions, one which tries to 'reduce' the objects of mathematics to something simpler (Russell and Wittgenstein), and another which claims that such objects are mere 'fictions' which have no reality (Field). | |
| From: David Bostock (Philosophy of Mathematics [2009], 9) |
| 18157 | Nominalism as based on application of numbers is no good, because there are too many applications [Bostock] |
| Full Idea: The style of nominalism which aims to reduce statements about numbers to statements about their applications does not work for the natural numbers, because they have many applications, and it is arbitrary to choose just one of them. | |
| From: David Bostock (Philosophy of Mathematics [2009], 9.B.5.iii) |
| 18150 | Actual measurement could never require the precision of the real numbers [Bostock] |
| Full Idea: We all know that in practice no physical measurement can be 100 per cent accurate, and so it cannot require the existence of a genuinely irrational number, rather than some of the rational numbers close to it. | |
| From: David Bostock (Philosophy of Mathematics [2009], 9.A.3) |
| 18158 | Ordinals are mainly used adjectively, as in 'the first', 'the second'... [Bostock] |
| Full Idea: The basic use of the ordinal numbers is their use as ordinal adjectives, in phrases such as 'the first', 'the second' and so on. | |
| From: David Bostock (Philosophy of Mathematics [2009], 9.5.iii) | |
| A reaction: That is because ordinals seem to attach to particulars, whereas cardinals seem to attach to groups. Then you say 'three is greater than four', it is not clear which type you are talking about. |
| 18127 | Simple type theory has 'levels', but ramified type theory has 'orders' [Bostock] |
| Full Idea: The simple theory of types distinguishes sets into different 'levels', but this is quite different from the distinction into 'orders' which is imposed by the ramified theory. | |
| From: David Bostock (Philosophy of Mathematics [2009], 8.1) | |
| A reaction: The ramified theory has both levels and orders (p.235). Russell's terminology is, apparently, inconsistent. |
| 18144 | Neo-logicists agree that HP introduces number, but also claim that it suffices for the job [Bostock] |
| Full Idea: The neo-logicists take up Frege's claim that Hume's Principle introduces a new concept (of a number), but unlike Frege they go on to claim that it by itself gives a complete account of that concept. | |
| From: David Bostock (Philosophy of Mathematics [2009], 9.A.2) | |
| A reaction: So the big difference between Frege and neo-logicists is the Julius Caesar problem. |
| 18147 | Neo-logicists meet the Caesar problem by saying Hume's Principle is unique to number [Bostock] |
| Full Idea: The response of neo-logicists to the Julius Caesar problem is to strengthen Hume's Principle in the hope of ensuring that only numbers will satisfy it. They say the criterion of identity provided by HP is essential to number, and not to anything else. | |
| From: David Bostock (Philosophy of Mathematics [2009], 9.A.2) |
| 18146 | If Hume's Principle is the whole story, that implies structuralism [Bostock] |
| Full Idea: If Hume's Principle is all we are given, by way of explanation of what the numbers are, the only conclusion to draw would seem to be the structuralists' conclusion, ...studying all systems that satisfy that principle. | |
| From: David Bostock (Philosophy of Mathematics [2009], 9.A.2) | |
| A reaction: Any approach that implies a set of matching interpretations will always imply structuralism. To avoid it, you need to pin the target down uniquely. |
| 18129 | Many crucial logicist definitions are in fact impredicative [Bostock] |
| Full Idea: Many of the crucial definitions in the logicist programme are in fact impredicative. | |
| From: David Bostock (Philosophy of Mathematics [2009], 8.2) |
| 18111 | Treating numbers as objects doesn't seem like logic, since arithmetic fixes their totality [Bostock] |
| Full Idea: If logic is neutral on the number of objects there are, then logicists can't construe numbers as objects, for arithmetic is certainly not neutral on the number of numbers there are. They must be treated in some other way, perhaps as numerical quantifiers. | |
| From: David Bostock (Philosophy of Mathematics [2009], 5.5) |
| 18159 | Higher cardinalities in sets are just fairy stories [Bostock] |
| Full Idea: In its higher reaches, which posit sets of huge cardinalities, set theory is just a fairy story. | |
| From: David Bostock (Philosophy of Mathematics [2009], 9.5.iii) | |
| A reaction: You can't say the higher reaches are fairy stories but the lower reaches aren't, if the higher is directly derived from the lower. The empty set and the singleton are fairy stories too. Bostock says the axiom of infinity triggers the fairy stories. |
| 18155 | A fairy tale may give predictions, but only a true theory can give explanations [Bostock] |
| Full Idea: A common view is that although a fairy tale may provide very useful predictions, it cannot provide explanations for why things happen as they do. In order to do that a theory must also be true (or, at least, an approximation to the truth). | |
| From: David Bostock (Philosophy of Mathematics [2009], 9.B.5) | |
| A reaction: Of course, fictionalism offers an explanation of mathematics as a whole, but not of the details (except as the implications of the initial fictional assumptions). |
| 18140 | The best version of conceptualism is predicativism [Bostock] |
| Full Idea: In my personal opinion, predicativism is the best version of conceptualism that we have yet discovered. | |
| From: David Bostock (Philosophy of Mathematics [2009], 8.4) | |
| A reaction: Since conceptualism is a major player in the field, this makes predicativism a very important view. I won't vote Predicativist quite yet, but I'm tempted. |
| 18138 | Conceptualism fails to grasp mathematical properties, infinity, and objective truth values [Bostock] |
| Full Idea: Three simple objections to conceptualism in mathematics are that we do not ascribe mathematical properties to our ideas, that our ideas are presumably finite, and we don't think mathematics lacks truthvalue before we thought of it. | |
| From: David Bostock (Philosophy of Mathematics [2009], 8.4) | |
| A reaction: [compressed; Bostock refers back to his Ch 2] Plus Idea 18134. On the whole I sympathise with conceptualism, so I will not allow myself to be impressed by any of these objections. (So, what's actually wrong with them.....?). |
| 18131 | If abstracta only exist if they are expressible, there can only be denumerably many of them [Bostock] |
| Full Idea: If an abstract object exists only when there is some suitable way of expressing it, then there are at most denumerably many abstract objects. | |
| From: David Bostock (Philosophy of Mathematics [2009], 8.2) | |
| A reaction: Fine by me. What an odd view, to think there are uncountably many abstract objects in existence, only a countable portion of which will ever be expressed! [ah! most people agree with me, p.243-4] |
| 18134 | Predicativism makes theories of huge cardinals impossible [Bostock] |
| Full Idea: Classical mathematicians say predicative mathematics omits areas of great interest, all concerning non-denumerable real numbers, such as claims about huge cardinals. There cannot be a predicative version of this theory. | |
| From: David Bostock (Philosophy of Mathematics [2009], 8.3) | |
| A reaction: I'm not sure that anyone will really miss huge cardinals if they are prohibited, though cryptography seems to flirt with such things. Are we ever allowed to say that some entity conjured up by mathematicians is actually impossible? |
| 18135 | If mathematics rests on science, predicativism may be the best approach [Bostock] |
| Full Idea: It has been claimed that only predicative mathematics has a justification through its usefulness to science (an empiricist approach). | |
| From: David Bostock (Philosophy of Mathematics [2009], 8.3) | |
| A reaction: [compressed. Quine is the obvious candidate] I suppose predicativism gives your theory roots, whereas impredicativism is playing an abstract game. |
| 18136 | If we can only think of what we can describe, predicativism may be implied [Bostock] |
| Full Idea: If we accept the initial idea that we can think only of what we ourselves can describe, then something like the theory of predicativism quite naturally results | |
| From: David Bostock (Philosophy of Mathematics [2009], 8.3) | |
| A reaction: I hate the idea that we can only talk of what falls under a sortal, but 'what we can describe' is much more plausible. Whether or not you agree with this approach (I'm pondering it), this makes predicativism important. |
| 18133 | The usual definitions of identity and of natural numbers are impredicative [Bostock] |
| Full Idea: The predicative approach cannot accept either the usual definition of identity or the usual definition of the natural numbers, for both of these definitions are impredicative. | |
| From: David Bostock (Philosophy of Mathematics [2009], 8.3) | |
| A reaction: [Bostock 237-8 gives details] |
| 18132 | The predicativity restriction makes a difference with the real numbers [Bostock] |
| Full Idea: It is with the real numbers that the restrictions imposed by predicativity begin to make a real difference. | |
| From: David Bostock (Philosophy of Mathematics [2009], 8.3) |
| 18775 | Russell showed that descriptions may not have ontological commitment [Russell, by Linsky,B] |
| Full Idea: Russell's theory of definite descriptions allows us to avoid being ontologically committed to objects simply by virtue of using descriptions which seemingly denote them. | |
| From: report of Bertrand Russell (On Denoting [1905]) by Bernard Linsky - Quantification and Descriptions 1.1.2 | |
| A reaction: This I take to be why Russell's theory is a famous landmark. I personally take ontological commitment to be independent of what we specifically say. Others, like Quine, prefer to trim what we say until the commitments seem sound. |
| 7533 | The Theory of Description dropped classes and numbers, leaving propositions, individuals and universals [Russell, by Monk] |
| Full Idea: The real Platonic entities left standing after the Theory of Descriptions were propositions (not classes or numbers), and their constituents did not include denoting concepts or classes, but only individuals (Socrates) and universals (mortality). | |
| From: report of Bertrand Russell (On Denoting [1905]) by Ray Monk - Bertrand Russell: Spirit of Solitude Ch.6 | |
| A reaction: Propositions look like being the problem here. If we identify them with facts, it is not clear how many facts there are in the universe, independent of human thought. Indeed, how many universals are there? Nay, how many individuals? See Idea 7534. |
| 6063 | Russell can't attribute existence to properties [McGinn on Russell] |
| Full Idea: Russell's view makes it impossible to attribute existence to properties, and this would have to be declared ill-formed and meaningless. | |
| From: comment on Bertrand Russell (On Denoting [1905]) by Colin McGinn - Logical Properties Ch.2 | |
| A reaction: This strikes me as a powerful criticism, used to support McGinn's view that existence cannot be analysed, using quantifiers or anything else. |
| 18777 | If the King of France is not bald, and not not-bald, this violates excluded middle [Linsky,B on Russell] |
| Full Idea: Russell says one won't find the present King of France on the list of bald things, nor on the list of things that are not bald. It would seem that this gives rise to a violation of the law of excluded middle. | |
| From: comment on Bertrand Russell (On Denoting [1905]) by Bernard Linsky - Quantification and Descriptions 2 | |
| A reaction: It's a bit hard to accuse the poor old King of violating a law when he doesn't exist. |
| 4567 | Russell argued with great plausibility that we rarely, if ever, refer with our words [Russell, by Cooper,DE] |
| Full Idea: Russell argued with great plausibility that we rarely, if ever, refer with our words. | |
| From: report of Bertrand Russell (On Denoting [1905]) by David E. Cooper - Philosophy and the Nature of Language §4 | |
| A reaction: I'm not sure if I understand this. Presumably phrases which appear to refer actually point at other parts of language rather than the world. |
| 5810 | Referring is not denoting, and Russell ignores the referential use of definite descriptions [Donnellan on Russell] |
| Full Idea: If I am right, referring is not the same as denoting and the referential use of definite descriptions is not recognised on Russell's view. | |
| From: comment on Bertrand Russell (On Denoting [1905]) by Keith Donnellan - Reference and Definite Descriptions §I | |
| A reaction: This introduces a new theory of reference, which goes beyond the mere contents of linguistic experessions. It says reference is an 'external' and 'causal' affair, and so a definite description is not sufficient to make a reference. |
| 16385 | A definite description 'denotes' an entity if it fits the description uniquely [Russell, by Recanati] |
| Full Idea: In Russell's definition of 'denoting', a definite description denotes an entity if that entity fits the description uniquely. | |
| From: report of Bertrand Russell (On Denoting [1905]) by François Recanati - Mental Files 17.2 | |
| A reaction: [Recanati cites Donnellan for this] Hence denoting is not the same thing as reference. A description can denote beautifully, but fail to refer. Donnellan says if denoting were reference, someone might refer without realising it. |
| 5774 | Denoting phrases are meaningless, but guarantee meaning for propositions [Russell] |
| Full Idea: Denoting phrases never have any meaning in themselves, but every proposition in whose verbal expression they occur has a meaning. | |
| From: Bertrand Russell (On Denoting [1905], p.43) | |
| A reaction: This is the important idea that the sentence is the basic unit of meaning, rather than the word. I'm not convinced that this dispute needs to be settled. Words are pretty pointless outside of propositions, and propositions are impossible without words. |
| 5775 | In 'Scott is the author of Waverley', denotation is identical, but meaning is different [Russell] |
| Full Idea: If we say 'Scott is the author of Waverley', we assert an identity of denotation with a difference of meaning. | |
| From: Bertrand Russell (On Denoting [1905], p.46) | |
| A reaction: This shows Russell picking up Frege's famous distinction, as shown in 'Hesperus is Phosphorus'. To distinguish the meaning from the reference was one of the greatest (and simplest) clarifications ever offered of how language works. |
| 16987 | By eliminating descriptions from primitive notation, Russell seems to reject 'sense' [Russell, by Kripke] |
| Full Idea: Russell, since he eliminates descriptions from his primitive notation, seems to hold in 'On Denoting' that the notion of 'sense' is illusory. | |
| From: report of Bertrand Russell (On Denoting [1905]) by Saul A. Kripke - Naming and Necessity notes and addenda note 04 | |
| A reaction: Presumably we can eliminate sense from formal languages, but natural languages are rich in connotations (or whatever we choose to call them). |
| 4570 | Russell assumes that expressions refer, but actually speakers refer by using expressions [Cooper,DE on Russell] |
| Full Idea: Russell assumes that it is expressions which refer if anything does, but strictly speaking it is WE who refer with the use of expressions. | |
| From: comment on Bertrand Russell (On Denoting [1905]) by David E. Cooper - Philosophy and the Nature of Language §4.1 | |
| A reaction: This sounds right. Russell is part of the overemphasis on language which plagued philosophy after Frege. Words are tools, like searchlights or pointing fingers. |
| 16349 | Russell rejected sense/reference, because it made direct acquaintance with things impossible [Russell, by Recanati] |
| Full Idea: Russell rejected Frege's sense/reference distinction, on the grounds that, if reference is mediated by sense, we lose the idea of direct acquaintance and succumb to Descriptivism. | |
| From: report of Bertrand Russell (On Denoting [1905]) by François Recanati - Mental Files 1.1 | |
| A reaction: [15,000th IDEA in the DB!! 23/3/2013, Weymouth] Recanati claims Russell made a mistake, because you can retain the sense/reference distinction, and still keep direct acquaintance (by means of 'non-descriptive senses'). |
| 7313 | 'Sense' is superfluous (rather than incoherent) [Russell, by Miller,A] |
| Full Idea: Russell does not claim that Frege's notion of sense is incoherent, but rather that it is superfluous. | |
| From: report of Bertrand Russell (On Denoting [1905]) by Alexander Miller - Philosophy of Language 2.9 | |
| A reaction: My initial reaction to this is that the notion of strict and literal meaning (see Idea 7309) is incredibly useful. Some of the best jokes depend on the gap between implications and strict meaning. How could metaphors be explained without it? |
| 7767 | The theory of definite descriptions aims at finding correct truth conditions [Russell, by Lycan] |
| Full Idea: Russell's theory of definite descriptions proceeds by sketching the truth conditions of sentences containing descriptions, and arguing on various grounds that they are the correct truth conditions. | |
| From: report of Bertrand Russell (On Denoting [1905]) by William Lycan - Philosophy of Language Ch.9 | |
| A reaction: It seems important to see both where Russell was going, and where Davidson has come from. The whole project of finding the logical form of sentences (which starts with Frege and Russell) implies that truth conditions is what matters. |
| 21726 | In graspable propositions the constituents are real entities of acquaintance [Russell] |
| Full Idea: In every proposition that we can apprehend, ...all the constituents are real entities with which we have immediate acquaintance. | |
| From: Bertrand Russell (On Denoting [1905], p.56), quoted by Bernard Linsky - Russell's Metaphysical Logic 7.2 | |
| A reaction: This is the clearest statement of the 'Russellian' concept of a proposition. It strikes me as entirely wrong. The examples are always nice concrete objects like Mont Blanc, but as an account of sophisticated general propositions it seem hopeless. |
| 18121 | In logic a proposition means the same when it is and when it is not asserted [Bostock] |
| Full Idea: In Modus Ponens where the first premise is 'P' and the second 'P→Q', in the first premise P is asserted but in the second it is not. Yet it must mean the same in both premises, or it would be guilty of the fallacy of equivocation. | |
| From: David Bostock (Philosophy of Mathematics [2009], 7.2) | |
| A reaction: This is Geach's thought (leading to an objection to expressivism in ethics, that P means the same even if it is not expressed). |
| 5773 | The ontological argument begins with an unproven claim that 'there exists an x..' [Russell] |
| Full Idea: 'There is one and only one entity x which is most perfect; that one has all perfections; existence is a perfection; therefore that one exists' fails as a proof because there is no proof of the first premiss. | |
| From: Bertrand Russell (On Denoting [1905], p.54) | |
| A reaction: This is the modern move of saying that existence (which is 'not a predicate', according to Kant) is actually a quantifier, which isolates the existence claim being made about a variable with a bunch of predicates. McGinn denies Russell's claim. |