82 ideas
| 6095 | The business of metaphysics is to describe the world [Russell] |
| Full Idea: It seems to me that the business of metaphysics is to describe the world. | |
| From: Bertrand Russell (The Philosophy of Logical Atomism [1918], §III) | |
| A reaction: At least he believed in metaphysics. Presumably he intends to describe the world in terms of its categories, rather than cataloguing every blade of grass. |
| 6106 | Reducing entities and premisses makes error less likely [Russell] |
| Full Idea: You diminish the risk of error with every diminution of entities and premisses. | |
| From: Bertrand Russell (The Philosophy of Logical Atomism [1918], §VIII) | |
| A reaction: If there are actually lots of entities, you would increase error if you reduced them too much. Ockham's Razor seems more to do with the limited capacity of the human mind than with the simplicity or complexity of reality. See Idea 4456. |
| 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) |
| 6090 | Facts make propositions true or false, and are expressed by whole sentences [Russell] |
| Full Idea: A fact is the kind of thing that makes a proposition true or false, …and it is the sort of thing that is expressed by a whole sentence, not by a single name like 'Socrates'. | |
| From: Bertrand Russell (The Philosophy of Logical Atomism [1918], §I) | |
| A reaction: It is important to note a point here which I consider vital - that Russell keeps the idea of a fact quite distinct from the language in which it is expressed. Facts are a 'sort of thing', of the kind which are now referred to as 'truth-makers'. |
| 18348 | Not only atomic truths, but also general and negative truths, have truth-makers [Russell, by Rami] |
| Full Idea: In 1918 Russell held that beside atomic truths, also general and negative truths have truth-makers. | |
| From: report of Bertrand Russell (The Philosophy of Logical Atomism [1918]) by Adolph Rami - Introduction: Truth and Truth-Making note 04 |
| 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) |
| 6103 | Normally a class with only one member is a problem, because the class and the member are identical [Russell] |
| Full Idea: With the ordinary view of classes you would say that a class that has only one member was the same as that one member; that will land you in terrible difficulties, because in that case that one member is a member of that class, namely, itself. | |
| From: Bertrand Russell (The Philosophy of Logical Atomism [1918], §VII) | |
| A reaction: The problem (I think) is that classes (sets) were defined by Frege as being identical with their members (their extension). With hindsight this may have been a mistake. The question is always 'why is that particular a member of that set?' |
| 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. |
| 6092 | In a logically perfect language, there will be just one word for every simple object [Russell] |
| Full Idea: In a logically perfect language, there will be one word and no more for every simple object. | |
| From: Bertrand Russell (The Philosophy of Logical Atomism [1918], §II) | |
| A reaction: In other words, there would be no universals, only names? All that matters is that a language can successfully refer (unambiguously) to anything it wishes to. There must be better ways than Russell's lexical explosion. |
| 6101 | Romulus does not occur in the proposition 'Romulus did not exist' [Russell] |
| Full Idea: Romulus does not occur in the proposition 'Romulus did not exist'. | |
| From: Bertrand Russell (The Philosophy of Logical Atomism [1918], §VI) | |
| A reaction: A very nice paradoxical assertion, which captures the problem of finding the logical form for negative existential statements. Presumably the proposition refers to the mythical founder of Rome, though. He is not, I suppose, rigidly designated. |
| 6102 | You can understand 'author of Waverley', but to understand 'Scott' you must know who it applies to [Russell] |
| Full Idea: If you understand English you would understand the phrase 'the author of Waverley' if you had not heard it before, whereas you would not understand the meaning of 'Scott', because to know the meaning of a name is to know who it is applied to. | |
| From: Bertrand Russell (The Philosophy of Logical Atomism [1918], §VI) | |
| A reaction: Actually, you would find 'Waverley' a bit baffling too. Would you understand "he was the author of his own destruction"? You can understand "Homer was the author of this" without knowing quite who 'Homer' applies to. All very tricky. |
| 10423 | There are a set of criteria for pinning down a logically proper name [Russell, by Sainsbury] |
| Full Idea: A logically proper name must be semantically simple, have just one referent, be understood by the user, be scopeless, is not a definite description, and rigidly designates. | |
| From: report of Bertrand Russell (The Philosophy of Logical Atomism [1918], 24th pg) by Mark Sainsbury - The Essence of Reference Intro | |
| A reaction: Famously, Russell's hopes of achieving this logically desirable end got narrower and narrower, and ended with 'this' or 'that'. Maybe pure language can't do the job. |
| 7744 | Treat description using quantifiers, and treat proper names as descriptions [Russell, by McCullogh] |
| Full Idea: Having proposed that descriptions should be treated in quantificational terms, Russell then went on to introduce the subsidiary injunction that proper names should be treated as descriptions. | |
| From: report of Bertrand Russell (The Philosophy of Logical Atomism [1918]) by Gregory McCullogh - The Game of the Name 2.18 | |
| A reaction: McCulloch says Russell 'has a lot to answer for' here. It became a hot topic with Kripke. Personally I find Lewis's notion of counterparts the most promising line of enquiry. |
| 10426 | A name has got to name something or it is not a name [Russell] |
| Full Idea: A name has got to name something or it is not a name. | |
| From: Bertrand Russell (The Philosophy of Logical Atomism [1918], 66th pg), quoted by Mark Sainsbury - The Essence of Reference 18.2 | |
| A reaction: This seems to be stipulative, since most people would say that a list of potential names for a baby counted as names. It may be wrong. There are fictional names, or mistakes. |
| 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) |
| 6104 | Numbers are classes of classes, and hence fictions of fictions [Russell] |
| Full Idea: Numbers are classes of classes, and classes are logical fictions, so that numbers are, as it were, fictions at two removes, fictions of fictions. | |
| From: Bertrand Russell (The Philosophy of Logical Atomism [1918], §VIII) | |
| A reaction: This summarises the findings of Russell and Whitehead's researches into logicism. Gödel may have proved that project impossible, but there is now debate about that. Personally I think of numbers as names of patterns. |
| 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). |
| 8714 | Fictionalists say 2+2=4 is true in the way that 'Oliver Twist lived in London' is true [Field,H] |
| Full Idea: The fictionalist can say that the sense in which '2+2=4' is true is pretty much the same as the sense in which 'Oliver Twist lived in London' is true. They are true 'according to a well-known story', or 'according to standard mathematics'. | |
| From: Hartry Field (Realism, Mathematics and Modality [1989], 1.1.1), quoted by Michčle Friend - Introducing the Philosophy of Mathematics 6.3 | |
| A reaction: The roots of this idea are in Carnap. Fictionalism strikes me as brilliant, but poisonous in large doses. Novels can aspire to artistic truth, or to documentary truth. We invent a fiction, and nudge it slowly towards reality. |
| 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) |
| 21708 | Russell's new logical atomist was of particulars, universals and facts (not platonic propositions) [Russell, by Linsky,B] |
| Full Idea: Russell's new logical atomist ontology was of particulars, universals and facts, replacing the ontology of 'platonic atomism' consisting just of propositions. | |
| From: report of Bertrand Russell (The Philosophy of Logical Atomism [1918]) by Bernard Linsky - Russell's Metaphysical Logic 1 | |
| A reaction: Linsky cites Peter Hylton as saying that the earlier view was never replaced. The earlier view required propositions to be 'unified'. I surmise that the formula 'Fa' combines a universal and a particular, to form an atomic fact. [...but Idea 6111!] |
| 19051 | Russell's atomic facts are actually compounds, and his true logical atoms are sense data [Russell, by Quine] |
| Full Idea: In 1918 Russell does not admit facts as fundamental; atomic facts are atomic as facts go, but they are compound objects. The atoms of Russell's logical atomism are not atomic facts but sense data. | |
| From: report of Bertrand Russell (The Philosophy of Logical Atomism [1918]) by Willard Quine - Russell's Ontological Development p.83 | |
| A reaction: By about 1921 Russell had totally given up sense-data, because he had been reading behaviourist psychology. |
| 6089 | Logical atomism aims at logical atoms as the last residue of analysis [Russell] |
| Full Idea: I call my doctrine logical atomism because, as the last residue of analysis, I wish to arrive at logical atoms and not physical atoms; some of them will be particulars, and others will be predicates and relations and so on. | |
| From: Bertrand Russell (The Philosophy of Logical Atomism [1918], §I) | |
| A reaction: However we judge it, logical atomism is a vital landmark in the history of 'analytical' philosophy, because it lays out the ideal for our assessment. It is fashionable to denigrate analysis, but I think it is simply the nearest to wisdom we will ever get. |
| 6100 | Once you have enumerated all the atomic facts, there is a further fact that those are all the facts [Russell] |
| Full Idea: When you have enumerated all the atomic facts in the world, it is a further fact about the world that those are all the atomic facts there are about the world. | |
| From: Bertrand Russell (The Philosophy of Logical Atomism [1918], §V) | |
| A reaction: There is obviously a potential regress of facts about facts here. This looks like one of the reasons why the original logical atomism had a short shelf-life. Personally I see this as an argument in favour of rationalism, in the way Bonjour argues for it. |
| 6105 | Logical atoms aims to get down to ultimate simples, with their own unique reality [Russell] |
| Full Idea: Logical atomism is the view that you can get down in theory, if not in practice, to ultimate simples, out of which the world is built, and that those simples have a kind of reality not belonging to anything else. | |
| From: Bertrand Russell (The Philosophy of Logical Atomism [1918], §VIII) | |
| A reaction: This dream is to empiricists what the Absolute is to rationalists - a bit silly, but an embodiment of the motivating dream. |
| 21709 | You can't name all the facts, so they are not real, but are what propositions assert [Russell] |
| Full Idea: Facts are the sort of things that are asserted or denied by propositions, and are not properly entities at all in the same sense in which their constituents are. That is shown by the fact that you cannot name them. | |
| From: Bertrand Russell (The Philosophy of Logical Atomism [1918], p.235), quoted by Bernard Linsky - Russell's Metaphysical Logic 2.2 | |
| A reaction: [ref to Papers vol.8] It is customary to specify a proposition by its capacity for T and F. So is a fact just 'a truth'? This contains the Fregean idea that things are only real if they can be picked out. I think of facts as independent of minds. |
| 18376 | Russell asserts atomic, existential, negative and general facts [Russell, by Armstrong] |
| Full Idea: Russell argues for atomic facts, and also for existential facts, negative facts and general facts. | |
| From: report of Bertrand Russell (The Philosophy of Logical Atomism [1918]) by David M. Armstrong - Truth and Truthmakers 05.1 | |
| A reaction: Armstrong says he overdoes it. I would even add disjunctive facts, which Russell rejects. 'Rain or snow will ruin the cricket match'. Rain can make that true, but it is a disjunctive fact about the match. |
| 5465 | Modern trope theory tries, like logical atomism, to reduce things to elementary states [Russell, by Ellis] |
| Full Idea: Russell and Wittgenstein sought to reduce everything to singular facts or states of affairs, and Armstrong and Keith Campbell have more recently advocated ontologies of tropes or elementary states of affairs. | |
| From: report of Bertrand Russell (The Philosophy of Logical Atomism [1918]) by Brian Ellis - The Philosophy of Nature: new essentialism Ch.3 n 11 | |
| A reaction: A very interesting historical link. Logical atomism strikes me as a key landmark in the history of philosophy, and not an eccentric cul-de-sac. It is always worth trying to get your ontology down to minimal small units, to see what happens. |
| 6060 | 'Existence' means that a propositional function is sometimes true [Russell] |
| Full Idea: When you take any propositional function and assert of it that it is possible, that it is sometimes true, that gives you the fundamental meaning of 'existence'. | |
| From: Bertrand Russell (The Philosophy of Logical Atomism [1918]), quoted by Colin McGinn - Logical Properties Ch.2 | |
| A reaction: Functions depend on variables, so this leads to Quine's slogan "to be is to be the value of a variable". Assertions of non-existence are an obvious problem, but Russell thought of all that. All of this makes existence too dependent on language. |
| 6099 | Modal terms are properties of propositional functions, not of propositions [Russell] |
| Full Idea: Traditional philosophy discusses 'necessary', 'possible' and 'impossible' as properties of propositions, whereas in fact they are properties of propositional functions; propositions are only true or false. | |
| From: Bertrand Russell (The Philosophy of Logical Atomism [1918], §V) | |
| A reaction: I am unclear how a truth could be known to be necessary if it is full of variables. 'x is human' seems to have no modality, but 'Socrates is human' could well be necessary. I like McGinn's rather adverbial account of modality. |
| 6098 | Perception goes straight to the fact, and not through the proposition [Russell] |
| Full Idea: I am inclined to think that perception, as opposed to belief, does go straight to the fact and not through the proposition. | |
| From: Bertrand Russell (The Philosophy of Logical Atomism [1918], §IV.4) | |
| A reaction: There seems to be a question of an intermediate stage, which is the formulation of concepts. Is full 'perception' (backed by attention and intellect) laden with concepts, which point to facts? Where are the facts in sensation without recognition? |
| 6097 | The theory of error seems to need the existence of the non-existent [Russell] |
| Full Idea: It is very difficult to deal with the theory of error without assuming the existence of the non-existent. | |
| From: Bertrand Russell (The Philosophy of Logical Atomism [1918], §IV.3) | |
| A reaction: This problem really bothered Russell (and Plato). I suspect that it was a self-inflicted problem because at this point Russell had ceased to believe in propositions. If we accept propositions as intentional objects, they can be as silly as you like. |
| 9022 | Russell uses 'propositional function' to refer to both predicates and to attributes [Quine on Russell] |
| Full Idea: Russell used the phrase 'propositional function' (adapted from Frege) to refer sometimes to predicates and sometimes to attributes. | |
| From: comment on Bertrand Russell (The Philosophy of Logical Atomism [1918]) by Willard Quine - Philosophy of Logic Ch.5 | |
| A reaction: He calls Russell 'confused' on this, and he would indeed be guilty of what now looks like a classic confusion, between the properties and the predicates that express them. Only a verificationist would hold such a daft view. |
| 6091 | Propositions don't name facts, because each fact corresponds to a proposition and its negation [Russell] |
| Full Idea: It is obvious that a proposition is not the name for a fact, from the mere circumstance that there are two propositions corresponding to each fact, one the negation of the other. | |
| From: Bertrand Russell (The Philosophy of Logical Atomism [1918], §I) | |
| A reaction: Russell attributes this point to Wittgenstein. Evidently you must add that the proposition is true before it will name a fact - which is bad news for the redundancy view of truth. Couldn't lots of propositions correspond to one fact? |
| 21702 | In 1918 still believes in nonlinguistic analogues of sentences, but he now calls them 'facts' [Russell, by Quine] |
| Full Idea: In 1918 Russell insists that the world does contain nonlinguistic things that are akin to sentences and are asserted by them; he merely does not call them propositions. He calls them facts. | |
| From: report of Bertrand Russell (The Philosophy of Logical Atomism [1918]) by Willard Quine - Russell's Ontological Development p.81 | |
| A reaction: Clarification! I have always been bewildered by the early Russell view of propositions as actual ingredients of the world. If we say that sentences assert facts, that makes more sense. Russell never believed in the mental entities I call 'propositions'. |
| 6094 | An inventory of the world does not need to include propositions [Russell] |
| Full Idea: It is quite clear that propositions are not what you might call 'real'; if you were making an inventory of the world, propositions would not come in. | |
| From: Bertrand Russell (The Philosophy of Logical Atomism [1918], §III) | |
| A reaction: I am not clear why this is "quite clear". Propositions might even turn up in our ontology as physical objects (brain states). He says beliefs are real, but if you can't have a belief without a proposition, and they aren't real, you are in trouble. |
| 6096 | I no longer believe in propositions, especially concerning falsehoods [Russell] |
| Full Idea: Time was when I thought there were propositions, but it does not seem to me very plausible to say that in addition to facts there are also these curious shadowy things going about as 'That today is Wednesday' when in fact it is Tuesday. | |
| From: Bertrand Russell (The Philosophy of Logical Atomism [1918], §IV.2) | |
| A reaction: You need to give some account of someone who thinks 'Today is Wednesday' when it is Tuesday. We can hardly avoid talking about something like an 'intentional object', which can be expressed in a sentence. Are there not possible (formulable) propositions? |
| 21712 | I know longer believe in shadowy things like 'that today is Wednesday' when it is actually Tuesday [Russell] |
| Full Idea: Time was when I thought there were propositions, but it does not seem to me very plausible to say that in addition to facts there are also these curious shadowy things going about such 'That today is Wednesday' when it is in fact Tuesday. | |
| From: Bertrand Russell (The Philosophy of Logical Atomism [1918], p.197), quoted by Bernard Linsky - Russell's Metaphysical Logic 3.1 | |
| A reaction: [Ref to Papers v8] I take Russell to have abandoned his propositions because his conception of them was mistaken. Presumably my thinking 'Today is Wednesay' conjures up a false proposition, which had not previously existed. |
| 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). |
| 6093 | The names in a logically perfect language would be private, and could not be shared [Russell] |
| Full Idea: A logically perfect language, if it could be constructed, would be, as regards its vocabulary, very largely private to one speaker; that is, all the names in it would be private to that speaker and could not enter into the language of another speaker. | |
| From: Bertrand Russell (The Philosophy of Logical Atomism [1918], §II) | |
| A reaction: Wittgenstein obviously thought there was something not quite right about this… See Idea 4147, for example. I presume Russell's thought is that you would have no means of explaining the 'meanings' of the names in the language. |
| 6119 | You can discuss 'God exists', so 'God' is a description, not a name [Russell] |
| Full Idea: The fact that you can discuss the proposition 'God exists' is a proof that 'God', as used in that proposition, is a description and not a name. If 'God' were a name, no question as to its existence could arise. | |
| From: Bertrand Russell (The Philosophy of Logical Atomism [1918], §VI) | |
| A reaction: Presumably 'a being than which none greater can be conceived' (Anselm's definition) is self-evidently a description, and doesn't claim to be a name. Aquinas caps each argument with a triumphant naming of the being he has proved. |