37 ideas
| 10308 | Questions about objects are questions about certain non-vacuous singular terms [Hale] |
| Full Idea: I understand questions about the Fregean notion of an object to be inseparable from questions in the philosophy of language - questions of the existence of objects are tantamount to questions about non-vacuous singular terms of a certain kind. | |
| From: Bob Hale (Abstract Objects [1987], Ch.1) | |
| A reaction: This view hovers somewhere between Quine and J.L. Austin, and Dummett is its originator. I am instinctively deeply opposed to the identification of metaphysics with semantics. |
| 13886 | Later Frege held that definitions must fix a function's value for every possible argument [Frege, by Wright,C] |
| Full Idea: Frege later became fastidious about definitions, and demanded that they must provide for every possible case, and that no function is properly determined unless its value is fixed for every conceivable object as argument. | |
| From: report of Gottlob Frege (Grundgesetze der Arithmetik 2 (Basic Laws) [1903]) by Crispin Wright - Frege's Concept of Numbers as Objects 3.xiv | |
| A reaction: Presumably definitions come in degrees of completeness, but it seems harsh to describe a desire for the perfect definition as 'fastidious', especially if we are talking about mathematics, rather than defining 'happiness'. |
| 9845 | We can't define a word by defining an expression containing it, as the remaining parts are a problem [Frege] |
| Full Idea: Given the reference (bedeutung) of an expression and a part of it, obviously the reference of the remaining part is not always determined. So we may not define a symbol or word by defining an expression in which it occurs, whose remaining parts are known | |
| From: Gottlob Frege (Grundgesetze der Arithmetik 2 (Basic Laws) [1903], §66) | |
| A reaction: Dummett cites this as Frege's rejection of contextual definitions, which he had employed in the Grundlagen. I take it not so much that they are wrong, as that Frege decided to set the bar a bit higher. |
| 10019 | Only what is logically complex can be defined; what is simple must be pointed to [Frege] |
| Full Idea: Only what is logically complex can be defined; what is simple can only be pointed to. | |
| From: Gottlob Frege (Grundgesetze der Arithmetik 2 (Basic Laws) [1903], §180), quoted by Harold Hodes - Logicism and Ontological Commits. of Arithmetic p.137 | |
| A reaction: Frege presumably has in mind his treasured abstract objects, such as cardinal numbers. It is hard to see how you could 'point to' anything in the phenomenal world that had atomic simplicity. Hodes calls this a 'desperate Kantian move'. |
| 10314 | An expression is a genuine singular term if it resists elimination by paraphrase [Hale] |
| Full Idea: An expression ... should be reckoned a genuine singular term only if it resists elimination by paraphrase. | |
| From: Bob Hale (Abstract Objects [1987], Ch.2.II) | |
| A reaction: This strikes me as extraordinarily optimistic. It will be relative to a language, and the resources of a given speaker, and seems open to the invention of new expressions to do the job (e.g. an equivalent adjective for every noun in the dictionary). |
| 10316 | We should decide whether singular terms are genuine by their usage [Hale] |
| Full Idea: The criteria for a genuine singular term should pick out not the singular terms themselves but their uses, since they may be genuine in one context and not another. | |
| From: Bob Hale (Abstract Objects [1987], Ch.2.II) | |
| A reaction: [rephrased] This will certainly meet problems with vagueness (e.g. as the reference of a singular term is gradually clarified). |
| 10312 | Often the same singular term does not ensure reliable inference [Hale] |
| Full Idea: In 'the whale is increasingly scarce' and 'the whale is much improved today' (our pet whale), we cannot infer that there is something that is much improved and increasingly scarce, so this singular term fails Dummett's criterion based on inference. | |
| From: Bob Hale (Abstract Objects [1987], Ch.2) | |
| A reaction: [much rephrased] This is not just a problem for a few cunningly selected examples. With contortions almost any singular term can be undermined in this way. Singular terms are simply not a useful guide to the existence of abstracta. |
| 10313 | Plenty of clear examples have singular terms with no ontological commitment [Hale] |
| Full Idea: Some examples where a definite singular noun phrase is not 'genuine' (giving ontological commitment): 'left us in the lurch'; 'for my mother's sake'; 'given the sack'; 'in the nick of time', 'the whereabouts of the PM', 'the identity of the murderer'. | |
| From: Bob Hale (Abstract Objects [1987], Ch.2.II) | |
| A reaction: These are not just freakish examples. If I 'go on a journey', that doesn't involve extra entities called 'journeys', just because the meaning is clearer and a more commonplace part of the language. |
| 10322 | If singular terms can't be language-neutral, then we face a relativity about their objects [Hale] |
| Full Idea: If we lack any general, language-neutral characterization of singular terms, must not a parallel linguistic relativity infect the objects which are to be thought of as their non-linguistic correlates? | |
| From: Bob Hale (Abstract Objects [1987], Ch.2.III) | |
| A reaction: Hale thinks he can answer this, but I would have thought that this problem dooms the linguistic approach from the start. There needs to be more imagination about how very different a language could be, while still qualifying as a language. |
| 9886 | Cardinals say how many, and reals give measurements compared to a unit quantity [Frege] |
| Full Idea: The cardinals and the reals are completely disjoint domains. The cardinal numbers answer the question 'How many objects of a given kind are there?', but the real numbers are for measurement, saying how large a quantity is compared to a unit quantity. | |
| From: Gottlob Frege (Grundgesetze der Arithmetik 2 (Basic Laws) [1903], §157), quoted by Michael Dummett - Frege philosophy of mathematics Ch.19 | |
| A reaction: We might say that cardinals are digital and reals are analogue. Frege is unusual in totally separating them. They map onto one another, after all. Cardinals look like special cases of reals. Reals are dreams about the gaps between cardinals. |
| 9889 | Real numbers are ratios of quantities [Frege, by Dummett] |
| Full Idea: Frege fixed on construing real numbers as ratios of quantities (in agreement with Newton). | |
| From: report of Gottlob Frege (Grundgesetze der Arithmetik 2 (Basic Laws) [1903]) by Michael Dummett - Frege philosophy of mathematics Ch.20 | |
| A reaction: If 3/4 is the same real number as 6/8, which is the correct ratio? Why doesn't the square root of 9/16 also express it? Why should irrationals be so utterly different from rationals? In what sense are they both 'numbers'? |
| 10553 | A number is a class of classes of the same cardinality [Frege, by Dummett] |
| Full Idea: For Frege, in 'Grundgesetze', a number is a class of classes of the same cardinality. | |
| From: report of Gottlob Frege (Grundgesetze der Arithmetik 2 (Basic Laws) [1903]) by Michael Dummett - Frege Philosophy of Language (2nd ed) Ch.14 |
| 10020 | Frege's biggest error is in not accounting for the senses of number terms [Hodes on Frege] |
| Full Idea: The inconsistency of Grundgesetze was only a minor flaw. Its fundamental flaw was its inability to account for the way in which the senses of number terms are determined. It leaves the reference-magnetic nature of the standard numberer a mystery. | |
| From: comment on Gottlob Frege (Grundgesetze der Arithmetik 2 (Basic Laws) [1903]) by Harold Hodes - Logicism and Ontological Commits. of Arithmetic p.139 | |
| A reaction: A point also made by Hofweber. As a logician, Frege was only concerned with the inferential role of number terms, and he felt he had captured their logical form, but it is when you come to look at numbers in natural language that he seem in trouble. |
| 9887 | Formalism misunderstands applications, metatheory, and infinity [Frege, by Dummett] |
| Full Idea: Frege's three main objections to radical formalism are that it cannot account for the application of mathematics, that it confuses a formal theory with its metatheory, and it cannot explain an infinite sequence. | |
| From: report of Gottlob Frege (Grundgesetze der Arithmetik 2 (Basic Laws) [1903], §86-137) by Michael Dummett - Frege philosophy of mathematics | |
| A reaction: The application is because we don't design maths randomly, but to be useful. The third objection might be dealt with by potential infinities (from formal rules). The second objection sounds promising. |
| 8751 | Only applicability raises arithmetic from a game to a science [Frege] |
| Full Idea: It is applicability alone which elevates arithmetic from a game to the rank of a science. | |
| From: Gottlob Frege (Grundgesetze der Arithmetik 2 (Basic Laws) [1903], §91), quoted by Stewart Shapiro - Thinking About Mathematics 6.1.2 | |
| A reaction: This is the basic objection to Formalism. It invites the question of why it is applicable, which platonists like Frege don't seem to answer (though Plato himself has reality modelled on the Forms). This is why I like structuralism. |
| 10512 | The abstract/concrete distinction is based on what is perceivable, causal and located [Hale] |
| Full Idea: The 'concrete/abstract' distinction has a strong intuitive feel, and can seem to be drawable by familiar contrasts, between what can/cannot be perceived, what can/cannot be involved in causal interactions, and is/is not located in space and time. | |
| From: Bob Hale (Abstract Objects [1987], Ch.3.I) | |
| A reaction: Problems arise, needless to say. The idea of an abstraction can be causal, and abstractions seem to change. If universals are abstract, we seem to perceive some of them. They can hardly be non-spatial if they have a temporal beginning and end. |
| 10517 | Colours and points seem to be both concrete and abstract [Hale] |
| Full Idea: It might seem that colours would qualify both as concrete and as abstract objects. ...and geometrical points also seem to be borderline. | |
| From: Bob Hale (Abstract Objects [1987], Ch.3.II) | |
| A reaction: The theory of tropes exploits this uncertainty. Dummett (1973:ch.14) notes that we can point to colours, but also slip from an adjectival to a noun usage of colour-terms. He concludes that colours are concrete. I think I agree. |
| 10519 | The abstract/concrete distinction is in the relations in the identity-criteria of object-names [Hale] |
| Full Idea: Noonan suggests that the distinction between abstract and concrete objects should be seen as derivative from a difference between the relations centrally involved in criteria of identity associated with names of objects. | |
| From: Bob Hale (Abstract Objects [1987], Ch.3.III) | |
| A reaction: [He cites Noonan 1976, but I've lost it] I don't understand this, but collect it as a lead to something that might be interesting. A careful reading of Hale might reveal what Noonan meant. |
| 10520 | Token-letters and token-words are concrete objects, type-letters and type-words abstract [Hale] |
| Full Idea: In familiar, though doubtless not wholly problematic jargon, token-letters and token-words are concrete objects, type-letters and type-words abstract. | |
| From: Bob Hale (Abstract Objects [1987], Ch.3.III) | |
| A reaction: This is indeed problematic. The marks may be tokens, but the preliminary to identifying the type is to see that the marks are in fact words. To grasp the concrete, grasp the abstraction. An excellent example of the blurring of the distinction. |
| 10524 | There is a hierarchy of abstraction, based on steps taken by equivalence relations [Hale] |
| Full Idea: The domain of the abstract can be seen as exemplifying a hierarchical structure, with differences of level reflecting the number of steps of abstraction, via appropriate equivalence relations, required for recognition at different levels. | |
| From: Bob Hale (Abstract Objects [1987], Ch.3.III) | |
| A reaction: I think this is right, and so does almost everyone else, since people cheerfully talk of 'somewhat' abstract and 'highly' abstract. Don't dream of a neat picture though. You might reach a level by two steps from one direction, and four from another. |
| 10521 | If F can't have location, there is no problem of things having F in different locations [Hale] |
| Full Idea: If Fs are incapable of spatial location, it is impossible for a and b to be at the same time in different places and yet be the same F. | |
| From: Bob Hale (Abstract Objects [1987], Ch.3.III) | |
| A reaction: A passing remark from Hale which strikes me as incredibly significant. The very idea of a 'one-over-many' is that there are many locations for the thing, so to conclude that the thing is therefore non-located seems to negate the original problem. |
| 10511 | It is doubtful if one entity, a universal, can be picked out by both predicates and abstract nouns [Hale] |
| Full Idea: The traditional conception of universals, resting as it does upon the idea that some single type of entity is picked out by expressions of such radically different logical types as predicates and abstract nouns, is of doubtful coherence. | |
| From: Bob Hale (Abstract Objects [1987], Ch.3 Intro) | |
| A reaction: A striking case of linguistic metaphysics in action. I don't believe in universals, but I don't find this persuasive, as our capacity to express the same proposition by means of extremely varied syntax is obvious. Is 'horse' an abstract noun? |
| 10318 | Realists take universals to be the referrents of both adjectives and of nouns [Hale] |
| Full Idea: On the traditional realist's view abstract qualities (universals) are the common referents of two quite different sorts of expression - of ordinary adjectives (predicates), and of abstract nouns referring to them. | |
| From: Bob Hale (Abstract Objects [1987], Ch.2.II) | |
| A reaction: This fact alone should make us suspicious, especially as there isn't an isomorphism between the nouns and the adjectives, and the match-up will vary between languages. |
| 10310 | Objections to Frege: abstracta are unknowable, non-independent, unstatable, unindividuated [Hale] |
| Full Idea: Objections to Frege's argument for abstract objects: that the objects would not have the right sort of independence; that we could have no knowledge of them; that the singular term statements can't be had; that thoughts of abstracta can't be identified. | |
| From: Bob Hale (Abstract Objects [1987], Ch.1) | |
| A reaction: [compressed] [See Idea 10309 for the original argument] It is helpful to have this list, even if Hale rejects them all. They are also created but then indestructible, and exist in unlimited profusion, and seem relative to a language. Etc! |
| 10518 | Shapes and directions are of something, but games and musical compositions are not [Hale] |
| Full Idea: While a shape or a direction is necessarily of something, games, musical compositions or dance routines are not of anything at all. | |
| From: Bob Hale (Abstract Objects [1987], Ch.3.II) | |
| A reaction: This seems important, because Frege's abstraction principle works nicely for abstractions 'of' some objects, but is not so clear for abstracta that are sui generis. |
| 10513 | Many abstract objects, such as chess, seem non-spatial, but are not atemporal [Hale] |
| Full Idea: There are many plausible example of abstract objects which, though non-spatial, do not appear to satisfy the suggested requirement of atemporality, such as chess, or the English language. | |
| From: Bob Hale (Abstract Objects [1987], Ch.3.1) | |
| A reaction: Given the point that modern physics is committed to 'space-time', with no conceivable separation of them, this looks dubious. Though I think the physics could be challenged. Try Idea 7621, for example. |
| 10514 | If the mental is non-spatial but temporal, then it must be classified as abstract [Hale] |
| Full Idea: If mental events are genuinely non-spatial, but not atemporal, its effect is to classify them as abstract; the distinction between the abstract and the mental simply collapses. | |
| From: Bob Hale (Abstract Objects [1987], Ch.3.1) | |
| A reaction: This is important. You can't discuss this sort of metaphysics in isolation from debates about the ontology of mind. Functionalists do treat mental events as abstractions. |
| 10523 | Being abstract is based on a relation between things which are spatially separated [Hale] |
| Full Idea: The abstract/concrete distinction is, roughly, between those sortals whose grounding relations can hold between abstract things which are spatially but not temporally separated, those concrete things whose grounding relations cannot so hold. | |
| From: Bob Hale (Abstract Objects [1987], Ch.3.III) | |
| A reaction: Thus being a father is based on 'begat', which does not involve spatial separation, and so is concrete. The relation is one of equivalence. |
| 10307 | The modern Fregean use of the term 'object' is much broader than the ordinary usage [Hale] |
| Full Idea: The notion of an 'object' first introduced by Frege is much broader than that of most comparable ordinary uses of 'object', and is now fairly standard and familiar. | |
| From: Bob Hale (Abstract Objects [1987], Ch.1) | |
| A reaction: This makes it very difficult to get to grips with the metaphysical issues involved, since the ontological claims disappear into a mist of semantic vagueness. |
| 10315 | We can't believe in a 'whereabouts' because we ask 'what kind of object is it?' [Hale] |
| Full Idea: Onotological outrage at such objects as the 'whereabouts of the Prime Minister' derives from the fact that we seem beggared for any convincing answer to the question 'What kind of objects are they?' | |
| From: Bob Hale (Abstract Objects [1987], Ch.2.II) | |
| A reaction: I go further and ask of any object 'what is it made of?' When I receive the answer that I am being silly, and that abstract objects are not 'made' of anything, I am tempted to become sarcastic, and say 'thank you - that makes it much clearer'. |
| 9891 | The first demand of logic is of a sharp boundary [Frege] |
| Full Idea: The first demand of logic is of a sharp boundary. | |
| From: Gottlob Frege (Grundgesetze der Arithmetik 2 (Basic Laws) [1903], §160), quoted by Michael Dummett - Frege philosophy of mathematics Ch.22 | |
| A reaction: Nothing I have read about vagueness has made me doubt Frege's view of this, although precisification might allow you to do logic with vague concepts without having to finally settle where the actual boundaries are. |
| 10522 | The relations featured in criteria of identity are always equivalence relations [Hale] |
| Full Idea: The relations which are featured in criteria of identity are always equivalence relations. | |
| From: Bob Hale (Abstract Objects [1987], Ch.3.III) | |
| A reaction: This will only apply to strict identity. If I say 'a is almost identical to b', this will obviously not be endlessly transitive (as when we get to k we may have lost the near-identity to a). Are 'two threes' identical to 'three twos'? |
| 10321 | We sometimes apply identity without having a real criterion [Hale] |
| Full Idea: Not every (apparent) judgement of identity involves application of anything properly describable as a criterion of identity, ...such as being able to pronounce that mercy is the quality of being merciful. | |
| From: Bob Hale (Abstract Objects [1987], Ch.2.II) | |
| A reaction: This suggests some distinction between internal criteria (e.g. grammatical, conceptual) and external criteria (existent, sensed). |
| 8840 | There are five possible responses to the problem of infinite regress in justification [Cleve] |
| Full Idea: Sceptics respond to the regress problem by denying knowledge; Foundationalists accept justifications without reasons; Positists say reasons terminate is mere posits; Coherentists say mutual support is justification; Infinitists accept the regress. | |
| From: James Van Cleve (Why coherence is not enough [2005], I) | |
| A reaction: A nice map of the territory. The doubts of Scepticism are not strong enough for anyone to embrace the view; Foundationalist destroy knowledge (?), as do Positists; Infinitism is a version of Coherentism - which is the winner. |
| 8841 | Modern foundationalists say basic beliefs are fallible, and coherence is relevant [Cleve] |
| Full Idea: Contemporary foundationalists are seldom of the strong Cartesian variety: they do not insist that basic beliefs be absolutely certain. They also tend to allow that coherence can enhance justification. | |
| From: James Van Cleve (Why coherence is not enough [2005], III) | |
| A reaction: It strikes me that they have got onto a slippery slope. How certain are the basic beliefs? How do you evaluate their certainty? Could incoherence in their implications undermine them? Skyscrapers need perfect foundations. |
| 9890 | The modern account of real numbers detaches a ratio from its geometrical origins [Frege] |
| Full Idea: From geometry we retain the interpretation of a real number as a ratio of quantities or measurement-number; but in more recent times we detach it from geometrical quantities, and from all particular types of quantity. | |
| From: Gottlob Frege (Grundgesetze der Arithmetik 2 (Basic Laws) [1903], §159), quoted by Michael Dummett - Frege philosophy of mathematics | |
| A reaction: Dummett glosses the 'recent' version as by Cantor and Dedekind in 1872. This use of 'detach' seems to me startlingly like the sort of psychological abstractionism which Frege was so desperate to avoid. |
| 11846 | If we abstract the difference between two houses, they don't become the same house [Frege] |
| Full Idea: If abstracting from the difference between my house and my neighbour's, I were to regard both houses as mine, the defect of the abstraction would soon be made clear. It may, though, be possible to obtain a concept by means of abstraction... | |
| From: Gottlob Frege (Grundgesetze der Arithmetik 2 (Basic Laws) [1903], §99) | |
| A reaction: Note the important concession at the end, which shows Frege could never deny the abstraction process, despite all the modern protests by Geach and Dummett that he totally rejected it. |