49 ideas
| 6161 | Structuralism is neo-Kantian idealism, with language playing the role of categories of understanding [Rowlands] |
| Full Idea: Structuralism is a form of neo-Kantian idealism, in which the job of creating Kant's phenomenal world has been taken over by language instead of forms of sensibility and categories of the understanding. | |
| From: Mark Rowlands (Externalism [2003], Ch.3) | |
| A reaction: A helpful connection, which explains my aversion to any attempt at understanding the world simply by analysing language, either in its ordinary usage, or in its underlying logical form. |
| 10633 | 'Some critics admire only one another' cannot be paraphrased in singular first-order [Linnebo] |
| Full Idea: The Geach-Kaplan sentence 'Some critics admire only one another' provably has no singular first-order paraphrase using only its predicates. | |
| From: Øystein Linnebo (Plural Quantification [2008], 1) | |
| A reaction: There seems to be a choice of either going second-order (picking out a property), or going plural (collectively quantifying), or maybe both. |
| 10779 | A comprehension axiom is 'predicative' if the formula has no bound second-order variables [Linnebo] |
| Full Idea: If φ contains no bound second-order variables, the corresponding comprehension axiom is said to be 'predicative'; otherwise it is 'impredicative'. | |
| From: Øystein Linnebo (Plural Quantification Exposed [2003], §1) | |
| A reaction: ['Predicative' roughly means that a new predicate is created, and 'impredicative' means that it just uses existing predicates] |
| 23445 | Naïve set theory says any formula defines a set, and coextensive sets are identical [Linnebo] |
| Full Idea: Naïve set theory is based on the principles that any formula defines a set, and that coextensive sets are identical. | |
| From: Øystein Linnebo (Philosophy of Mathematics [2017], 4.2) | |
| A reaction: The second principle is a standard axiom of ZFC. The first principle causes the trouble. |
| 10781 | A 'pure logic' must be ontologically innocent, universal, and without presuppositions [Linnebo] |
| Full Idea: I offer these three claims as a partial analysis of 'pure logic': ontological innocence (no new entities are introduced), universal applicability (to any realm of discourse), and cognitive primacy (no extra-logical ideas are presupposed). | |
| From: Øystein Linnebo (Plural Quantification Exposed [2003], §1) |
| 10638 | A pure logic is wholly general, purely formal, and directly known [Linnebo] |
| Full Idea: The defining features of a pure logic are its absolute generality (the objects of discourse are irrelevant), and its formality (logical truths depend on form, not matter), and its cognitive primacy (no extra-logical understanding is needed to grasp it). | |
| From: Øystein Linnebo (Plural Quantification [2008], 3) | |
| A reaction: [compressed] This strikes me as very important. The above description seems to contain no ontological commitment at all, either to the existence of something, or to two things, or to numbers, or to a property. Pure logic seems to be 'if-thenism'. |
| 6163 | If bivalence is rejected, then excluded middle must also be rejected [Rowlands] |
| Full Idea: If you reject the principle of bivalence (that a proposition is either determinately true or false), then statements are also not subject to the Law of Excluded Middle (P or not-P). | |
| From: Mark Rowlands (Externalism [2003], Ch.3) | |
| A reaction: I think Rowlands is wrong about this. Excluded Middle could be purely syntacti, or its semantics could be 'True or Not-True'. Only bivalent excluded middle introduces 'True or False'. Compare Idea 4752. |
| 10783 | Plural quantification depends too heavily on combinatorial and set-theoretic considerations [Linnebo] |
| Full Idea: If my arguments are correct, the theory of plural quantification has no right to the title 'logic'. ...The impredicative plural comprehension axioms depend too heavily on combinatorial and set-theoretic considerations. | |
| From: Øystein Linnebo (Plural Quantification Exposed [2003], §4) |
| 10635 | Second-order quantification and plural quantification are different [Linnebo] |
| Full Idea: Second-order quantification and plural quantification are generally regarded as different forms of quantification. | |
| From: Øystein Linnebo (Plural Quantification [2008], 2) |
| 10641 | Traditionally we eliminate plurals by quantifying over sets [Linnebo] |
| Full Idea: The traditional view in analytic philosophy has been that all plural locutions should be paraphrased away by quantifying over sets, though Boolos and other objected that this is unnatural and unnecessary. | |
| From: Øystein Linnebo (Plural Quantification [2008], 5) |
| 10640 | Instead of complex objects like tables, plurally quantify over mereological atoms tablewise [Linnebo] |
| Full Idea: Plural quantification can be used to eliminate the commitment of science and common sense to complex objects. We can use plural quantification over mereological atoms arranged tablewise or chairwise. | |
| From: Øystein Linnebo (Plural Quantification [2008], 4.5) | |
| A reaction: [He cites Hossack and van Ingwagen] |
| 10778 | Can second-order logic be ontologically first-order, with all the benefits of second-order? [Linnebo] |
| Full Idea: According to its supporters, second-order logic allow us to pay the ontological price of a mere first-order theory and get the corresponding monadic second-order theory for free. | |
| From: Øystein Linnebo (Plural Quantification Exposed [2003], §0) |
| 10636 | Plural plurals are unnatural and need a first-level ontology [Linnebo] |
| Full Idea: Higher-order plural quantification (plural plurals) is often rejected because plural quantification is supposedly ontological innocent, with no plural things to be plural, and because it is not found in ordinary English. | |
| From: Øystein Linnebo (Plural Quantification [2008], 2.4) | |
| A reaction: [Summary; he cites Boolos as a notable rejector] Linnebo observes that Icelandic contains a word 'tvennir' which means 'two pairs of'. |
| 10639 | Plural quantification may allow a monadic second-order theory with first-order ontology [Linnebo] |
| Full Idea: Plural quantification seems to offer ontological economy. We can pay the price of a mere first-order theory and then use plural quantification to get for free the corresponding monadic second-order theory, which would be an ontological bargain. | |
| From: Øystein Linnebo (Plural Quantification [2008], 4.4) | |
| A reaction: [He mentions Hellman's modal structuralism in mathematics] |
| 23447 | In classical semantics singular terms refer, and quantifiers range over domains [Linnebo] |
| Full Idea: In classical semantics the function of singular terms is to refer, and that of quantifiers, to range over appropriate domains of entities. | |
| From: Øystein Linnebo (Philosophy of Mathematics [2017], 7.1) |
| 23443 | The axioms of group theory are not assertions, but a definition of a structure [Linnebo] |
| Full Idea: Considered in isolation, the axioms of group theory are not assertions but comprise an implicit definition of some abstract structure, | |
| From: Øystein Linnebo (Philosophy of Mathematics [2017], 3.5) | |
| A reaction: The traditional Euclidean approach is that axioms are plausible assertions with which to start. The present idea sums up the modern approach. In the modern version you can work backwards from a structure to a set of axioms. |
| 23444 | To investigate axiomatic theories, mathematics needs its own foundational axioms [Linnebo] |
| Full Idea: Mathematics investigates the deductive consequences of axiomatic theories, but it also needs its own foundational axioms in order to provide models for its various axiomatic theories. | |
| From: Øystein Linnebo (Philosophy of Mathematics [2017], 4.1) | |
| A reaction: This is a problem which faces the deductivist (if-then) approach. The deductive process needs its own grounds. |
| 13007 | Archimedes defined a straight line as the shortest distance between two points [Archimedes, by Leibniz] |
| Full Idea: Archimedes gave a sort of definition of 'straight line' when he said it is the shortest line between two points. | |
| From: report of Archimedes (fragments/reports [c.240 BCE]) by Gottfried Leibniz - New Essays on Human Understanding 4.13 | |
| A reaction: Commentators observe that this reduces the purity of the original Euclidean axioms, because it involves distance and measurement, which are absent from the purest geometry. |
| 23446 | You can't prove consistency using a weaker theory, but you can use a consistent theory [Linnebo] |
| Full Idea: If the 2nd Incompleteness Theorem undermines Hilbert's attempt to use a weak theory to prove the consistency of a strong one, it is still possible to prove the consistency of one theory, assuming the consistency of another theory. | |
| From: Øystein Linnebo (Philosophy of Mathematics [2017], 4.6) | |
| A reaction: Note that this concerns consistency, not completeness. |
| 23448 | Mathematics is the study of all possible patterns, and is thus bound to describe the world [Linnebo] |
| Full Idea: Philosophical structuralism holds that mathematics is the study of abstract structures, or 'patterns'. If mathematics is the study of all possible patterns, then it is inevitable that the world is described by mathematics. | |
| From: Øystein Linnebo (Philosophy of Mathematics [2017], 11.1) | |
| A reaction: [He cites the physicist John Barrow (2010) for this] For me this is a major idea, because the concept of a pattern gives a link between the natural physical world and the abstract world of mathematics. No platonism is needed. |
| 14085 | 'Deductivist' structuralism is just theories, with no commitment to objects, or modality [Linnebo] |
| Full Idea: The 'deductivist' version of eliminativist structuralism avoids ontological commitments to mathematical objects, and to modal vocabulary. Mathematics is formulations of various (mostly categorical) theories to describe kinds of concrete structures. | |
| From: Øystein Linnebo (Structuralism and the Notion of Dependence [2008], 1) | |
| A reaction: 'Concrete' is ambiguous here, as mathematicians use it for the actual working maths, as opposed to the metamathematics. Presumably the structures are postulated rather than described. He cites Russell 1903 and Putnam. It is nominalist. |
| 14084 | Non-eliminative structuralism treats mathematical objects as positions in real abstract structures [Linnebo] |
| Full Idea: The 'non-eliminative' version of mathematical structuralism takes it to be a fundamental insight that mathematical objects are really just positions in abstract mathematical structures. | |
| From: Øystein Linnebo (Structuralism and the Notion of Dependence [2008], I) | |
| A reaction: The point here is that it is non-eliminativist because it is committed to the existence of mathematical structures. I oppose this view, since once you are committed to the structures, you may as well admit a vast implausible menagerie of abstracta. |
| 14086 | 'Modal' structuralism studies all possible concrete models for various mathematical theories [Linnebo] |
| Full Idea: The 'modal' version of eliminativist structuralism lifts the deductivist ban on modal notions. It studies what necessarily holds in all concrete models which are possible for various theories. | |
| From: Øystein Linnebo (Structuralism and the Notion of Dependence [2008], I) | |
| A reaction: [He cites Putnam 1967, and Hellman 1989] If mathematical truths are held to be necessary (which seems to be right), then it seems reasonable to include modal notions, about what is possible, in its study. |
| 14087 | 'Set-theoretic' structuralism treats mathematics as various structures realised among the sets [Linnebo] |
| Full Idea: 'Set-theoretic' structuralism rejects deductive nominalism in favour of a background theory of sets, and mathematics as the various structures realized among the sets. This is often what mathematicians have in mind when they talk about structuralism. | |
| From: Øystein Linnebo (Structuralism and the Notion of Dependence [2008], I) | |
| A reaction: This is the big shift from 'mathematics can largely be described in set theory' to 'mathematics just is set theory'. If it just is set theory, then which version of set theory? Which axioms? The safe iterative conception, or something bolder? |
| 14089 | Structuralism differs from traditional Platonism, because the objects depend ontologically on their structure [Linnebo] |
| Full Idea: Structuralism can be distinguished from traditional Platonism in that it denies that mathematical objects from the same structure are ontologically independent of one another | |
| From: Øystein Linnebo (Structuralism and the Notion of Dependence [2008], III) | |
| A reaction: My instincts strongly cry out against all versions of this. If you are going to be a platonist (rather as if you are going to be religious) you might as well go for it big time and have independent objects, which will then dictate a structure. |
| 14083 | Structuralism is right about algebra, but wrong about sets [Linnebo] |
| Full Idea: Against extreme views that all mathematical objects depend on the structures to which they belong, or that none do, I defend a compromise view, that structuralists are right about algebraic objects (roughly), but anti-structuralists are right about sets. | |
| From: Øystein Linnebo (Structuralism and the Notion of Dependence [2008], Intro) |
| 14090 | In mathematical structuralism the small depends on the large, which is the opposite of physical structures [Linnebo] |
| Full Idea: If objects depend on the other objects, this would mean an 'upward' dependence, in that they depend on the structure to which they belong, where the physical realm has a 'downward' dependence, with structures depending on their constituents. | |
| From: Øystein Linnebo (Structuralism and the Notion of Dependence [2008], III) | |
| A reaction: This nicely captures an intuition I have that there is something wrong with a commitment primarily to 'structures'. Our only conception of such things is as built up out of components. Not that I am committing to mathematical 'components'! |
| 23441 | Logical truth is true in all models, so mathematical objects can't be purely logical [Linnebo] |
| Full Idea: Modern logic requires that logical truths be true in all models, including ones devoid of any mathematical objects. It follows immediately that the existence of mathematical objects can never be a matter of logic alone. | |
| From: Øystein Linnebo (Philosophy of Mathematics [2017], 2) | |
| A reaction: Hm. Could there not be a complete set of models for a theory which all included mathematical objects? (I can't answer that). |
| 23442 | Game Formalism has no semantics, and Term Formalism reduces the semantics [Linnebo] |
| Full Idea: Game Formalism seeks to banish all semantics from mathematics, and Term Formalism seeks to reduce any such notions to purely syntactic ones. | |
| From: Øystein Linnebo (Philosophy of Mathematics [2017], 3.3) | |
| A reaction: This approach was stimulated by the need to justify the existence of the imaginary number i. Just say it is a letter! |
| 14091 | There may be a one-way direction of dependence among sets, and among natural numbers [Linnebo] |
| Full Idea: We can give an exhaustive account of the identity of the empty set and its singleton without mentioning infinite sets, and it might be possible to defend the view that one natural number depends on its predecessor but not vice versa. | |
| From: Øystein Linnebo (Structuralism and the Notion of Dependence [2008], V) | |
| A reaction: Linnebo uses this as one argument against mathematical structuralism, where the small seems to depend on the large. The view of sets rests on the iterative conception, where each level is derived from a lower level. He dismisses structuralism of sets. |
| 6155 | Supervenience is a one-way relation of dependence or determination between properties [Rowlands] |
| Full Idea: Supervenience is essentially a one-way relation of dependence or determination, …which holds, in the first instance, between properties. | |
| From: Mark Rowlands (Externalism [2003], Ch.2) | |
| A reaction: This definition immediately shows why supervenient properties are in danger of being epiphenomenal (i.e. causally irrelevant). Carefully thought about the notion of a 'one-way' relation will, I think, make it more obscure rather than clearer. |
| 10643 | We speak of a theory's 'ideological commitments' as well as its 'ontological commitments' [Linnebo] |
| Full Idea: Some philosophers speak about a theory's 'ideological commitments' and not just about its 'ontological commitments'. | |
| From: Øystein Linnebo (Plural Quantification [2008], 5.4) | |
| A reaction: This is a third strategy for possibly evading one's ontological duty, along with fiddling with the words 'exist' or 'object'. An ideological commitment to something to which one is not actually ontologically committed conjures up stupidity and dogma. |
| 10637 | Ordinary speakers posit objects without concern for ontology [Linnebo] |
| Full Idea: Maybe ordinary speakers aren't very concerned about their ontological commitments, and sometimes find it convenient to posit objects. | |
| From: Øystein Linnebo (Plural Quantification [2008], 2.4) | |
| A reaction: I think this is the whole truth about the ontological commitment of ordinary language. We bring abstraction under control by pretending it is a world of physical objects. The 'left wing' in politics, 'dark deeds', a 'huge difference'. |
| 14088 | An 'intrinsic' property is either found in every duplicate, or exists independent of all externals [Linnebo] |
| Full Idea: There are two main ways of spelling out an 'intrinsic' property: if and only if it is shared by every duplicate of an object, ...and if and only if the object would have this property even if the rest of the universe were removed or disregarded. | |
| From: Øystein Linnebo (Structuralism and the Notion of Dependence [2008], II) | |
| A reaction: [He cites B.Weatherson's Stanford Encyclopaedia article] How about an intrinsic property being one which explains its identity, or behaviour, or persistence conditions? |
| 10782 | The modern concept of an object is rooted in quantificational logic [Linnebo] |
| Full Idea: Our modern general concept of an object is given content only in connection with modern quantificational logic. | |
| From: Øystein Linnebo (Plural Quantification Exposed [2003], §2) | |
| A reaction: [He mentions Frege, Carnap, Quine and Dummett] This is the first thing to tell beginners in modern analytical metaphysics. The word 'object' is very confusing. I think I prefer 'entity'. |
| 6154 | It is argued that wholes possess modal and counterfactual properties that parts lack [Rowlands] |
| Full Idea: Some have argued that a mereological whole should not be identified with the sum of its parts on the grounds that the former possess certain properties - specifically modal and (perhaps) counterfactual properties - that the latter lacks. | |
| From: Mark Rowlands (Externalism [2003], Ch.2) | |
| A reaction: I am not convinced that modal and counterfactual claims should count as properties. If my pen is heated it melts (a property), but if my pen were intelligent it could do philosophy. Intelligence is a property, but the situation isn't. |
| 6157 | Tokens are dated, concrete particulars; types are their general properties or kinds [Rowlands] |
| Full Idea: Tokens are dated, concrete, particular occurrences or instances; types are the general properties that these occurrences exemplify or the kinds to which they belong. | |
| From: Mark Rowlands (Externalism [2003], Ch.2) | |
| A reaction: It might be said that types are sets, of which tokens are the members. The question of 'general properties' raises the question of whether universals must exist to make kinds possible. |
| 6159 | Strong idealism is the sort of mess produced by a Cartesian separation of mind and world [Rowlands] |
| Full Idea: Neo-Kantian idealism, and the excesses of recent versions of it, are precisely the sort of mess one can get oneself into through an uncritical acceptance of the dichotomizing of mind and world along Cartesian internalist lines. | |
| From: Mark Rowlands (Externalism [2003], Ch.3) | |
| A reaction: I am unconvinced that internalism about the mind (that its contents can be defined without reference to anything external) leads to this disastrous split. We don't have to abandon the links between an internal mind and the world. |
| 6152 | Minds are rational, conscious, subjective, self-knowing, free, meaningful and self-aware [Rowlands] |
| Full Idea: The apparent features of mind which are not obviously physical include: rationality, thought, consciousness, subjectivity, infallible first-person knowledge, freedom, meaning and self-awareness. | |
| From: Mark Rowlands (Externalism [2003], Ch.2) | |
| A reaction: A helpful list, some of which can be challenged. Ryle challenges first-person infallibility. Hume challenges self-awareness. Quine challenges meaning. Lots of people (e.g. Spinoza) challenge freedom. The Churchlands seem to challenge consciousness. |
| 6173 | Content externalism implies that we do not have privileged access to our own minds [Rowlands] |
| Full Idea: Content externalism threatens the idea of first-person authority in all its forms, and does so because it calls into question the idea that the access we have to our own mental states is privileged in the way required for such authority. | |
| From: Mark Rowlands (Externalism [2003], Ch.7) | |
| A reaction: I am inclined to respond by saying that since we clearly have privileged access to our own minds, that means there must be something wrong with content externalism. |
| 6174 | If someone is secretly transported to Twin Earth, others know their thoughts better than they do [Rowlands] |
| Full Idea: If someone knew that a thinker had, without realising it, been transported to Twin Earth, they would almost certainly be a higher authority on the content of the thinker's thoughts than would the thinker. | |
| From: Mark Rowlands (Externalism [2003], Ch.8) | |
| A reaction: They would certainly be a higher authority on the truth of the thinker's thoughts, but only in the way that you might think I hold a diamond when I know it is a club. If the thinker believes it is H2O, the fact that it isn't is irrelevant to content. |
| 6158 | Supervenience of mental and physical properties often comes with token-identity of mental and physical particulars [Rowlands] |
| Full Idea: One often finds a supervenience thesis concerning the relation between mental and physical properties combined with a token identity theory concerning the relation between mental and physical particulars. | |
| From: Mark Rowlands (Externalism [2003], Ch.2) | |
| A reaction: This brings out the important clarifying point that supervenience is said to be between properties, not substances. The point is that supervenience will always cry out for an explanation, preferably a sensible one. |
| 6168 | The content of a thought is just the meaning of a sentence [Rowlands] |
| Full Idea: The content of the thought that the sky is blue is simply the meaning of the sentence "The sky is blue". | |
| From: Mark Rowlands (Externalism [2003], Ch.5) | |
| A reaction: This seems to imply that it is logically impossible for a non-language-speaker, such as a chimpanzee, to think that the sky is the same colour as the water. If we allow propositions, we might be able to keep meanings without the sentences. |
| 10634 | Predicates are 'distributive' or 'non-distributive'; do individuals do what the group does? [Linnebo] |
| Full Idea: The predicate 'is on the table' is 'distributive', since some things are on the table if each one is, whereas the predicate 'form a circle' is 'non-distributive', since it is not analytic that when some things form a circle, each one forms a circle. | |
| From: Øystein Linnebo (Plural Quantification [2008], 1.1) | |
| A reaction: The first predicate can have singular or plural subjects, but the second requires a plural subject? Hm. 'The rope forms a circle'. The second is example is not true, as well as not analytic. |
| 6167 | Action is bodily movement caused by intentional states [Rowlands] |
| Full Idea: An action is a bodily movement that is caused by intentional states such as beliefs, desires and so on. | |
| From: Mark Rowlands (Externalism [2003], Ch.5) | |
| A reaction: A useful definition, and clearly one that has no truck with attempts at giving behaviourist definitions of action. The definition of a 'moral action' needs to be built on this one. Particular types of belief and desire, presumably. |
| 6177 | Moral intuition seems unevenly distributed between people [Rowlands] |
| Full Idea: The faculty of moral intuition seems to be unevenly distributed between people. | |
| From: Mark Rowlands (Externalism [2003], Ch.11) | |
| A reaction: This would be a good argument if it was thought that the source of moral intuitions was divine, but people vary enormously in their intuitions about maths, about character, about danger. If you believe in any intuition at all, you must accept its variety. |
| 6156 | The 17th century reintroduced atoms as mathematical modes of Euclidean space [Rowlands] |
| Full Idea: The seventeenth century revolution reintroduced the classical concept of the atom in somewhat new attire as an essentially mathematical entity whose primary qualities could be precisely quantified as modes or aspects of Euclidean space. | |
| From: Mark Rowlands (Externalism [2003], Ch.2) | |
| A reaction: Obviously this very abstract view of atoms didn't last, once they began to identify specific physical atoms, such as oxygen. This view fits in with Newton's use of pure (abstract) points such as the 'centre of gravity'. |
| 6170 | Natural kinds are defined by their real essence, as in gold having atomic number 79 [Rowlands] |
| Full Idea: Part of what it means to be a natural kind is that they are defined by a real essence, a constitution that marks them out as the substance they are (as water is essentially H2O, and gold essentially has atomic number 79). | |
| From: Mark Rowlands (Externalism [2003], Ch.6) | |
| A reaction: A 'real essence' would be the opposite of a 'conventional essence', which is just a human way of seeing things. |
| 6178 | It is common to see the value of nature in one feature, such as life, diversity, or integrity [Rowlands] |
| Full Idea: In recent environmental philosophy it is common to see the value of nature identified with one or another natural feature of the environment: life, diversity, ecosystemic integrity and so on. | |
| From: Mark Rowlands (Externalism [2003], Ch.11) | |
| A reaction: This thought seems to be asking for the Open Question argument. What is so good about life, or diversity? Our strongest intuition must be that the survival of the ecosystem, and whatever makes that possible, is the highest value. |