40 ideas
| 16943 | Philosophy is continuous with science, and has no external vantage point [Quine] |
| Full Idea: I see philosophy not as an a priori propaedeutic or groundwork for science, but as continuous with science. I see philosophy and science as in the same boat. …There is no external vantage point, no first philosophy. | |
| From: Willard Quine (Natural Kinds [1969], p.126) | |
| A reaction: Philosophy is generalisation. Science holds the upper hand, because it settles the subject-matter to be generalised. |
| 8250 | So-called 'free logic' operates without existence assumptions [Meinong, by George/Van Evra] |
| Full Idea: Meinong has recently been credited with inspiring 'free logic': a logic without existence assumptions. | |
| From: report of Alexius Meinong (The Theory of Objects [1904]) by George / Van Evra - The Rise of Modern Logic 8 | |
| A reaction: This would appear to be a bold escape from the quandries concerning the existential implications of quantifiers. I immediately find it very appealing. It seems to spell disaster for the Quinean program of deducing ontology from language. |
| 8729 | Intuitionists deny excluded middle, because it is committed to transcendent truth or objects [Shapiro] |
| Full Idea: Intuitionists in mathematics deny excluded middle, because it is symptomatic of faith in the transcendent existence of mathematical objects and/or the truth of mathematical statements. | |
| From: Stewart Shapiro (Thinking About Mathematics [2000], 1.2) | |
| A reaction: There are other problems with excluded middle, such as vagueness, but on the whole I, as a card-carrying 'realist', am committed to the law of excluded middle. |
| 16949 | Klein summarised geometry as grouped together by transformations [Quine] |
| Full Idea: Felix Klein's so-called 'Erlangerprogramm' in geometry involved characterizing the various branches of geometry by what transformations were irrelevant to each. | |
| From: Willard Quine (Natural Kinds [1969], p.137) |
| 8763 | The number 3 is presumably identical as a natural, an integer, a rational, a real, and complex [Shapiro] |
| Full Idea: It is surely wise to identify the positions in the natural numbers structure with their counterparts in the integer, rational, real and complex number structures. | |
| From: Stewart Shapiro (Thinking About Mathematics [2000], 10.2) | |
| A reaction: The point is that this might be denied, since 3, 3/1, 3.00.., and -3*i^2 are all arrived at by different methods of construction. Natural 3 has a predecessor, but real 3 doesn't. I agree, intuitively, with Shapiro. Russell (1919) disagreed. |
| 18249 | Cauchy gave a formal definition of a converging sequence. [Shapiro] |
| Full Idea: A sequence a1,a2,... of rational numbers is 'Cauchy' if for each rational number ε>0 there is a natural number N such that for all natural numbers m, n, if m>N and n>N then -ε < am - an < ε. | |
| From: Stewart Shapiro (Thinking About Mathematics [2000], 7.2 n4) | |
| A reaction: The sequence is 'Cauchy' if N exists. |
| 8764 | Categories are the best foundation for mathematics [Shapiro] |
| Full Idea: There is a dedicated contingent who hold that the category of 'categories' is the proper foundation for mathematics. | |
| From: Stewart Shapiro (Thinking About Mathematics [2000], 10.3 n7) | |
| A reaction: He cites Lawvere (1966) and McLarty (1993), the latter presenting the view as a form of structuralism. I would say that the concept of a category will need further explication, and probably reduce to either sets or relations or properties. |
| 8762 | Two definitions of 3 in terms of sets disagree over whether 1 is a member of 3 [Shapiro] |
| Full Idea: Zermelo said that for each number n, its successor is the singleton of n, so 3 is {{{null}}}, and 1 is not a member of 3. Von Neumann said each number n is the set of numbers less than n, so 3 is {null,{null},{null,{null}}}, and 1 is a member of 3. | |
| From: Stewart Shapiro (Thinking About Mathematics [2000], 10.2) | |
| A reaction: See Idea 645 - Zermelo could save Plato from the criticisms of Aristotle! These two accounts are cited by opponents of the set-theoretical account of numbers, because it seems impossible to arbitrate between them. |
| 8760 | Numbers do not exist independently; the essence of a number is its relations to other numbers [Shapiro] |
| Full Idea: The structuralist vigorously rejects any sort of ontological independence among the natural numbers; the essence of a natural number is its relations to other natural numbers. | |
| From: Stewart Shapiro (Thinking About Mathematics [2000], 10.1) | |
| A reaction: This seems to place the emphasis on ordinals (what order?) rather than on cardinality (how many?). I am strongly inclined to think that this is the correct view, though you can't really have relations if there is nothing to relate. |
| 8761 | A 'system' is related objects; a 'pattern' or 'structure' abstracts the pure relations from them [Shapiro] |
| Full Idea: A 'system' is a collection of objects with certain relations among them; a 'pattern' or 'structure' is the abstract form of a system, highlighting the interrelationships and ignoring any features they do not affect how they relate to other objects. | |
| From: Stewart Shapiro (Thinking About Mathematics [2000], 10.1) | |
| A reaction: Note that 'ignoring' features is a psychological account of abstraction, which (thanks to Frege and Geach) is supposed to be taboo - but which I suspect is actually indispensable in any proper account of thought and concepts. |
| 8744 | Logicism seems to be a non-starter if (as is widely held) logic has no ontology of its own [Shapiro] |
| Full Idea: The thesis that principles of arithmetic are derivable from the laws of logic runs against a now common view that logic itself has no ontology. There are no particular logical objects. From this perspective logicism is a non-starter. | |
| From: Stewart Shapiro (Thinking About Mathematics [2000], 5.1) | |
| A reaction: This criticism strikes me as utterly devastating. There are two routes to go: prove that logic does have an ontology of objects (what would they be?), or - better - deny that arithmetic contains any 'objects'. Or give up logicism. |
| 8749 | Term Formalism says mathematics is just about symbols - but real numbers have no names [Shapiro] |
| Full Idea: Term Formalism is the view that mathematics is just about characters or symbols - the systems of numerals and other linguistic forms. ...This will cover integers and rational numbers, but what are real numbers supposed to be, if they lack names? | |
| From: Stewart Shapiro (Thinking About Mathematics [2000], 6.1.1) | |
| A reaction: Real numbers (such as pi and root-2) have infinite decimal expansions, so we can start naming those. We could also start giving names like 'Harry' to other reals, though it might take a while. OK, I give up. |
| 8750 | Game Formalism is just a matter of rules, like chess - but then why is it useful in science? [Shapiro] |
| Full Idea: Game Formalism likens mathematics to chess, where the 'content' of mathematics is exhausted by the rules of operating with its language. ...This, however, leaves the problem of why the mathematical games are so useful to the sciences. | |
| From: Stewart Shapiro (Thinking About Mathematics [2000], 6.1.2) | |
| A reaction: This thought pushes us towards structuralism. It could still be a game, but one we learned from observing nature, which plays its own games. Chess is, after all, modelled on warfare. |
| 8752 | Deductivism says mathematics is logical consequences of uninterpreted axioms [Shapiro] |
| Full Idea: The Deductivist version of formalism (sometimes called 'if-thenism') says that the practice of mathematics consists of determining logical consequences of otherwise uninterpreted axioms. | |
| From: Stewart Shapiro (Thinking About Mathematics [2000], 6.2) | |
| A reaction: [Hilbert is the source] More plausible than Term or Game Formalism (qv). It still leaves the question of why it seems applicable to nature, and why those particular axioms might be chosen. In some sense, though, it is obviously right. |
| 8753 | Critics resent the way intuitionism cripples mathematics, but it allows new important distinctions [Shapiro] |
| Full Idea: Critics commonly complain that the intuitionist restrictions cripple the mathematician. On the other hand, intuitionist mathematics allows for many potentially important distinctions not available in classical mathematics, and is often more subtle. | |
| From: Stewart Shapiro (Thinking About Mathematics [2000], 7.1) | |
| A reaction: The main way in which it cripples is its restriction on talk of infinity ('Cantor's heaven'), which was resented by Hilbert. Since high-level infinities are interesting, it would be odd if we were not allowed to discuss them. |
| 8731 | Conceptualist are just realists or idealist or nominalists, depending on their view of concepts [Shapiro] |
| Full Idea: I classify conceptualists according to what they say about properties or concepts. If someone classified properties as existing independent of language I would classify her as a realist in ontology of mathematics. Or they may be idealists or nominalists. | |
| From: Stewart Shapiro (Thinking About Mathematics [2000], 2.2.1) | |
| A reaction: In other words, Shapiro wants to eliminate 'conceptualist' as a useful label in philosophy of mathematics. He's probably right. All thought involves concepts, but that doesn't produce a conceptualist theory of, say, football. |
| 8730 | 'Impredicative' definitions refer to the thing being described [Shapiro] |
| Full Idea: A definition of a mathematical entity is 'impredicative' if it refers to a collection that contains the defined entity. The definition of 'least upper bound' is impredicative as it refers to upper bounds and characterizes a member of this set. | |
| From: Stewart Shapiro (Thinking About Mathematics [2000], 1.2) | |
| A reaction: The big question is whether mathematics can live with impredicative definitions, or whether they threaten to be viciously circular, and undermine the whole enterprise. |
| 16939 | Mass terms just concern spread, but other terms involve both spread and individuation [Quine] |
| Full Idea: 'Yellow' and 'water' are mass terms, concerned only with spread; 'apple' and 'square' are terms of divided reference, concerned with both spread and individuation. | |
| From: Willard Quine (Natural Kinds [1969], p.124) | |
| A reaction: Would you like some apple? Pass me that water. It is helpful to see that it is a requirement of 'individuation' that is missing from terms for stuff. |
| 16948 | Once we know the mechanism of a disposition, we can eliminate 'similarity' [Quine] |
| Full Idea: Once we can legitimize a disposition term by defining the relevant similarity standard, we are apt to know the mechanism of the disposition, and so by-pass the similarity. | |
| From: Willard Quine (Natural Kinds [1969], p.135) | |
| A reaction: I love mechanisms, but can we characterise mechanisms without mentioning powers and dispositions? Quine's dream is to eliminate 'similarity'. |
| 16945 | We judge things to be soluble if they are the same kind as, or similar to, things that do dissolve [Quine] |
| Full Idea: Intuitively, what qualifies a thing as soluble though it never gets into water is that it is of the same kind as the things that actually did or will dissolve; it is similar to them. | |
| From: Willard Quine (Natural Kinds [1969], p.130) | |
| A reaction: If you can judge that the similar things 'will' dissolve, you can cut to the chase and judge that this thing will dissolve. |
| 8719 | There can be impossible and contradictory objects, if they can have properties [Meinong, by Friend] |
| Full Idea: Meinong (and Priest) leave room for impossible objects (like a mountain made entirely of gold), and even contradictory objects (such as a round square). This would have a property, of 'being a contradictory object'. | |
| From: report of Alexius Meinong (The Theory of Objects [1904]) by Michèle Friend - Introducing the Philosophy of Mathematics 6.8 | |
| A reaction: This view is only possible with a rather lax view of properties. Personally I don't take 'being a pencil' to be a property of a pencil. It might be safer to just say that 'round squares' are possible linguistic subjects of predication. |
| 8971 | There are objects of which it is true that there are no such objects [Meinong] |
| Full Idea: There are objects of which it is true that there are no such objects. | |
| From: Alexius Meinong (The Theory of Objects [1904]), quoted by Peter van Inwagen - Existence,Ontological Commitment and Fictions p.131 | |
| A reaction: Van Inwagen say this idea is 'infamous', but Meinong is undergoing a revival, and commitment to non-existent objects may be the best explanation of some ways of talking. |
| 8718 | Meinong says an object need not exist, but must only have properties [Meinong, by Friend] |
| Full Idea: Meinong distinguished between 'existing objects' and 'subsisting objects', and being an object does not imply existence, but only 'having properties'. | |
| From: report of Alexius Meinong (The Theory of Objects [1904]) by Michèle Friend - Introducing the Philosophy of Mathematics 6.8 | |
| A reaction: Meinong is treated as a joke (thanks to Russell), but this is good. "Father Christmas does not exist, but he has a red coat". He'd better have some sort of existy aspect if he is going to have a property. So he's 'an object'. 'Insubstantial'? |
| 7756 | Meinong said all objects of thought (even self-contradictions) have some sort of being [Meinong, by Lycan] |
| Full Idea: Meinong insisted (à la Anselm) that any possible object of thought - even a self-contradictory one - has being of a sort even though only a few such things are so lucky as to exist in reality as well. | |
| From: report of Alexius Meinong (The Theory of Objects [1904]) by William Lycan - Philosophy of Language Ch.1 | |
| A reaction: ['This idea gave Russell fits' says Lycan]. In the English-speaking world this is virtually the only idea for which Meinong is remembered. Russell (Idea 5409) was happy for some things to merely 'subsist' as well as others which could 'exist'. |
| 15781 | The objects of knowledge are far more numerous than objects which exist [Meinong] |
| Full Idea: The totality of what exists, including what has existed and what will exist, is infinitely small in comparison with the totality of Objects of knowledge. | |
| From: Alexius Meinong (The Theory of Objects [1904]), quoted by William Lycan - The Trouble with Possible Worlds 01 | |
| A reaction: This is rather profound, but the word 'object' doesn't help. I would say 'What we know concerns far more than what merely exists'. |
| 8725 | Rationalism tries to apply mathematical methodology to all of knowledge [Shapiro] |
| Full Idea: Rationalism is a long-standing school that can be characterized as an attempt to extend the perceived methodology of mathematics to all of knowledge. | |
| From: Stewart Shapiro (Thinking About Mathematics [2000], 1.1) | |
| A reaction: Sometimes called 'Descartes's Dream', or the 'Enlightenment Project', the dream of proving everything. Within maths, Hilbert's Programme aimed for the same certainty. Idea 22 is the motto for the opposition to this approach. |
| 16944 | Science is common sense, with a sophisticated method [Quine] |
| Full Idea: Sciences differ from common sense only in the degree of methodological sophistication. | |
| From: Willard Quine (Natural Kinds [1969], p.129) | |
| A reaction: Science is normal thinking about the world, but it is teamwork, with the bar set very high. |
| 16940 | Induction is just more of the same: animal expectations [Quine] |
| Full Idea: Induction is essentially only more of the same: animal expectation or habit formation. | |
| From: Willard Quine (Natural Kinds [1969], p.125) | |
| A reaction: My working definition of induction is 'learning from experience', but that doesn't disagree with Quine. Lipton has a richer account of different types of induction. Quine's point is that it rests on resemblance. |
| 16941 | Induction relies on similar effects following from each cause [Quine] |
| Full Idea: Induction expresses our hopes that similar causes will have similar effects. | |
| From: Willard Quine (Natural Kinds [1969], p.125) | |
| A reaction: Some top philosophers are also top teachers, and Quine was one of them, in his writings. He boils it down for the layman. Once again, he is pointing to the fundamental role of the similarity relation. |
| 16933 | Grue is a puzzle because the notions of similarity and kind are dubious in science [Quine] |
| Full Idea: What makes Goodman's example a puzzle is the dubious scientific standing of a general notion of similarity, or of kind. | |
| From: Willard Quine (Natural Kinds [1969], p.116) | |
| A reaction: Illuminating. It might be best expressed as revealing a problem with sortal terms, as employed by Geach, or by Wiggins. Grue is a bit silly, but sortals are subject to convention and culture. 'Natural' properties seem needed. |
| 16934 | General terms depend on similarities among things [Quine] |
| Full Idea: The usual general term, whether a common noun or a verb or an adjective, owes its generality to some resemblance among the things referred to. | |
| From: Willard Quine (Natural Kinds [1969], p.116) | |
| A reaction: Quine has a nice analysis of the basic role of similarity in a huge amount of supposedly strict scientific thought. |
| 16938 | To learn yellow by observation, must we be told to look at the colour? [Quine] |
| Full Idea: According to the 'respects' view, our learning of yellow by ostension would have depended on our first having been told or somehow apprised that it was going to be a question of color. | |
| From: Willard Quine (Natural Kinds [1969], p.122) | |
| A reaction: Quine suggests there is just one notion of similarity, and respects can be 'abstracted' afterwards. Even the ontologically ruthless Quine admits psychological abstraction! |
| 8486 | Standards of similarity are innate, and the spacing of qualities such as colours can be mapped [Quine] |
| Full Idea: A standard of similarity is in some sense innate. The spacing of qualities (such as red, pink and blue) can be explored and mapped in the laboratory by experiments. They are needed for all learning. | |
| From: Willard Quine (Natural Kinds [1969], p.123) | |
| A reaction: This reasserts Hume's original point in more scientific terms. It is one of the undeniable facts about our perceptions of qualities and properties, no matter how platonist your view of universals may be. |
| 16947 | Similarity is just interchangeability in the cosmic machine [Quine] |
| Full Idea: Things are similar to the extent that they are interchangeable parts of the cosmic machine. | |
| From: Willard Quine (Natural Kinds [1969], p.134) | |
| A reaction: This is a major idea for Quine, because it is a means to gradually eliminate the fuzzy ideas of 'resemblance' or 'similarity' or 'natural kind' from science. I love it! Two tigers are same insofar as they are substitutable. |
| 16932 | Projectible predicates can be universalised about the kind to which they refer [Quine] |
| Full Idea: 'Projectible' predicates are predicates F and G whose shared instances all do count, for whatever reason, towards confirmation of 'All F are G'. ….A projectible predicate is one that is true of all and only the things of a kind. | |
| From: Willard Quine (Natural Kinds [1969], p.115-6) | |
| A reaction: Both Quine and Goodman are infuriatingly brief about the introduction of this concept. 'Red' is true of all ripe tomatoes, but not 'only' of them. Hardly any predicates are true only of one kind. Is that a scholastic 'proprium'? |
| 7375 | Quine probably regrets natural kinds now being treated as essences [Quine, by Dennett] |
| Full Idea: The concept of natural kinds was reintroduced by Quine, who may now regret the way it has become a stand-in for the dubious but covertly popular concept of essences. | |
| From: report of Willard Quine (Natural Kinds [1969]) by Daniel C. Dennett - Consciousness Explained 12.2 n2 | |
| A reaction: He is right that Quine would regret it, and he is right that we can't assume that there are necessary essences just because there seem to be stable natural kinds, but personally I am an essentialist, so I'm not that bothered. |
| 16935 | If similarity has no degrees, kinds cannot be contained within one another [Quine] |
| Full Idea: If similarity has no degrees there is no containing of kinds within broader kinds. If colored things are a kind, they are similar, but red things are too narrow for a kind. If red things are a kind, colored things are not similar, and it's too broad. | |
| From: Willard Quine (Natural Kinds [1969], p.118) | |
| A reaction: [compressed] I'm on Quine's side with this. We glibly talk of 'kinds', but the criteria for sorting things into kinds seems to be a mess. Quine goes on to offer a better account than the (diadic, yes-no) one rejected here. |
| 16936 | Comparative similarity allows the kind 'colored' to contain the kind 'red' [Quine] |
| Full Idea: With the triadic relation of comparative similarity, kinds can contain one another, as well as overlapping. Red and colored things can both count as kinds. Colored things all resemble one another, even though less than red things do. | |
| From: Willard Quine (Natural Kinds [1969], p.119) | |
| A reaction: [compressed] Quine claims that comparative similarity is necessary for kinds - that there be some 'foil' in a similarity - that A is more like C than B is. |
| 16937 | You can't base kinds just on resemblance, because chains of resemblance are a muddle [Quine] |
| Full Idea: If kinds are based on similarity, this has the Imperfect Community problem. Red round, red wooden and round wooden things all resemble one another somehow. There may be nothing outside the set resembling them, so it meets the definition of kind. | |
| From: Willard Quine (Natural Kinds [1969], p.120) | |
| A reaction: [ref. to Goodman 'Structure' 2nd 163- , which attacks Carnap on this] This suggests an invocation of Wittgenstein's family resemblance, which won't be much help for natural kinds. |
| 16942 | It is hard to see how regularities could be explained [Quine] |
| Full Idea: Why there have been regularities is an obscure question, for it is hard to see what would count as an answer. | |
| From: Willard Quine (Natural Kinds [1969], p.126) | |
| A reaction: This is the standard pessimism of the 20th century Humeans, but it strikes me as comparable to the pessimism about science found in Locke and Hume. Regularities are explained all the time by scientists, though the lowest level may be hopeless. |