26 ideas
| 2510 | Traditionally philosophy is an a priori enquiry into general truths about reality [Katz] |
| Full Idea: The traditional conception of philosophy is that it is an a priori enquiry into the most general facts about reality. | |
| From: Jerrold J. Katz (Realistic Rationalism [2000], Int.xi) | |
| A reaction: I think this still defines philosophy, though it also highlights the weakness of the subject, which is over-confidence about asserting necessary truths. How could the most god-like areas of human thought be about anything else? |
| 2516 | Most of philosophy begins where science leaves off [Katz] |
| Full Idea: Philosophy, or at least one large part of it, is subsequent to science; it begins where science leaves off. | |
| From: Jerrold J. Katz (Realistic Rationalism [2000], Int.xxi) | |
| A reaction: In some sense this has to be true. Without metaphysics there couldn't be any science. Rationalists should not forget, though, the huge impact which Darwin's science has (or should have) on fairly abstract philosophy (e.g. epistemology). |
| 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. |
| 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. |
| 2521 | 'Real' maths objects have no causal role, no determinate reference, and no abstract/concrete distinction [Katz] |
| Full Idea: Three objections to realism in philosophy of mathematics: mathematical objects have no space/time location, and so no causal role; that such objects are determinate, but reference to numbers aren't; and that there is no abstract/concrete distinction. | |
| From: Jerrold J. Katz (Realistic Rationalism [2000], Int.xxix) |
| 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. |
| 13074 | Only natural kinds and their members have real essences [Suárez, by Cover/O'Leary-Hawthorne] |
| Full Idea: On Suarez's account, only natural kinds and their members have real essences. | |
| From: report of Francisco Suárez (works [1588]) by Cover,J/O'Leary-Hawthorne,J - Substance and Individuation in Leibniz 1.3.1 n21 | |
| A reaction: Interesting. Rather than say that everything is a member of some kind, we leave quirky individuals out, with no essence at all. What is the status of the very first exemplar of a given kind? |
| 2513 | We don't have a clear enough sense of meaning to pronounce some sentences meaningless or just analytic [Katz] |
| Full Idea: Linguistic meaning is not rich enough to show either that all metaphysical sentences are meaningless or that all alleged synthetic a priori propositions are just analytic a priori propositions. | |
| From: Jerrold J. Katz (Realistic Rationalism [2000], Int.xx) |
| 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. |
| 2522 | Experience cannot teach us why maths and logic are necessary [Katz] |
| Full Idea: The Leibniz-Kant criticism of empiricism is that experience cannot teach us why mathematical and logical facts couldn't be otherwise than they are. | |
| From: Jerrold J. Katz (Realistic Rationalism [2000], Int.xxxi) |
| 2517 | Structuralists see meaning behaviouristically, and Chomsky says nothing about it [Katz] |
| Full Idea: In linguistics there are two schools of thought: Bloomfieldian structuralism (favoured by Quine) conceives of sentences acoustically and meanings behaviouristically; and Chomskian generative grammar (which is silent about semantics). | |
| From: Jerrold J. Katz (Realistic Rationalism [2000], Int.xxiv) | |
| A reaction: They both appear to be wrong, so there is (or was) something rotten in the state of linguistics. Are the only options for meaning either behaviourist or eliminativist? |
| 2519 | It is generally accepted that sense is defined as the determiner of reference [Katz] |
| Full Idea: There is virtually universal acceptance of Frege's definition of sense as the determiner of reference. | |
| From: Jerrold J. Katz (Realistic Rationalism [2000], Int.xxvi) | |
| A reaction: Not any more, since Kripke and Putnam. It is one thing to say sense determines reference, and quite another to say that this is the definition of sense. |
| 2520 | Sense determines meaning and synonymy, not referential properties like denotation and truth [Katz] |
| Full Idea: Pace Frege, sense determines sense properties and relations, like meaningfulness and synonymy, rather than determining referential properties, like denotation and truth. | |
| From: Jerrold J. Katz (Realistic Rationalism [2000], Int.xxvi) | |
| A reaction: This leaves room for Fregean 'sense', after Kripke has demolished the idea that sense determines reference. |
| 2518 | Sentences are abstract types (like musical scores), not individual tokens [Katz] |
| Full Idea: Sentences are types, not utterance tokens or mental/neural tokens, and hence sentences are abstract objects (like musical scores). | |
| From: Jerrold J. Katz (Realistic Rationalism [2000], Int.xxvi) | |
| A reaction: If sentences are abstract types, then two verbally indistinguishable sentences are the same sentence. But if I say 'I am happy', that isn't the same as you saying it. |
| 7563 | The old 'influx' view of causation says it is a flow of accidental properties from A to B [Suárez, by Jolley] |
| Full Idea: The 'influx' model of causation says that causes involve a process of contagion, as it were; when the kettle boils, the gas infects the water inside the kettle with its own 'individual accident' of heat, which literally flows from one to the other. | |
| From: report of Francisco Suárez (works [1588]) by Nicholas Jolley - Leibniz Ch.2 | |
| A reaction: This nicely captures the scholastic target of Hume's sceptical thinking on the subject. However, see Idea 2542, where the idea of influx has had a revival. It is hard to see how the water could change if it didn't 'catch' something from the gas. |