26 ideas
| 9560 | S5 provides the correct logic for necessity in the broadly logical sense [Fine,K] |
| Full Idea: S5 provides the correct logic for necessity in the broadly logical sense. | |
| From: Kit Fine (Model Theory for Modal Logic I [1978], 151), quoted by Charles Chihara - A Structural Account of Mathematics | |
| A reaction: I have no view on this, but I am prejudiced in favour of the idea that there is a correct logic for such things, whichever one it may be. Presumably the fact that S5 has no restrictions on accessibility makes it more comprehensive and 'metaphysical'. |
| 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. |
| 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. |
| 14329 | Some dispositional properties (such as mental ones) may have no categorical base [Price,HH] |
| Full Idea: There is no a priori necessity for supposing that all disposition properties must have a 'categorical base'. In particular, there may be some mental dispositions which are ultimate. | |
| From: H.H. Price (Thinking and Experience [1953], Ch.XI) | |
| A reaction: I take the notion that mental dispositions could be ultimate as rather old-fashioned, but I agree with the notion that dispositions might be more fundamental that categorical (actual) properties. Personally I like 'powers'. |
| 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. |
| 9032 | Before we can abstract from an instance of violet, we must first recognise it [Price,HH] |
| Full Idea: Abstraction is preceded by an earlier stage, in which we learn to recognize instances; before I can conceive of the colour violet in abstracto, I must learn to recognize instances of this colour when I see them. | |
| From: H.H. Price (Thinking and Experience [1953], Ch.II) | |
| A reaction: The problem here might be one of circularity. If you are actually going to identify something as violet, you seem to need the abstract concept of 'violet' in advance. See Idea 9034 for Price's attempt to deal with the problem. |
| 9035 | If judgement of a characteristic is possible, that part of abstraction must be complete [Price,HH] |
| Full Idea: If we are to 'judge' - rightly or not - that this object has a specific characteristic, it would seem that so far as the characteristic is concerned the process of abstraction must already be completed. | |
| From: H.H. Price (Thinking and Experience [1953], Ch.III) | |
| A reaction: Personally I think Price is right, despite the vicious attack from Geach that looms. We all know the experiences of familiarity, recognition, and identification that go on when see a person or picture. 'What animal is that, in the distance?' |
| 9034 | There may be degrees of abstraction which allow recognition by signs, without full concepts [Price,HH] |
| Full Idea: If abstraction is a matter of degree, and the first faint beginnings of it are already present as soon as anything has begun to feel familiar to us, then recognition by means of signs can occur long before the process of abstraction has been completed. | |
| From: H.H. Price (Thinking and Experience [1953], Ch.III) | |
| A reaction: I like this, even though it is unscientific introspective psychology, for which no proper evidence can be adduced - because it is right. Neuroscience confirms that hardly any mental life has an all-or-nothing form. |
| 9036 | There is pre-verbal sign-based abstraction, as when ice actually looks cold [Price,HH] |
| Full Idea: We must still insist that some degree of abstraction, and even a very considerable degree of it, is present in sign-cognition, pre-verbal as it is. ...To us, who are familiar with northern winters, the ice actually looks cold. | |
| From: H.H. Price (Thinking and Experience [1953], Ch.IV) | |
| A reaction: Price may be in the weak position of doing armchair psychology, but something like his proposal strikes me as correct. I'm much happier with accounts of thought that talk of 'degrees' of an activity, than with all-or-nothing cut-and-dried pictures. |
| 9037 | Intelligent behaviour, even in animals, has something abstract about it [Price,HH] |
| Full Idea: Though it may sound odd to say so, intelligent behaviour has something abstract about it no less than intelligent cognition; and indeed at the animal level it is unrealistic to separate the two. | |
| From: H.H. Price (Thinking and Experience [1953], Ch.IV) | |
| A reaction: This elusive thought strikes me as being a key one for understanding human existence. To think is to abstract. Brains are abstraction machines. Resemblance and recognition require abstaction. |
| 9033 | Recognition must precede the acquisition of basic concepts, so it is the fundamental intellectual process [Price,HH] |
| Full Idea: Recognition is the first stage towards the acquisition of a primary or basic concept. It is, therefore, the most fundamental of all intellectual processes. | |
| From: H.H. Price (Thinking and Experience [1953], Ch.II) | |
| A reaction: An interesting question is whether it is an 'intellectual' process. Animals evidently recognise things, though it is a moot point whether slugs 'recognise' tasty leaves. |
| 9030 | Abstractions can be interpreted dispositionally, as the ability to recognise or imagine an item [Price,HH] |
| Full Idea: An abstract idea may have a dispositional as well as an occurrent interpretation. ..A man who possesses the concept Dog, when he is actually perceiving a dog can recognize that it is one, and can think about dogs when he is not perceiving any dog. | |
| From: H.H. Price (Thinking and Experience [1953], Ch.IX) | |
| A reaction: Ryle had just popularised the 'dispositional' account of mental events. Price is obviously right. The man may also be able to use the word 'dog' in sentences, but presumably dogs recognise dogs, and probably dream about dogs too. |
| 9029 | If ideas have to be images, then abstract ideas become a paradoxical problem [Price,HH] |
| Full Idea: There used to be a 'problem of Abstract Ideas' because it was assumed that an idea ought, somehow, to be a mental image; if some of our ideas appeared not to be images, this was a paradox and some solution must be found. | |
| From: H.H. Price (Thinking and Experience [1953], Ch.VIII) | |
| A reaction: Berkeley in particular seems to be struck by the fact that we are incapable of thinking of a general triangle, simply because there is no image related to it. Most conversations go too fast for images to form even of very visual things. |
| 9031 | The basic concepts of conceptual cognition are acquired by direct abstraction from instances [Price,HH] |
| Full Idea: Basic concepts are acquired by direct abstraction from instances; unless there were some concepts acquired in this way by direct abstraction, there would be no conceptual cognition at all. | |
| From: H.H. Price (Thinking and Experience [1953], Ch.II) | |
| A reaction: This seems to me to be correct. A key point is that not only will I acquire the concept of 'dog' in this direct way, from instances, but also the concept of 'my dog Spot' - that is I can acquire the abstract concept of an instance from an instance. |