35 ideas
21459 | Kant halted rationalism, and forced empiricists to worry about foundations [Gardner] |
Full Idea: Kant's Critique swiftly brought rationalism to a halt, and after Kant empiricism has displayed a nervousness regarding its foundations, and been forced to assume more sophisticated forms. | |
From: Sebastian Gardner (Kant and the Critique of Pure Reason [1999], 10 Intro) | |
A reaction: See the ideas of Laurence Bonjour for a modern revival of rationalism. After Kant philosophers either went existential, or stared gloomily into the obscure depths. Formal logic was seen as a possible rope ladder down. |
21463 | Hamann, Herder and Jacobi were key opponents of the Enlightenment [Gardner] |
Full Idea: Hamann, Herder and Jacobi are central figues in the reaction against Enlightenment. | |
From: Sebastian Gardner (Kant and the Critique of Pure Reason [1999], 10 'immediate') | |
A reaction: From a British perspective I would see Hume as the leading such figure. Hamann emphasised the neglect of the role of language. Jacobi was a Christian. |
21460 | Only Kant and Hegel have united nature, morals, politics, aesthetics and religion [Gardner] |
Full Idea: Apart from Hegel, no later philosophical system equals in stature Kant's attempt to weld together the diverse fields of natural science, morality, politics, aesthetics and religion into a systematic overarching epistemological and metaphysical unity. | |
From: Sebastian Gardner (Kant and the Critique of Pure Reason [1999], 10) | |
A reaction: Earlier candidate are Plato and Aristotle. Earlier Enlightenment figures say little about morality or aesthetics. Hobbes ranges widely. Aquinas covered most things. |
21443 | Transcendental proofs derive necessities from possibilities (e.g. possibility of experiencing objects) [Gardner] |
Full Idea: A transcendental proof converts a possibility into a necessity: by saying under what conditions experience of objects is possible, transcendental proofs show those conditions to be necessary for us to the extent that we have any experience of objects. | |
From: Sebastian Gardner (Kant and the Critique of Pure Reason [1999], 02 'Transc') | |
A reaction: They appear to be hypothetical necessities, rather than true metaphysical necessities. Gardner is discussing Kant, but seems to be generalising. Hypothetical necessities are easy: if it is flying, it is necessarily above the ground. |
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. |
21444 | Modern geoemtry is either 'pure' (and formal), or 'applied' (and a posteriori) [Gardner] |
Full Idea: There is now 'pure' geometry, consisting of formal systems based on axioms for which truth is not claimed, and which are consequently not synthetic; and 'applied', a branch of physics, the truth of which is empirical, and therefore not a priori. | |
From: Sebastian Gardner (Kant and the Critique of Pure Reason [1999], 03 'Maths') | |
A reaction: His point is that there is no longer any room for a priori geometry. Might the same division be asserted of arithmetic, or analysis, or set theory? |
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. |
21453 | Leibnizian monads qualify as Kantian noumena [Gardner] |
Full Idea: Leibnizian monads clearly satisfy Kant's definition of noumena. | |
From: Sebastian Gardner (Kant and the Critique of Pure Reason [1999], 06 'Noumena') | |
A reaction: This needs qualifying, because Leibniz clearly specifies the main attributes of monads, where Kant is adamant that we can saying virtually nothing about noumena. |
4444 | One moderate nominalist view says that properties and relations exist, but they are particulars [Armstrong] |
Full Idea: There is a 'moderate' nominalism (found in G.F.Stout, for example) which says that properties and relations do exist, but that they are particulars rather than universals. | |
From: David M. Armstrong (Universals [1995], p.504) | |
A reaction: Both this view and the 'mereological' view seem to be ducking the problem. If you have two red particulars and a green one, how do we manage to spot the odd one out? |
4445 | If properties and relations are particulars, there is still the problem of how to classify and group them [Armstrong] |
Full Idea: The view that properties exist, but are particulars rather than universals, is still left with the problem of classification. On what basis do we declare that different things have the same property? | |
From: David M. Armstrong (Universals [1995], p.504) | |
A reaction: This seems like a fairly crucial objection. The original problem was how we manage to classify things (group them into sets), and it looks as if this theory leaves the problem untouched. |
4448 | Should we decide which universals exist a priori (through words), or a posteriori (through science)? [Armstrong] |
Full Idea: Should we decide what universals exist a priori (probably on semantic grounds, identifying them with the meanings of general words), or a posteriori (looking to our best general theories about nature to give revisable conjectures about universals)? | |
From: David M. Armstrong (Universals [1995], p.505) | |
A reaction: Nice question for a realist. Although the problem is first perceived in the use of language, if we think universals are a real feature of nature, we should pursue them scientifically, say I. |
4446 | It is claimed that some universals are not exemplified by any particular, so must exist separately [Armstrong] |
Full Idea: There are some who claim that there can be uninstantiated universals, which are not exemplified by any particular, past, present or future; this would certainly imply that those universals have a Platonic transcendent existence outside time and space. | |
From: David M. Armstrong (Universals [1995], p.504) | |
A reaction: Presumably this is potentially circular or defeasible, because one can deny the universal simply because there is no particular. |
4439 | 'Resemblance Nominalism' says properties are resemblances between classes of particulars [Armstrong] |
Full Idea: Resemblance Nominalists say that to have a property is to be a member of a class which is part of a network of resemblance relations with other classes of particulars. ..'Resemblance' is taken to be a primitive notion, though one that admits of degrees. | |
From: David M. Armstrong (Universals [1995], p.503) | |
A reaction: Intuition suggests that this proposal has good prospects, as properties are neither identical, nor just particulars, but have a lot in common, which 'resemblance' captures. Hume saw resemblance as a 'primitive' process. |
4440 | 'Resemblance Nominalism' finds that in practice the construction of resemblance classes is hard [Armstrong] |
Full Idea: It is difficult for Resemblance Nominalists to construct their interconnected classes in practice. | |
From: David M. Armstrong (Universals [1995], p.503) | |
A reaction: Given the complexity of the world this is hardly surprising, but it doesn't seem insuperable for the theory. It is hard to decide whether an object is white, or hot, whatever your theory of universals. |
4431 | 'Predicate Nominalism' says that a 'universal' property is just a predicate applied to lots of things [Armstrong] |
Full Idea: For a Predicate Nominalist different things have the same property, or belong to the same kind, if the same predicates applies to, or is 'true of', the different things. | |
From: David M. Armstrong (Universals [1995], p.503) | |
A reaction: This immediately strikes me as unlikely, because I think the action is at the proposition level, not the sentence level. And why do some predicates seem to be synonymous? |
4433 | Concept and predicate nominalism miss out some predicates, and may be viciously regressive [Armstrong] |
Full Idea: The standard objections to Predicate and Concept Nominalism are that some properties have no predicates or concepts, and that predicates and concepts seem to be types rather than particulars, and it is types the theory is seeking to analyse. | |
From: David M. Armstrong (Universals [1995], p.503) | |
A reaction: The claim that some properties have no concepts is devastating if true, but may not be. The regress problem is likely to occur in any explanation of universals, I suspect. |
4432 | 'Concept Nominalism' says a 'universal' property is just a mental concept applied to lots of things [Armstrong] |
Full Idea: Concept Nominalism says different things have the same property, or belong to the same kind, if the same concept in the mind is applied to different things. | |
From: David M. Armstrong (Universals [1995], p.503) | |
A reaction: This is more appealing than Predicate Nominalism, and may be right. Our perception of the 'properties' of a thing may be entirely dictated by human interests, not by nature. |
4436 | 'Class Nominalism' may explain properties if we stick to 'natural' sets, and ignore random ones [Armstrong] |
Full Idea: Class Nominalism can be defended (by Quinton) against the problem of random sets (with nothing in common), by giving an account of properties in terms of 'natural' classes, where 'natural' comes in degrees, but is fundamental and unanalysable. | |
From: David M. Armstrong (Universals [1995], p.503) | |
A reaction: This still seems to beg the question, because you still have to decide whether two things have anything 'naturally' in common before you assign them to a set. |
4435 | 'Class Nominalism' cannot explain co-extensive properties, or sets with random members [Armstrong] |
Full Idea: Class Nominalism cannot explain co-extensive properties (which qualify the same things), and also a random (non-natural) set has particulars with nothing in common, thus failing to capture an essential feature of a general property. | |
From: David M. Armstrong (Universals [1995], p.503) | |
A reaction: These objections strike me as conclusive, since we can assign things to a set quite arbitrarily, so membership of a set may signify no shared property at all (except, say, 'owned by me', which is hardly a property). |
4434 | 'Class Nominalism' says that properties or kinds are merely membership of a set (e.g. of white things) [Armstrong] |
Full Idea: Class Nominalists substitute classes or sets for properties or kinds, so that being white is just being a member of the set of white things; relations are treated as ordered sets. | |
From: David M. Armstrong (Universals [1995], p.503) | |
A reaction: This immediately seems wrong, because it invites the question of why something is a member of a set (unless membership is arbitrary and whimsical - which it usually isn't). |
4437 | 'Mereological Nominalism' sees whiteness as a huge white object consisting of all the white things [Armstrong] |
Full Idea: Mereological Nominalism views a property as the omnitemporal whole or aggregate of all the things said to have the property, so whiteness is a huge white object whose parts are all the white things. | |
From: David M. Armstrong (Universals [1995], p.503) | |
A reaction: A charming proposal, in which bizarre and beautiful unities thread themselves across the universe, but white objects may also be soft and warm. |
4438 | 'Mereological Nominalism' may work for whiteness, but it doesn't seem to work for squareness [Armstrong] |
Full Idea: Mereological Nominalism has some plausibility for a case like whiteness, but breaks down completely for other universals, such as squareness. | |
From: David M. Armstrong (Universals [1995], p.503) | |
A reaction: A delightful request that you attempt a hopeless feat of imagination, by seeing all squares as parts of one supreme square. A nice objection. |
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. |