31 ideas
| 15457 | Interdefinition is useless by itself, but if we grasp one separately, we have them both [Lewis] |
| Full Idea: All circles of interdefinition are useless by themselves. But if we reach one of the interdefined pair, then we have them both. | |
| From: David Lewis (Defining 'Intrinsic' (with Rae Langton) [1998], IV) |
| 8820 | Rules of reasoning precede the concept of truth, and they are what characterize it [Pollock] |
| Full Idea: Rather than truth being fundamental and rules for reasoning being derived from it, the rules for reasoning come first and truth is characterized by the rules for reasoning about truth. | |
| From: John L. Pollock (Epistemic Norms [1986], 'Cog.Mach') | |
| A reaction: This nicely disturbs our complacency about such things. There is plenty of reasoning in Homer, but I bet there is no talk of 'truth'. Pontius Pilate seems to have been a pioneer (Idea 8821). Do the truth tables define or describe logical terms? |
| 8819 | We need the concept of truth for defeasible reasoning [Pollock] |
| Full Idea: It might be wondered why we even have a concept of truth. The answer is that this concept is required for defeasible reasoning. | |
| From: John L. Pollock (Epistemic Norms [1986], 'Cog.Mach') | |
| A reaction: His point is that we must be able to think critically about our beliefs ('is p true?') if we are to have any knowledge at all. An excellent point. Give that man a teddy bear. |
| 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. |
| 15400 | We must avoid circularity between what is intrinsic and what is natural [Lewis, by Cameron] |
| Full Idea: Lewis revised his analysis of duplication because he had assumed that as a matter of necessity perfectly natural properties are intrinsic, and that necessarily how a thing is intrinsically is determined completely by the natural properties it has. | |
| From: report of David Lewis (Defining 'Intrinsic' (with Rae Langton) [1998]) by Ross P. Cameron - Intrinsic and Extrinsic Properties 'Analysis' | |
| A reaction: [This compares Lewis 1986:61 with Langton and Lewis 1998] I am keen on both intrinsic and on natural properties, but I have not yet confronted this little problem. Time for a displacement activity, I think.... |
| 15458 | A property is 'intrinsic' iff it can never differ between duplicates [Lewis] |
| Full Idea: A property is 'intrinsic' iff it never can differ between duplicates; iff whenever two things (actual or possible) are duplicates, either both of them have the property or both of them lack it. | |
| From: David Lewis (Defining 'Intrinsic' (with Rae Langton) [1998], IV) | |
| A reaction: This leaves me wondering how one could arrive at a precise definition of 'duplicates'. Can it be done without mentioning that they have the same intrinsic properties? |
| 15459 | Ellipsoidal stars seem to have an intrinsic property which depends on other objects [Lewis] |
| Full Idea: The property of being an ellipsoidal star would seem offhand to be a basic intrinsic property, but it is incompatible (nomologically) with being an isolated object. | |
| From: David Lewis (Defining 'Intrinsic' (with Rae Langton) [1998], V) | |
| A reaction: Another nice example from Lewis. It makes you wonder whether the intrinsic/extrinsic distinction should go. Modern physics, with its 'entanglements', doesn't seem to suit the distinction. |
| 8822 | Statements about necessities need not be necessarily true [Pollock] |
| Full Idea: True statements about the necessary properties of things need not be necessarily true. The well-known example is that the number of planets (9) is necessarily an odd number. The necessity is de re, but not de dicto. | |
| From: John L. Pollock (Epistemic Norms [1986], 'Nat.Internal') | |
| A reaction: This would be a matter of the scope (the placing of the brackets) of the 'necessarily' operator in a formula. The quick course in modal logic should eradicate errors of this kind in your budding philosopher. |
| 8818 | Defeasible reasoning requires us to be able to think about our thoughts [Pollock] |
| Full Idea: Defeasible reasoning requires us to be able to think about our thoughts. | |
| From: John L. Pollock (Epistemic Norms [1986], 'Cog.Mach') | |
| A reaction: This is why I do not think animals 'know' anything, though they seem to have lots of true beliefs about their immediate situation. |
| 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. |
| 8811 | What we want to know is - when is it all right to believe something? [Pollock] |
| Full Idea: When we ask whether a belief is justified, we want to know whether it is all right to believe it. The question we must ask is 'when is it permissible (epistemically) to believe P?'. | |
| From: John L. Pollock (Epistemic Norms [1986], 'Ep.Norms') | |
| A reaction: Nice to see someone trying to get the question clear. The question clearly points to the fact that there must at least be some sort of social aspect to criteria of justification. I can't cheerfully follow my intuitions if everyone else laughs at them. |
| 8817 | Logical entailments are not always reasons for beliefs, because they may be irrelevant [Pollock] |
| Full Idea: Epistemologists have noted that logical entailments do not always constitute reasons. P may entail Q without the connection between P and Q being at all obvious. | |
| From: John L. Pollock (Epistemic Norms [1986], 'Ref.of Extern') | |
| A reaction: Graham Priest and others try to develop 'relevance logic' to deal with this. This would deny the peculiar classical claim that everything is entailed by a falsehood. A belief looks promising if it entails lots of truths about the world. |
| 8814 | Epistemic norms are internalised procedural rules for reasoning [Pollock] |
| Full Idea: Epistemic norms are to be understood in terms of procedural knowledge involving internalized rules for reasoning. | |
| From: John L. Pollock (Epistemic Norms [1986], 'How regulate?') | |
| A reaction: He offers analogies with bicycly riding, but the simple fact that something is internalized doesn't make it a norm. Some mention of truth is needed, equivalent to 'don't crash the bike'. |
| 8823 | Reasons are always for beliefs, but a perceptual state is a reason without itself being a belief [Pollock] |
| Full Idea: When one makes a perceptual judgement on the basis of a perceptual state, I want to say that the perceptual state itself is one's reason. ..Reason are always reasons for beliefs, but the reasons themselves need not be beliefs. | |
| From: John L. Pollock (Epistemic Norms [1986], 'Dir.Realism') | |
| A reaction: A crucial issue. I think I prefer the view of Davidson, in Ideas 8801 and 8804. Three options: a pure perception counts as a reason, or perceptions involve some conceptual content, or you only acquire a reason when a proposition is formulated. |
| 8813 | If we have to appeal explicitly to epistemic norms, that will produce an infinite regress [Pollock] |
| Full Idea: If we had to make explicit appeal to epistemic norms for justification (the 'intellectualist model') we would find ourselves in an infinite regress. The norms, their existence and their application would themselves have to be justified. | |
| From: John L. Pollock (Epistemic Norms [1986], 'How regulate?') | |
| A reaction: This is counter to the 'space of reasons' picture, where everything is rationally assessed. There are regresses for both reasons and for experiences, when they are offered as justifications. |
| 8812 | Norm Externalism says norms must be internal, but their selection is partly external [Pollock] |
| Full Idea: Norm Externalism acknowledges that the content of our epistemic norms must be internalist, but employs external considerations in the selection of the norms themselves. | |
| From: John L. Pollock (Epistemic Norms [1986], 'Ep.Norms') | |
| A reaction: It can't be right that you just set your own norms, so this must contain some truth. Equally, even the most hardened externalist can't deny that what goes on in the head of the person concerned must have some relevance. |
| 8816 | Externalists tend to take a third-person point of view of epistemology [Pollock] |
| Full Idea: Externalists tend to take a third-person point of view in discussing epistemology. | |
| From: John L. Pollock (Epistemic Norms [1986], 'Ref.of Extern') | |
| A reaction: Pollock's point, quite reasonably, is that the first-person aspect must precede any objective assessment of whether someone knows. External facts, such as unpublicised information, can undermine high quality internal justification. |
| 8815 | Belief externalism is false, because external considerations cannot be internalized for actual use [Pollock] |
| Full Idea: External considerations of reliability could not be internalized. Consequently, it is in principle impossible for us to actually employ externalist norms. I take this to be a conclusive refutation of belief externalism. | |
| From: John L. Pollock (Epistemic Norms [1986], 'Ref.of Extern') | |
| A reaction: Not so fast. He earlier rejected the 'intellectualist model' (Idea 8813), so he doesn't think norms have to be fully conscious and open to criticism. So they could be innate, or the result of indoctrination (sorry, teaching), or just forgotten. |