46 ideas
| 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. |
| 2730 | Because 'gold is malleable' is necessary does not mean that it is analytic [Audi,R] |
| Full Idea: Taking the proposition that gold is malleable to be necessary does not commit one to considering it analytic. | |
| From: Robert Audi (Epistemology: contemporary introduction [1998], IV p.116) |
| 2715 | Beliefs are based on perception, memory, introspection or reason [Audi,R] |
| Full Idea: The four basic kinds of belief are rooted in perception, memory, introspective consciousness, and reason. | |
| From: Robert Audi (Epistemology: contemporary introduction [1998], Intr.p.7) |
| 2735 | Could you have a single belief on its own? [Audi,R] |
| Full Idea: Could one have just a single belief? | |
| From: Robert Audi (Epistemology: contemporary introduction [1998], VII p.198) |
| 2736 | We can make certain of what we know, so knowing does not entail certainty [Audi,R] |
| Full Idea: The possibility of making certain of what we already know suggests that knowing a proposition does not entail its being certain. | |
| From: Robert Audi (Epistemology: contemporary introduction [1998], VIII p.220) |
| 2721 | If you gradually remove a book's sensory properties, what is left at the end? [Audi,R] |
| Full Idea: If you imagine subtracting a book's sensory properties one by one, what is left of it? | |
| From: Robert Audi (Epistemology: contemporary introduction [1998], I p.42) |
| 2722 | Sense-data theory is indirect realism, but phenomenalism is direct irrealism [Audi,R] |
| Full Idea: Where the sense-datum theory is an indirect realism, phenomenalism is a direct irrealism. | |
| From: Robert Audi (Epistemology: contemporary introduction [1998], I p.43) |
| 2728 | The concepts needed for a priori thought may come from experience [Audi,R] |
| Full Idea: I may well need experience to acquire the concepts needed for knowledge of the a priori, such as the concept of a colour. | |
| From: Robert Audi (Epistemology: contemporary introduction [1998], IV p.103) |
| 2727 | Red and green being exclusive colours seems to be rationally graspable but not analytic [Audi,R] |
| Full Idea: The proposition that nothing is red and green all over at once is not analytic, but it is rationally graspable, so it seems to be an a priori synthetic proposition. | |
| From: Robert Audi (Epistemology: contemporary introduction [1998], IV p.100) |
| 2717 | How could I see a field and believe nothing regarding it? [Audi,R] |
| Full Idea: How could I see a field and believe nothing regarding it? | |
| From: Robert Audi (Epistemology: contemporary introduction [1998], I p.20) |
| 2716 | To see something as a field, I obviously need the concept of a field [Audi,R] |
| Full Idea: The propositional belief which portrays what I see in front of me AS a field requires my having a concept of one. | |
| From: Robert Audi (Epistemology: contemporary introduction [1998], I p.17) | |
| A reaction: To me this immediately invites the question of what a cow or horse experiences when they look at a familiar field. They know how to leave and enter it, and register its boundaries and qualities. Concepts? |
| 2719 | Sense data imply representative realism, possibly only representing primary qualities [Audi,R] |
| Full Idea: A sense-datum theory might be called a representative realism because it conceives perception as a relation in which sense-data represent perceived external (hence real) objects to us. For Locke they were resemblances only of primary qualities. | |
| From: Robert Audi (Epistemology: contemporary introduction [1998], I p.33) |
| 2720 | Sense-data (and the rival 'adverbial' theory) are to explain illusions and hallucinations [Audi,R] |
| Full Idea: The sense-datum theory is mainly to explain hallucinations and illusions, though there might be other theories, such as the 'adverbial' theory. | |
| From: Robert Audi (Epistemology: contemporary introduction [1998], I p.36) |
| 2718 | Perception is first simple, then objectual (with concepts) and then propositional [Audi,R] |
| Full Idea: Simple perceiving gives rise to objectual perceiving (attaching concepts to the object), which gives rise to propositional perceiving. | |
| From: Robert Audi (Epistemology: contemporary introduction [1998], I p.23) |
| 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. |
| 2741 | The principles of justification have to be a priori [Audi,R] |
| Full Idea: The crucial principles of justification are a priori. | |
| From: Robert Audi (Epistemology: contemporary introduction [1998], X p.311) |
| 2729 | Virtually all rationalists assert that we can have knowledge of synthetic a priori truths [Audi,R] |
| Full Idea: Rationalists virtually always assert or imply that, in addition to knowledge of analytic truths, there is knowledge of synthetic a priori truths. | |
| From: Robert Audi (Epistemology: contemporary introduction [1998], IV p.105) |
| 2725 | To remember something is to know it [Audi,R] |
| Full Idea: Remembering something is so entails knowing that it is so. | |
| From: Robert Audi (Epistemology: contemporary introduction [1998], II p.68) | |
| A reaction: Clearly I can say I "remember" x, but be wrong. Presumably we then say that I didn't really remember, which requires success, like "I know". It is true (as with "know") that as soon as I say that the something is false, I can't claim to remember it. |
| 2724 | I might remember someone I can't recall or image, by recognising them on meeting [Audi,R] |
| Full Idea: If I can neither recall nor image Jane I can still remember her, for on seeing her I might recognise her, and might remember, and even recall, our last meeting. | |
| From: Robert Audi (Epistemology: contemporary introduction [1998], II p.66) | |
| A reaction: Hm. I can hardly claim to remember her if I have no concept of her, and don't recall our last meeting. If seeing her triggers recognition, I would say that I NOW remember her, but I didn't before. Memory is more conscious than Audi claims. |
| 2731 | Justification is either unanchored (infinite or circular), or anchored (in knowledge or non-knowledge) [Audi,R] |
| Full Idea: There are four possible kinds of epistemic chain: infinite and unanchored, circular and unanchored, anchored in a belief which is not knowledge, and anchored in a belief which is bedrock knowledge. | |
| From: Robert Audi (Epistemology: contemporary introduction [1998], VII p.183) | |
| A reaction: About right, though I don't think 'chain' is the right word for what is proposed if justification is to be coherent. The justifications float like lilies in the pond of reason, and a Self (Monet?) seems needed to assess the picture |
| 2739 | Internalism about justification implies that there is a right to believe something [Audi,R] |
| Full Idea: Internalists about justification tend to conceive of it as a matter of having a right to believe something. | |
| From: Robert Audi (Epistemology: contemporary introduction [1998], VIII p.234) | |
| A reaction: I'm an internalist, but I don't understand this, unless it refers to the social aspect of justification. Can I grant myself internal rights? I can justify my belief to other people. |
| 2732 | Maths may be consistent with observations, but not coherent [Audi,R] |
| Full Idea: That 7+5=12 and that carrots are nourishing are mutually consistent, but do not exhibit coherence. | |
| From: Robert Audi (Epistemology: contemporary introduction [1998], VII p.192) | |
| A reaction: This shows how difficult it would be to define 'coherent'. Is 'carrots are nourishing' coherent with 'fish are nourishing'? Is the battle of Hastings coherent with the battle of Waterloo? |
| 2733 | It is very hard to show how much coherence is needed for justification [Audi,R] |
| Full Idea: It is very difficult to specify when an explanatory relation generates enough coherence to create justification. | |
| From: Robert Audi (Epistemology: contemporary introduction [1998], VII p.193) | |
| A reaction: I take coherence to be the key concept in epistemology, and quite impossible to define. This is why the 'space of reasons' is a useful concept. It is a courtroom, in which each case is different. |
| 2734 | A consistent madman could have a very coherent belief system [Audi,R] |
| Full Idea: A schizophrenic who thinks he is Napoleon, if he has a completely consistent story with enough interlocking details, may have a belief system that is superbly coherent. | |
| From: Robert Audi (Epistemology: contemporary introduction [1998], VII p.194) | |
| A reaction: This is an exaggeration, but the fact is that one isolated lie is totally coherent, so coherence can only emerge when a system is large. Sense experience must be central to coherence. |
| 2738 | Consistent accurate prediction looks like knowledge without justified belief [Audi,R] |
| Full Idea: If someone consistently prophesied the winners of horse races, it appears that this man knows who will win the races, but surely he does not have justified beliefs as to who will win? | |
| From: Robert Audi (Epistemology: contemporary introduction [1998], VIII p.229) | |
| A reaction: This is where internalists and externalists (notably reliabilists) sharply part company. IF a reliable clairvoyant appeared, we would eventually accept them as a knower. But they DON'T appear, because knowledge needs justification! |
| 2740 | A reliability theory of knowledge seems to involve truth as correspondence [Audi,R] |
| Full Idea: If one favours a reliability theory of knowledge (which is externalist) the correspondence theory of truth seems the most appropriate. | |
| From: Robert Audi (Epistemology: contemporary introduction [1998], VIII p.243) | |
| A reaction: Sounds right. Coherence implies some sort of internal assessment, whereas correspondence just needs to plugged into the facts. I like coherence justification and correspondence truth. |
| 2737 | 'Reliable' is a very imprecise term, and may even mean 'justified' [Audi,R] |
| Full Idea: Reliabilism cannot specify how reliable a process must be before it grounds knowledge, and it cannot specify what is reliable in the first place. 'Reliable' may become circular, and may mean 'justified'. | |
| From: Robert Audi (Epistemology: contemporary introduction [1998], VIII p.225) | |
| A reaction: The first time you ever read an instrument, or talk to a stranger, you have no indication of reliability. Circularity looks like a big problem. Knowledge must precede reliability? |
| 2726 | We can be ignorant about ourselves, for example, our desires and motives [Audi,R] |
| Full Idea: We can have false beliefs, or some degree of ignorance, about our own mental lives. For example, about our own dispositions, such as not believing that we have a certain ignoble desire. | |
| From: Robert Audi (Epistemology: contemporary introduction [1998], III p.83) | |
| A reaction: This idea, that we don't know ourselves, has become a commonplace of recent philosophy, but I am unconvinced. Mostly we know only too well that we harbour a base desire, and we feel a creeping sense of shame. Total ignorance is very rare. |
| 22252 | Capitalism may actually be the best way to foster community [Conway,D] |
| Full Idea: Not only is there no good reason for supposing capitalism inimical to community, but there is reason to think it more conducive to community than the feasible alternatives to it. | |
| From: David Conway (Capitalism and Community [1996], I) | |
| A reaction: Conway is defending an obviously unorthodox view, while attacking the hopes of communitarians. |
| 22253 | Capitalism is just the market, with optional limited government, and perhaps democracy [Conway,D] |
| Full Idea: There are three types of capitalism: 1) the market - private ownership, labor contracts and profit, 2) limited government - the state provides goods the market cannot do, 3) limited government with democracy - with political freedom and elections. | |
| From: David Conway (Capitalism and Community [1996], II) | |
| A reaction: [compressed] I would have thought that capitalism is compatible with a fair degree of workplace democracy, which would make a fourth type. |
| 22256 | Capitalism prefers representative democracy, which avoids community decision-making [Conway,D] |
| Full Idea: By opting for representative rather than direct democracy, capitalism is said to preclude political community, for which the citizens of a state must possess a common will, which needs their direct participation in decisions. | |
| From: David Conway (Capitalism and Community [1996], V) | |
| A reaction: Conway does not accept this claim. I'm beginning to wonder whether the famous British electoral system is actually a capitalist conspiracy against the people. |
| 22255 | Capitalism breaks up extended families, and must then provide welfare for the lonely people [Conway,D] |
| Full Idea: It is said that capitalism encourages the breakup of extended families, which creates the need for extensive state welfare for those indigent members of society who can no longer rely on their own family to take care of them. | |
| From: David Conway (Capitalism and Community [1996], V) | |
| A reaction: Conway does not accept this claim. It seems to simplistic to say that capitalism is the sole culprit. Any rise of mechanisation in agriculture would break up rural extended families. |
| 22257 | Capitalism is anti-community, by only valuing individuals, and breaking up families [Conway,D] |
| Full Idea: Communitarns say capitalism is inimical to family community, because it encourages an individualistic mentality which only values self-fulfilment, and because it demands labour mobility which is disruptive of families. | |
| From: David Conway (Capitalism and Community [1996], VI) | |
| A reaction: Chicken-and-egg with the first one. Small entrepreneurs are individualists who seek their own gain. It is big capitalism that sucks in the others. Traditional community is based on labour-intensive agriculture. |