36 ideas
| 19463 | Induction assumes some uniformity in nature, or that in some respects the future is like the past [Ayer] |
| Full Idea: In all inductive reasoning we make the assumption that there is a measure of uniformity in nature; or, roughly speaking, that the future will, in the appropriate respects, resemble the past. | |
| From: A.J. Ayer (The Problem of Knowledge [1956], 2.viii) | |
| A reaction: I would say that nature is 'stable'. Nature changes, so a global assumption of total uniformity is daft. Do we need some global uniformity assumptions, if the induction involved is local? I would say yes. Are all inductions conditional on this? |
| 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. |
| 9476 | If dispositions are more fundamental than causes, then they won't conceptually reduce to them [Bird on Lewis] |
| Full Idea: Maybe a disposition is a more fundamental notion than a cause, in which case Lewis has from the very start erred in seeking a causal analysis, in a traditional, conceptual sense, of disposition terms. | |
| From: comment on David Lewis (Causation [1973]) by Alexander Bird - Nature's Metaphysics 2.2.8 | |
| A reaction: Is this right about Lewis? I see him as reducing both dispositions and causes to a set of bald facts, which exist in possible and actual worlds. Conditionals and counterfactuals also suffer the same fate. |
| 8425 | For true counterfactuals, both antecedent and consequent true is closest to actuality [Lewis] |
| Full Idea: A counterfactual is non-vacuously true iff it takes less of a departure from actuality to make the consequent true along with the antecedent than it does to make the antecedent true without the consequent. | |
| From: David Lewis (Causation [1973], p.197) | |
| A reaction: Almost every theory proposed by Lewis hangs on the meaning of the word 'close', as used here. If you visited twenty Earth-like worlds (watch Startrek?), it would be a struggle to decide their closeness to ours in rank order. |
| 19461 | Knowing I exist reveals nothing at all about my nature [Ayer] |
| Full Idea: To know that one exists is not to know anything about oneself any more than knowing that 'this' exists is knowing anything about 'this'. | |
| From: A.J. Ayer (The Problem of Knowledge [1956], 2.iii) | |
| A reaction: Descartes proceeds to define himself as a 'thinking thing', inferring that thinking is his essence. Ayer casts nice doubt on that. |
| 19459 | To say 'I am not thinking' must be false, but it might have been true, so it isn't self-contradictory [Ayer] |
| Full Idea: To say 'I am not thinking' is self-stultifying since if it is said intelligently it must be false: but it is not self-contradictory. The proof that it is not self-contradictory is that it might have been false. | |
| From: A.J. Ayer (The Problem of Knowledge [1956], 2.iii) | |
| A reaction: If it doesn't imply a contradiction, then it is not a necessary truth, which is what it is normally taken to be. Is 'This is a sentence' necessarily true? It might not have been one, if the rules of English syntax changed recently. |
| 19460 | 'I know I exist' has no counterevidence, so it may be meaningless [Ayer] |
| Full Idea: If there is no experience at all of finding out that one is not conscious, or that one does not exist, ..it is tempting to say that sentences like 'I exist', 'I am conscious', 'I know that I exist' do not express genuine propositions. | |
| From: A.J. Ayer (The Problem of Knowledge [1956], 2.iii) | |
| A reaction: This is, of course, an application of the somewhat discredited verification principle, but the fact that strictly speaking the principle has been sort of refuted does not mean that we should not take it seriously, and be influenced by it. |
| 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. |
| 19464 | We only discard a hypothesis after one failure if it appears likely to keep on failing [Ayer] |
| Full Idea: Why should a hypothesis which has failed the test be discarded unless this shows it to be unreliable; that is, having failed once it is likely to fail again? There is no contradiction in a hypothesis that was falsified being more likely to pass in future. | |
| From: A.J. Ayer (The Problem of Knowledge [1956], 2.viii) | |
| A reaction: People may become more likely to pass a test after they have failed at the first attempt. Birds which fail to fly at the first attempt usually achieve total mastery of it. There are different types of hypothesis here. |
| 19462 | Induction passes from particular facts to other particulars, or to general laws, non-deductively [Ayer] |
| Full Idea: Inductive reasoning covers all cases in which we pass from a particular statement of fact, or set of them, to a factual conclusion which they do not formally entail. The inference may be to a general law, or by analogy to another particular instance. | |
| From: A.J. Ayer (The Problem of Knowledge [1956], 2.viii) | |
| A reaction: My preferred definition is 'learning from experience' - which I take to be the most rational behaviour you could possibly imagine. I don't think a definition should be couched in terms of 'objects' or 'particulars'. |
| 8424 | Determinism says there can't be two identical worlds up to a time, with identical laws, which then differ [Lewis] |
| Full Idea: By determinism I mean that the prevailing laws of nature are such that there do not exist any two possible worlds which are exactly alike up to that time, which differ thereafter, and in which those laws are never violated. | |
| From: David Lewis (Causation [1973], p.196) | |
| A reaction: This would mean that the only way an action of free will could creep in would be if it accepted being a 'violation' of the laws of nature. Fans of free will would probably prefer to call it a 'natural' phenomenon. I'm with Lewis. |
| 8420 | A proposition is a set of possible worlds where it is true [Lewis] |
| Full Idea: I identify a proposition with the set of possible worlds where it is true. | |
| From: David Lewis (Causation [1973], p.193) | |
| A reaction: As it stands, I'm baffled by this. How can 'it is raining' be a set of possible worlds? I assume it expands to refer to the truth-conditions, among possibilities as well as actualities. 'It is raining' fits all worlds where it is raining. |
| 8405 | A theory of causation should explain why cause precedes effect, not take it for granted [Lewis, by Field,H] |
| Full Idea: Lewis thinks it is a major defect in a theory of causation that it builds in the condition that the time of the cause precede that of the effect: that cause precedes effect is something we ought to explain (which his counterfactual theory claims to do). | |
| From: report of David Lewis (Causation [1973]) by Hartry Field - Causation in a Physical World | |
| A reaction: My immediate reaction is that the chances of explaining such a thing are probably nil, and that we might as well just accept the direction of causation as a given. Even philosophers balk at the question 'why doesn't time go backwards?' |
| 8427 | I reject making the direction of causation axiomatic, since that takes too much for granted [Lewis] |
| Full Idea: One might stipulate that a cause must always precede its effect, but I reject this solution. It won't solve the problem of epiphenomena, it rejects a priori any backwards causation, and it trivializes defining time-direction through causation. | |
| From: David Lewis (Causation [1973], p.203) | |
| A reaction: [compressed] Not strong arguments, I would say. Maybe apparent causes are never epiphenomenal. Maybe backwards causation is impossible. Maybe we must use time to define causal direction, and not vice versa. |
| 10392 | It is just individious discrimination to pick out one cause and label it as 'the' cause [Lewis] |
| Full Idea: We sometimes single out one among all the causes of some event and call it 'the' cause. ..We may select the abnormal causes, or those under human control, or those we deem good or bad, or those we want to talk about. This is invidious discrimination. | |
| From: David Lewis (Causation [1973]) | |
| A reaction: This is the standard view expressed by Mill - presumably the obvious empiricist line. But if we specify 'the pre-conditions' for an event, we can't just mention ANY fact prior to the effect - there is obvious relevance. So why not for 'the' cause as well? |
| 8419 | The modern regularity view says a cause is a member of a minimal set of sufficient conditions [Lewis] |
| Full Idea: In present-day regularity analyses, a cause is defined (roughly) as any member of any minimal set of actual conditions that are jointly sufficient, given the laws, for the existence of the effect. | |
| From: David Lewis (Causation [1973], p.193) | |
| A reaction: This is the view Lewis is about to reject. It seem to summarise the essence of the Mackie INUS theory. This account would make the presence of oxygen a cause of almost every human event. |
| 8421 | Regularity analyses could make c an effect of e, or an epiphenomenon, or inefficacious, or pre-empted [Lewis] |
| Full Idea: In the regularity analysis of causes, instead of c causing e, c might turn out to be an effect of e, or an epiphenomenon, or an inefficacious effect of a genuine cause, or a pre-empted cause (by some other cause) of e. | |
| From: David Lewis (Causation [1973], p.194) | |
| A reaction: These are Lewis's reasons for rejecting the general regularity account, in favour of his own particular counterfactual account. It is unlikely that c would be regularly pre-empted or epiphenomenal. If we build time's direction in, it won't be an effect. |
| 17524 | Lewis has basic causation, counterfactuals, and a general ancestral (thus handling pre-emption) [Lewis, by Bird] |
| Full Idea: Lewis's basic account has a basic causal relation, counterfactual dependence, and the general causal relation is the ancestral of this basic one. ...This is motivated by counterfactual dependence failing to be general because of the pre-emption problem. | |
| From: report of David Lewis (Causation [1973]) by Alexander Bird - Causation and the Manifestation of Powers p.161 | |
| A reaction: It is so nice when you struggle for ages with a topic, and then some clever person summarises it clearly for you. |
| 8397 | Counterfactual causation implies all laws are causal, which they aren't [Tooley on Lewis] |
| Full Idea: Some counterfactuals are based on non-causal laws, such as Newton's Third Law of Motion. 'If no force one way, then no force the other'. Lewis's counterfactual analysis implies that one force causes the other, which is not the case. | |
| From: comment on David Lewis (Causation [1973]) by Michael Tooley - Causation and Supervenience 5.2 | |
| A reaction: So what exactly does 'cause' my punt to move forwards? Basing causal laws on counterfactual claims looks to me like putting the cart before the horse. |
| 8423 | My counterfactual analysis applies to particular cases, not generalisations [Lewis] |
| Full Idea: My (counterfactual) analysis is meant to apply to causation in particular cases; it is not an analysis of causal generalizations. Those presumably quantify over particulars, but it is hard to match natural language to the quantifiers. | |
| From: David Lewis (Causation [1973], p.195) | |
| A reaction: What authority could you have for asserting a counterfactual claim, if you only had one observation? Isn't the counterfactual claim the hallmark of a generalisation? For one case, 'if not-c, then not-e' is just a speculation. |
| 8426 | One event causes another iff there is a causal chain from first to second [Lewis] |
| Full Idea: One event is the cause of another iff there exists a causal chain leading from the first to the second. | |
| From: David Lewis (Causation [1973], p.200) | |
| A reaction: It will be necessary to both explain and identify a 'chain'. Some chains are extremely tenuous (Alexander could stop a barrel of beer). Go back a hundred years, and the cause of any present event is everything back then. |
| 17525 | The counterfactual view says causes are necessary (rather than sufficient) for their effects [Lewis, by Bird] |
| Full Idea: The Humean idea, developed by Lewis, is that rather than being sufficient for their effects, causes are (counterfactual) necessary for their effects. | |
| From: report of David Lewis (Causation [1973]) by Alexander Bird - Causation and the Manifestation of Powers p.162 |
| 4795 | Lewis's account of counterfactuals is fine if we know what a law of nature is, but it won't explain the latter [Cohen,LJ on Lewis] |
| Full Idea: Lewis can elucidate the logic of counterfactuals on the assumption that you are not at all puzzled about what a law of nature is. But if you are puzzled about this, it cannot contribute anything towards resolving your puzzlement. | |
| From: comment on David Lewis (Causation [1973]) by L. Jonathan Cohen - The Problem of Natural Laws p.219 | |
| A reaction: This seems like a penetrating remark. The counterfactual theory is wrong, partly because it is epistemological instead of ontological, and partly because it refuses to face the really difficult problem, of what is going on out there. |