27 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. |
| 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'. |
| 8353 | Freedom involves acting according to an idea [Anscombe] |
| Full Idea: Freedom at least involves the power of acting according to an idea. | |
| From: G.E.M. Anscombe (Causality and Determinism [1971], §2) | |
| A reaction: Since 'you' presumably have to sit above the idea and pass a judgement on it, then the same principle should apply to acting on a desire, which presumably 'you' could reject because it just wasn't attractive enough. |
| 8352 | To believe in determinism, one must believe in a system which determines events [Anscombe] |
| Full Idea: 'The ball's path is determined' must mean 'there is only one possible path for the ball (assuming no air currents)', but what ground could one have for believing this, if one does not believe in some system for which it is a consequence? | |
| From: G.E.M. Anscombe (Causality and Determinism [1971], §2) | |
| A reaction: This seems right, but it doesn't follow that one has to know the full details of the system. The system might just be the best explanation, or even a matter of vague faith. It might, though, be just that you can't imagine any other outcome. |
| 8351 | With diseases we easily trace a cause from an effect, but we cannot predict effects [Anscombe] |
| Full Idea: It is much easier to trace effects back to causes with certainty than to predict effects from causes. If I have one contact with someone with a disease and I get it, we suppose I got it from him, but a doctor cannot predict a disease from one contact. | |
| From: G.E.M. Anscombe (Causality and Determinism [1971], §1) | |
| A reaction: An interesting, and obviously correct, observation. Her point is that we get more certainty of causes from observing a singular effect than we get certainty of effects from regularities or laws. |
| 4777 | The word 'cause' is an abstraction from a group of causal terms in a language (scrape, push..) [Anscombe] |
| Full Idea: The word "cause" can be added to a language in which are already represented many causal concepts; a small selection: scrape, push, wet, carry, eat, burn, knock over, keep off, squash, make, hurt. | |
| From: G.E.M. Anscombe (Causality and Determinism [1971], p.93) | |
| A reaction: An interesting point, perhaps reinforcing the Humean idea of causation as a 'natural belief', or the Kantian view of it as a category of thought. Or maybe causation is built into language because it is a feature of reality… |
| 10363 | Causation is relative to how we describe the primary relata [Anscombe, by Schaffer,J] |
| Full Idea: Anscombe has inspired the view that causation is an intensional relation, and takes it to be relative to the descriptions of the primary relata. | |
| From: report of G.E.M. Anscombe (Causality and Determinism [1971], 1) by Jonathan Schaffer - The Metaphysics of Causation 1 | |
| A reaction: It seems too linguistic to say that there is nothing more to it. It seems relevant in human examples, but if a landslide crushes a tree, what difference does the description make? 'It was just a few rocks and some miserable little tree'. No excuse! |
| 8350 | Since Mill causation has usually been explained by necessary and sufficient conditions [Anscombe] |
| Full Idea: Since Mill it has been fairly common to explain causation one way or another in terms of 'necessary' and 'sufficient' conditions. | |
| From: G.E.M. Anscombe (Causality and Determinism [1971], §1) | |
| A reaction: Interesting to see what Hume implies about these criteria. Anscombe is going to propose that causal events are fairly self-evident and self-explanatory, and don't need analyses of conditions. Another approach is regularities and laws. |