28 ideas
| 23025 | Philosophers should be more inductive, and test results by their conclusions, not their self-evidence [Russell] |
| Full Idea: The progress of philosophy seems to demand that, like science, it should learn to practise induction, to test its premisses by the conclusions to which they lead, and not merely by their apparent self-evidence. | |
| From: Bertrand Russell (Explanations in reply to Mr Bradley [1899], nr end) | |
| A reaction: [from Twitter] Love this. It is 'one person's modus ponens is another person's modus tollens'. I think all philosophical conclusions, without exception, should be reached by evaluating the final result fully, and not just following a line of argument. |
| 17070 | Coherence is consilience, simplicity, analogy, and fitting into a web of belief [Smart] |
| Full Idea: I shall make use of the admittedly imprecise notions of consilience, simplicity, analogy and fitting into a web of belief, or in short of 'coherence'. | |
| From: J.J.C. Smart (Explanation - Opening Address [1990], p.06) | |
| A reaction: Coherence sounds like a family of tests, rather than a single unified concept. I still like coherence, though. |
| 17072 | We need comprehensiveness, as well as self-coherence [Smart] |
| Full Idea: Not mere self-coherence, but comprehensiveness belongs to the notion of coherence. | |
| From: J.J.C. Smart (Explanation - Opening Address [1990], p.07) |
| 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. |
| 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. |
| 17073 | I simply reject evidence, if it is totally contrary to my web of belief [Smart] |
| Full Idea: The simplest way of fitting the putative observed phenomena of telepathy or clairvoyance into my web of belief is to refuse to take them at face value. | |
| From: J.J.C. Smart (Explanation - Opening Address [1990], p.07-8) | |
| A reaction: Love it. It is very disconcerting for the sceptical naturalist to be faced with adamant claims that the paranormal has occurred, but my response is exactly the same as Smart's. I reject the reports, no matter how passionately they are asserted. |
| 17077 | The height of a flagpole could be fixed by its angle of shadow, but that would be very unusual [Smart] |
| Full Idea: You could imagine a person using the angle from a theodolite to decide a suitable spot to cut the height of the flagpole, …but since such circumstances would be very unusual we naturally say the flagpole subtends the angle because of its height. | |
| From: J.J.C. Smart (Explanation - Opening Address [1990], p.14) | |
| A reaction: [compressed; he mentions Van Fraassen 1980:132-3 for a similar point] As a response this seems a bit lame, if the direction is fixed by what is 'usual'. I think the key point is that the direction of explanation is one way or the other, not both. |
| 17078 | Universe expansion explains the red shift, but not vice versa [Smart] |
| Full Idea: The theory of the expansion of the universe renders the red shift no longer puzzling, whereas he expansion of the universe is hardly rendered less puzzling by facts about the red shift. | |
| From: J.J.C. Smart (Explanation - Opening Address [1990], p.15) | |
| A reaction: The direction of explanation is, I take it, made obvious by the direction of causation, with questions about what is 'puzzling' as mere side-effects. |
| 17061 | Explanation of a fact is fitting it into a system of beliefs [Smart] |
| Full Idea: I want to characterise explanation of some fact as a matter of fitting belief in this fact into a system of beliefs. | |
| From: J.J.C. Smart (Explanation - Opening Address [1990], p.02) | |
| A reaction: Sounds good to me. Simple facts slot straight into daily beliefs, and deep obscure facts are explained when we hook them up to things we have already grasped. Quark theory fits into prior physics of forces, properties etc. |
| 17074 | Explanations are bad by fitting badly with a web of beliefs, or fitting well into a bad web [Smart] |
| Full Idea: An explanation may be bad if it fits only into a bad web of belief. It can also be bad if it fits into a (possibly good) web of belief in a bad sort of way. | |
| From: J.J.C. Smart (Explanation - Opening Address [1990], p.09) | |
| A reaction: Nice. If you think someone has an absurd web of beliefs, then it counts against some belief (for you) if it fits beautifully into the other person's belief system. Judgement of coherence comes in at different levels. |
| 17076 | Deducing from laws is one possible way to achieve a coherent explanation [Smart] |
| Full Idea: The Hempelian deductive-nomological model of explanation clearly fits in well with the notion of explanation in terms of coherence. One way of fitting a belief into a system is to show that it is deducible from other beliefs. | |
| From: J.J.C. Smart (Explanation - Opening Address [1990], p.13) | |
| A reaction: Smart goes on to reject the law-based deductive approach, for familiar reasons, but at least it has something in common with the Smart view of explanation, which is the one I like. |
| 17071 | An explanation is better if it also explains phenomena from a different field [Smart] |
| Full Idea: One explanation will be a better explanation that another if it also explains a set of phenomena from a different field ('consilience'). | |
| From: J.J.C. Smart (Explanation - Opening Address [1990], p.07) | |
| A reaction: This would count as 'unexpected accommodation', rather than prediction. It is a nice addition to Lipton's comparison of mere accommodation versus prediction as criteria. It sounds like a strong criterion for a persuasive explanation. |
| 17062 | If scientific explanation is causal, that rules out mathematical explanation [Smart] |
| Full Idea: I class mathematical explanation with scientific explanation. This would be resisted by those who, unlike me, regard the notion of causation as essential to scientific explanation. | |
| From: J.J.C. Smart (Explanation - Opening Address [1990], p.02-3) | |
| A reaction: I aim to champion mathematical explanation, in terms of axioms etc., so I am realising that my instinctive attraction to exclusively causal explanation won't do. What explanation needs is a direction of dependence. |
| 17075 | Scientific explanation tends to reduce things to the unfamiliar (not the familiar) [Smart] |
| Full Idea: The history of science suggests that most often explanation is reduction to the unfamiliar. | |
| From: J.J.C. Smart (Explanation - Opening Address [1990], p.11) | |
| A reaction: Boyle was keen to reduce things to the familiar, but that was early days for science, and some nasty shocks were coming our way. What would Boyle make of quantum non-locality? |
| 17063 | Unlike Newton, Einstein's general theory explains the perihelion of Mercury [Smart] |
| Full Idea: Newtonian celestial mechanics does not explain the advance of the perihelion of Mercury, while Einstein's general theory of relativity does. | |
| From: J.J.C. Smart (Explanation - Opening Address [1990], p.03) | |
| A reaction: A perfect example of why explanation is the central concept in science, and probably in all epistemological activity. The desire to know is the desire for an explanation. Once the explanation is obvious, we know. |