40 ideas
| 8251 | The logical space of reasons is a natural phenomenon, and it is the realm of freedom [McDowell] |
| Full Idea: The logical space of reasons is just part of the logical space of nature. ...And, in a Kantian slogan, the space of reasons is the realm of freedom. | |
| From: John McDowell (Mind and World [1994], Intro 7) | |
| A reaction: [second half on p.5] This is a modern have-your-cake-and-eat-it view of which I am becoming very suspicious. The modern Kantians (Davidson, Nagel, McDowell) are struggling to naturalise free will, but it won't work. Just dump it! |
| 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. |
| 13065 | Understanding is an extremely vague concept [Salmon] |
| Full Idea: Understanding is an extremely vague concept. | |
| From: Wesley Salmon (Four Decades of Scientific Explanation [1989], 4.3) | |
| A reaction: True, I suppose, but we usually recognise understanding when we encounter it, and everybody has a pretty clear notion of an 'increase' in understanding. I suspect that the concept is perfectly clear, but we lack any scale for measuring it. |
| 13047 | It is knowing 'why' that gives scientific understanding, not knowing 'that' [Salmon] |
| Full Idea: Knowledge 'that' is descriptive, and knowledge 'why' is explanatory, and it is the latter that provides scientific understanding of our world. | |
| From: Wesley Salmon (Four Decades of Scientific Explanation [1989], Intro) | |
| A reaction: I agree, but of course, knowing 'why' may require a lot of knowing 'that'. People with extensive knowledge 'that' things are so tend to understand why something happens more readily than the rest of us ignoramuses. |
| 8128 | Representation must be propositional if it can give reasons and be epistemological [McDowell, by Burge] |
| Full Idea: McDowell has claimed that one cannot make sense of representation that plays a role in epistemology unless one takes the representation to be propositional, and thus capable of yielding reasons. | |
| From: report of John McDowell (Mind and World [1994]) by Tyler Burge - Philosophy of Mind: 1950-2000 p.456 | |
| A reaction: A transcendental argument leads back to a somewhat implausible conclusion. I suspect that McDowell has a slightly inflated (Kantian) notion of the purity of the 'space of reasons'. Do philosophers just imagine their problems? |
| 19092 | There is no pure Given, but it is cultured, rather than entirely relative [McDowell, by Macbeth] |
| Full Idea: McDowell argues that the Myth of the Given shows not that there is no content to a concept that is not a matter of its inferential relations to other concepts but only that awareness of the sort that we enjoy ...is acquired in the course of acculturation. | |
| From: report of John McDowell (Mind and World [1994]) by Danielle Macbeth - Pragmatism and Objective Truth p.185 | |
| A reaction: The first view is of Wilfred Sellars, who derives pragmatic relativism from his rejection of the Myth. This idea is helpful is seeing why McDowell has a good proposal. As I look out of my window, my immediate experience seems 'cultured'. |
| 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. |
| 8253 | Sense impressions already have conceptual content [McDowell] |
| Full Idea: The world's impressions on our senses are already possessed of conceptual content. | |
| From: John McDowell (Mind and World [1994], I.6) | |
| A reaction: This is a key idea of McDowell's, which challenges most traditional empiricist views, and (maybe) offers a solution to the rationalist/empiricist debate. His commitment to the 'space of reasons' strikes me as an optional extra. |
| 13054 | Correlations can provide predictions, but only causes can give explanations [Salmon] |
| Full Idea: Various kinds of correlations exist that provide excellent bases for prediction, but because no suitable causal relations exist (or are known), these correlations do not furnish explanation. | |
| From: Wesley Salmon (Four Decades of Scientific Explanation [1989], 2.3) | |
| A reaction: There may be problem cases for the claim that all explanations are causal, but I certainly think that this idea is essentially right. Prediction can come from induction, but inductions may be true and yet baffling. |
| 13067 | For the instrumentalists there are no scientific explanations [Salmon] |
| Full Idea: There is a centuries-old philosophical tradition, sometimes referred to by the name of 'instrumentalism', that has denied the claim that science has explanatory power. For the instrumentalists there are no scientific explanations. | |
| From: Wesley Salmon (Four Decades of Scientific Explanation [1989], 4.3) | |
| A reaction: [He quotes Coffa] Presumably it is just a matter of matching the world to the readings on the instruments, aiming at van Fraassen's 'empirical adequacy'. If there are no scientific explanations, does that mean that there are no explanations at all? Daft! |
| 13055 | Good induction needs 'total evidence' - the absence at the time of any undermining evidence [Salmon] |
| Full Idea: Inductive logicians have a 'requirement of total evidence': induction is strong if 1) it has true premises, 2) it has correct inductive form, and 3) no additional evidence that would change the degree of support is available at the time. | |
| From: Wesley Salmon (Four Decades of Scientific Explanation [1989], 2.4.2) | |
| A reaction: The evidence might be very close at hand, but not quite 'available' to the person doing the induction. |
| 13046 | Scientific explanation is not reducing the unfamiliar to the familiar [Salmon] |
| Full Idea: I reject the view that scientific explanation involves reduction of the unfamiliar to the familiar. | |
| From: Wesley Salmon (Four Decades of Scientific Explanation [1989], Pref) | |
| A reaction: Aristotle sometimes seems to imply this account of explanation, and I would have to agree with Salmon's view of it. Aristotle is also, though, aware of real explanations, definitions and essences. People are 'familiar' with some peculiar things. |
| 13058 | Why-questions can seek evidence as well as explanation [Salmon] |
| Full Idea: There are evidence-seeking why-questions, as well as explanation-seeking why-questions. | |
| From: Wesley Salmon (Four Decades of Scientific Explanation [1989], 3.2) | |
| A reaction: Surely we would all prefer an explanation to mere evidence? It seems to me that they are all explanation-seeking, but that we are grateful for some evidence when no full explanation is available. Explanation renders evidence otiose. |
| 13050 | The 'inferential' conception is that all scientific explanations are arguments [Salmon] |
| Full Idea: The 'inferential' conception of scientific explanation is the thesis that all legitimate scientific explanations are arguments of one sort or another. | |
| From: Wesley Salmon (Four Decades of Scientific Explanation [1989], 1.1) | |
| A reaction: This seems to imply that someone has to be persuaded of something, and hence seems a rather too pragmatic view. I presume an explanation might be no more than dumbly pointing at conclusive evidence of a cause. Man with smoking gun. |
| 13059 | Ontic explanations can be facts, or reports of facts [Salmon] |
| Full Idea: Proponents of the ontic conception of explanation can say that explanations exist in the world as facts, or that they are reports of such facts (as opposed to the view of explanations as arguments, or as speech acts). | |
| From: Wesley Salmon (Four Decades of Scientific Explanation [1989], 3.2) | |
| A reaction: [compressed] I am strongly drawn to the ontic approach, but not sure whether we want facts, or reports of them. The facts are the causal nexus, but which parts of the nexus provide the main aspect of explanation? I'll vote for reports, for now. |
| 13064 | The three basic conceptions of scientific explanation are modal, epistemic, and ontic [Salmon] |
| Full Idea: There are three basic conceptions of scientific explanation - modal, epistemic, and ontic - which can be discerned in Aristotle, and that have persisted down the ages. | |
| From: Wesley Salmon (Four Decades of Scientific Explanation [1989], 4.1) |
| 13051 | Deductive-nomological explanations will predict, and their predictions will explain [Salmon] |
| Full Idea: The deductive-nomological view has an explanation/prediction symmetry thesis - that a correct explanation could be a scientific prediction, and that any deductive prediction could serve as a deductive-nomological explanation. | |
| From: Wesley Salmon (Four Decades of Scientific Explanation [1989], 1.1) | |
| A reaction: Of course, not all predictions will explain, or vice versa. Weird regularities become predictable but remain baffling. Good explanations may be of unrepeatable events. It is the 'law' in the account that ties the two ends together. |
| 13053 | A law is not enough for explanation - we need information about what makes a difference [Salmon] |
| Full Idea: To provide an adequate explanation of any given fact, we need to provide information that is relevant to the occurrence of that fact - information that makes a difference to its occurrence. It is not enough to subsume it under a general law. | |
| From: Wesley Salmon (Four Decades of Scientific Explanation [1989], 2.2) | |
| A reaction: [He cites Bromberger for this idea] Salmon is identifying this idea as the beginnings of trouble for the covering-law account of explanation, and it sounds exactly right. |
| 13049 | We must distinguish true laws because they (unlike accidental generalizations) explain things [Salmon] |
| Full Idea: The problem is to distinguish between laws and accidental generalizations, for laws have explanatory force while accidental generalizations, even if they are true, do not. | |
| From: Wesley Salmon (Four Decades of Scientific Explanation [1989], 1.1) | |
| A reaction: [He is discussing Hempel and Oppenheim 1948] This seems obviously right, but I can only make sense of the explanatory power if we have identified the mechanism which requires the generalisation to continue in future cases. |
| 13061 | Flagpoles explain shadows, and not vice versa, because of temporal ordering [Salmon] |
| Full Idea: The height of the flagpole explains the length of the shadow because the interaction between the sunlight and the flagpole occurs before the interaction between the sunlight and the ground. | |
| From: Wesley Salmon (Four Decades of Scientific Explanation [1989], 3.6) | |
| A reaction: [Bromberger produced the flagpole example] This seems to be correct, and would apply to all physical cases, but there may still be cases of explanation which are not causal (in mathematics, for example). |
| 13045 | Explanation at the quantum level will probably be by entirely new mechanisms [Salmon] |
| Full Idea: My basic feeling about explanation in the quantum realm is that it will involve mechanisms, but mechanisms that are quite different from those that seem to work in the macrocosm. | |
| From: Wesley Salmon (Four Decades of Scientific Explanation [1989], Pref) | |
| A reaction: Since I take most explanation to be by mechanisms (or some abstraction analogous to mechanisms), then I think this is probably right (rather than being by new 'laws'). |
| 13062 | Does an item have a function the first time it occurs? [Salmon] |
| Full Idea: In functional explanation, there is a disagreement over whether an item has a function the first time it occurs. | |
| From: Wesley Salmon (Four Decades of Scientific Explanation [1989], 3.8) | |
| A reaction: This question arises particularly in evolutionary contexts, and would obviously not generally arise in the case of human artefacts. |
| 13063 | Explanations reveal the mechanisms which produce the facts [Salmon] |
| Full Idea: I favour an ontic conception of explanation, that explanations reveal the mechanisms, causal or other, that produce the facts we are trying to explain. | |
| From: Wesley Salmon (Four Decades of Scientific Explanation [1989], 4.1) | |
| A reaction: [He also cites Coffa and Peter Railton] A structure may explain, and only be supported by causal powers, but it doesn't seem to be the causal powers that do the explaining. Is a peg fitting a hole explained causally? |
| 13060 | Can events whose probabilities are low be explained? [Salmon] |
| Full Idea: Can events whose probabilities are low be explained? | |
| From: Wesley Salmon (Four Decades of Scientific Explanation [1989], 3.6) | |
| A reaction: I take this to be one of the reasons why explanation must ultimately reside at the level of individual objects and events, rather than residing with generalisations and laws. |
| 13056 | Statistical explanation needs relevance, not high probability [Salmon] |
| Full Idea: Statistical relevance, not high probability, is the key desideratum in statistical explanation. | |
| From: Wesley Salmon (Four Decades of Scientific Explanation [1989], 2.5) | |
| A reaction: I suspect that this is because the explanation will not ultimately be probabilistic at all, but mechanical and causal. Hence the link is what counts, which is the relevance. He notes that relevance needs two values instead of one high value. |
| 13057 | Think of probabilities in terms of propensities rather than frequencies [Salmon] |
| Full Idea: Perhaps we should think of probabilities in terms of propensities rather than frequencies. | |
| From: Wesley Salmon (Four Decades of Scientific Explanation [1989], 3.2) | |
| A reaction: [He cites Coffa 1974 for this] I find this suggestion very appealing, as it connects up with dispositions and powers, which I take to be the building blocks of all explanation. It is, of course, easier to render frequencies numerically. |
| 8254 | Forming concepts by abstraction from the Given is private definition, which the Private Lang. Arg. attacks [McDowell] |
| Full Idea: The idea that concepts can be formed by abstraction from the Given just is the idea of private ostensive definition. So the Private Language Argument just is the rejection of the Given, in so far as it bears on the possibilities for language. | |
| From: John McDowell (Mind and World [1994], I.7) | |
| A reaction: I'm not clear why the process of abstraction from raw impressions shouldn't be a matter of public, explicit, community negotiation. We seem to be able to share and compare fairly raw impressions without much trouble (discussing sunsets). |