29 ideas
| 20768 | Like spiderswebs, dialectical arguments are clever but useless [Ariston, by Diog. Laertius] |
| Full Idea: He said that dialectical arguments were like spiderswebs: although they seem to indicate craftsmanlike skill, they are useless. | |
| From: report of Ariston (fragments/reports [c.250 BCE]) by Diogenes Laertius - Lives of Eminent Philosophers 07.161 | |
| A reaction: Useful for the spider, but useless to Ariston. |
| 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. |
| 12790 | Generalisations must be invariant to explain anything [Leuridan] |
| Full Idea: A generalisation is explanatory if and only if it is invariant. | |
| From: Bert Leuridan (Can Mechanisms Replace Laws of Nature? [2010], §4) | |
| A reaction: [He cites Jim Woodward 2003] I dislike the idea that generalisations and regularities explain anything at all, but this rule sounds like a bare minimum for being taken seriously in the space of explanations. |
| 12789 | Biological functions are explained by disposition, or by causal role [Leuridan] |
| Full Idea: The main alternative to the dispositional theory of biological functions (which confer a survival-enhancing propensity) is the etiological theory (effects are functions if they play a role in the causal history of that very component). | |
| From: Bert Leuridan (Can Mechanisms Replace Laws of Nature? [2010], §3) | |
| A reaction: [Bigelow/Pargetter 1987 for the first, Mitchell 2003 for the second] The second one sounds a bit circular, but on the whole a I prefer causal explanations to dispositional explanations. |
| 14388 | Mechanisms must produce macro-level regularities, but that needs micro-level regularities [Leuridan] |
| Full Idea: Nothing can count as a mechanism unless it produces some macro-level regular behaviour. To produce macro-level regular behaviour, it has to rely on micro-level regularities. | |
| From: Bert Leuridan (Can Mechanisms Replace Laws of Nature? [2010], §5) | |
| A reaction: This is the core of Leuridan's argument that regularities are more basic than mechanisms. It doesn't follow, though, that the more basic a thing is the more explanatory work it can do. I say mechanisms explain more than low-level regularities do. |
| 14386 | Mechanisms are ontologically dependent on regularities [Leuridan] |
| Full Idea: Mechanisms are ontologically dependent on the existence of regularities. | |
| From: Bert Leuridan (Can Mechanisms Replace Laws of Nature? [2010], §3) | |
| A reaction: This seems to be the Humean rearguard action in favour of the regularity account of laws. Wrong, but a nice paper. This point shows why only powers (despite their vagueness!) are the only candidate for the bottom level of explanation. |
| 12787 | Mechanisms can't explain on their own, as their models rest on pragmatic regularities [Leuridan] |
| Full Idea: To model a mechanism one must incorporate pragmatic laws. ...As valuable as the concept of mechanism and mechanistic explanation are, they cannot replace regularities nor undermine their relevance for scientific explanation. | |
| From: Bert Leuridan (Can Mechanisms Replace Laws of Nature? [2010], §1) | |
| A reaction: [See Idea 12786 for 'pragmatic laws'] I just don't see how the observation of a regularity is any sort of explanation. I just take a regularity to be something interesting which needs to be explained. |
| 14384 | We can show that regularities and pragmatic laws are more basic than mechanisms [Leuridan] |
| Full Idea: Summary: mechanisms depend on regularities, there may be regularities without mechanisms, models of mechanisms must incorporate pragmatic laws, and pragmatic laws do not depend epistemologically on mechanistic models. | |
| From: Bert Leuridan (Can Mechanisms Replace Laws of Nature? [2010], §1) | |
| A reaction: See Idea 14382 for 'pragmatic' laws. I'm quite keen on mechanisms, so this is an arrow close to the heart, but at this point I say that my ultimate allegiance is to powers, not to mechanisms. |
| 14389 | There is nothing wrong with an infinite regress of mechanisms and regularities [Leuridan] |
| Full Idea: I see nothing metaphysically wrong in an infinite ontological regress of mechanisms and regularities. | |
| From: Bert Leuridan (Can Mechanisms Replace Laws of Nature? [2010], §5) | |
| A reaction: This is a pretty unusual view, and I can't accept it. My revulsion at this regress is precisely the reason why I believe in powers, as the bottom level of explanation. |
| 3049 | The chief good is indifference to what lies midway between virtue and vice [Ariston, by Diog. Laertius] |
| Full Idea: The chief good is to live in perfect indifference to all those things which are of an intermediate character between virtue and vice. | |
| From: report of Ariston (fragments/reports [c.250 BCE]) by Diogenes Laertius - Lives of Eminent Philosophers 07.2.1 |
| 3549 | Ariston says rules are useless for the virtuous and the non-virtuous [Ariston, by Annas] |
| Full Idea: Ariston says that rules are useless if you are virtuous, and useless if you are not. | |
| From: report of Ariston (fragments/reports [c.250 BCE]) by Julia Annas - The Morality of Happiness 2.4 |
| 14387 | Rather than dispositions, functions may be the element that brought a thing into existence [Leuridan] |
| Full Idea: The dispositional theory of biological functions is not unquestioned. The main alternative is the etiological theory: a component's effect is a function of that component if it has played an essential role in the causal history of its existence. | |
| From: Bert Leuridan (Can Mechanisms Replace Laws of Nature? [2010], §3) | |
| A reaction: [He cites S.D. Mitchell 2003] Presumably this account is meant to fit into a theory of evolution in biology. The obvious problem is where something comes into existence for one reason, and then acquires a new function (such as piano-playing). |
| 14382 | Pragmatic laws allow prediction and explanation, to the extent that reality is stable [Leuridan] |
| Full Idea: A generalization is a 'pragmatic law' if it allows of prediction, explanation and manipulation, even if it fails to satisfy the traditional criteria. To this end, it should describe a stable regularity, but not necessarily a universal and necessary one. | |
| From: Bert Leuridan (Can Mechanisms Replace Laws of Nature? [2010], §1) | |
| A reaction: I am tempted to say of this that all laws are pragmatic, given that it is rather hard to know whether reality is stable. The universal laws consist of saying that IF reality stays stable in certain ways, certain outcomes will ensue necessarily. |
| 14385 | Strict regularities are rarely discovered in life sciences [Leuridan] |
| Full Idea: Strict regularities are rarely if ever discovered in the life sciences. | |
| From: Bert Leuridan (Can Mechanisms Replace Laws of Nature? [2010], §2) | |
| A reaction: This is elementary once it is pointed out, but too much philosophy have science has aimed at the model provided by the equations of fundamental physics. Science is a broad church, to employ an entertaining metaphor. |
| 14383 | A 'law of nature' is just a regularity, not some entity that causes the regularity [Leuridan] |
| Full Idea: By 'law of nature' or 'natural law' I mean a generalization describing a regularity, not some metaphysical entity that produces or is responsible for that regularity. | |
| From: Bert Leuridan (Can Mechanisms Replace Laws of Nature? [2010], §1 n1) | |
| A reaction: I take the second version to be a relic of a religious world view, and having no place in a naturalistic metaphysic. The regularity view is then the only player in the field, and the question is, can we do more? Can't we explain regularities? |