25 ideas
| 17325 | Truth-maker theory can't cope with non-causal dependence [Liggins] |
| Full Idea: My charge is that truth-maker theory cannot be integrated into an attractive general account of non-causal dependence. | |
| From: David Liggins (Truth-makers and dependence [2012], 10.6) | |
| A reaction: [You'll have to read Liggins to see why] |
| 17318 | Truthmakers for existence is fine; otherwise maybe restrict it to synthetic truths? [Liggins] |
| Full Idea: Many philosophers agree that true existential propositions have a truth-maker, but some go further, claiming that every true proposition has a truth-maker. More cautious theorists specify a class of truths, such as synthetic propositions. | |
| From: David Liggins (Truth-makers and dependence [2012], 10.1) | |
| A reaction: [compressed; Armstrong is the ambitious one, and Rodriguez-Pereyra proposes the synthetic propositions] Presumably synthetic propositions can make negative assertions, which are problematic for truth-makers. |
| 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. |
| 17320 | Either p is true or not-p is true, so something is true, so something exists [Liggins] |
| Full Idea: Either p or not-p. If p, then the proposition 'p' is true. If not p, then the proposition 'not p' is true. Either way, something is true. Thus something exists. | |
| From: David Liggins (Truth-makers and dependence [2012], 10.3 n5) | |
| A reaction: Liggins offers this dodgy argument as an objection to conceptual truths having truth-makers. |
| 17326 | The dependence of {Socrates} on Socrates involves a set and a philosopher, not facts [Liggins] |
| Full Idea: The dependence of {Socrates} on Socrates appears to involve a set and a philosopher, neither of which is a fact. | |
| From: David Liggins (Truth-makers and dependence [2012], 10.6) | |
| A reaction: He points out that defenders of facts as the basis of dependence could find a suitable factual paraphrase here. Socrates is just Socrates, but the singleton has to be understood in a particular way to generate the dependence. |
| 17327 | Non-causal dependence is at present only dimly understood [Liggins] |
| Full Idea: Non-causal dependence is at present only dimly understood. | |
| From: David Liggins (Truth-makers and dependence [2012], 10.8) | |
| A reaction: Not very helpful, you may be thinking, but it is always helpful to know where we have got to in the enquiry. |
| 17322 | Necessities supervene on everything, but don't depend on everything [Liggins] |
| Full Idea: Necessities supervene upon everything, but they do not depend on everything. | |
| From: David Liggins (Truth-makers and dependence [2012], 10.4) | |
| A reaction: I'm not sure if merely existing together counts as sufficiently close to be 'supervenience'. If 2+2 necessitates 4, that hardly seems to 'supervene' on the Eiffel Tower. If so, how close must things be to qualify for supervenience? |
| 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. |
| 17324 | 'Because' can signal an inference rather than an explanation [Liggins] |
| Full Idea: 'Because' can signal an inference rather than an explanation. | |
| From: David Liggins (Truth-makers and dependence [2012], 10.5) | |
| A reaction: Aristotle starts from words like 'why?', but it can be a deceptive approach to explanation. |
| 17321 | Value, constitution and realisation are non-causal dependences that explain [Liggins] |
| Full Idea: 'It is wrong because it produces pain for fun', and 'these constitute a table because they are arranged tablewise', and 'tea is poisonous because it contains arsenic' are clearly non-causal uses of 'because', and neither are they conceptual. | |
| From: David Liggins (Truth-makers and dependence [2012], 10.4) | |
| A reaction: The general line seems to be that any form of determination will underwrite an explanation. He talks later of the 'wrongmaker' and 'poisonmaker' relationships to add to the 'truthmaker'. The table example is the 'object-maker' dependence relation. |
| 17323 | If explanations track dependence, then 'determinative' explanations seem to exist [Liggins] |
| Full Idea: If explanation often tracks dependence, then we have a theoretical reason to expect such explanations to exist. Let us call such explanations 'determinative'. | |
| From: David Liggins (Truth-makers and dependence [2012], 10.4) | |
| A reaction: There seems to be an emerging understanding that this 'determination' relation is central to all of explanation - with causal explanations, for example, being a particular instance of it. I like it. These are real, not conventional, explanations. |
| 20662 | The biology of societies: kin selection, parenting, mating; status, territory, contracts [Wilson,EO] |
| Full Idea: Societies are ordered around six sociobiological principles: kin selection; parental investment; mating strategy; status; territorial expansion and defence; contractual agreement. | |
| From: Edmund O. Wilson (Consilience [1998], 19 'Intro'), quoted by Peter Watson - Convergence | |
| A reaction: I'm not sure I trust such a precise list. Personally I'm in society because I'm too frightend to drop out. So where is 'defence'? Still, I like attempts at assembling such a list. Politics needs grounding. |