30 ideas
| 20771 | Six parts: dialectic, rhetoric, ethics, politics, physics, theology [Cleanthes, by Diog. Laertius] |
| Full Idea: Cleanthes says there are six parts: dialectic, rhetoric, ethics, politics, physics, and theology. | |
| From: report of Cleanthes (fragments/reports [c.270 BCE]) by Diogenes Laertius - Lives of Eminent Philosophers 07.41 | |
| A reaction: This was a minority view, as most stoics agreed with Zeno and Chrysippus that there are three main topics. Nowadays there is little discussion of the 'parts' of philosophy, but the recent revival of meta-philosophy should encourage 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. |
| 16405 | To understand a name (unlike a description) picking the thing out is sufficient? [Stalnaker] |
| Full Idea: If we ask 'what must you know to understand a name?', the naïve answer is that one must know who or what it names - nothing more. (But no one would give this answer about what is needed to understand a definite description). | |
| From: Robert C. Stalnaker (Reference and Necessity [1997], 4) | |
| A reaction: Presumably this is naive because names can be full of meaning ('the Empress'), or description and reference together ('there's the man who robbed me') and so on. It's a nice starting point though. A number can serve as a name. |
| 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. |
| 16407 | Possible worlds allow separating all the properties, without hitting a bare particular [Stalnaker] |
| Full Idea: The possible worlds framework suggests a way to express the idea that a particular is conceptually separable from its properties without relying on the rejected picture of a bare particular. | |
| From: Robert C. Stalnaker (Reference and Necessity [1997], 5) | |
| A reaction: As I read him, Stalnaker's proposal just comes down to replacing each property in turn with a different one. 'Strip away' red by making it green. It being green in w1 doesn't throw extra light. Can it be a bare particular in w37? |
| 16397 | If it might be true, it might be true in particular ways, and possible worlds describe such ways [Stalnaker] |
| Full Idea: A clarifying assumption is that if something might be true, then it might be true in some particular way. …Possible worlds begin from this, and the assumption that what might be true can be described as how a possibility might be realised. | |
| From: Robert C. Stalnaker (Reference and Necessity [1997], 2) | |
| A reaction: This is a leading practitioner giving his best shot at explaining the rationale of the possible worlds approach, addressed to many sceptics. Most sceptics, I think, don't understand the qualifications the practitioners apply to their game. |
| 16399 | Possible worlds are ontologically neutral, but a commitment to possibilities remains [Stalnaker] |
| Full Idea: I argue for the metaphysical neutrality of the possible worlds framework, but I do not suggest that its use is free of ontological commitment to possibilities (ways things might be, counterfactual situations, possible states of worlds). | |
| From: Robert C. Stalnaker (Reference and Necessity [1997], 2) | |
| A reaction: Glad to hear this, as I have always been puzzled at possible aspirations to eliminate modality (such as possibility) by introducing 'possible' worlds. Commitment to possibilities I take to be basic and unavoidable. |
| 16398 | Possible worlds allow discussion of modality without controversial modal auxiliaries [Stalnaker] |
| Full Idea: The main benefit of the possible worlds move is to permit one to paraphrase modal claims in an extensional language that has quantifiers, but no modal auxiliaries, so the semantic stucture of modal discourse can be discussed without the controversies. | |
| From: Robert C. Stalnaker (Reference and Necessity [1997], 2) | |
| A reaction: The strategy introduces the controversy of possible worlds instead, but since they just boil down to collections of objects with properties, classical logic can reign. Possible worlds are one strategy alongside many others. |
| 16396 | Kripke's possible worlds are methodological, not metaphysical [Stalnaker] |
| Full Idea: The possible worlds framework that Kripke introduces should be understood not as a metaphysical theory, but as a methodological framework. | |
| From: Robert C. Stalnaker (Reference and Necessity [1997], Intro) | |
| A reaction: That's certainly how I see possible worlds. I lose no sleep over whether they exist. I just take a set of possible worlds to be like cells in a spreadsheet, or records in a database. |
| 16408 | Rigid designation seems to presuppose that differing worlds contain the same individuals [Stalnaker] |
| Full Idea: A rigid designator is a designator that denotes the same individual in all possible worlds; doesn't this presuppose that the same individuals can be found in differing possible worlds? | |
| From: Robert C. Stalnaker (Reference and Necessity [1997], 5) | |
| A reaction: This is part of Stalnaker's claim that Kripke already has a metaphysics in place when he starts on his semantics and his theory of reference. Kripke needs a global domain, not a variable domain. Possibilities suggest variable domains to me. |
| 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. |
| 6028 | Bodies interact with other bodies, and cuts cause pain, and shame causes blushing, so the soul is a body [Cleanthes, by Nemesius] |
| Full Idea: Cleanthes says no incorporeal interacts with a body, but one body interacts with another body; the soul interacts with the body when it is sick and being cut, and the body feels shame and fear, and turns red or pale, so the soul is a body. | |
| From: report of Cleanthes (fragments/reports [c.270 BCE]) by Nemesius - De Natura Hominis 78,7 | |
| A reaction: This is precisely the interaction problem with dualism, or, as we might now say, the problem of mental causation. The standard Stoic view is that the soul is a sort of rarefied fire, which disperses at death. |
| 20831 | The soul suffers when the body hurts, creates redness from shame, and pallor from fear [Cleanthes] |
| Full Idea: Nothing incorporeal shares an experience with a body …but the soul suffers with the body when it is ill and when it is cut, and the body suffers with the soul - when the soul is ashamed the body turns red, and pale when the soul is frightened. | |
| From: Cleanthes (fragments/reports [c.270 BCE]), quoted by Nemesius - De Natura Hominis 2 | |
| A reaction: Aha - my favourite example of the corporeal nature of the mind - blushing! It is the conscious content of the thought which brings blood to the cheeks. |
| 16406 | If you don't know what you say you can't mean it; what people say usually fits what they mean [Stalnaker] |
| Full Idea: If you don't know what you are saying then you don't mean what you say, and also speakers generally mean what they say (in that what they say coincides with what they mean). | |
| From: Robert C. Stalnaker (Reference and Necessity [1997], 4) | |
| A reaction: Both these thoughts seem completely acceptable and correct, but rely on something called 'meaning' that is distinct from saying. I would express this in terms of propositions, which I take to be mental events. |
| 16404 | In the use of a name, many individuals are causally involved, but they aren't all the referent [Stalnaker] |
| Full Idea: The causal theory of reference is criticised for vagueness. Causal connections are ubiquitous, and there are obviously many individuals that are causally implicated in the speaker's use of a name, but they aren't all plausible candidates for the referent. | |
| From: Robert C. Stalnaker (Reference and Necessity [1997], 4) | |
| A reaction: This seems to be a very good objection. Among all the causal links back to some baptised object, we have to pick out the referential link, which needs a criterion. |
| 16403 | 'Descriptive' semantics gives a system for a language; 'foundational' semantics give underlying facts [Stalnaker] |
| Full Idea: 'Descriptive' semantics gives a semantics for the language without saying how practice explains why the semantics is right; …'foundational' semantics concerns the facts that give expressions their semantic values. | |
| From: Robert C. Stalnaker (Reference and Necessity [1997], §1) | |
| A reaction: [compressed] Sounds parallel to the syntax/semantics distinction, or proof-theoretical and semantic validity. Or the sense/reference distinction! Or object language/metalanguage. Shall I go on? |
| 16401 | To understand an utterance, you must understand what the world would be like if it is true [Stalnaker] |
| Full Idea: To understand what is said in an utterance of 'The first dog born at sea was a basset hound', one needs to know what the world would have been like in order for what was said in that utterance to be true. | |
| From: Robert C. Stalnaker (Reference and Necessity [1997], 3) | |
| A reaction: Put like that, the idea is undeniable. Understanding involves truth conditions. Does mean involve the understanding of the meaning. What do you understand when you understand a sentence? Just facts about dogs? Or something in the sentence? |
| 5993 | The ascending scale of living creatures requires a perfect being [Cleanthes, by Tieleman] |
| Full Idea: Cleanthes tried to prove the existence of God, arguing that the ascending scale of living creatures requires there to be a perfect being. | |
| From: report of Cleanthes (fragments/reports [c.270 BCE]) by Teun L. Tieleman - Cleanthes | |
| A reaction: Not a very good argument. Even if you accept its basic claim, it is not clear what has to exist. A perfect tree? If the being transcends the physical (in order to achieve perfection), does it cease to be a 'being'? |