32 ideas
3972 | Truth and objectivity depend on a community of speakers to interpret what they mean [Davidson] |
Full Idea: The basis on which the concepts of truth and objectivity depend for application is a community of understanding, agreement among speakers on how each is to be understood. | |
From: Donald Davidson (Davidson on himself [1994], p.233) | |
A reaction: Obviously all understanding is, in practice, an interpretation by a community, but that isn't what 'truth' means. We mean 'true independently of any community'. |
3969 | There are no ultimate standards of rationality, since we only assess others by our own standard [Davidson] |
Full Idea: It makes no sense to speak of comparing or agreeing on ultimate standards of rationality, since it is our own standards in each case to which we must turn in interpreting others. This is not a failure of objectivity, but where 'questions come to an end'. | |
From: Donald Davidson (Davidson on himself [1994], p.232) | |
A reaction: This seems wrong, given the commitment to truth and charity in interpretation. He could have said the same about perception, but I doubt if he would. |
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. |
10990 | Conditionals are truth-functional, but unassertable in tricky cases? [Grice, by Read] |
Full Idea: The 'conversational defence' of the truth-functional view of conditionals is that a conditional may not be assertible in difficult cases. | |
From: report of H. Paul Grice (Presupposition and Conversational Implicature [1977]) by Stephen Read - Thinking About Logic Ch.3 |
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. |
3960 | There are no such things as minds, but people have mental properties [Davidson] |
Full Idea: There are no such things as minds, but people have mental properties. | |
From: Donald Davidson (Davidson on himself [1994], p.231) | |
A reaction: I think this is right. It fits with Searle's notion of consciousness as a property, like the liquidity of water. I don't panic if I think "I have no mind, but I have extraordinary properties". |
3964 | If the mind is an anomaly, this makes reduction of the mental to the physical impossible [Davidson] |
Full Idea: If there are no strict psychophysical laws, this rules out reductionism, either by definition of mental predicates in physical terms, or by way of bridging laws. | |
From: Donald Davidson (Davidson on himself [1994], p.231) | |
A reaction: But it is by no means clear that there are no psycho-physical laws. How could this be known a priori? |
3961 | Obviously all mental events are causally related to physical events [Davidson] |
Full Idea: All mental events are causally related to physical events. ..This seems obvious. | |
From: Donald Davidson (Davidson on himself [1994], p.231) | |
A reaction: All mental events are physically caused. Some bodily physical events result from mental events. Probably all mental events have some effect of other mental events (all of which are in some sense physical). |
3963 | There are no strict psychophysical laws connecting mental and physical events [Davidson] |
Full Idea: There are no strict psychophysical laws (that is, laws connecting mental events under their mental descriptions with physical events under their physical descriptions). | |
From: Donald Davidson (Davidson on himself [1994], p.231) | |
A reaction: This is clearly open to question. It may be just that no human mind could ever grasp such laws, given their probable complexity. |
3965 | Mental entities do not add to the physical furniture of the world [Davidson] |
Full Idea: Mental entities do not add to the physical furniture of the world. | |
From: Donald Davidson (Davidson on himself [1994], p.231) | |
A reaction: This seems to me clearly true, however we propose to characterise mental events. |
3966 | The correct conclusion is ontological monism combined with conceptual dualism [Davidson] |
Full Idea: My basic premises lead to the conclusion of ontological monism coupled with conceptual dualism (like Spinoza, except that he denied mental causation). | |
From: Donald Davidson (Davidson on himself [1994], p.231) | |
A reaction: 'Conceptual dualism' implies no real difference, but 'property dualism' is better, suggesting different properties when viewed from different angles. |
3967 | Absence of all rationality would be absence of thought [Davidson] |
Full Idea: To imagine a totally irrational animal is to imagine an animal without thought. | |
From: Donald Davidson (Davidson on himself [1994], p.232) | |
A reaction: This wouldn't be so clear without the theory of evolution, which suggests that only the finders of truth last long enough to breed. |
3974 | Our meanings are partly fixed by events of which we may be ignorant [Davidson] |
Full Idea: What we mean by what we say is partly fixed by events of which we may be ignorant. | |
From: Donald Davidson (Davidson on himself [1994], p.235) | |
A reaction: There is 'strict and literal meaning', which is fixed by the words, even if I don't know what I am saying. But 'speaker's meaning' is surely a pure matter of a state of mind? |
3968 | Propositions explain nothing without an explanation of how sentences manage to name them [Davidson] |
Full Idea: The idea of a proposition is unhelpful, until it is explained how exactly the words in the contained sentence manage to name or describe a proposition (which even Frege failed to achieve). | |
From: Donald Davidson (Davidson on himself [1994], p.232) | |
A reaction: It seems obvious to me that there are brain events best labelled as propositions, even if their fit with language is puzzling. |
3970 | Thought is only fully developed if we communicate with others [Davidson] |
Full Idea: We would have no fully-fledge thoughts if we were not in communication with others. | |
From: Donald Davidson (Davidson on himself [1994], p.233) | |
A reaction: This seems a plausible empirical observation, though I would doubt any a priori proof of it. If animals could speak, they would become intellectuals? |
10991 | Key conversational maxims are 'quality' (assert truth) and 'quantity' (leave nothing out) [Grice, by Read] |
Full Idea: Grice particularly identified two maxims as guiding conversation: the maxim of 'quality' (that one should assert only what one believes to be true and justified), and of 'quantity' (one should not assert less than one can). | |
From: report of H. Paul Grice (Presupposition and Conversational Implicature [1977]) by Stephen Read - Thinking About Logic Ch.3 | |
A reaction: I think it would be very foolish to boldly embrace the second maxim when talking to strangers. If white lies are occasionally acceptable, then what is the status of the first 'maxim'? Is it a moral maxim? |
3971 | There is simply no alternative to the 'principle of charity' in interpreting what others do [Davidson] |
Full Idea: The 'principle of charity' is a misleading term, since there is no alternative if we want to make sense of the attitudes and actions of the agents around us. | |
From: Donald Davidson (Davidson on himself [1994], p.233) | |
A reaction: I suppose so, but only with a background of evolutionary theory. I would necessarily assume charity if a robot spoke to me. |
3973 | Without a teacher, the concept of 'getting things right or wrong' is meaningless [Davidson] |
Full Idea: Without a 'teacher', nothing would give content to the idea that there is a difference between getting things right and getting them wrong. | |
From: Donald Davidson (Davidson on himself [1994], p.234) | |
A reaction: Seems right. A group of speculators with no one in the role of 'teacher' would seem to be paralysed with uncertain (except where judgements are very obvious). |
3962 | Cause and effect relations between events must follow strict laws [Davidson] |
Full Idea: If two events are related as cause and effect, there is a strict law under which they may be subsumed. | |
From: Donald Davidson (Davidson on himself [1994], p.231) | |
A reaction: Davidson admits that this is open to challenge (though Hume and Kant supported it). It does seem to be central to our understanding of nature. |