31 ideas
| 2557 | Analytical philosophy seems to have little interest in how to tell a good analysis from a bad one [Rorty] |
| Full Idea: There is nowadays little attempt to bring "analytic philosophy" to self-consciousness by explaining how to tell a successful from an unsuccessful analysis. | |
| From: Richard Rorty (Philosophy and the Mirror of Nature [1980], 4.1) |
| 2556 | Rational certainty may be victory in argument rather than knowledge of facts [Rorty] |
| Full Idea: We can think of "rational certainty" as a matter of victory in argument rather than relation to an object known. | |
| From: Richard Rorty (Philosophy and the Mirror of Nature [1980], 3.4) |
| 4726 | Rorty seems to view truth as simply being able to hold one's view against all comers [Rorty, by O'Grady] |
| Full Idea: Rorty seems to view truth as simply being able to hold one's view against all comers. | |
| From: report of Richard Rorty (Philosophy and the Mirror of Nature [1980]) by Paul O'Grady - Relativism Ch.4 | |
| A reaction: This may be a caricature of Rorty, but he certainly seems to be in the business of denying truth as much as possible. This strikes me as the essence of pragmatism, and as a kind of philosophical nihilism. |
| 2549 | For James truth is "what it is better for us to believe" rather than a correct picture of reality [Rorty] |
| Full Idea: Truth is, in James' phrase, "what it is better for us to believe", rather than "the accurate representation of reality". | |
| From: Richard Rorty (Philosophy and the Mirror of Nature [1980], Intro) |
| 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. |
| 13132 | A snowball's haecceity is the property of being identical with itself [Plantinga, by Westerhoff] |
| Full Idea: Plantinga assumes that being identical with that snowball names a property which is that snowball's haecceity. | |
| From: report of Alvin Plantinga (De Essentia [1979]) by Jan Westerhoff - Ontological Categories §52 | |
| A reaction: Only a philosopher would suggest such a bizarre way of establishing the unique individuality of a given snowball. You could hardly keep track of the snowball with just that criterion. How do you decide whether something has Plantinga's property? |
| 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. |
| 2548 | If knowledge is merely justified belief, justification is social [Rorty] |
| Full Idea: If we have a Deweyan conception of knowledge, as what we are justified in believing, we will see "justification" as a social phenomenon. | |
| From: Richard Rorty (Philosophy and the Mirror of Nature [1980], Intro) | |
| A reaction: I find this observation highly illuminating (though I probably need to study Dewey to understand it). There just is no absolute about whether someone is justified. How justified do you want to be? |
| 6599 | Knowing has no definable essence, but is a social right, found in the context of conversations [Rorty] |
| Full Idea: If we see knowing not as having an essence, described by scientists or philosophers, but rather as a right, by current standards, to believe, then we see conversation as the ultimate context within which knowledge is to be understood. | |
| From: Richard Rorty (Philosophy and the Mirror of Nature [1980], Ch.5), quoted by Robert Fogelin - Walking the Tightrope of Reason Ch.5 | |
| A reaction: This teeters towards ridiculous relativism (e.g. what if the conversation is among a group of fools? - Ah, there are no fools! Politically incorrect!). However, knowledge can be social, provided we are healthily elitist. Scientists know more than us. |
| 2566 | You can't debate about whether to have higher standards for the application of words [Rorty] |
| Full Idea: The decision about whether to have higher than usual standards for the application of words like "true" or "good" or "red" is, as far as I can see, not a debatable issue. | |
| From: Richard Rorty (Philosophy and the Mirror of Nature [1980], 6.6) |
| 2553 | The mind is a property, or it is baffling [Rorty] |
| Full Idea: All that is needed for the mind-body problem to be unintelligible is for us to be nominalist, to refuse firmly to hypostasize individual properties. | |
| From: Richard Rorty (Philosophy and the Mirror of Nature [1980], 1.3) | |
| A reaction: Edelman says the mind is a process rather than a property. It might vanish if the clockspeed was turned right down? Nominalism here sounds like behaviourism or instrumentalism. Would Dennett plead guilty? |
| 2550 | Pain lacks intentionality; beliefs lack qualia [Rorty] |
| Full Idea: We can't define the mental as intentional because pains aren't about anything, and we can't define it as phenomenal because beliefs don't feel like anything. | |
| From: Richard Rorty (Philosophy and the Mirror of Nature [1980], 1.2) | |
| A reaction: Nice, but simplistic? There is usually an intentional object for a pain, and the concepts which we use to build beliefs contain the residue of remembered qualia. It seems unlikely that any mind could have one without the other (even a computer). |
| 2554 | Is intentionality a special sort of function? [Rorty] |
| Full Idea: Following Wittgenstein, we shall treat the intentional as merely a subspecies of the functional. | |
| From: Richard Rorty (Philosophy and the Mirror of Nature [1980], 1.3) | |
| A reaction: Intriguing but obscure. Sounds wrong to me. The intentional refers to the content of thoughts, but function concerns their role. They have roles because they have content, so they can't be the same. |
| 2565 | Nature has no preferred way of being represented [Rorty] |
| Full Idea: Nature has no preferred way of being represented. | |
| From: Richard Rorty (Philosophy and the Mirror of Nature [1980], 6.5) | |
| A reaction: Tree rings accidentally represent the passing of the years. If God went back and started again would she or he opt for a 'preferred way'? |
| 2560 | Can meanings remain the same when beliefs change? [Rorty] |
| Full Idea: For cooler heads there must be some middle view between "meanings remain and beliefs change" and "meanings change whenever beliefs do". | |
| From: Richard Rorty (Philosophy and the Mirror of Nature [1980], 6.2) | |
| A reaction: The second one seems blatanty false. How could we otherwise explain a change in belief? But obviously some changes in belief (e.g. about electrons) produce a change in meaning. |
| 2562 | A theory of reference seems needed to pick out objects without ghostly inner states [Rorty] |
| Full Idea: The need to pick out objects without the help of definitions, essences, and meanings of terms produced, philosophers thought, a need for a "theory of reference". | |
| From: Richard Rorty (Philosophy and the Mirror of Nature [1980], 6.3) | |
| A reaction: Frege's was very perceptive in noting that meaning and reference are not the same. Whether we need a 'theory' of reference is unclear. It is worth describing how it occurs. |
| 2559 | Davidson's theory of meaning focuses not on terms, but on relations between sentences [Rorty] |
| Full Idea: A theory of meaning, for Davidson, is not an assemblage of "analyses" of the meanings of individual terms, but rather an understanding of the inferential relations between sentences. | |
| From: Richard Rorty (Philosophy and the Mirror of Nature [1980], 6.1) | |
| A reaction: Put that way, the influence of Frege on Davidson is obvious. Purely algebraic expressions can have inferential relations, using variables and formal 'sentences'. |
| 2558 | Since Hegel we have tended to see a human as merely animal if it is outside a society [Rorty] |
| Full Idea: Only since Hegel have philosophers begun toying with the idea that the individual apart from his society is just one more animal. | |
| From: Richard Rorty (Philosophy and the Mirror of Nature [1980], 4.3) |