26 ideas
| 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. |
| 9169 | A statement can be metaphysically necessary and epistemologically contingent [Putnam] |
| Full Idea: A statement can be (metaphysically) necessary and epistemologically contingent. Human intuition has no privileged access to metaphysical necessity. | |
| From: Hilary Putnam (Meaning and Reference [1973], p.160) | |
| A reaction: The terminology here is dangerously confusing. 'Contingent' is a term which (as Kripke insists) refers to reality, not to our epistemological abilities. The locution of adding the phrase "for all I know" seems to handle the problem better. |
| 5819 | Conceivability is no proof of possibility [Putnam] |
| Full Idea: Conceivability is no proof of possibility. | |
| From: Hilary Putnam (Meaning and Reference [1973], p.159) | |
| A reaction: This strikes me as a really basic truth which all novice philosophers should digest. It led many philosophers, especially rationalists, into all sorts of ill-founded claims about what is possible or necessary. Zombies, for instance… |
| 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. |
| 9168 | I can't distinguish elm trees, but I mean by 'elm' the same set of trees as everybody else [Putnam] |
| Full Idea: My concept of an elm tree is exactly the same as my concept of a beech tree (I blush to confess). ..We still say that the extension of 'elm' in my idiolect is the same as the extension of 'elm' in anyone else's, viz. the set of all elm trees. | |
| From: Hilary Putnam (Meaning and Reference [1973], p.154) | |
| A reaction: This example is clearer and less open to hair-splitting than his water/XYZ example. You could, with Putnam, say that his meaning of 'elm' is outside his head, but you could also say that he doesn't understand the word very well. |
| 5820 | 'Water' has an unnoticed indexical component, referring to stuff around here [Putnam] |
| Full Idea: Our theory can be summarized as saying that words like 'water' have an unnoticed indexical component: "water" is stuff that bears a certain similarity relation to the water around here. | |
| From: Hilary Putnam (Meaning and Reference [1973], p.160) | |
| A reaction: This is the causal theory of reference, which leads to externalism about concepts, which leads to an externalist view of thought, which undermines internal accounts of the mind like functionalism, and leaves little room for scepticism… Etc. |
| 9170 | We need to recognise the contribution of society and of the world in determining reference [Putnam] |
| Full Idea: Traditional semantic theory leaves out two contributions to the determination of reference - the contribution of society and the contribution of the real world; a better semantic theory must encompass both. | |
| From: Hilary Putnam (Meaning and Reference [1973], p.161) | |
| A reaction: I strongly agree that there is a social aspect to reference-fixing, but I am much more dubious about the world 'determining' anything. The whole of his Twin Earth point could be mopped up by a social account, with 'experts' as the key idea. |
| 5817 | Language is more like a cooperative steamship than an individual hammer [Putnam] |
| Full Idea: There are tools like a hammer used by one person, and there are tools like a steamship which require cooperative activity; words have been thought of too much on the model of the first sort of tool. | |
| From: Hilary Putnam (Meaning and Reference [1973], p.156) | |
| A reaction: This clear thought strikes me as the most fruitful and sensible consequence of Wittgenstein's later ideas (as opposed to the relativistic 'language game' ideas). I am unconvinced that a private language is logically impossible, but it would be feeble. |
| 22375 | Moral judgements need more than the relevant facts, if the same facts lead to 'x is good' and 'x is bad' [Foot] |
| Full Idea: It is suggested that anyone who has considered all the facts which could bear on his moral position has ipso facto produced a 'well founded' moral judgement, ...How 'x is good' can be well founded when 'x is bad' is equally well founded is hard to see. | |
| From: Philippa Foot (Moral Arguments [1958], p.96) | |
| A reaction: This seems to be a warning to particularists, if they hope that moral judgements just emerge from the facts. It doesn't rule out physicalist naturalism about morality, if the attitudes we bring to the facts have arisen out of further facts. |
| 22378 | We can't affirm a duty without saying why it matters if it is not performed [Foot] |
| Full Idea: I do not know what could be meant by saying it was someone's duty to do something unless there was an attempt to show why it mattered if this sort of thing was not done. | |
| From: Philippa Foot (Moral Arguments [1958], p.105) | |
| A reaction: The Kantian idea assumes that duty is an absolute, and yet each duty rests on a particular maxim which is going to be universalised. So why should that maxim be universalised, and not some other? |
| 22377 | Whether someone is rude is judged by agreed criteria, so the facts dictate the value [Foot] |
| Full Idea: Whether a man is speaking of behaviour as rude or not rude, he must use the same criteria as anyone else. ...We have here an example of a non-evaluative premise from which an evaluative conclusion can be deduced. | |
| From: Philippa Foot (Moral Arguments [1958], p.104) | |
| A reaction: We would now call 'rude' a 'thick' ethical concept (where 'good' is 'thin'). Her powerful point is, I take it, that evidence is always relevant to judgements of thick concepts, so there is no fact-value gap. 'Rude' has criteria, but 'good' may not. |
| 22376 | Facts and values are connected if we cannot choose what counts as evidence of rightness [Foot] |
| Full Idea: To show that facts and values are connected we must show that some things do and some things don't count in favour of a moral conclusion, and that no one can choose what counts as evidence for rightness or wrongness. | |
| From: Philippa Foot (Moral Arguments [1958], p.99) | |
| A reaction: But what sort of facts might do the job? I can only think of right functioning and health as facts which seem to imply value. Pleasure and misery don't quite get there. |
| 5818 | If water is H2O in the actual world, there is no possible world where it isn't H2O [Putnam] |
| Full Idea: Once we have discovered that water (in the actual world) is H2O, nothing counts as a possible world in which water isn't H2O. | |
| From: Hilary Putnam (Meaning and Reference [1973], p.159) | |
| A reaction: Presumably there could be a possible world in which water is a bit cloudy, so the fact that it is H2O is being judged as essential. Presumably the scientists in the possible world might discover that we are wrong about the chemistry of water? |