27 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. |
| 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. |
| 8239 | If the King likes music then there is hope for the state [Mengzi (Mencius)] |
| Full Idea: If the King has a great fondness for music, then perhaps there is hope for the state of Ch'i. | |
| From: Mengzi (Mencius) (The Mengzi (Mencius) [c.332 BCE], 1.B.1) | |
| A reaction: This seems to be Shakespeare's attitude to music as well. The general idea must be that love of music requires a selfless state of mind, where the mind revels in the beauty of something outside of itself. Respect is the desirable result. |
| 23398 | Human nature is naturally compassionate and good (as a 'sprout'), but people may not be good [Mengzi (Mencius), by Norden] |
| Full Idea: Mengzi does not claim that humans are innately good; he claims that human nature is innately good. …He says that 'the heart of compassion' (manifested when anyone sees a child about to fall into a well) is the 'sprout of benevolence'. | |
| From: report of Mengzi (Mencius) (The Mengzi (Mencius) [c.332 BCE]) by Bryan van Norden - Intro to Classical Chinese Philosophy 6.II | |
| A reaction: There is a nice distinction here between the 'sprout' of human nature and the finished product. Seeds have the potential to produce tall healthy plants, but circumstances can warp them. |
| 23400 | Righteousness is extending the unthinkable, to reveal what must be done [Mengzi (Mencius)] |
| Full Idea: People all have things they will not do. To extend this reaction to that which they will do is righteousness. | |
| From: Mengzi (Mencius) (The Mengzi (Mencius) [c.332 BCE], 7B31), quoted by Bryan van Norden - Intro to Classical Chinese Philosophy 6.IV | |
| A reaction: Very nice! Kekes points out the enormous importance of unthinkable deeds. Depravity is when the unthinkable gradually begins to look possible, which is probably a social phenomenon, a creeping cancer in a culture. |
| 23399 | Each correct feeling relies on an underlying virtue [Mengzi (Mencius)] |
| Full Idea: The heart of compassion is benevolence. The heart of disdain is righteousness. The heart of respect is propriety. The heart of approval and disapproval is wisdom. | |
| From: Mengzi (Mencius) (The Mengzi (Mencius) [c.332 BCE], 6A6), quoted by Bryan van Norden - Intro to Classical Chinese Philosophy 6.III | |
| A reaction: 'Disdain' seems to be the response to anyone who is disrespectful. Note that wisdom concerns judgements. Respect seems to be more of a social convention than an actual concern for others. |
| 8235 | Should a coward who ran fifty paces from a battle laugh at another who ran a hundred? [Mengzi (Mencius)] |
| Full Idea: If two soldiers were fleeing from a battle, and one stopped after a hundred paces and the other stopped after a fifty paces, what would you think if the latter, as one who only ran fifty paces, were to laugh at the former who ran a hundred? | |
| From: Mengzi (Mencius) (The Mengzi (Mencius) [c.332 BCE], 1.A.3) | |
| A reaction: A nice illustration, in my view, of the universality of truths about human virtue. In no culture would this laughter be appropriate. Nevertheless, there must be degrees of dishonour. Better to flee than join in with the likely winners. |
| 8240 | A true king shares his pleasure with the people [Mengzi (Mencius)] |
| Full Idea: If you shared your enjoyment of music or of hunting with the people, you would be a true King. | |
| From: Mengzi (Mencius) (The Mengzi (Mencius) [c.332 BCE], 1.B.1) | |
| A reaction: I suspect that this is a great truth for dictators and traditional monarchs. One pictures the successful ones attending public entertainments, and allowing the public to see their own. Tyrants keep entertainment private. Nero is a counterexample! |
| 8237 | Extend the treatment of the old and young in your family to the rest of society [Mengzi (Mencius)] |
| Full Idea: Treat the aged of your own family in a manner befitting their venerable age and extend this treatment to the aged of other families. Treat your own young in a manner befitting their tender age, and extend this to the young of other families. | |
| From: Mengzi (Mencius) (The Mengzi (Mencius) [c.332 BCE], 1.A.7) | |
| A reaction: This seems to me to articulate the ideal of communitarianism very nicely. Morality is not just about healthy adults in war and peace. It must include the children and the old. The values of the family are above the values of contracts and calculations. |
| 8241 | Only put someone to death if the whole population believes it is deserved [Mengzi (Mencius)] |
| Full Idea: When close attendants say a man deserves death, do not listen; when all the councillors say so, do not listen; when everyone says so, have the case investigated. If he is guilty, put him to death; he was put to death by the whole country. | |
| From: Mengzi (Mencius) (The Mengzi (Mencius) [c.332 BCE], 1.B.7) | |
| A reaction: The jury system is a gesture in this direction. Compare Idea 95. In Mencius's time, no doubt, everyone believed that capital punishment was sometimes right. Nowadays, when many people (e.g. me) reject it, the procedure won't work. |
| 8238 | Seeking peace through war is like looking for fish up a tree [Mengzi (Mencius)] |
| Full Idea: Your desire to extend your territory by war, in order to bring peace, is like looking for fish by climbing a tree. | |
| From: Mengzi (Mencius) (The Mengzi (Mencius) [c.332 BCE], 1.A.7) | |
| A reaction: Mencius had a flair for analogies. Just occasionally I suppose he might be wrong on this point, but I would think that experiments in the laboratory of history have shown that he is right in nearly all cases. |
| 8236 | Avoid the animals you are going to eat, as it is hard once you have got to know them [Mengzi (Mencius)] |
| Full Idea: Once a gentleman has seen animals alive, he cannot bear to see them die, and once having heard their cry, he cannot bear to eat their flesh. That is why the gentleman keeps his distance from the kitchen. | |
| From: Mengzi (Mencius) (The Mengzi (Mencius) [c.332 BCE], 1.A.7) | |
| A reaction: If you applied this to a Gestapo officer and his victims, it would obviously be the epitome of wickedness. But it is complex. Compassion is expected when we encounter suffering, but we are not obliged to seek out suffering. Or are we? |
| 1748 | Archelaus was the first person to say that the universe is boundless [Archelaus, by Diog. Laertius] |
| Full Idea: Archelaus was the first person to say that the universe is boundless. | |
| From: report of Archelaus (fragments/reports [c.450 BCE]) by Diogenes Laertius - Lives of Eminent Philosophers 02.Ar.3 |
| 5989 | Archelaus said life began in a primeval slime [Archelaus, by Schofield] |
| Full Idea: Archelaus wrote that life on Earth began in a primeval slime. | |
| From: report of Archelaus (fragments/reports [c.450 BCE]) by Malcolm Schofield - Archelaus | |
| A reaction: This sounds like a fairly clearcut assertion of the production of life by evolution. Darwin's contribution was to propose the mechanism for achieving it. We should honour the name of Archelaus for this idea. |