25 ideas
| 9271 | Human knowledge may not produce well-being; the examined life may not be worth living [Gray] |
| Full Idea: Human knowledge is one thing, human well-being another. There is no predetermined harmony between the two. The examined life may not be worth living. | |
| From: John Gray (Straw Dogs [2002], 1.9) | |
| A reaction: John Gray has set himself up as the Eeyore of modern times, but this point may obviously be correct. Presumably Socrates meant that the examined life was better even if the result was less 'well-being'. Even Gray doesn't want a lobotomy. |
| 21677 | How can the not-true fail to be false, or the not-false fail to be true? [Cicero] |
| Full Idea: How can something that is not true not be false, or how can something that is not false not be true? | |
| From: M. Tullius Cicero (On Fate ('De fato') [c.44 BCE], 16.38) | |
| A reaction: We must at least distinguish between whether the contrary thing is not actually true, or whether we are prepared to assert that it is not true. The disjunction may seem to be a false dichotomy. 'He isn't good' may not entail 'he is evil'. |
| 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. |
| 9275 | Knowledge does not need minds or nervous systems; it is found in all living things [Gray] |
| Full Idea: Knowledge does not need minds, or even nervous systems. It is found in all living things. | |
| From: John Gray (Straw Dogs [2002], 2.10) | |
| A reaction: I consider it a misnomer to call such things 'knowledge', for which I have much higher standards. Gray is talking about 'information'. Knowledge needs reasons, and possibility of error, not just anticipatory behaviour. |
| 9276 | The will hardly ever does anything; most of our life just happens to us [Gray] |
| Full Idea: We think our actions express our decisions, but in nearly all of our life, willing decides nothing. We cannot wake up or fall asleep, remember or forget our dreams, summon or banish our thoughts, by deciding to do so. | |
| From: John Gray (Straw Dogs [2002], 2.12) | |
| A reaction: Gray's point does not rule out occasional total control over mental life, but his point is important. The traditional picture is of a life controlled, so the will is seen as at the centre of a person, but it just isn't the case. |
| 21667 | Oratory and philosophy are closely allied; orators borrow from philosophy, and ornament it [Cicero] |
| Full Idea: There is a close alliance between the orator and the philosophical system of which I am a follower, since the orator borrows subtlely from the Academy, and repays the loan by giving to it a copious and flowing style and rhetorical ornament. | |
| From: M. Tullius Cicero (On Fate ('De fato') [c.44 BCE], 02.03) | |
| A reaction: It is a misundertanding to think that rhetoric and philosophy are seen as in necessary opposition. Philosophers just seemed to think that oratory works a lot better if it is truthful. |
| 21678 | If desire is not in our power then neither are choices, so we should not be praised or punished [Cicero] |
| Full Idea: If the cause of desire is not situated within us, even desire itself is also not in our power. ...It follows that neither assent nor action is in our power. Hence there is no justice in either praise or blame, either honours or punishments. | |
| From: M. Tullius Cicero (On Fate ('De fato') [c.44 BCE], 17.40) | |
| A reaction: This is the view of 'old philosophers', but I'm unsure which ones. Cicero spurns this view. It is obvious that the causes of our desires are largely out of our control. Responsibility seems to concern what we do about our desires. |
| 9278 | Nowadays we identify the free life with the good life [Gray] |
| Full Idea: We do not value freedom more than people did in earlier times, but we have identified the good life with the chosen life. | |
| From: John Gray (Straw Dogs [2002], 3.13) | |
| A reaction: Interesting. This is Enlightenment liberalism gradually filtering down into common consciousness, especially via the hegemony of American culture. I sympathise the Gray; don't get me wrong, but I think freedom is overrated. |
| 9280 | Over forty percent of the Earth's living tissue is human [Gray] |
| Full Idea: Humans co-opt over forty per cent of the Earth's living tissue. | |
| From: John Gray (Straw Dogs [2002], 4.15) | |
| A reaction: If you add our domestic animals, I understand that the figure goes up to 95 per cent! I take this to be virtually the only significant ecological fact - population, population, population. Why are there so many cars? So many carbon footprints? |
| 9272 | Without Christianity we lose the idea that human history has a meaning [Gray] |
| Full Idea: For Christians, it is because they occur in history that the lives of humans have a meaning that the lives of other animals do not. ..If we truly leave Christianity behind, we must give up the idea that human history has a meaning. | |
| From: John Gray (Straw Dogs [2002], 2.3) | |
| A reaction: Interesting. Compare the dispute between 'whig' and 'tory' historians, the former of whom believe that history is going somewhere. |
| 9279 | What was our original sin, and how could Christ's suffering redeem it? [Gray] |
| Full Idea: No one can say what was humankind's original sin, and no one understands how the suffering of Christ can redeem it. | |
| From: John Gray (Straw Dogs [2002], 4.1) | |
| A reaction: This nicely articulates a problem that has half bothered me, but I have never put into words. I always assumed Eve committed the sin, and Adam cops the blame for not controlling his woman. Dying for our sins has always puzzled me. |