32 ideas
| 14052 | Begin philosophy when you are young, and keep going when you are old [Epicurus] |
| Full Idea: Let no one delay the study of philosophy while young nor weary of it when old; for no one is either too young or too old for the health of the soul. | |
| From: Epicurus (Letter to Menoeceus [c.291 BCE], 122) | |
| A reaction: I agree with this on both accounts. I think the correct age to begin the study of philosophy is four, and it is vital to continue its study up to the point where you can no longer remember your own name. 'Health of the soul' sounds right too. |
| 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. |
| 13128 | 'Ultimate sortals' cannot explain ontological categories [Westerhoff on Wiggins] |
| Full Idea: 'Ultimate sortals' are said to be non-subordinated, disjoint from one another, and uniquely paired with each object. Because of this, the ultimate sortal cannot be a satisfactory explication of the notion of an ontological category. | |
| From: comment on David Wiggins (Identity and Spatio-Temporal Continuity [1971], p.75) by Jan Westerhoff - Ontological Categories §26 | |
| A reaction: My strong intuitions are that Wiggins is plain wrong, and Westerhoff gives the most promising reasons for my intuition. The simplest point is that objects can obviously belong to more than one category. |
| 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. |
| 14062 | Sooner follow mythology, than accept the 'fate' of natural philosophers [Epicurus] |
| Full Idea: It would be better to follow the stories told about the gods than to be a slave to the fate of the natural philosophers. | |
| From: Epicurus (Letter to Menoeceus [c.291 BCE], 134) | |
| A reaction: At this point in history there is a blurring between autonomous decisions and what we now call free will, and also between fate and determinism, which we try to keep distinct. |
| 1837 | We should not refer things to irresponsible necessity, but either to fortune or to our own will [Epicurus] |
| Full Idea: The best men have no belief in necessity (set up by some as mistress of all), but refer some things to fortune, some to ourselves, because necessity is irresponsible, and fortune is unstable, while our own will is free. | |
| From: Epicurus (Letter to Menoeceus [c.291 BCE], 133), quoted by Diogenes Laertius - Lives of Eminent Philosophers 10.27 |
| 1836 | Prudence is more valuable than philosophy, because it avoids confusions of the soul [Epicurus] |
| Full Idea: The greatest good in avoiding confusion of the soul is prudence [phronesis], on which account prudence is something more valuable than even philosophy. | |
| From: Epicurus (Letter to Menoeceus [c.291 BCE], 132), quoted by Diogenes Laertius - Lives of Eminent Philosophers 10.27 |
| 14061 | Our own choices are autonomous, and the basis for praise and blame [Epicurus] |
| Full Idea: What occurs by our own agency is autonomous, and it is to this that praise and blame are attached. | |
| From: Epicurus (Letter to Menoeceus [c.291 BCE], 133) | |
| A reaction: I don't think this should be understand as an assertion of free will in the modern sense. The 'swerve' of the atoms just means that decisions can arise out of us - not that they are somehow outside of nature. |
| 14054 | Fearing death is absurd, because we are not present when it occurs [Epicurus] |
| Full Idea: Death, the most frightening of bad things, is nothing to us; since when we exist, death is not yet present, and when death is present, then we do not exist. | |
| From: Epicurus (Letter to Menoeceus [c.291 BCE], 125) | |
| A reaction: This is a fairly accurate observation. To fear not being in this life is a bit like fearing not being in Vancouver next Tuesday. It also involves the paradox of the present moment. E.g. Idea 1904. |
| 14053 | It is absurd to fear the pain of death when you are not even facing it [Epicurus] |
| Full Idea: He is a fool who says that he fears death not because it will be painful when present but because it is painful when it is still to come. | |
| From: Epicurus (Letter to Menoeceus [c.291 BCE], 125) | |
| A reaction: Not very plausible, I'm afraid. It provides a good argument in favour of smoking, if the lung cancer is far in the future. Paralysing fear is daft, but some remote fears should be heeded. |
| 14055 | The wisdom that produces a good life also produces a good death [Epicurus] |
| Full Idea: The same kind of practice produces a good life and a good death. | |
| From: Epicurus (Letter to Menoeceus [c.291 BCE], 126) | |
| A reaction: This is the kind of old fashioned observation which we would do well to hang on to. The ideal of dying well has vanished from our culture. |
| 14057 | All pleasures are good, but it is not always right to choose them [Epicurus] |
| Full Idea: Every pleasure is a good thing, since it has a nature congenial to us, but not every one is to be chosen, just as every pain is a bad thing, but not every one is such as to be always avoided. | |
| From: Epicurus (Letter to Menoeceus [c.291 BCE], 129) | |
| A reaction: This kind of sensible remark would be wholly endorsed by Bentham and Mill. This fits in with the excellent distinction between what is right and what is good. |
| 14058 | Pleasure is the goal, but as lack of pain and calm mind, not as depraved or greedy pleasure [Epicurus] |
| Full Idea: When we say that pleasure is the goal we do not mean the pleasures of the profligate or the pleasures of consumption, but rather the lack of pain in the body and disturbance in the soul. | |
| From: Epicurus (Letter to Menoeceus [c.291 BCE], 131) | |
| A reaction: I don't really understand the aspiration to a 'calm mind'. No one likes stress, but total calmness sounds close to non-existence. The mean! There is no achievement without pain. |
| 1833 | Pleasure is the first good in life [Epicurus] |
| Full Idea: Pleasure is the beginning and end of living happily, and we recognise this as the first good. | |
| From: Epicurus (Letter to Menoeceus [c.291 BCE], 128) | |
| A reaction: We might enquire what we would live for if our capacities for pleasure were surgically removed. Would we still experience intellectual curiosity, or an aspiration to some cold and remote goodness? |
| 14063 | Sooner a good decision going wrong, than a bad one turning out for the good [Epicurus] |
| Full Idea: It is better for a good decision not to turn out right in action than for a bad decision to turn out right because of chance. | |
| From: Epicurus (Letter to Menoeceus [c.291 BCE], 135) | |
| A reaction: This sounds right, and on the whole the law agrees. Notice that what we need is a 'good decision', and not just to 'mean well'. The well-meaning fool is wicked. I am opposed to consequentialism, and agree with this idea. |
| 14059 | The best life is not sensuality, but rational choice and healthy opinion [Epicurus] |
| Full Idea: It is not drinking bouts or enjoying boys and women or consuming fish which produces the pleasant life, but sober calculation which searches out reasons for every choice, and drives out opinions which produce turmoil of the soul. | |
| From: Epicurus (Letter to Menoeceus [c.291 BCE], 132) | |
| A reaction: This more or less sums up what I would call the philosophical life. Spontaneity is good, and some pleasures are killed by excessive thought, but on the whole actions are always better if good reasons are found, and error brings chaos. |
| 1835 | True pleasure is not debauchery, but freedom from physical and mental pain [Epicurus] |
| Full Idea: When we say that pleasure is the chief good, we do not mean debauchery, but freedom of the body from pain, and of the soul from confusion…. which requires sober contemplation. | |
| From: Epicurus (Letter to Menoeceus [c.291 BCE], 131), quoted by Diogenes Laertius - Lives of Eminent Philosophers 10.27 | |
| A reaction: I'm not clear how lack of pain and confusion counts as pleasure. Also the concepts of debauchery held by the puritan and the sybarite are wildly different. |
| 14056 | We only need pleasure when we have the pain of desire [Epicurus] |
| Full Idea: We are in need of pleasure only when we are in pain because of the absence of pleasure, and when we are not in pain, then we no longer need pleasure. | |
| From: Epicurus (Letter to Menoeceus [c.291 BCE], 128) | |
| A reaction: This Buddhist aspiration to eliminate desire has no appeal for me. It just sounds like a recipe for boredom, and an aversion to risk-taking. Start by asking what is best in life; it inevitably involves pleasure of some sort. Anyway, desire isn't painful. |
| 14060 | Prudence is the greatest good, and more valuable than philosophy, because it produces virtue [Epicurus] |
| Full Idea: Prudence is the principle of the rational life and is the greatest good. That is why prudence is more valuable than philosophy, for prudence is the source of all the other virtues. | |
| From: Epicurus (Letter to Menoeceus [c.291 BCE], 132) | |
| A reaction: ['prudence' will be Greek 'phronesis']The interest of this is that it is almost copied straight out of Aristotle's Ethics. Epicurus was an opponent of the Peripatetics, but greatly influenced by them. |