28 ideas
| 14519 | It is a great good to show reverence for a wise man [Epicurus] |
| Full Idea: To show reverence for a wise man is itself a great good for him who reveres [the wise man]. | |
| From: Epicurus (Principle Doctrines ('Kuriai Doxai') (frags) [c.290 BCE], 32) | |
| A reaction: It is characteristic of Epicurus to move up a level in his thinking, and not merely respect wisdom, but ask after the value of his own respect. Compare Idea 14517. Nice. |
| 14518 | In the study of philosophy, pleasure and knowledge arrive simultaneously [Epicurus] |
| Full Idea: In philosophy the pleasure accompanies the knowledge. For the enjoyment does not come after the learning but the learning and the enjoyment are simultaneous. | |
| From: Epicurus (Principle Doctrines ('Kuriai Doxai') (frags) [c.290 BCE], 27) |
| 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. |
| 14524 | Bodies are combinations of shape, size, resistance and weight [Epicurus] |
| Full Idea: Epicurus said that body was conceived as an aggregate of shape and size and resistance and weight. | |
| From: Epicurus (Principle Doctrines ('Kuriai Doxai') (frags) [c.290 BCE]) | |
| A reaction: [Source Sextus 'Adversus Mathematicos' 10.257] Note that this is how we 'conceive' them. They might be intrinsically different, except that Epicurus is pretty much a phenomenalist. |
| 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. |
| 14521 | If everything is by necessity, then even denials of necessity are by necessity [Epicurus] |
| Full Idea: He who claims that everything occurs by necessity has no complaint against him who claims that everything does not occur by necessity. For he makes the very claim in question by necessity. | |
| From: Epicurus (Principle Doctrines ('Kuriai Doxai') (frags) [c.290 BCE], 40) |
| 14522 | What happens to me if I obtain all my desires, and what if I fail? [Epicurus] |
| Full Idea: One should bring this question to bear on all one's desires: what will happen to me if what is sought by desire is achieved, and what will happen if it is not? | |
| From: Epicurus (Principle Doctrines ('Kuriai Doxai') (frags) [c.290 BCE], 71) | |
| A reaction: Yet another example of Epicurus moving up a level in his thinking about ethical issues, as in Idea 14517 and Idea 14519. The mark of a true philosopher. This seems to be a key idea for wisdom - to think further ahead than merely what you desire. |
| 3563 | Pleasure and virtue entail one another [Epicurus] |
| Full Idea: It is not possible to live pleasantly without living intelligently and finely and justly, nor to live intelligently and finely and justly without living pleasantly. | |
| From: Epicurus (Principle Doctrines ('Kuriai Doxai') (frags) [c.290 BCE], 5), quoted by Julia Annas - The Morality of Happiness Ch.16 | |
| A reaction: A person with all these virtues might still suffer from depression. And I don't see why having limited intelligence should stop someone from living pleasantly. Just be warm-hearted. |
| 3560 | Justice is merely a contract about not harming or being harmed [Epicurus] |
| Full Idea: There is no such things as justice in itself; in people's relations with one another in any place and at any time it is a contract about not harming or being harmed. | |
| From: Epicurus (Principle Doctrines ('Kuriai Doxai') (frags) [c.290 BCE], 33), quoted by Julia Annas - The Morality of Happiness 13.2 |
| 14517 | We value our own character, whatever it is, and we should respect the characters of others [Epicurus] |
| Full Idea: We value our characters as our own personal possessions, whether they are good and envied by men or not. We must regard our neighbours' characters thus too, if they are respectable. | |
| From: Epicurus (Principle Doctrines ('Kuriai Doxai') (frags) [c.290 BCE], 15) | |
| A reaction: I like this because it introduces a metaethical dimension to the whole problem of virtue. We should value our own character - so should we try to improve it? Should we improve so much as to become unrecognisable? |
| 14513 | Justice is a pledge of mutual protection [Epicurus] |
| Full Idea: The justice of nature is a pledge of reciprocal usefulness, neither to harm one another nor to be harmed. | |
| From: Epicurus (Principle Doctrines ('Kuriai Doxai') (frags) [c.290 BCE], 31) | |
| A reaction: Notice that justice is not just reciprocal usefulness, but a 'pledge' to that effect. This implies a metaethical value of trust and honesty in keeping the pledge. Is it better to live by the pledge, or to be always spontaneously useful? |
| 14515 | A law is not just if it is not useful in mutual associations [Epicurus] |
| Full Idea: If someone passes a law and it does not turn out to be in accord with what is useful in mutual associations, this no longer possesses the nature of justice. | |
| From: Epicurus (Principle Doctrines ('Kuriai Doxai') (frags) [c.290 BCE], 37) |
| 14520 | It is small-minded to find many good reasons for suicide [Epicurus] |
| Full Idea: He is utterly small-minded for whom there are many plausible reasons for committing suicide. | |
| From: Epicurus (Principle Doctrines ('Kuriai Doxai') (frags) [c.290 BCE], 38) | |
| A reaction: It is a pity that the insult of 'small-minded' has slipped out of philosophy. The Greeks use it all the time, and know exactly what it means. We all recognise small-mindedness, and it is a great (and subtle) vice. |
| 1474 | Moral evil may be acceptable to God because it allows free will (even though we don't see why this is necessary) [Plantinga, by PG] |
| Full Idea: Moral evil may be acceptable to a benevolent God because it is the only way to allow genuine free will, which may have a supreme value in creation (even if we are unsure what it is). | |
| From: report of Alvin Plantinga (Free Will Defence [1965], Pref.) by PG - Db (ideas) |
| 1475 | It is logically possible that natural evil like earthquakes is caused by Satan [Plantinga, by PG] |
| Full Idea: Physical evil (e.g. earthquakes) may be attributable to a fallen angel (Satan), who is the enemy of God, and this is enough to retain the idea that God is omnipotent and benevolent, and yet evil exists. | |
| From: report of Alvin Plantinga (Free Will Defence [1965], III) by PG - Db (ideas) |