53 ideas
| 1887 | You cannot divide anything into many parts, because after the first division you are no longer dividing the original [Sext.Empiricus] |
| Full Idea: You cannot divide anything (such as the decad) into many parts, because as soon as you separate the first part, you are no longer dividing the original. | |
| From: Sextus Empiricus (Outlines of Pyrrhonism [c.180], II.215) |
| 1885 | Proof moves from agreed premises to a non-evident inference [Sext.Empiricus] |
| Full Idea: Dogmatists define proof as "an argument which, by means of agreed premises, reveals by way of deduction a nonevident inference". | |
| From: Sextus Empiricus (Outlines of Pyrrhonism [c.180], II.135) |
| 12196 | A valid hypothetical syllogism is 'that which does not begin with a truth and end with a falsehood' [Sext.Empiricus] |
| Full Idea: Philo (of Megara) says that a valid hypothetical syllogism is 'that which does not begin with a truth and end with a falsehood,' as for instance the syllogism 'If it is day, I converse,' when in fact it is day and I am conversing. | |
| From: Sextus Empiricus (Outlines of Pyrrhonism [c.180], II.110) | |
| A reaction: Russell endorses this, and Rumfitt quotes it as the classic case of denying that there is any modal aspect (such as 'logical necessity') involved in logical consequence. He labels it 'material or Philonian consequence'. |
| 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. |
| 1902 | Since Socrates either died when he was alive (a contradiction) or died when he was dead (meaningless), he didn't die [Sext.Empiricus] |
| Full Idea: If Socrates died, he died either when he lived or when he died; so he was either dead when he was alive, or he was twice dead when he was dead. So he didn't die. | |
| From: Sextus Empiricus (Outlines of Pyrrhonism [c.180], III.111) | |
| A reaction: One of my favourites. Of all the mysteries facing us, the one that boggles me most is how anything can happen in the 'present' moment, if the present is just the overlap point between past and future. |
| 10245 | One geometry cannot be more true than another [Poincaré] |
| Full Idea: One geometry cannot be more true than another; it can only be more convenient. | |
| From: Henri Poincaré (Science and Method [1908], p.65), quoted by Stewart Shapiro - Philosophy of Mathematics | |
| A reaction: This is the culminating view after new geometries were developed by tinkering with Euclid's parallels postulate. |
| 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. |
| 1889 | If an argument has an absurd conclusion, we should not assent to the absurdity, but avoid the absurd argument [Sext.Empiricus] |
| Full Idea: If an argument leads to confessedly absurd conclusions, we should not assent to the absurdity just because of the argument, but avoid the argument because of the absurdity. | |
| From: Sextus Empiricus (Outlines of Pyrrhonism [c.180], II.252) | |
| A reaction: cf. G.E.Moore. Denying that you have a hand seems to be an absurdity, but I'm not sure if I can give a criterion for absurdity in such a case. One person's modus ponens is another person's modus tollens. |
| 1871 | Whether honey is essentially sweet may be doubted, as it is a matter of judgement rather than appearance [Sext.Empiricus] |
| Full Idea: Honey appears to sceptics to be sweet, but whether it is also sweet in its essence is for us a matter of doubt, since this is not an appearance but a judgement. | |
| From: Sextus Empiricus (Outlines of Pyrrhonism [c.180], I.20) |
| 1883 | How can the intellect know if sensation is reliable if it doesn't directly see external objects? [Sext.Empiricus] |
| Full Idea: Just as you can't know if a portrait of Socrates is good without seeing the man, so when the intellect gazes on sensations but not the external objects it cannot know whether they are similar. | |
| From: Sextus Empiricus (Outlines of Pyrrhonism [c.180], II.75) |
| 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. |
| 1890 | We distinguish ambiguities by seeing what is useful [Sext.Empiricus] |
| Full Idea: It is the experience of what is useful in each affair that brings about the distinguishing of ambiguities. | |
| From: Sextus Empiricus (Outlines of Pyrrhonism [c.180], II.258) |
| 1870 | The basis of scepticism is the claim that every proposition has an equal opposing proposition [Sext.Empiricus] |
| Full Idea: The main basic principle of the sceptic system is that of opposing to every proposition an equal proposition. | |
| From: Sextus Empiricus (Outlines of Pyrrhonism [c.180], I.12) |
| 1872 | The same tower appears round from a distance, but square close at hand [Sext.Empiricus] |
| Full Idea: The same tower appears round from a distance, but square close at hand. | |
| From: Sextus Empiricus (Outlines of Pyrrhonism [c.180], I.32) |
| 1881 | The same oar seems bent in water and straight when out of it [Sext.Empiricus] |
| Full Idea: The same oar seems bent when in the water but straight when out of the water. | |
| From: Sextus Empiricus (Outlines of Pyrrhonism [c.180], I.119) |
| 1882 | The necks of doves appear different in colour depending on the angle of viewing [Sext.Empiricus] |
| Full Idea: The necks of doves appear different in hue according to the differences in the angle of inclination. | |
| From: Sextus Empiricus (Outlines of Pyrrhonism [c.180], I.120) |
| 1873 | If we press the side of an eyeball, objects appear a different shape [Sext.Empiricus] |
| Full Idea: When we press the eyeball at one side the forms, figures and sizes of the objects appear oblong and narrow. | |
| From: Sextus Empiricus (Outlines of Pyrrhonism [c.180], I.47) |
| 1874 | How can we judge between our impressions and those of other animals, when we ourselves are involved? [Sext.Empiricus] |
| Full Idea: We cannot judge between our own impressions and those of other animals, because we ourselves are involved in the dispute. | |
| From: Sextus Empiricus (Outlines of Pyrrhonism [c.180], I.59) |
| 1879 | Sickness is perfectly natural to the sick, so their natural perceptions should carry some weight [Sext.Empiricus] |
| Full Idea: Health is natural for the healthy but unnatural for the sick, and sickness is unnatural for the healthy but natural for the sick, so we must give credence to the natural perceptions of the sick. | |
| From: Sextus Empiricus (Outlines of Pyrrhonism [c.180], I.103) |
| 1876 | If we enjoy different things, presumably we receive different impressions [Sext.Empiricus] |
| Full Idea: The enjoyment of different things is an indication that we get varying impressions from the underlying objects. | |
| From: Sextus Empiricus (Outlines of Pyrrhonism [c.180], I.80) |
| 1877 | If we had no hearing or sight, we would assume no sound or sight exists, so there may be unsensed qualities [Sext.Empiricus] |
| Full Idea: A man with touch, taste and smell, but no hearing or sight, will assume nothing audible or visible exists, so maybe an apple has qualities which we have no senses to perceive. | |
| From: Sextus Empiricus (Outlines of Pyrrhonism [c.180], I.96) |
| 1880 | Some actions seem shameful when sober but not when drunk [Sext.Empiricus] |
| Full Idea: Actions which seem shameful to us when sober do not seem shameful when drunk. | |
| From: Sextus Empiricus (Outlines of Pyrrhonism [c.180], I.109) |
| 1878 | Water that seems lukewarm can seem very hot on inflamed skin [Sext.Empiricus] |
| Full Idea: The same water which seems very hot when poured on inflamed spots seems lukewarm to us. | |
| From: Sextus Empiricus (Outlines of Pyrrhonism [c.180], I.101) |
| 1911 | Even if all known nations agree on a practice, there may be unknown nations which disagree [Sext.Empiricus] |
| Full Idea: Even among practices on which all known cultures are agreed, disagreement about them may possibly exist amongst some of the nations which are unknown to us. | |
| From: Sextus Empiricus (Outlines of Pyrrhonism [c.180], III.234) |
| 1910 | With us it is shameful for men to wear earrings, but among Syrians it is considered noble [Sext.Empiricus] |
| Full Idea: It is a shameful thing with us for men to wear earrings, but among some of the barbarians, such as the Syrians, it is a token of nobility. | |
| From: Sextus Empiricus (Outlines of Pyrrhonism [c.180], III.203) |
| 1886 | If you don't view every particular, you may miss the one which disproves your universal induction [Sext.Empiricus] |
| Full Idea: Induction cannot establish the universal by means of the particular, since limited particulars may omit crucial examples which disprove the universal, and infinite particulars are impossible to know. | |
| From: Sextus Empiricus (Outlines of Pyrrhonism [c.180], II.204) |
| 1884 | If we utter three steps of a logical argument, they never exist together [Sext.Empiricus] |
| Full Idea: If we say "If day exists, lights exists", and then "day exists", and then "light exists", then parts of the judgement never exist together, and so the whole judgement will have no real existence. | |
| From: Sextus Empiricus (Outlines of Pyrrhonism [c.180], II.109) |
| 1894 | Some say that causes are physical, some say not [Sext.Empiricus] |
| Full Idea: Some affirm cause to be corporeal, some incorporeal. | |
| From: Sextus Empiricus (Outlines of Pyrrhonism [c.180], III.14) |
| 1898 | Cause can't exist before effect, or exist at the same time, so it doesn't exist [Sext.Empiricus] |
| Full Idea: If cause neither subsists before its effect, nor subsists along with it, nor does the effect precede the cause, it would seem that it has no substantial existence at all. | |
| From: Sextus Empiricus (Outlines of Pyrrhonism [c.180], III.27) |
| 1896 | If there were no causes then everything would have been randomly produced by everything [Sext.Empiricus] |
| Full Idea: If causes were non-existent everything would have been produced by everything, and at random. | |
| From: Sextus Empiricus (Outlines of Pyrrhonism [c.180], III.18) |
| 1897 | Knowing an effect results from a cause means knowing that the cause belongs with the effect, which is circular [Sext.Empiricus] |
| Full Idea: To know an effect belongs to a cause, we must also know that that cause belongs to that effect, and this is circular. | |
| From: Sextus Empiricus (Outlines of Pyrrhonism [c.180], III.21) |
| 1895 | Causes are either equal to the effect, or they link equally with other causes, or they contribute slightly [Sext.Empiricus] |
| Full Idea: The majority say causes are immediate (when they are directly proportional to effects), or associate (making an equal contribution to effects), or cooperant (making a slight contribution). | |
| From: Sextus Empiricus (Outlines of Pyrrhonism [c.180], III.15) |
| 1901 | If all atoms, times and places are the same, everything should move with equal velocity [Sext.Empiricus] |
| Full Idea: If objects are reducible to atoms, and each thing passes in an atomic time with its own first atom into an atomic point of space, then all moving things are of equal velocity. | |
| From: Sextus Empiricus (Outlines of Pyrrhonism [c.180], III.77) |
| 1899 | Does the original self-mover push itself from behind, or pull itself from in front? [Sext.Empiricus] |
| Full Idea: Self-movement must move in some particular direction, but if it pushes it will be behind itself, and if it pulls it will be in front of itself. | |
| From: Sextus Empiricus (Outlines of Pyrrhonism [c.180], III.68) | |
| A reaction: This is the same as Aquinas's First Way of proving God's existence. |
| 1900 | If time and place are infinitely divided, it becomes impossible for movement ever to begin [Sext.Empiricus] |
| Full Idea: If bodies, and the places and times when they are said to move, are divided into infinity, motion will not occur, it being impossible to find anything which will initiate the first movement. | |
| From: Sextus Empiricus (Outlines of Pyrrhonism [c.180], III.76) |
| 1903 | If motion and rest are abolished, so is time [Sext.Empiricus] |
| Full Idea: Since time does not seem to subsist without motion or even rest, if motion is abolished, and likewise rest, time is abolished. | |
| From: Sextus Empiricus (Outlines of Pyrrhonism [c.180], III.141) |
| 1904 | Time must be unlimited, but past and present can't be non-existent, and can't be now, so time does not exist [Sext.Empiricus] |
| Full Idea: There can't be a time when there was no time, so time is not limited; but unlimited time means past and present are non-existent (so time is limited to the present), or they exist (which means they are present). Time does not exist. | |
| From: Sextus Empiricus (Outlines of Pyrrhonism [c.180], III.142) |
| 1905 | How can time be divisible if we can't compare one length of time with another? [Sext.Empiricus] |
| Full Idea: Time is clearly divisible (into past, present and future), but it can't be, because a divisible thing is measured by some part of itself (divisions of length), but the two parts must coincide to make the measurement (e.g. present must coincide with past). | |
| From: Sextus Empiricus (Outlines of Pyrrhonism [c.180], III.143) |
| 1891 | How can we agree on the concept of God, unless we agree on his substance or form or place? [Sext.Empiricus] |
| Full Idea: How shall we be able to reach a conception of God when we have no agreement about his substance or his form or his place of abode? | |
| From: Sextus Empiricus (Outlines of Pyrrhonism [c.180], III.3) |
| 1892 | The existence of God can't be self-evident or everyone would have agreed on it, so it needs demonstration [Sext.Empiricus] |
| Full Idea: The existence of God is not pre-evident, for if it was the dogmatists would have agreed about it, whereas their disagreements show it is non-evident, and in need of demonstration. | |
| From: Sextus Empiricus (Outlines of Pyrrhonism [c.180], III.6) |
| 1893 | If God foresaw evil he would presumably prevent it, and if he only foresees some things, why those things? [Sext.Empiricus] |
| Full Idea: If God had forethought for all, there would be no evil in the world, yet they say the world is full of evil. And if he forethinks some things, why those and not others? | |
| From: Sextus Empiricus (Outlines of Pyrrhonism [c.180], III.9) |