30 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. |
| 6685 | 'Subjectivism' is an extension of relativism from the social group to the individual [Graham] |
| Full Idea: What is called 'subjectivism' is really just an extension of relativism from the level of the social group to the level of the individual. | |
| From: Gordon Graham (Eight Theories of Ethics [2004], Ch.1) | |
| A reaction: Personally I prefer to stick with 'relativism', at any level. 'Relative' is a two-place predicate, so we should always specify what is relative to what, unless it is obvious from context. Morality might be relative to God, for example. |
| 15282 | Facts should be deducible from the theory and initial conditions, and prefer the simpler theory [Osiander, by Harré/Madden] |
| Full Idea: The two positivist criteria for a scientific theory are that the facts must be deducible from the theory together with initial conditions, and if there is more than one theory the simplest must be chosen. | |
| From: report of Andreas Osiander (Preface to 'De Revolutionibus' [1543]) by Harré,R./Madden,E.H. - Causal Powers 7.I | |
| A reaction: Harré and Madden cite this as a famous early statement of positivism. It seems to combine Hempel and Lewis very concisely. Wrong, of course. It does not, though, appear to mention 'laws'. |
| 6699 | The chain of consequences may not be the same as the chain of responsibility [Graham] |
| Full Idea: From a utilitarian point of view, the error of Archduke Ferdinand's driver (he turned up a cul-de-sac) was the worst in history, ...but the chain of consequences may not be the same as the chain of responsibility. | |
| From: Gordon Graham (Eight Theories of Ethics [2004], Ch.7) | |
| A reaction: Can you cause something, and yet not be responsible for it? The driver was presumably fully conscious, rational and deliberate. He must share the responsibility for catastrophe, just as he shares in the causing of all the consequences. |
| 6698 | Negative consequences are very hard (and possibly impossible) to assess [Graham] |
| Full Idea: Negative consequences make the extension of the consequences of our actions indefinite, and this means that it is difficult to assess them; it may make it impossible, since there is now no clear sense to the idea of THE consequences of an action at all. | |
| From: Gordon Graham (Eight Theories of Ethics [2004], Ch.7) | |
| A reaction: The general slogan of 'Do your best' covers most objections to the calculation of consequences. It is no excuse for stealing a wallet that 'at least I wasn't committing genocide'. How easy were the alternative actions to do? |
| 6700 | We can't criticise people because of unforeseeable consequences [Graham] |
| Full Idea: It is unreasonable to say that people have acted badly because of consequences which were not merely unforeseen but unforeseeable. | |
| From: Gordon Graham (Eight Theories of Ethics [2004], Ch.7) | |
| A reaction: Interesting, and it sounds right. A key question in moral philosophy is how much effort people should make to assess the consequences of their actions. We must surely absolve them of the truly 'unforeseeable' consequence. |
| 6704 | Egoism submits to desires, but cannot help form them [Graham] |
| Full Idea: Egoism is inadequate as a guide to good living. Though it tells us what to do, given pre-existent desires, it cannot help us critically form those desires. | |
| From: Gordon Graham (Eight Theories of Ethics [2004], Ch.9) | |
| A reaction: A crucial point in morality. It also applies to utilitarianism (should I change my capacity for pleasure?), and virtue theory (how should I genetically engineer 'human nature'?). I think these problems push us towards Platonism. See Idea 4840. |
| 6701 | Rescue operations need spontaneous benevolence, not careful thought [Graham] |
| Full Idea: If more lives are to be saved in natural disasters, what is needed is spontaneity on the part of the rescuers, a willingness not to stop and think but to act spontaneously. | |
| From: Gordon Graham (Eight Theories of Ethics [2004], Ch.7) | |
| A reaction: This seems right, but must obviously be applied with caution, as when people are drowned attempting hopeless rescues. The most valuable person in an earthquake may be the thinker, not the digger. |
| 6693 | 'What if everybody did that?' rather misses the point as an objection to cheating [Graham] |
| Full Idea: I can object to your walking on the grass by asking 'What if everybody did that?', but the advantages of cheating depend upon the fact that most people don't cheat, so justifying my own cheating must involve special pleading. | |
| From: Gordon Graham (Eight Theories of Ethics [2004], Ch.6) | |
| A reaction: It is, of course, reasonable to ask 'What if everybody cheated?', but it is also reasonable to reply that 'the whole point of cheating is that it exploits the honesty of others'. This shows that Kant cannot simply demolish the 'free rider'. |
| 6691 | It is more plausible to say people can choose between values, than that they can create them [Graham] |
| Full Idea: To say that individuals are free to choose their own values is more naturally interpreted as meaning that they are free to choose between pre-existent values. | |
| From: Gordon Graham (Eight Theories of Ethics [2004], Ch.5) | |
| A reaction: Existentialism seems absurdly individualistic in its morality. Nietzsche was the best existentialist, who saw that most people have to be sheep. Strong personalities can promote or demote the old values on the great scale of what is good. |
| 6688 | Life is only absurd if you expected an explanation and none turns up [Graham] |
| Full Idea: If 'life is absurd' just means 'there is no logical explanation for human existence', we have no reason for anguish, unless we think there should be such an explanation. | |
| From: Gordon Graham (Eight Theories of Ethics [2004], Ch.5) | |
| A reaction: This is aimed at Kierkegaard and Camus. 'Absurd' certainly seems to be a relative notion, and we have nothing to compare life with. However, life does strike us as a bit odd sometimes, don't you think? |
| 6705 | Existentialism may transcend our nature, unlike eudaimonism [Graham] |
| Full Idea: It is the freedom to transcend our nature which eudaimonism seems to ignore and existentialism brings to the fore. | |
| From: Gordon Graham (Eight Theories of Ethics [2004], Ch.9) | |
| A reaction: It is wildly exciting to 'transcend our nature', and very dreary to polish up the nature which is given to us. In this I am a bit conservative. We should not go against the grain, but we shouldn't assume current living is the correct line of the grain. |
| 6690 | A standard problem for existentialism is the 'sincere Nazi' [Graham] |
| Full Idea: A standard problem for existentialism is the 'sincere Nazi'; there were undoubtedly some true believers, who saw in Nazism a creed that they wanted to believe, and who freely chose to endorse it. | |
| From: Gordon Graham (Eight Theories of Ethics [2004], Ch.5) | |
| A reaction: The failing of Nazis was that they were not good citizens. They might have been good members of a faction, but they were (in my opinion) poor citizens of Germany, and (obviously) appalling citizens of Europe. The objection to existentialism is good. |
| 6689 | The key to existentialism: the way you make choices is more important than what you choose [Graham] |
| Full Idea: The chief implication of existentialism is this: what you choose to do, how you choose to spend your life, is not as important as the way you choose it. | |
| From: Gordon Graham (Eight Theories of Ethics [2004], Ch.5) | |
| A reaction: While existentialists place emphasis on some notion of 'pure' choice, this is very close to the virtue theory idea that in a dilemma there may be several different choices which could all be rightly made by virtuous people. Integrity is a central virtue. |
| 6706 | The great religions are much more concerned with the religious life than with ethics [Graham] |
| Full Idea: The fact is that the great religions of the world are not principally concerned with ethics at all, but with the religious life for its own sake. ..The Sermon on the Mount, for example, is mainly concerned with how to pray and worship. | |
| From: Gordon Graham (Eight Theories of Ethics [2004], Ch.9) | |
| A reaction: This seems to me a highly significant point, given that most people nowadays seem to endorse religion precisely because they wish to endorse morality, and think religion is its essential underpinning. See Idea 336 for the core problem ('Euthyphro'). |
| 6709 | Western religion saves us from death; Eastern religion saves us from immortality [Graham] |
| Full Idea: For Western minds, religion entails the belief and hope that we will be saved from death and live forever, but the belief of Eastern religions is that we do live forever, and it is from this dreadful fate that we must look to spirituality to save us. | |
| From: Gordon Graham (Eight Theories of Ethics [2004], Ch.9) | |
| A reaction: Nice. I have certainly come to prefer the Eastern view, simply on the grounds that human beings have a limited capacity. I quite fancy three hundred years of healthy life, but after that I am sure that any potential I have will be used up. |