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. |
| 12727 | It's impossible, but imagine a body carrying on normally, but with no mind [Leibniz] |
| Full Idea: If it could be supposed that a body exists without a mind, then a man would do everything in the same way as if he did not have a mind, and men would speak and write the same things, without knowing what they do. ...But this supposition is impossible. | |
| From: Gottfried Leibniz (Paper of December 1676 [1676], A6.3.400), quoted by Daniel Garber - Leibniz:Body,Substance,Monad 5 | |
| A reaction: This is clearly the zombie dream, three centuries before Robert Kirk's modern invention of the idea. Leibniz's reason for denying the possibility of zombies won't be the modern physicalist reason. |
| 21059 | General rules of action also need a judgement about when to apply them [Kant] |
| Full Idea: A concept of the understanding, which contains the general rule, must be supplemented by an act of judgement whereby the practitioner distinguishes instances where the rule applies from those where it does not. | |
| From: Immanuel Kant (True in Theory, but not in Practice [1792], Intro) | |
| A reaction: This is Aristotle's phronesis, and Hart's 'rules of recognition' in law courts. So is the link between theory and practice an intellectual one, or a sort of inarticulate intuition? I like 'common sense' for this ability. |
| 21061 | Duty does not aim at an end, but gives rise to universal happiness as aim of the will [Kant] |
| Full Idea: My conception of duty does not need to be based on any particular end, but rather itself occasions a new end for the human will, that of striving with all one's power towards the highest good possible on earth, the universal happiness of the whole world. | |
| From: Immanuel Kant (True in Theory, but not in Practice [1792], 1B) | |
| A reaction: I see nothing in the categorical imperative that demands 'all one's power', and nothing that specifies happiness as what has to be universalised. Nietzsche, for one, thinks happiness is overrated. |
| 21060 | It can't be a duty to strive after the impossible [Kant] |
| Full Idea: It would not be a duty to strive after a certain effect of our will if this effect were impossible in experience. | |
| From: Immanuel Kant (True in Theory, but not in Practice [1792], Intro) | |
| A reaction: 'Ought implies can' has become a familiar slogan. The quickest way to get shot of a tiresome duty is to persuade yourself that it is impossible. The seemingly impossible is occasionally achieved. |
| 21062 | The will's motive is the absolute law itself, and moral feeling is receptivity to law [Kant] |
| Full Idea: The will must have motives. But these are not objects of physical feeling as predetermined ends in themselves. They are none other than the absolute law itself, and the will's receptivity to it as an absolute compulsion is known as moral feeling. | |
| From: Immanuel Kant (True in Theory, but not in Practice [1792], 1Bb) | |
| A reaction: This sounds like our natural motivation to get the right answer when doing arithmetic, which is the innate motivation towards truth. I once heard it said that truth is the only value. So why does Donald Trump fail to value truth? |
| 21071 | There can be no restraints on freedom if reason does not reveal some basic rights [Kant] |
| Full Idea: If there is nothing which commands immediate respect through reason, such as the basic rights of man, no influence can prevail upon man's arbitrary will and restrain his freedom. | |
| From: Immanuel Kant (True in Theory, but not in Practice [1792], 2-Concl) | |
| A reaction: I think this is the nearest Kant gets to natural rights. It is hard to see how basic rights can be identified by pure reason, without some inbuilt human values. Kant's usual move is to say denial of them leads to a contradiction, but I'm going off that. |
| 21063 | Personal contracts are for some end, but a civil state contract involves a duty to share [Kant] |
| Full Idea: In all social contracts, we find a union of many individuals for some common end which they all share. But a union as an end in itself which they all ought to share …is only found in a society insofar as it constitutes a civil state i.e. a commonwealth. | |
| From: Immanuel Kant (True in Theory, but not in Practice [1792], 2 Intro) | |
| A reaction: This makes a nice link between the contractarian individual morality of Hobbes and his social contract view of society. Kant seems to reject the first but accept the second. Presumably because the first implies benefit and the second implies duty. |
| 21068 | There must be a unanimous contract that citizens accept majority decisions [Kant] |
| Full Idea: The actual principle of being content with majority decisions must be accepted unanimously and embodied in a contract, and this itself must be the ultimate basis on which a civil constitution is established. | |
| From: Immanuel Kant (True in Theory, but not in Practice [1792], 2-3) | |
| A reaction: This is the contract which combines a social contract with democracy. We unanimously agree not to be unanimous? Cf Idea 21065. What should the minority do when the majority elect criminal Nazi leaders? |
| 21069 | A contract is theoretical, but it can guide rulers to make laws which the whole people will accept [Kant] |
| Full Idea: The original contract …is merely an idea of reason, which nonetheless has undoubted practical reality; for it can oblige every legislator to frame his laws in such a way that they could have been produced by the united will of a whole nation. | |
| From: Immanuel Kant (True in Theory, but not in Practice [1792], 2-Concl) | |
| A reaction: The contractualist theory of morality of Thomas Scanlon approaches this. Note that Kant says it 'can' oblige the legislators. Nothing would compel them to follow such a principle. |
| 21070 | A law is unjust if the whole people could not possibly agree to it [Kant] |
| Full Idea: If the law is such that a whole people could not possibly agree to it …it is unjust. | |
| From: Immanuel Kant (True in Theory, but not in Practice [1792], 2-Concl) | |
| A reaction: Kant is explicitly trying to approximate Rousseau's general will. The categorical imperative was greatly influenced by Rousseau. The key point is not whether they accept it, but that unanimous acceptance is unthinkable. Unfair laws will fail. |
| 21067 | A citizen must control his own life, and possess property or an important skill [Kant] |
| Full Idea: The only qualification required by a citizen (apart, of course, from being an adult male) is that he must be his own master, and must have some property (which can include any skill, trade, fine art or science) to support himself. | |
| From: Immanuel Kant (True in Theory, but not in Practice [1792], 2-3) | |
| A reaction: Of course! Being one's own master evidently allows for being an employee, as long as this is a free contract, and not exploitation. Invites lots of interesting test cases. We need a Marxist commentary on this idea. |
| 21064 | A lawful civil state must embody freedom, equality and independence for its members [Kant] |
| Full Idea: The civil state, regarded purely as a lawful state, is based on the following a priori principles. 1) the freedom of every member as a human being, 2) the equality of each as a subject, 3) the independence of each as a subject. | |
| From: Immanuel Kant (True in Theory, but not in Practice [1792], 2 Intro) | |
| A reaction: Written in 1792, three years after the start of the French Revolution. He says that a state with an inbuilt hierarchy or aristocracy is unlawful. Which freedoms, equality in what respects, and independence from what? |
| 21066 | Citizens can rise to any rank that talent, effort and luck can achieve [Kant] |
| Full Idea: Every member of the commonwealth must be entitled to reach any degree of rank which a subject can earn through his talent, his industry and his good fortune. | |
| From: Immanuel Kant (True in Theory, but not in Practice [1792], 2-2) | |
| A reaction: This is equality of opportunity, which is a mantra for liberals, but has been subjected to good criticisms in modern times. The main question is whether there is formal and legal equality, or actual practical equality. |
| 21065 | You can't make a contract renouncing your right to make contracts! [Kant] |
| Full Idea: No one can voluntarily renounce his rights by a contract ..to the effect that he has no rights but only duties, for such a contract would deprive him of the right to make a contract, and would thus invalidate the one he had already made. | |
| From: Immanuel Kant (True in Theory, but not in Practice [1792], 2-2) | |
| A reaction: Kant tries to establish all of his principles by showing that their denial is contradictory. But this example is blatantly wrong. King Lear didn't nullify his previous legislation when he abdicated, and his two daughters legally kept their territories. |
| 21072 | The people (who have to fight) and not the head of state should declare a war [Kant] |
| Full Idea: Each state must be organised so that the head of state, for whom the war costs nothing (for he wages it at the expense of the people) must no longer have the deciding vote on whether war is to be declared or not, for the people who pay for it must decide. | |
| From: Immanuel Kant (True in Theory, but not in Practice [1792], 3) | |
| A reaction: I would guess that he has Louis XIV particularly in mind. Imagine if Kant's proposal had been implemented in 1914. A referendum takes ages, and the people would need the facts (from the intelligence agencies). |