36 ideas
| 21032 | Speak truth only to those who deserve the truth [Sandel] |
| Full Idea: The duty to tell the truth applies only to those who deserve the truth. | |
| From: Michael J. Sandel (Justice: What's the right thing to do? [2009], 05) | |
| A reaction: [from Benjamin Constant, in opposition to Kant] I prefer the idea that we should use people 'after our own honour and dignity' (Hamlet), which means speaking the truth even to Donald Trump (for those of you who remember 2018). But not always. |
| 21033 | Careful evasions of truth at least show respect for it [Sandel] |
| Full Idea: A carefully crafted evasion pays homage to truth-telling in a way that an outright lie does not. | |
| From: Michael J. Sandel (Justice: What's the right thing to do? [2009], 05) | |
| A reaction: Nicely put. He refers to an incident in Kant's life. I think of the great equivocation controversy at the time of the 1605 Gunpowder Plot. See the porter in Macbeth. All I ask is that people care about the truth. Many people don't. Why? |
| 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. |
| 594 | Speusippus suggested underlying principles for every substance, and ended with a huge list [Speussipus, by Aristotle] |
| Full Idea: Speusippus suggested principles for each substance, including principles for numbers, magnitude and the soul. He thus arrived at no mean list of substances. | |
| From: report of Speussipus (thirty titles (lost) [c.367 BCE]) by Aristotle - Metaphysics 1028b |
| 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. |
| 21036 | Not all deals are fair deals [Sandel] |
| Full Idea: The mere fact that you and I make a deal is not enough to make it fair. | |
| From: Michael J. Sandel (Justice: What's the right thing to do? [2009], 06) |
| 21038 | Does consent create the obligation, or must there be some benefit? [Sandel] |
| Full Idea: Legal thinkers have debated this question for a long time: can consent create an obligation on its own, or is some element of benefit or reliance required? | |
| From: Michael J. Sandel (Justice: What's the right thing to do? [2009], 06) | |
| A reaction: Clearly mere consent could be under some compulsion, either by the other party, or by some other forces. Keeping a deathbed promise usually brings no benefit, but is a matter of honour. Ah, honour! Can anyone remember what that is? |
| 21039 | Moral contracts involve both consent and reciprocity; making the deal, and keeping it [Sandel] |
| Full Idea: Despite a tendency to read consent into moral claims, it is hard to make sense of our morality without acknowledging the independent weight of reciprocity. If my wife is unfaithful I have two different grounds of outrage: our promise, and my loyalty. | |
| From: Michael J. Sandel (Justice: What's the right thing to do? [2009], 06) | |
| A reaction: The point is that Hobbes and co over-simplify what a contract is. Compare a contract with a promise. One must be two-sided, the other can be one-sided. |
| 21030 | The categorical imperative is not the Golden Rule, which concerns contingent desires [Sandel] |
| Full Idea: The Golden Rule depends on contingent facts about how people like to be treated. The categorical imperative asks that we abstract from such contingencies and respect persons as rational beings, regardless of what they might want in particular situations. | |
| From: Michael J. Sandel (Justice: What's the right thing to do? [2009], 05) | |
| A reaction: I think the Golden Rule is wrong for a different reason. It assumes that we all want similar things, which we don't. Focus on other people's needs, not yours. |
| 21031 | Man cannot dispose of himself, because he is not a thing to be owned [Sandel] |
| Full Idea: Man cannot dispose over himself because he is not a thing; he is not his own property. | |
| From: Michael J. Sandel (Justice: What's the right thing to do? [2009], 05) | |
| A reaction: [Kant lecture note] This is an important qualification to persons as ends. If a person owned themselves, that would separate the person from what they owned. Sandel mentions selling your own organs. Kant is considering prostitution. Why is slavery wrong? |
| 21035 | Just visiting (and using roads) is hardly ratifying the Constitution [Sandel] |
| Full Idea: It is hard to see how just passing through town is morally akin to ratifying the Constitution. | |
| From: Michael J. Sandel (Justice: What's the right thing to do? [2009], 06) | |
| A reaction: They say that philosophical ideas are never refuted, and no progress is made, but this sure put paid to John Locke. |
| 21037 | A ratified constitution may not be a just constitution [Sandel] |
| Full Idea: The fact that a constitution is ratified by the people does not prove that its provisions are just. | |
| From: Michael J. Sandel (Justice: What's the right thing to do? [2009], 06) | |
| A reaction: Yes indeed. And the fact that a majority won a referendum does not make their decision wise. Hence all constitutions must be open to evaluation. Gun laws in the US are the obvious example. |
| 21034 | A just constitution harmonises the different freedoms [Sandel] |
| Full Idea: As Kant sees it, a just constitution aims at harmonising each individual's freedom with that of everyone else. | |
| From: Michael J. Sandel (Justice: What's the right thing to do? [2009], 05) | |
| A reaction: [source?] Nice statement of the project. I increasingly see political philosophy as constitution design. I say philosophers have got fifty years to design an optimum constitution, and they should then down tools and promote it, in simple language. |
| 21049 | Liberal freedom was a response to assigned destinies like caste and class [Sandel] |
| Full Idea: Liberal freedom developed as an antidote to political theories that consigned persons to destinies fixed by caste or class, station or rank, custom, tradition or inherited status. | |
| From: Michael J. Sandel (Justice: What's the right thing to do? [2009], 09) | |
| A reaction: Virtually all human beings before modern times found that they had been 'assigned destinies'. The huge exception is war, especially civil war, which must be a huge liberation for many people, despite the danger. |
| 21040 | Libertarians just want formal equality in a free market; the meritocratic view wants fair equality [Sandel] |
| Full Idea: The libertarian view of distributive justice is a free market with formal equality of opportunity. The meritocratic view is a free market with fair equality of opportunity. | |
| From: Michael J. Sandel (Justice: What's the right thing to do? [2009], 06) | |
| A reaction: The obvious question is what has to be done, by intervention, to make the market fair. There are two major rival views of equality here. Is the starting point fair, and is the race itself fair? |
| 21028 | We can approach justice through welfare, or freedom, or virtue [Sandel] |
| Full Idea: We have identified three ways of approaching the distribution of goods: welfare, freedom and virtue. ...and these are three ways of thinking about justice. | |
| From: Michael J. Sandel (Justice: What's the right thing to do? [2009], 01) | |
| A reaction: Virtue is Sandel's distinctively Aristotelian contribution to the problem. The best known instance of justice is punishment, which is a distribution of harms. |
| 21027 | Justice concerns how a society distributes what it prizes - wealth, rights, power and honours [Sandel] |
| Full Idea: To ask whether a society is just is to ask how we distribute the things we prize - income and wealth, duties and rights, powers and opportunities, offices and honours. | |
| From: Michael J. Sandel (Justice: What's the right thing to do? [2009], 01) | |
| A reaction: There is, of course, the prior question of what things should be controlled by a society for distribution. But there is also justice in the promotional and pay structure of institutions within a society, including private institutions. |
| 21042 | Should we redress wrongs done by a previous generation? [Sandel] |
| Full Idea: Can we ever have a moral responsibility to redress wrongs committed by a previous generation? | |
| From: Michael J. Sandel (Justice: What's the right thing to do? [2009], 07) | |
| A reaction: Just asking for a friend. It seems to depend on how close we feel to the previous generation. Regretting the crime committed by a beloved parent is one thing. Despising the crime committed by some right bastard who shares my nationality is another. |
| 21043 | Distributive justice concern deserts, as well as who gets what [Sandel] |
| Full Idea: Debates about distributive justice are about not only who gets what but also what qualities are worthy of honour and reward. | |
| From: Michael J. Sandel (Justice: What's the right thing to do? [2009], 07) | |
| A reaction: So the 'undeserving poor' get nuffink? Does just being a human being deserve anything? Obviously yes. That said, we can't deny the concept of 'appropriate reward'. |
| 21052 | Justice is about how we value things, and not just about distributions [Sandel] |
| Full Idea: Justice is not only about the right way to distribute things. It is also about the right way to value things. | |
| From: Michael J. Sandel (Justice: What's the right thing to do? [2009], 10) | |
| A reaction: This is Sandel's distinctively Aristotelian contribution to the justice debate - with which I have great sympathy. And, as he argues, the values of things arise out of assessing their essential natures. |
| 21048 | Work is not fair if it is negotiated, even in a fair situation, but if it suits the nature of the worker [Sandel] |
| Full Idea: For the libertarian free exchange for labour is fair; for Rawls it requires fair background conditions; for Aristotle, for the work to be just it must be suited to the nature of the workers who perform it. | |
| From: Michael J. Sandel (Justice: What's the right thing to do? [2009], 08) | |
| A reaction: [compressed] Aristotle's idea is powerful, and Sandel performs a great service in drawing attention to it. Imagine the key negotiation in an interview being whether this particular work suits your nature! |
| 21045 | Teleological thinking is essential for social and political issues [Sandel] |
| Full Idea: It is not easy to dispense with teleological reasoning in thinking about social institutions and political practices. | |
| From: Michael J. Sandel (Justice: What's the right thing to do? [2009], 08) | |
| A reaction: I think teleological thinking is also indispensable in biology. You can't understand an ear or an eye if you don't know what it is FOR. If it relates to a mind, it is teleological. The eye of a dead person is for nothing. |
| 2632 | Speusippus said things were governed by some animal force rather than the gods [Speussipus, by Cicero] |
| Full Idea: Speusippus, following his uncle Plato, held that all things were governed by some kind of animal force, and tried to eradicate from our minds any notion of the gods. | |
| From: report of Speussipus (thirty titles (lost) [c.367 BCE]) by M. Tullius Cicero - On the Nature of the Gods ('De natura deorum') I.33 |