41 ideas
| 9901 | Numbers can't be sets if there is no agreement on which sets they are [Benacerraf] |
| Full Idea: The fact that Zermelo and Von Neumann disagree on which particular sets the numbers are is fatal to the view that each number is some particular set. | |
| From: Paul Benacerraf (What Numbers Could Not Be [1965], II) | |
| A reaction: I agree. A brilliantly simple argument. There is the possibility that one of the two accounts is correct (I would vote for Zermelo), but it is not actually possible to prove it. |
| 9912 | There are no such things as numbers [Benacerraf] |
| Full Idea: There are no such things as numbers. | |
| From: Paul Benacerraf (What Numbers Could Not Be [1965], IIIC) | |
| A reaction: Mill said precisely the same (Idea 9794). I think I agree. There has been a classic error of reification. An abstract pattern is not an object. If I coin a word for all the three-digit numbers in our system, I haven't created a new 'object'. |
| 9151 | Benacerraf says numbers are defined by their natural ordering [Benacerraf, by Fine,K] |
| Full Idea: Benacerraf thinks of numbers as being defined by their natural ordering. | |
| From: report of Paul Benacerraf (What Numbers Could Not Be [1965]) by Kit Fine - Cantorian Abstraction: Recon. and Defence §5 | |
| A reaction: My intuition is that cardinality is logically prior to ordinality, since that connects better with the experienced physical world of objects. Just as the fact that people have different heights must precede them being arranged in height order. |
| 13891 | To understand finite cardinals, it is necessary and sufficient to understand progressions [Benacerraf, by Wright,C] |
| Full Idea: Benacerraf claims that the concept of a progression is in some way the fundamental arithmetical notion, essential to understanding the idea of a finite cardinal, with a grasp of progressions sufficing for grasping finite cardinals. | |
| From: report of Paul Benacerraf (What Numbers Could Not Be [1965]) by Crispin Wright - Frege's Concept of Numbers as Objects 3.xv | |
| A reaction: He cites Dedekind (and hence the Peano Axioms) as the source of this. The interest is that progression seems to be fundamental to ordianls, but this claims it is also fundamental to cardinals. Note that in the first instance they are finite. |
| 17904 | A set has k members if it one-one corresponds with the numbers less than or equal to k [Benacerraf] |
| Full Idea: Any set has k members if and only if it can be put into one-to-one correspondence with the set of numbers less than or equal to k. | |
| From: Paul Benacerraf (What Numbers Could Not Be [1965], I) | |
| A reaction: This is 'Ernie's' view of things in the paper. This defines the finite cardinal numbers in terms of the finite ordinal numbers. He has already said that the set of numbers is well-ordered. |
| 17906 | To explain numbers you must also explain cardinality, the counting of things [Benacerraf] |
| Full Idea: I would disagree with Quine. The explanation of cardinality - i.e. of the use of numbers for 'transitive counting', as I have called it - is part and parcel of the explication of number. | |
| From: Paul Benacerraf (What Numbers Could Not Be [1965], I n2) | |
| A reaction: Quine says numbers are just a progression, with transitive counting as a bonus. Interesting that Benacerraf identifies cardinality with transitive counting. I would have thought it was the possession of numerical quantity, not ascertaining it. |
| 9898 | We can count intransitively (reciting numbers) without understanding transitive counting of items [Benacerraf] |
| Full Idea: Learning number words in the right order is counting 'intransitively'; using them as measures of sets is counting 'transitively'. ..It seems possible for someone to learn the former without learning the latter. | |
| From: Paul Benacerraf (What Numbers Could Not Be [1965], I) | |
| A reaction: Scruton's nice question (Idea 3907) is whether you could be said to understand numbers if you could only count intransitively. I would have thought such a state contained no understanding at all of numbers. Benacerraf agrees. |
| 17903 | Someone can recite numbers but not know how to count things; but not vice versa [Benacerraf] |
| Full Idea: It seems that it is possible for someone to learn to count intransitively without learning to count transitively. But not vice versa. | |
| From: Paul Benacerraf (What Numbers Could Not Be [1965], I) | |
| A reaction: Benacerraf favours the priority of the ordinals. It is doubtful whether you have grasped cardinality properly if you don't know how to count things. Could I understand 'he has 27 sheep', without understanding the system of natural numbers? |
| 9897 | The application of a system of numbers is counting and measurement [Benacerraf] |
| Full Idea: The application of a system of numbers is counting and measurement. | |
| From: Paul Benacerraf (What Numbers Could Not Be [1965], I) | |
| A reaction: A simple point, but it needs spelling out. Counting seems prior, in experience if not in logic. Measuring is a luxury you find you can indulge in (by imagining your quantity) split into parts, once you have mastered counting. |
| 9900 | For Zermelo 3 belongs to 17, but for Von Neumann it does not [Benacerraf] |
| Full Idea: Ernie's number progression is [φ],[φ,[φ]],[φ,[φ],[φ,[φ,[φ]]],..., whereas Johnny's is [φ],[[φ]],[[[φ]]],... For Ernie 3 belongs to 17, not for Johnny. For Ernie 17 has 17 members; for Johnny it has one. | |
| From: Paul Benacerraf (What Numbers Could Not Be [1965], II) | |
| A reaction: Benacerraf's point is that there is no proof-theoretic way to choose between them, though I am willing to offer my intuition that Ernie (Zermelo) gives the right account. Seventeen pebbles 'contains' three pebbles; you must pass 3 to count to 17. |
| 9899 | The successor of x is either x and all its members, or just the unit set of x [Benacerraf] |
| Full Idea: For Ernie, the successor of a number x was the set consisting of x and all the members of x, while for Johnny the successor of x was simply [x], the unit set of x - the set whose only member is x. | |
| From: Paul Benacerraf (What Numbers Could Not Be [1965], II) | |
| A reaction: See also Idea 9900. Benacerraf's famous point is that it doesn't seem to make any difference to arithmetic which version of set theory you choose as its basis. I take this to conclusively refute the idea that numbers ARE sets. |
| 8697 | Disputes about mathematical objects seem irrelevant, and mathematicians cannot resolve them [Benacerraf, by Friend] |
| Full Idea: If two children were brought up knowing two different set theories, they could entirely agree on how to do arithmetic, up to the point where they discuss ontology. There is no mathematical way to tell which is the true representation of numbers. | |
| From: report of Paul Benacerraf (What Numbers Could Not Be [1965]) by Michèle Friend - Introducing the Philosophy of Mathematics | |
| A reaction: Benacerraf ends by proposing a structuralist approach. If mathematics is consistent with conflicting set theories, then those theories are not shedding light on mathematics. |
| 8304 | No particular pair of sets can tell us what 'two' is, just by one-to-one correlation [Benacerraf, by Lowe] |
| Full Idea: Hume's Principle can't tell us what a cardinal number is (this is one lesson of Benacerraf's well-known problem). An infinity of pairs of sets could actually be the number two (not just the simplest sets). | |
| From: report of Paul Benacerraf (What Numbers Could Not Be [1965]) by E.J. Lowe - The Possibility of Metaphysics 10.3 | |
| A reaction: The drift here is for numbers to end up as being basic, axiomatic, indefinable, universal entities. Since I favour patterns as the basis of numbers, I think the basis might be in a pre-verbal experience, which even a bird might have, viewing its eggs. |
| 9906 | If ordinal numbers are 'reducible to' some set-theory, then which is which? [Benacerraf] |
| Full Idea: If a particular set-theory is in a strong sense 'reducible to' the theory of ordinal numbers... then we can still ask, but which is really which? | |
| From: Paul Benacerraf (What Numbers Could Not Be [1965], IIIB) | |
| A reaction: A nice question about all reductions. If we reduce mind to brain, does that mean that brain is really just mind. To have a direction (up/down?), reduction must lead to explanation in a single direction only. Do numbers explain sets? |
| 9907 | If any recursive sequence will explain ordinals, then it seems to be the structure which matters [Benacerraf] |
| Full Idea: If any recursive sequence whatever would do to explain ordinal numbers suggests that what is important is not the individuality of each element, but the structure which they jointly exhibit. | |
| From: Paul Benacerraf (What Numbers Could Not Be [1965], IIIC) | |
| A reaction: This sentence launched the whole modern theory of Structuralism in mathematics. It is hard to see what properties a number-as-object could have which would entail its place in an ordinal sequence. |
| 9908 | The job is done by the whole system of numbers, so numbers are not objects [Benacerraf] |
| Full Idea: 'Objects' do not do the job of numbers singly; the whole system performs the job or nothing does. I therefore argue that numbers could not be objects at all. | |
| From: Paul Benacerraf (What Numbers Could Not Be [1965], IIIC) | |
| A reaction: This thought is explored by structuralism - though it is a moot point where mere 'nodes' in a system (perhaps filled with old bits of furniture) will do the job either. No one ever explains the 'power' of numbers (felt when you do a sudoku). Causal? |
| 9909 | The number 3 defines the role of being third in a progression [Benacerraf] |
| Full Idea: Any object can play the role of 3; that is, any object can be the third element in some progression. What is peculiar to 3 is that it defines that role, not by being a paradigm, but by representing the relation of any third member of a progression. | |
| From: Paul Benacerraf (What Numbers Could Not Be [1965], IIIC) | |
| A reaction: An interesting early attempt to spell out the structuralist idea. I'm thinking that the role is spelled out by the intersection of patterns which involve threes. |
| 9911 | Number words no more have referents than do the parts of a ruler [Benacerraf] |
| Full Idea: Questions of the identification of the referents of number words should be dismissed as misguided in just the way that a question about the referents of the parts of a ruler would be seen as misguided. | |
| From: Paul Benacerraf (What Numbers Could Not Be [1965], IIIC) | |
| A reaction: What a very nice simple point. It would be very strange to insist that every single part of the continuum of a ruler should be regarded as an 'object'. |
| 8925 | Mathematical objects only have properties relating them to other 'elements' of the same structure [Benacerraf] |
| Full Idea: Mathematical objects have no properties other than those relating them to other 'elements' of the same structure. | |
| From: Paul Benacerraf (What Numbers Could Not Be [1965], p.285), quoted by Fraser MacBride - Structuralism Reconsidered §3 n13 | |
| A reaction: Suppose we only had one number - 13 - and we all cried with joy when we recognised it in a group of objects. Would that be a number, or just a pattern, or something hovering between the two? |
| 9938 | How can numbers be objects if order is their only property? [Benacerraf, by Putnam] |
| Full Idea: Benacerraf raises the question how numbers can be 'objects' if they have no properties except order in a particular ω-sequence. | |
| From: report of Paul Benacerraf (What Numbers Could Not Be [1965], p.301) by Hilary Putnam - Mathematics without Foundations | |
| A reaction: Frege certainly didn't think that order was their only property (see his 'borehole' metaphor in Grundlagen). It might be better to say that they are objects which only have relational properties. |
| 9910 | Number-as-objects works wholesale, but fails utterly object by object [Benacerraf] |
| Full Idea: The identification of numbers with objects works wholesale but fails utterly object by object. | |
| From: Paul Benacerraf (What Numbers Could Not Be [1965], IIIC) | |
| A reaction: This seems to be a glaring problem for platonists. You can stare at 1728 till you are blue in the face, but it only begins to have any properties at all once you examine its place in the system. This is unusual behaviour for an object. |
| 9903 | Number words are not predicates, as they function very differently from adjectives [Benacerraf] |
| Full Idea: The unpredicative nature of number words can be seen by noting how different they are from, say, ordinary adjectives, which do function as predicates. | |
| From: Paul Benacerraf (What Numbers Could Not Be [1965], II) | |
| A reaction: He points out that 'x is seventeen' is a rare construction in English, unlike 'x is happy/green/interesting', and that numbers outrank all other adjectives (having to appear first in any string of them). |
| 9904 | The set-theory paradoxes mean that 17 can't be the class of all classes with 17 members [Benacerraf] |
| Full Idea: In no consistent theory is there a class of all classes with seventeen members. The existence of the paradoxes is a good reason to deny to 'seventeen' this univocal role of designating the class of all classes with seventeen members. | |
| From: Paul Benacerraf (What Numbers Could Not Be [1965], II) | |
| A reaction: This was Frege's disaster, and seems to block any attempt to achieve logicism by translating numbers into sets. It now seems unclear whether set theory is logic, or mathematics, or sui generis. |
| 9905 | Identity statements make sense only if there are possible individuating conditions [Benacerraf] |
| Full Idea: Identity statements make sense only in contexts where there exist possible individuating conditions. | |
| From: Paul Benacerraf (What Numbers Could Not Be [1965], III) | |
| A reaction: He is objecting to bizarre identifications involving numbers. An identity statement may be bizarre even if we can clearly individuate the two candidates. Winston Churchill is a Mars Bar. Identifying George Orwell with Eric Blair doesn't need a 'respect'. |
| 23110 | Human injustice is not a permanent feature of communities [Rawls] |
| Full Idea: Men's propensity to injustice is not a permanent aspect of community life. | |
| From: John Rawls (A Theory of Justice [1972], p.245), quoted by John Kekes - Against Liberalism | |
| A reaction: This attitude is dismissed by Kekes, with some justification, as naïve optimism. What could be Rawls's grounds for making such a claim? It couldn't be the facts of human history. |
| 6000 | The goal is rationality in the selection of things according to nature [Diogenes of Babylon, by Blank] |
| Full Idea: Diogenes of Babylon defined the goal to be rationality in the selection and rejection of the things according to nature. | |
| From: report of Diogenes (Bab) (fragments/reports [c.180 BCE]) by D.L. Blank - Diogenes of Babylon | |
| A reaction: This captures the central Stoic idea quite nicely. 'Live according to nature', but this always meant 'live according to reason', because that is (as Aristotle had taught) the essence of our nature. This only makes sense if reason and nature coincide. |
| 5999 | The good is what is perfect by nature [Diogenes of Babylon, by Blank] |
| Full Idea: Diogenes of Babylon defined the good as what is perfect by nature. | |
| From: report of Diogenes (Bab) (fragments/reports [c.180 BCE]) by D.L. Blank - Diogenes of Babylon | |
| A reaction: This might come close to G.E. Moore's Ideal Utilitarianism, but its dependence on the rather uneasy of concept of 'perfection' makes it questionable. Personally I find it appealing. I wish we had Diogenes' explanation. |
| 15676 | Rawls defends the priority of right over good [Rawls, by Finlayson] |
| Full Idea: Rawls defends the thesis of the priority of the right over the good. | |
| From: report of John Rawls (A Theory of Justice [1972]) by James Gordon Finlayson - Habermas Ch.7:100 | |
| A reaction: It depends whether you are talking about actions, or about states of affairs. I don't see how any state of affairs can be preferred to the good one. It may be that the highest duty of action is to do what is right, rather than to achieve what is good. |
| 4123 | A fair arrangement is one that parties can agree to without knowing how it will benefit them personally [Rawls, by Williams,B] |
| Full Idea: Rawls's theory is an elaboration of a simple idea: a fair system of arrangements is one that the parties can agree to without knowing how it will benefit them personally. | |
| From: report of John Rawls (A Theory of Justice [1972]) by Bernard Williams - Ethics and the Limits of Philosophy Ch.5 | |
| A reaction: The essence of modern Kantian contractualism. It is an appealing principle for building a rational world, but I hear Nietzsche turning in his grave. |
| 6001 | Justice is a disposition to distribute according to desert [Diogenes of Babylon, by Blank] |
| Full Idea: Diogenes of Babylon defined justice as the disposition which distributes to everyone what he deserves. | |
| From: report of Diogenes (Bab) (fragments/reports [c.180 BCE]) by D.L. Blank - Diogenes of Babylon | |
| A reaction: The questions that arise would be 'what does a new-born baby deserve?', and 'what do animals deserve?', and 'does the lowest and worst of criminals deserve anything at all?' |
| 3279 | Utilitarianism inappropriately scales up the individual willingness to make sacrifices [Rawls, by Nagel] |
| Full Idea: Rawls claims that utilitarianism applies to the problem of many interests a method appropriate for one individual. A single person may accept disadvantages in exchange for benefits, but in society other people get the benefits. | |
| From: report of John Rawls (A Theory of Justice [1972], p.74,104) by Thomas Nagel - Equality §7 |
| 3280 | Why does the rational agreement of the 'Original Position' in Rawls make it right? [Nagel on Rawls] |
| Full Idea: Why does what it is rational to agree to in Rawls' 'Original Position' determine what is right? | |
| From: comment on John Rawls (A Theory of Justice [1972]) by Thomas Nagel - Equality §7 |
| 20552 | The original position models the idea that citizens start as free and equal [Rawls, by Swift] |
| Full Idea: The original position is presented by Rawls as modelling the sense in which citizens are to be understood as free and equal. | |
| From: report of John Rawls (A Theory of Justice [1972]) by Adam Swift - Political Philosophy (3rd ed) 3 'Strikes' | |
| A reaction: In other words, Rawls's philosophy is not a demonstration of why we should be liberals, but a guidebook for how liberals should go about organising society. |
| 18636 | Choose justice principles in ignorance of your own social situation [Rawls] |
| Full Idea: The principles of justice are chosen behind a veil of ignorance. ...Since all are similarly situated and no one is able to design principles to favor his particular condition, the principles of justice are the rest of a fair agreement or bargain. | |
| From: John Rawls (A Theory of Justice [1972], §03) | |
| A reaction: A famous idea. It tries to impose a Kantian impartiality onto the assessment of political principles. It is a beautifully simple idea, and saying that such impartiality never occurs is no objection to it. Think of a planet far far away. |
| 18631 | All desirable social features should be equal, unless inequality favours the disadvantaged [Rawls] |
| Full Idea: All social primary goods - liberty and opportunity, income and wealth, and the bases of self-respect - are to be distributed equally unless an unequal distribution of any or all of these goods is to the advantage of the least favoured. | |
| From: John Rawls (A Theory of Justice [1972], §46) | |
| A reaction: In the wholehearted capitalism of the 21st century this sounds like cloud-cuckoo land. As an 'initial position' (just as in the 'Republic') the clean slate brings out some interesting principles. Actual politics takes vested interests as axiomatic. |
| 20538 | Utilitarians lump persons together; Rawls somewhat separates them; Nozick wholly separates them [Swift on Rawls] |
| Full Idea: Rawls objects to utilitarianism because it fails to take seriously the separateness of persons (because there is no overall person to enjoy the overall happiness). But Nozick thinks Rawls does not take the separateness of persons seriously enough. | |
| From: comment on John Rawls (A Theory of Justice [1972]) by Adam Swift - Political Philosophy (3rd ed) 1 'Nozick' | |
| A reaction: In this sense, Nozick seems to fit our picture of a liberal more closely than Rawls does. I think they both exaggerate the separateness of persons, based on a false concept of human nature. |
| 9277 | Rawls's account of justice relies on conventional fairness, avoiding all moral controversy [Gray on Rawls] |
| Full Idea: Rawls's account of justice works only with widely accepted intuitions of fairness and relies at no point on controversial positions in ethics. The fruit of this modesty is a pious commentary on conventional moral beliefs. | |
| From: comment on John Rawls (A Theory of Justice [1972]) by John Gray - Straw Dogs 3.6 | |
| A reaction: Presumably this is the thought which provoked Nozick to lob his grenade on the subject. It resembles the charges of Schopenhauer and Nietzsche against Kant, that he was just dressing up conventional morality. Are 'controversial' ethics good? |
| 20527 | Liberty Principle: everyone has an equal right to liberties, if compatible with others' liberties [Rawls] |
| Full Idea: First Principle [Liberty]: Each person is to have an equal right to the most extensive total system of equal basic liberties compatible with a similar system of liberty for all. | |
| From: John Rawls (A Theory of Justice [1972], 46) | |
| A reaction: This is the result of consensus after the initial ignorant position of assessment. It is characteristic of liberalism. I'm struggling to think of a disagreement. |
| 21018 | The social contract has problems with future generations, national boundaries, disabilities and animals [Rawls, by Nussbaum] |
| Full Idea: Rawls saw four difficulties for justice in the social contract approach: future generations; justice across national boundaries; fair treatment of people with disabilities; and moral issues involving non-human animals. | |
| From: report of John Rawls (A Theory of Justice [1972]) by Martha Nussbaum - Creating Capabilities 4 | |
| A reaction: These are all classic examples of groups who do not have sufficient power to negotiate contracts. |
| 21041 | Justice concerns not natural distributions, or our born location, but what we do about them [Rawls] |
| Full Idea: The natural distribution is neither just nor unjust; nor is it unjust that persons are born into society at some particular position. These are simply natural facts. What is just and unjust is the way that institutions deal with these facts. | |
| From: John Rawls (A Theory of Justice [1972], 17) | |
| A reaction: Lovely quotation. There is no point in railing against the given, and that includes what is given by history, as well as what is given by nature. It comes down to intervening, in history and in nature. How much intervention will individuals tolerate? |
| 23583 | If an aggression is unjust, the constraints on how it is fought are much stricter [Rawls] |
| Full Idea: When a country's right to war is questionable and uncertain, the constraints on the means it can use are all the more severe. | |
| From: John Rawls (A Theory of Justice [1972], p.379), quoted by Michael Walzer - Just and Unjust Wars 14 | |
| A reaction: This is Rawls opposing the idea that combatants are moral equals. The restraints are, of course, moral. In practice aggressors are usually the worst behaved. |