37 ideas
15395 | Give up objects necessitating truths, and say their natures cause the truths? [Cameron] |
Full Idea: We could abandon the view that truthmakers necessitate the truth of that which makes them true, and say that an object makes a truth when its intrinsic nature suffices for that truth. The object would have a different intrinsic nature if the truth failed. | |
From: Ross P. Cameron (Intrinsic and Extrinsic Properties [2009], 'Truthmakers') | |
A reaction: [He cites Josh Parsons 1999, 2005 for this] This approach seems closely related to Kit Fine's proposal that necessities arise from the natures of things. It sounds to me as if an object with that intrinsic nature would necessitate that truth. |
15394 | Truthmaker requires a commitment to tropes or states of affairs, for contingent truths [Cameron] |
Full Idea: The most popular view is that an object is a truthmaker if the object couldn't exist and the truth be false. But contingent predications are also held to need truthmakers. Socrates is not necessarily snub-nosed, so a trope or state of affairs is needed. | |
From: Ross P. Cameron (Intrinsic and Extrinsic Properties [2009], 'Truthmakers') | |
A reaction: Cameron calls this 'some heavy ontological commitments'. If snub-nosedness is necessitated by the trope of 'being snub-nosed', what is the truthmaker for Socrates having that trope? |
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'. |
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. |
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. |
15401 | Essentialists say intrinsic properties arise from what the thing is, irrespective of surroundings [Cameron] |
Full Idea: The essentialist approach would be to say that an intrinsic property is one such that it is no part of what it is to instantiate that property that the bearer stands in some relation to its surroundings. | |
From: Ross P. Cameron (Intrinsic and Extrinsic Properties [2009], 'Analysis') | |
A reaction: This is offered as an alternative to the David Lewis account in terms of duplicates across possible worlds. You will have gathered by now, if you have spent days poring over my stuff, that I favour the essentialist approach. |
15393 | An object's intrinsic properties are had in virtue of how it is, independently [Cameron] |
Full Idea: Intrinsic properties are those that an object has solely in virtue of how it is, independently of its surroundings. | |
From: Ross P. Cameron (Intrinsic and Extrinsic Properties [2009], 'Intro') | |
A reaction: Better not mention quantum mechanics and fields if you want to talk of objects being independent of their surroundings. Am I 'independent' of gravity, or is gravity 'independent' of me? |
15396 | Most criteria for identity over time seem to leave two later objects identical to the earlier one [Cameron] |
Full Idea: Criteria for identity across times have proven hard to give. Whatever criteria we lay down, it seems that there are possible situations in which two later objects bear the relevant relation to one earlier object, though only one of them can be identical. | |
From: Ross P. Cameron (Intrinsic and Extrinsic Properties [2009], 'Personal') | |
A reaction: We only have to think of twins, amoebae that fission, and the Ship of Theseus. We seem to end up inventing a dubious criterion in order to break the tie. |
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'. |
7357 | People who control others with fluent language often end up being hated [Kongzi (Confucius)] |
Full Idea: Of what use is eloquence? He who engages in fluency of words to control men often finds himself hated by them. | |
From: Kongzi (Confucius) (The Analects (Lunyu) [c.511 BCE], V.5) | |
A reaction: I don't recall Socrates making this very good point to any of the sophists (such as Gorgias). The idea that if you battle or connive your way to dominance over others then you are successful is false. Life is a much longer game than that. |
7358 | All men prefer outward appearance to true excellence [Kongzi (Confucius)] |
Full Idea: I have yet to meet a man as fond of excellence as he is of outward appearances. | |
From: Kongzi (Confucius) (The Analects (Lunyu) [c.511 BCE], IX.18) | |
A reaction: Interestingly, this cynical view of the love of virtue is put by Plato into the mouths of Glaucon and Adeimantus (in Bk II of 'Republic', e.g. Idea 12), and not into the mouth of Socrates, who goes on to defend the possibility of true virtue. |
7362 | Humans are similar, but social conventions drive us apart (sages and idiots being the exceptions) [Kongzi (Confucius)] |
Full Idea: In our natures we approximate one another; habits put us further and further apart. The only ones who do not change are sages and idiots. | |
From: Kongzi (Confucius) (The Analects (Lunyu) [c.511 BCE], XVII.2) | |
A reaction: I find most of Confucius rather uninteresting, but this is a splendid remark about the influence of social conventions on human nature. Sages can achieve universal morality if they rise above social convention, and seek the true virtues of human nature. |
7360 | Do not do to others what you would not desire yourself [Kongzi (Confucius)] |
Full Idea: Do not do to others what you would not desire yourself. Then you will have no enemies, either in the state or in your home. | |
From: Kongzi (Confucius) (The Analects (Lunyu) [c.511 BCE], XII.2) | |
A reaction: The Golden Rule, but note the second sentence. Logically, it leads to the absurdity of not giving someone an Elvis record for Christmas because you yourself don't like Elvis. Kant (Idea 3733) and Nietzsche (Idea 4560) offer good criticisms. |
7359 | Excess and deficiency are equally at fault [Kongzi (Confucius)] |
Full Idea: Excess and deficiency are equally at fault. | |
From: Kongzi (Confucius) (The Analects (Lunyu) [c.511 BCE], XI.16) | |
A reaction: This is the sort of wisdom we admire in Aristotle (and in any sensible person), but it may also be the deepest motto of conservatism, and it is a long way from romantic philosophy, and the clarion call of Nietzsche to greater excitement in life. |
7363 | The virtues of the best people are humility, maganimity, sincerity, diligence, and graciousness [Kongzi (Confucius)] |
Full Idea: He who in this world can practise five things may indeed be considered Man-at-his-best: humility, maganimity, sincerity, diligence, and graciousness. | |
From: Kongzi (Confucius) (The Analects (Lunyu) [c.511 BCE], XVII.5) | |
A reaction: A very nice list. Who could resist working with a colleague who had such virtues? Who could go wrong if they married a person who had them? I can't think of anything important that is missing. |
7361 | Men of the highest calibre avoid political life completely [Kongzi (Confucius)] |
Full Idea: Men of the highest calibre avoid political life completely. | |
From: Kongzi (Confucius) (The Analects (Lunyu) [c.511 BCE], XIV.37) | |
A reaction: Plato notes that such people tend to avoid political life (and a left sheltering, as if from a wild storm!), but he thinks they should be dragged into the political arena for the common good. Confucius seems to approve of the avoidance. Plato is right. |
23393 | Confucianism assumes that all good developments have happened, and there is only one Way [Norden on Kongzi (Confucius)] |
Full Idea: The two major limitations of Confucianism are that it assumes that all worthwhile cultural, social and ethical innovation has already occurred, and that it does not recognise the plurality of worthwhile ways of life. | |
From: comment on Kongzi (Confucius) (The Analects (Lunyu) [c.511 BCE]) by Bryan van Norden - Intro to Classical Chinese Philosophy 3.III | |
A reaction: In modern liberal terms that is about as conservative as it is possible to get. We think of it as the state of mind of an old person who can only long for the way things were when they were young. But 'hold fast to that which is good'! |