44 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. |
| 6472 | Continuity is a sufficient criterion for the identity of a rock, but not for part of a smooth fluid [Russell] |
| Full Idea: Continuity is not a sufficient criterion of material identity; it is sufficient in many cases, such as rocks and tables, where the appearances change slowly, but in others, such as the parts of an approximately homogeneous fluid, it fails us utterly. | |
| From: Bertrand Russell (The Relation of Sense-Data to Physics [1914], §XI) | |
| A reaction: It might be debatable to what extent the 'parts' of a homogeneous fluid have identity. How many 'parts' are there in a glass of water? This seems, now, a problem for internalists; externalists can define the identity by the unseen molecules. |
| 6473 | Physical things are series of appearances whose matter obeys physical laws [Russell] |
| Full Idea: We may lay down the following definition: Physical things are those series of appearances whose matter obeys the laws of physics. | |
| From: Bertrand Russell (The Relation of Sense-Data to Physics [1914], §XI) | |
| A reaction: We will then have to define the laws of physic without making any reference to 'physical things'. There is an obvious suspicion of circularity somewhere here. I find it very odd to define objects just in terms of their appearances. |
| 6465 | We need not deny substance, but there seems no reason to assert it [Russell] |
| Full Idea: It is not necessary to deny a substance or substratum underlying appearances; it is merely expedient (by the application of Occam's Razor) to abstain from asserting this unnecessary entity. | |
| From: Bertrand Russell (The Relation of Sense-Data to Physics [1914], §V) | |
| A reaction: Russell then goes on to struggle heroically in attempts to give accounts of 'matter' and 'objects' entirely in terms of 'sense-data'. If he failed, as many think he did, should we go back to belief in Aristotelian substance? |
| 6471 | The assumption by physicists of permanent substance is not metaphysically legitimate [Russell] |
| Full Idea: The assumption of permanent substance, which technically underlies the procedure of physics, cannot of course be regarded as metaphysically legitimate. | |
| From: Bertrand Russell (The Relation of Sense-Data to Physics [1914], §XI) | |
| A reaction: It is a moot point whether physicists still thought this way after the full arrival of quantum theory in 1926. Russell raises all sorts of nice questions about the relationship between physics and philosophy here. I'm on Russell's side. |
| 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'. |
| 6466 | Where possible, logical constructions are to be substituted for inferred entities [Russell] |
| Full Idea: The supreme maxim in scientific philosophising is this: Wherever possible, logical constructions are to be substituted for inferred entities. | |
| From: Bertrand Russell (The Relation of Sense-Data to Physics [1914], §VI) | |
| A reaction: This seems to represent Russell's first move (in 1914) into what looks like phenomenalism. One might ask what is the difference between 'logical constructions' and 'inferred entities'. The latter appear to have unity, so I prefer them. |
| 6483 | Russell held that we are aware of states of our own brain [Russell, by Robinson,H] |
| Full Idea: Russell held that we are aware of states of our own brain. | |
| From: report of Bertrand Russell (The Relation of Sense-Data to Physics [1914]) by Howard Robinson - Perception 1.1 | |
| A reaction: I can't say that I had ever intepreted Russell in this way, but it is a wonderfully thought-provoking idea. All the time that I thought I was looking at a table, I was just looking at my own brain, and drawing an unspoken inference that a table caused it. |
| 8244 | Sense-data are qualities devoid of subjectivity, which are the basis of science [Russell, by Deleuze/Guattari] |
| Full Idea: Rather than oppose sensory knowledge and scientific knowledge, we should identify the sensibilia that are peculiar to science. This is what Russell did when he evoked sense-data, qualities devoid of all subjectivity. | |
| From: report of Bertrand Russell (The Relation of Sense-Data to Physics [1914]) by G Deleuze / F Guattari - What is Philosophy? 2.5 | |
| A reaction: An interesting observation. Russell is striking for his lack of interest in theories of arts and ethics, and his whole work focuses on understanding the scientific view. What is involved in sensibilia is a key modern issue (e.g. McDowell). |
| 6462 | Sense-data are not mental, but are part of the subject-matter of physics [Russell] |
| Full Idea: I regard sense-data as not mental, and as being, in fact, part of the actual subject-matter of physics. | |
| From: Bertrand Russell (The Relation of Sense-Data to Physics [1914], §III) | |
| A reaction: Russell had clearly given himself an ontological problem with the introduction of sense-data, and this is his drastic solution. In 1912 his account seems ambiguous between sense-data being mental and being physical. |
| 6463 | Sense-data are objects, and do not contain the subject as part, the way beliefs do [Russell] |
| Full Idea: Logically a sense-datum is an object, a particular of which the subject is aware; it does not contain the subject as a part, as for example beliefs and volitions do. | |
| From: Bertrand Russell (The Relation of Sense-Data to Physics [1914], §IV) | |
| A reaction: This very firmly rejects any notion that a sense-datum is mental. It is a left as a strange sort of object which gets as close as it is possible to get to the 'borders' of the mind, without actually becoming part of it. |
| 6464 | Sense-data are usually objects within the body, but are not part of the subject [Russell] |
| Full Idea: The sense-datum is an external object of which in sensation the subject is aware; it is true that the sense-datum is in many cases in the subject's body, but the subject's body is as distinct from the subject as tables and chairs are. | |
| From: Bertrand Russell (The Relation of Sense-Data to Physics [1914], §IV) | |
| A reaction: This is probably Russell's clearest statement of the nature of sense-data, which are objects within the subjects body, but are not part of the mind. So once again we come up against the question of their ontology. Are they made of neurons? |
| 6467 | No sensibile is ever a datum to two people at once [Russell] |
| Full Idea: No sensibile is ever a datum to two people at once. | |
| From: Bertrand Russell (The Relation of Sense-Data to Physics [1914], §VII) | |
| A reaction: So a loud bang has to broken down into an almost infinite number of sound sensibilia - each one presumably the size of the apperture of a small ear. This is beginning to sound a bit silly. |
| 6459 | We do not know whether sense-data exist as objects when they are not data [Russell] |
| Full Idea: We do not know, except by means of more or less precarious inferences, whether the objects which are at one time sense-data continue to exist at times when they are not data. | |
| From: Bertrand Russell (The Relation of Sense-Data to Physics [1914], §II) | |
| A reaction: Note that he actually refers to sense-data as 'objects'. It shows how thoroughly reified they are in his theory if they have the possibility of independent existence. This invites the question 'what are they made of?' |
| 6460 | 'Sensibilia' are identical to sense-data, without actually being data for any mind [Russell] |
| Full Idea: I shall give the name 'sensibilia' to those objects which have the same metaphysical and physical status as sense-data without necessarily being data to any mind. | |
| From: Bertrand Russell (The Relation of Sense-Data to Physics [1914], §III) | |
| A reaction: This is his response to the problem of whether sense-data can exist independently of experience, which was unclear in 1912. Presumably sensibilia are objects which are possible sources of experience, but that seems to cover most objects. |
| 6461 | Ungiven sense-data can no more exist than unmarried husbands [Russell] |
| Full Idea: We cannot ask, 'Can sense-data exist without being given?' for that is like asking, 'Can husbands exist without being married?' | |
| From: Bertrand Russell (The Relation of Sense-Data to Physics [1914], §III) | |
| A reaction: This follows hard on Idea 6460, which introduces the idea of 'sensibilia' for things which are like sense-data, but are not 'given'. This is a new distinction in 1914, which he had not made in 1912. |
| 6458 | Individuating sense-data is difficult, because they divide when closely attended to [Russell] |
| Full Idea: There is some difficulty in deciding what is to be considered one sense-datum: often attention causes divisions to appear where, so far as can be discovered, there were no divisions before. | |
| From: Bertrand Russell (The Relation of Sense-Data to Physics [1914], §II) | |
| A reaction: This was, I suspect, why Russell had dropped the idea of sense-data by 1921. He does, however, say that they are the last unit in analysis, rather than being the most basic unit of perception. In other words, they are purely theoretical. |
| 6469 | Sense-data may be subjective, if closing our eyes can change them [Russell] |
| Full Idea: One reason often alleged for the subjectivity of sense-data is that the appearance of a thing itself may change when we find it hard to suppose that the thing itself has changed - as when we shut our eyes, or screw them up to make things look double. | |
| From: Bertrand Russell (The Relation of Sense-Data to Physics [1914], §VIII) | |
| A reaction: Russell firmly denies that they are subjective. These examples are also said to support to proposed existence of sense-data in the first place, since they show the gap between appearance and reality. |
| 1748 | Archelaus was the first person to say that the universe is boundless [Archelaus, by Diog. Laertius] |
| Full Idea: Archelaus was the first person to say that the universe is boundless. | |
| From: report of Archelaus (fragments/reports [c.450 BCE]) by Diogenes Laertius - Lives of Eminent Philosophers 02.Ar.3 |
| 6470 | Matter is the limit of appearances as distance from the object diminishes [Russell] |
| Full Idea: We offer the following tentative definition: The matter of a given thing is the limit of its appearances as their distance from the thing diminishes. | |
| From: Bertrand Russell (The Relation of Sense-Data to Physics [1914], §IX) | |
| A reaction: This strikes me as empiricism gone mad. Russell is famous for being a 'realist', but you would hardly know it at this point. Personally I put emphasis on 'best explanation', which fairly simply delivers most of our commonsense understandings of reality. |
| 6468 | There is 'private space', and there is also the 'space of perspectives' [Russell] |
| Full Idea: In addition to the private spaces, ..there is the 'space of perspectives', since each private world may be regarded as the appearance which the universe presents from a certain point of view. | |
| From: Bertrand Russell (The Relation of Sense-Data to Physics [1914], §VII) | |
| A reaction: This replaces his concept of 'public space', which he introduced in 1912. Russell gradually dropped this, but I like the idea that we somehow directly perceive space in two ways simultaneously (which led him to say that space is six-dimensional). |
| 5989 | Archelaus said life began in a primeval slime [Archelaus, by Schofield] |
| Full Idea: Archelaus wrote that life on Earth began in a primeval slime. | |
| From: report of Archelaus (fragments/reports [c.450 BCE]) by Malcolm Schofield - Archelaus | |
| A reaction: This sounds like a fairly clearcut assertion of the production of life by evolution. Darwin's contribution was to propose the mechanism for achieving it. We should honour the name of Archelaus for this idea. |