37 ideas
13886 | Later Frege held that definitions must fix a function's value for every possible argument [Frege, by Wright,C] |
Full Idea: Frege later became fastidious about definitions, and demanded that they must provide for every possible case, and that no function is properly determined unless its value is fixed for every conceivable object as argument. | |
From: report of Gottlob Frege (Grundgesetze der Arithmetik 2 (Basic Laws) [1903]) by Crispin Wright - Frege's Concept of Numbers as Objects 3.xiv | |
A reaction: Presumably definitions come in degrees of completeness, but it seems harsh to describe a desire for the perfect definition as 'fastidious', especially if we are talking about mathematics, rather than defining 'happiness'. |
9845 | We can't define a word by defining an expression containing it, as the remaining parts are a problem [Frege] |
Full Idea: Given the reference (bedeutung) of an expression and a part of it, obviously the reference of the remaining part is not always determined. So we may not define a symbol or word by defining an expression in which it occurs, whose remaining parts are known | |
From: Gottlob Frege (Grundgesetze der Arithmetik 2 (Basic Laws) [1903], §66) | |
A reaction: Dummett cites this as Frege's rejection of contextual definitions, which he had employed in the Grundlagen. I take it not so much that they are wrong, as that Frege decided to set the bar a bit higher. |
10019 | Only what is logically complex can be defined; what is simple must be pointed to [Frege] |
Full Idea: Only what is logically complex can be defined; what is simple can only be pointed to. | |
From: Gottlob Frege (Grundgesetze der Arithmetik 2 (Basic Laws) [1903], §180), quoted by Harold Hodes - Logicism and Ontological Commits. of Arithmetic p.137 | |
A reaction: Frege presumably has in mind his treasured abstract objects, such as cardinal numbers. It is hard to see how you could 'point to' anything in the phenomenal world that had atomic simplicity. Hodes calls this a 'desperate Kantian move'. |
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'. |
9886 | Cardinals say how many, and reals give measurements compared to a unit quantity [Frege] |
Full Idea: The cardinals and the reals are completely disjoint domains. The cardinal numbers answer the question 'How many objects of a given kind are there?', but the real numbers are for measurement, saying how large a quantity is compared to a unit quantity. | |
From: Gottlob Frege (Grundgesetze der Arithmetik 2 (Basic Laws) [1903], §157), quoted by Michael Dummett - Frege philosophy of mathematics Ch.19 | |
A reaction: We might say that cardinals are digital and reals are analogue. Frege is unusual in totally separating them. They map onto one another, after all. Cardinals look like special cases of reals. Reals are dreams about the gaps between cardinals. |
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. |
9889 | Real numbers are ratios of quantities [Frege, by Dummett] |
Full Idea: Frege fixed on construing real numbers as ratios of quantities (in agreement with Newton). | |
From: report of Gottlob Frege (Grundgesetze der Arithmetik 2 (Basic Laws) [1903]) by Michael Dummett - Frege philosophy of mathematics Ch.20 | |
A reaction: If 3/4 is the same real number as 6/8, which is the correct ratio? Why doesn't the square root of 9/16 also express it? Why should irrationals be so utterly different from rationals? In what sense are they both 'numbers'? |
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. |
10553 | A number is a class of classes of the same cardinality [Frege, by Dummett] |
Full Idea: For Frege, in 'Grundgesetze', a number is a class of classes of the same cardinality. | |
From: report of Gottlob Frege (Grundgesetze der Arithmetik 2 (Basic Laws) [1903]) by Michael Dummett - Frege Philosophy of Language (2nd ed) Ch.14 |
10020 | Frege's biggest error is in not accounting for the senses of number terms [Hodes on Frege] |
Full Idea: The inconsistency of Grundgesetze was only a minor flaw. Its fundamental flaw was its inability to account for the way in which the senses of number terms are determined. It leaves the reference-magnetic nature of the standard numberer a mystery. | |
From: comment on Gottlob Frege (Grundgesetze der Arithmetik 2 (Basic Laws) [1903]) by Harold Hodes - Logicism and Ontological Commits. of Arithmetic p.139 | |
A reaction: A point also made by Hofweber. As a logician, Frege was only concerned with the inferential role of number terms, and he felt he had captured their logical form, but it is when you come to look at numbers in natural language that he seem in trouble. |
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. |
9887 | Formalism misunderstands applications, metatheory, and infinity [Frege, by Dummett] |
Full Idea: Frege's three main objections to radical formalism are that it cannot account for the application of mathematics, that it confuses a formal theory with its metatheory, and it cannot explain an infinite sequence. | |
From: report of Gottlob Frege (Grundgesetze der Arithmetik 2 (Basic Laws) [1903], §86-137) by Michael Dummett - Frege philosophy of mathematics | |
A reaction: The application is because we don't design maths randomly, but to be useful. The third objection might be dealt with by potential infinities (from formal rules). The second objection sounds promising. |
8751 | Only applicability raises arithmetic from a game to a science [Frege] |
Full Idea: It is applicability alone which elevates arithmetic from a game to the rank of a science. | |
From: Gottlob Frege (Grundgesetze der Arithmetik 2 (Basic Laws) [1903], §91), quoted by Stewart Shapiro - Thinking About Mathematics 6.1.2 | |
A reaction: This is the basic objection to Formalism. It invites the question of why it is applicable, which platonists like Frege don't seem to answer (though Plato himself has reality modelled on the Forms). This is why I like structuralism. |
9891 | The first demand of logic is of a sharp boundary [Frege] |
Full Idea: The first demand of logic is of a sharp boundary. | |
From: Gottlob Frege (Grundgesetze der Arithmetik 2 (Basic Laws) [1903], §160), quoted by Michael Dummett - Frege philosophy of mathematics Ch.22 | |
A reaction: Nothing I have read about vagueness has made me doubt Frege's view of this, although precisification might allow you to do logic with vague concepts without having to finally settle where the actual boundaries are. |
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'. |
9890 | The modern account of real numbers detaches a ratio from its geometrical origins [Frege] |
Full Idea: From geometry we retain the interpretation of a real number as a ratio of quantities or measurement-number; but in more recent times we detach it from geometrical quantities, and from all particular types of quantity. | |
From: Gottlob Frege (Grundgesetze der Arithmetik 2 (Basic Laws) [1903], §159), quoted by Michael Dummett - Frege philosophy of mathematics | |
A reaction: Dummett glosses the 'recent' version as by Cantor and Dedekind in 1872. This use of 'detach' seems to me startlingly like the sort of psychological abstractionism which Frege was so desperate to avoid. |
11846 | If we abstract the difference between two houses, they don't become the same house [Frege] |
Full Idea: If abstracting from the difference between my house and my neighbour's, I were to regard both houses as mine, the defect of the abstraction would soon be made clear. It may, though, be possible to obtain a concept by means of abstraction... | |
From: Gottlob Frege (Grundgesetze der Arithmetik 2 (Basic Laws) [1903], §99) | |
A reaction: Note the important concession at the end, which shows Frege could never deny the abstraction process, despite all the modern protests by Geach and Dummett that he totally rejected it. |
7903 | The six perfections are giving, morality, patience, vigour, meditation, and wisdom [Nagarjuna] |
Full Idea: The six perfections are of giving, morality, patience, vigour, meditation, and wisdom. | |
From: Nagarjuna (Mahaprajnaparamitashastra [c.120], 88) | |
A reaction: What is 'morality', if giving is not part of it? I like patience and vigour being two of the virtues, which immediately implies an Aristotelian mean (which is always what is 'appropriate'). |