display all the ideas for this combination of texts
43 ideas
| 16883 | Arithmetical statements can't be axioms, because they are provable [Frege, by Burge] |
| Full Idea: For Frege, no arithmetical statement is an axiom, because all are provable. | |
| From: report of Gottlob Frege (Grundlagen der Arithmetik (Foundations) [1884]) by Tyler Burge - Frege on Knowing the Foundations 1 | |
| A reaction: This is Frege's logicism, in which the true and unprovable axioms are all found in the logic, not in the arithmetic. Compare that view with the Dedekind/Peano axioms. |
| 17460 | A statement of number contains a predication about a concept [Frege] |
| Full Idea: A statement of number [Zahlangabe] contains a predication about a concept. | |
| From: Gottlob Frege (Grundlagen der Arithmetik (Foundations) [1884], §46), quoted by Ian Rumfitt - Concepts and Counting Intro | |
| A reaction: See Rumfitt 'Concepts and Counting' for a discussion. |
| 16890 | Frege's problem is explaining the particularity of numbers by general laws [Frege, by Burge] |
| Full Idea: The worry with the attempt to derive arithmetic from general logical laws (which is required for it to be analytic apriori) is that it is incompatible with the particularity of numbers. | |
| From: report of Gottlob Frege (Grundlagen der Arithmetik (Foundations) [1884], §13) by Tyler Burge - Frege on Apriority (with ps) 1 | |
| A reaction: Burge cites §13 (end) of Grundlagen, and then the doomed Basic Law V as his attempt to bridge the gap from general to particular. |
| 8630 | Individual numbers are best derived from the number one, and increase by one [Frege] |
| Full Idea: The individual numbers are best derived from the number one together with increase by one. | |
| From: Gottlob Frege (Grundlagen der Arithmetik (Foundations) [1884], §18) | |
| A reaction: Frege rejects the empirical approach partly because of the intractability of zero, but this approach has the same problem. I suggest a combination of empiricism for simple numbers, and pure formalism for extensions into complexity, and zero. |
| 11029 | 'Exactly ten gallons' may not mean ten things instantiate 'gallon' [Rumfitt on Frege] |
| Full Idea: To the question 'How many gallons of water are in the tank', the correct answer might be 'exactly ten'. But this does not mean that exactly ten things instantiate the concept 'gallon of water in the tank'. | |
| From: comment on Gottlob Frege (Grundlagen der Arithmetik (Foundations) [1884], §46) by Ian Rumfitt - Concepts and Counting p.43 | |
| A reaction: The difficulty for Frege that is being raised is that whole numbers are used to designate quantities of stuff, as well as for counting denumerable things. Rumfitt notes that 'ten' answers 'how much?' as well as Frege's 'how many?'. |
| 10013 | Numerical statements have first-order logical form, so must refer to objects [Frege, by Hodes] |
| Full Idea: Summary: numerical terms are singular terms designating objects; numerical predicates are level 1 concepts and relations; quantification over mathematics is referential; hence arithmetic has first-order form, and mathematical objects exist, non-spatially. | |
| From: report of Gottlob Frege (Grundlagen der Arithmetik (Foundations) [1884], §55?) by Harold Hodes - Logicism and Ontological Commits. of Arithmetic p.123 | |
| A reaction: [compressed] So the heart of Frege is his translation of 'Jupiter has four moons' into a logical form which only refers to numerical objects. Commentators seem vague as to whether the theory is first-order or second-order. |
| 18181 | The Number for F is the extension of 'equal to F' (or maybe just F itself) [Frege] |
| Full Idea: My definition is as follows: the Number which belongs to the concept F is the extension of the concept 'equal to the concept F' [note: I believe that for 'extension of the concept' we could simply write 'concept']. | |
| From: Gottlob Frege (Grundlagen der Arithmetik (Foundations) [1884], §68) | |
| A reaction: The note has caused huge discussion [Maddy 1997:24]. No wonder I am confused about whether a Fregean number is a concept, or a property of a concept, or a collection of things, or an object. Or all four. Or none of the above. |
| 18103 | Numbers are objects because they partake in identity statements [Frege, by Bostock] |
| Full Idea: One can always say 'the number of Jupiter's moons is 4', which is explicitly a statement of identity, and for Frege identity is always to be construed as a relation between objects. This is really all he gives to argue that numbers are objects. | |
| From: report of Gottlob Frege (Grundlagen der Arithmetik (Foundations) [1884], 55-57) by David Bostock - Philosophy of Mathematics | |
| A reaction: I struggle to understand why numbers turn out to be objects for Frege, given that they are defined in terms of sets of equinumerous sets. Is the number not a property of that meta-set. Bostock confirms my uncertainty. Paraphrase as solution? |
| 10625 | Frege had a motive to treat numbers as objects, but not a justification [Hale/Wright on Frege] |
| Full Idea: It has been observed that Frege has a motive to treat numbers as objects, but not a justification for doing so. | |
| From: comment on Gottlob Frege (Grundlagen der Arithmetik (Foundations) [1884]) by B Hale / C Wright - Intro to 'The Reason's Proper Study' §3.2 |
| 13871 | Frege claims that numbers are objects, as opposed to them being Fregean concepts [Frege, by Wright,C] |
| Full Idea: When Frege urges that numbers are to be thought of as objects, the content of this claim has to be derived from its opposition to the claim that numbers are Fregean concepts. | |
| From: report of Gottlob Frege (Grundlagen der Arithmetik (Foundations) [1884]) by Crispin Wright - Frege's Concept of Numbers as Objects 1.ii |
| 13872 | Numbers are second-level, ascribing properties to concepts rather than to objects [Frege, by Wright,C] |
| Full Idea: Frege had the insight that statements of number, like statements of existence, are in a sense second-level. That is, they are most fruitfully and least confusingly seen as ascribing a property not to an object, but to a concept. | |
| From: report of Gottlob Frege (Grundlagen der Arithmetik (Foundations) [1884]) by Crispin Wright - Frege's Concept of Numbers as Objects 1.iii | |
| A reaction: This sounds neat, but I'm immediately wondering whether he is just noticing how languages work, rather than how things are. If I say red is a bright colour, I am saying something about red objects. |
| 9816 | For Frege, successor was a relation, not a function [Frege, by Dummett] |
| Full Idea: Frege was operating with a successor relation, rather than a successor function. | |
| From: report of Gottlob Frege (Grundlagen der Arithmetik (Foundations) [1884]) by Michael Dummett - Frege philosophy of mathematics Ch.2 | |
| A reaction: That is, succession is a given fact, not a construction. 4 may be the successor of 3 in natural numbers, but not in rational or real numbers, so we can't take the relation for granted. |
| 9953 | Numbers are more than just 'second-level concepts', since existence is also one [Frege, by George/Velleman] |
| Full Idea: Frege needs more than just saying that numbers are second-level concepts under which first-level concepts fall, because they can fall under many second-level concepts, such as that of existence. | |
| From: report of Gottlob Frege (Grundlagen der Arithmetik (Foundations) [1884]) by A.George / D.J.Velleman - Philosophies of Mathematics Ch.2 | |
| A reaction: This marks the end of the first stage of Frege's theory, which leads him on to objects and Hume's Principle. After you have written 'level' a few times, you begin to wonder whether thought and world really are carved up in such a neat way. |
| 9954 | "Number of x's such that ..x.." is a functional expression, yielding a name when completed [Frege, by George/Velleman] |
| Full Idea: We can view "the number of x's such that ...x..." as a functional expression that is completed by a first-level predicate and yields a name (which is of the right kind to denote an object). | |
| From: report of Gottlob Frege (Grundlagen der Arithmetik (Foundations) [1884]) by A.George / D.J.Velleman - Philosophies of Mathematics Ch.2 | |
| A reaction: This is how Frege gets, in his account, from numbers being predicates to numbers being objects. He was a clever lad. |
| 10139 | Frege gives an incoherent account of extensions resulting from abstraction [Fine,K on Frege] |
| Full Idea: Frege identifies each conceptual abstract with the corresponding extension of concepts. But the extensions themselves are among the abstracts, so each extension is identical with the class of all concepts that have that extension, which is absurd. | |
| From: comment on Gottlob Frege (Grundlagen der Arithmetik (Foundations) [1884]) by Kit Fine - The Limits of Abstraction I.2 | |
| A reaction: Fine says this point is 'from the standpoint of a general theory of abstracts', which presumably was implied in Frege, but not actually spelled out. |
| 10028 | For Frege the number of F's is a collection of first-level concepts [Frege, by George/Velleman] |
| Full Idea: Frege defines 'the number of F's' as the extension of the concept 'equinumerous with F'. The extension of such a concept will be a collection of first-level concepts, namely, just those that are equinumerous with F. | |
| From: report of Gottlob Frege (Grundlagen der Arithmetik (Foundations) [1884]) by A.George / D.J.Velleman - Philosophies of Mathematics Ch.2 | |
| A reaction: This must be reconciled with Frege's platonism, which tells us that numbers are objects, so the objects are second-level sets. Would there be third-level object/sets, such as the set of all the second-level sets perfectly divisible by three? |
| 17636 | A cardinal number may be defined as a class of similar classes [Frege, by Russell] |
| Full Idea: Frege showed that a cardinal number may be defined as a class of similar classes. | |
| From: report of Gottlob Frege (Grundlagen der Arithmetik (Foundations) [1884]) by Bertrand Russell - Regressive Method for Premises in Mathematics p.277 |
| 10029 | Numbers need to be objects, to define the extension of the concept of each successor to n [Frege, by George/Velleman] |
| Full Idea: The fact that numbers are objects guarantees the availability of a supply of n+1 objects, which can be used to define the concept F for the successor of n, by defining the objects which fall under F. | |
| From: report of Gottlob Frege (Grundlagen der Arithmetik (Foundations) [1884]) by A.George / D.J.Velleman - Philosophies of Mathematics Ch.2 | |
| A reaction: [compressed] This is the key step which takes from from numbers being adjectival to numbers being objectual. One wonders whether physical objects might do the necessary job at the next level down. Numbers need countables. |
| 9973 | The number of F's is the extension of the second level concept 'is equipollent with F' [Frege, by Tait] |
| Full Idea: Frege's definition is that the number N F(x) of F's, where F is a concept, is the extension of the second level concept 'is equipollent with F'. | |
| From: report of Gottlob Frege (Grundlagen der Arithmetik (Foundations) [1884]) by William W. Tait - Frege versus Cantor and Dedekind III | |
| A reaction: In trying to pin Frege down precisely, we must remember that an extension can be a collection of sets, as well as a collection of concrete particulars. |
| 16500 | Frege showed that numbers attach to concepts, not to objects [Frege, by Wiggins] |
| Full Idea: It was a justly celebrated insight of Frege that numbers attach to the concepts under which objects fall, and not to the objects themselves. | |
| From: report of Gottlob Frege (Grundlagen der Arithmetik (Foundations) [1884]) by David Wiggins - Sameness and Substance 1.6 | |
| A reaction: A combination of this idea, and Aristotle's 'Categories', give us the roots of the philosophy of David Wiggins. Frege's example of two boots (or one 'pair' of boots) is the clearest example. …But the world dictates our concepts. |
| 9990 | Frege replaced Cantor's sets as the objects of equinumerosity attributions with concepts [Frege, by Tait] |
| Full Idea: Frege's contribution with respect to the definition of equinumerosity was to replace Cantor's sets as the objects of number attributions by concepts. | |
| From: report of Gottlob Frege (Grundlagen der Arithmetik (Foundations) [1884]) by William W. Tait - Frege versus Cantor and Dedekind XII | |
| A reaction: This pinpoints Frege's big idea, which is a powerful one, and may be right. The difficulty seems to be that the extension is ultimately what counts (because that is where plurality resides), and it is tricky getting the concept to determine the extension. |
| 7738 | Zero is defined using 'is not self-identical', and one by using the concept of zero [Frege, by Weiner] |
| Full Idea: Zero is the extension of 'is equinumerous with the concept "is not self-identical"' (which holds of no objects); ..one is defined as the extension of 'is equinumerous with the concept "is identical to zero"'. | |
| From: report of Gottlob Frege (Grundlagen der Arithmetik (Foundations) [1884]) by Joan Weiner - Frege Ch.4 | |
| A reaction: It sounds like some sort of cheating to define zero in terms of objects, but one in terms of concepts. |
| 23456 | Frege said logical predication implies classes, which are arithmetical objects [Frege, by Morris,M] |
| Full Idea: Frege's idea is that the logical notion of predication is enough to generate appropriate objects. Every predicate defines a class, which is in turn an object to which predicates apply; and the notion of a class can be used to generate arithmetic. | |
| From: report of Gottlob Frege (Grundlagen der Arithmetik (Foundations) [1884]) by Michael Morris - Guidebook to Wittgenstein's Tractatus 2H | |
| A reaction: At last, a lovely clear account of what Frege was doing - and why Russell's paradox was Frege's disaster. Logicism must take the ingredients of logic, and generate arithmetical 'objects' from them alone. But do we need 'objects'? |
| 13887 | Frege started with contextual definition, but then switched to explicit extensional definition [Frege, by Wright,C] |
| Full Idea: Frege abandoned contextual definition of numerical singular terms, and decided to go for explicit definitions in terms of extension-denoting terms instead. | |
| From: report of Gottlob Frege (Grundlagen der Arithmetik (Foundations) [1884]) by Crispin Wright - Frege's Concept of Numbers as Objects 3.xiv |
| 13897 | Each number, except 0, is the number of the concept of all of its predecessors [Frege, by Wright,C] |
| Full Idea: In Frege's definition of numbers, each number, except 0, is defined as the number belonging to the concept under which just its predecessors fall. | |
| From: report of Gottlob Frege (Grundlagen der Arithmetik (Foundations) [1884]) by Crispin Wright - Frege's Concept of Numbers as Objects 4.xvii | |
| A reaction: This would make the numbers dependent on all of the predecessors, just as Dedekind's numbers do. Dedekind's progression has to continue, but why should Frege's? Frege's are just there, where Dedekind's are constructed. Why are Frege's ordered? |
| 9856 | Frege's account of cardinals fails in modern set theory, so they are now defined differently [Dummett on Frege] |
| Full Idea: In standard set theory, Frege's cardinals could not be members of classes, and his proof of the infinity of natural numbers fails. Nowadays they are defined as sets each representative of its cardinality, comprising ordinals of lower cardinality. | |
| From: comment on Gottlob Frege (Grundlagen der Arithmetik (Foundations) [1884]) by Michael Dummett - Frege philosophy of mathematics Ch.14 | |
| A reaction: Pinning something down in a unique way is not the same as telling you its intrinsic nature. But a completely successful definition seems to have locked on to some deep truth about its target. |
| 9902 | Frege's incorrect view is that a number is an equivalence class [Benacerraf on Frege] |
| Full Idea: Frege view (which has little to commend it) was that the number 3 is the extension of the concept 'equivalent with some 3-membered set'; that is, for Frege a number was an equivalence class - the class of all classes equivalent with a given class. | |
| From: comment on Gottlob Frege (Grundlagen der Arithmetik (Foundations) [1884]) by Paul Benacerraf - What Numbers Could Not Be II | |
| A reaction: Frege is a platonist, who takes numbers to be objects, so this equivalence class must be identical with an object. What exactly was Frege claiming? I mean, really exactly? |
| 17814 | The natural number n is the set of n-membered sets [Frege, by Yourgrau] |
| Full Idea: Frege defines the natural number n in terms of the set of n-membered sets. | |
| From: report of Gottlob Frege (Grundlagen der Arithmetik (Foundations) [1884]) by Palle Yourgrau - Sets, Aggregates and Numbers 'Two' | |
| A reaction: He says this view 'has been treated rudely by history', because Frege's view of sets was naive, and because independence results have undermined set-theoretic platonism. |
| 17819 | A set doesn't have a fixed number, because the elements can be seen in different ways [Yourgrau on Frege] |
| Full Idea: Given the set {Carter, Reagan} ...I still want to know How many what? Members? 2. Sets? 1. Feet of members? 4. Relatives of members? 44. | |
| From: comment on Gottlob Frege (Grundlagen der Arithmetik (Foundations) [1884]) by Palle Yourgrau - Sets, Aggregates and Numbers 'New Problem' | |
| A reaction: This is his 'new problem' for Frege. Frege want a concept to divide a pack of cards, by cards, suits or pips. You choose 'pips' and form a set, but then the pips may have a number of corners. Solution: pick your 'objects' or 'units', and stick to it. |
| 17820 | If you can subdivide objects many ways for counting, you can do that to set-elements too [Yourgrau on Frege] |
| Full Idea: If we are allowed in the case of sets to construe the number question as 'really': How many (elements)?, then we could just as well construe Frege's famous question about the deck of cards as: How many (cards)? | |
| From: comment on Gottlob Frege (Grundlagen der Arithmetik (Foundations) [1884]) by Palle Yourgrau - Sets, Aggregates and Numbers 'New Problem' | |
| A reaction: My view is that counting is not entirely relative to the concept employed, but that the world imposes objects on us which thus impose their concepts and their counting. This is 'natural', but we can then counter nature with pragmatics and whimsy. |
| 9956 | 'The number of Fs' is the extension (a collection of first-level concepts) of the concept 'equinumerous with F' [Frege, by George/Velleman] |
| Full Idea: Frege defines 'the number of Fs' as equal to the extension of the concept 'equinumerous with F'. The extension of such a concept will be a collection of first-level concepts, namely those that are equinumerous with F. | |
| From: report of Gottlob Frege (Grundlagen der Arithmetik (Foundations) [1884]) by A.George / D.J.Velleman - Philosophies of Mathematics Ch.2 | |
| A reaction: Presumably this means equinumerous with 'instances' of F, if F is a predicate. The problem of universals looms. I was clear about this idea until I tried to draw a diagram illustrating it. You try! |
| 13527 | Frege's cardinals (equivalences of one-one correspondences) is not permissible in ZFC [Frege, by Wolf,RS] |
| Full Idea: Frege defined a cardinal as an equivalence class of one-one correspondences. The cardinal 3 is the class of all sets with three members. This definition is intuitively appealing, but it is not permissible in ZFC. | |
| From: report of Gottlob Frege (Grundlagen der Arithmetik (Foundations) [1884]) by Robert S. Wolf - A Tour through Mathematical Logic 2.2 | |
| A reaction: This is why Frege's well known definition of cardinals no longer figures in standard discussions of the subject. His definition is acceptable in Von Neumann-Bernays-Gödel set theory (Wolf p.73). |
| 22292 | Hume's Principle fails to implicitly define numbers, because of the Julius Caesar [Frege, by Potter] |
| Full Idea: Frege rejected Hume's Principle as an implicit definition of number terms, because of the Julius Caesar problem. ....[128] Instead Frege adopted an explicit definition of the number-of function. | |
| From: report of Gottlob Frege (Grundlagen der Arithmetik (Foundations) [1884]) by Michael Potter - The Rise of Analytic Philosophy 1879-1930 19 'Uniq' |
| 17442 | Frege thinks number is fundamentally bound up with one-one correspondence [Frege, by Heck] |
| Full Idea: Frege's answer is that the concept of number is fundamentally bound up with the notion of one-one correspondence. | |
| From: report of Gottlob Frege (Grundlagen der Arithmetik (Foundations) [1884]) by Richard G. Heck - Cardinality, Counting and Equinumerosity 1 | |
| A reaction: Birds seem to find a mate with virtually no concept of number. I'm beginning to think that the essence of numbers is that they are both ordinals and cardinals. Frege, of course, thinks identity is basic to metaphysics. |
| 11030 | The words 'There are exactly Julius Caesar moons of Mars' are gibberish [Rumfitt on Frege] |
| Full Idea: The word 'Julius Caesar is prime' may well involve some kind of category error, but the still compose a grammatical sentence. The words 'There are exactly Julius Caesar moons of Mars', by contrast, are gibberish. | |
| From: comment on Gottlob Frege (Grundlagen der Arithmetik (Foundations) [1884]) by Ian Rumfitt - Concepts and Counting p.48 |
| 10030 | 'Julius Caesar' isn't a number because numbers inherit properties of 0 and successor [Frege, by George/Velleman] |
| Full Idea: 'Julius Caesar' is not a natural number in Frege's account because he does not fall under every concept under which 0 falls and which is hereditary with respect to successor. | |
| From: report of Gottlob Frege (Grundlagen der Arithmetik (Foundations) [1884]) by A.George / D.J.Velleman - Philosophies of Mathematics Ch.2 | |
| A reaction: Significant for structuralist views. One might say that any object can occupy the structural place of '17', but if you derive your numbers from 0, successor and induction, then the 17-object must also inherit the properties of zero and successors. |
| 8690 | From within logic, how can we tell whether an arbitrary object like Julius Caesar is a number? [Frege, by Friend] |
| Full Idea: The 'Julius Caesar problem' in Frege's theory is that from within logic we cannot tell if an arbitrary objects such as Julius Caesar is a number or not. Logic itself cannot tell us enough to distinguish numbers from other sorts of objects. | |
| From: report of Gottlob Frege (Grundlagen der Arithmetik (Foundations) [1884]) by Michčle Friend - Introducing the Philosophy of Mathematics 3.4 | |
| A reaction: What a delightful problem (raised by Frege himself). A theory can look beautiful till you ask a question like this. Only a logician would, I suspect, get into this mess. Numbers can be used to count or order things! "I've got Caesar pencils"? |
| 10219 | Frege said 2 is the extension of all pairs (so Julius Caesar isn't 2, because he's not an extension) [Frege, by Shapiro] |
| Full Idea: Frege proposed that the number 2 is a certain extension, the collection of all pairs. Thus, 2 is not Julius Caesar because, presumably, persons are not extensions. | |
| From: report of Gottlob Frege (Grundlagen der Arithmetik (Foundations) [1884]) by Stewart Shapiro - Philosophy of Mathematics 3.2 | |
| A reaction: Unfortunately, as Shapiro notes, Frege's account of extension went horribly wrong. Nevertheless, this seems to show why the Julius Caesar problem does not matter for Frege, though it might matter for the neo-logicists. |
| 13889 | Fregean numbers are numbers, and not 'Caesar', because they correlate 1-1 [Frege, by Wright,C] |
| Full Idea: We cannot reasonably suppose that any numerical singular term has the same reference as 'Caesar', because Frege's numbers (unlike persons) are to be identified and distinguished by appeal to facts about 1-1 correlation among concepts. | |
| From: report of Gottlob Frege (Grundlagen der Arithmetik (Foundations) [1884]) by Crispin Wright - Frege's Concept of Numbers as Objects 3.xiv |
| 18142 | One-one correlations imply normal arithmetic, but don't explain our concept of a number [Frege, by Bostock] |
| Full Idea: Frege inferred from the Julius Caesar problem that even though Hume's Principle sufficed as a single axiom for deducing the arithmetic of the finite cardinal numbers, still it does not explain our ordinary understanding of those numbers. | |
| From: report of Gottlob Frege (Grundlagen der Arithmetik (Foundations) [1884]) by David Bostock - Philosophy of Mathematics 9.A.2 |
| 9046 | Our definition will not tell us whether or not Julius Caesar is a number [Frege] |
| Full Idea: We can never decide by means of our definitions whether any concept has the number JULIUS CAESAR belonging to it, or whether that same familiar conqueror of Gaul is a number or not. | |
| From: Gottlob Frege (Grundlagen der Arithmetik (Foundations) [1884], §56) | |
| A reaction: This has become a famous modern problem. The point is that the definition of a number must explain why this is a number, and not something else. Must you mention that you could use it to count? Count you count using emperors? |
| 16896 | If numbers can be derived from logic, then set theory is superfluous [Frege, by Burge] |
| Full Idea: Frege thought that if one could derive the existence of numbers from logical concepts, one would not need set theory to explain number theory, or for any other good purpose. | |
| From: report of Gottlob Frege (Grundlagen der Arithmetik (Foundations) [1884]) by Tyler Burge - Frege on Apriority (with ps) 2 | |
| A reaction: Note that we have two possible routes to 'explain' numbers. I'm inclined to see set theory as modelling numbers rather than explaining them. Frege did better at explanation, but I suspect he is wrong too. |
| 8639 | If numbers are supposed to be patterns, each number can have many patterns [Frege] |
| Full Idea: Patterns can be completely different while the number of their elements remains the same, so that here we would have different distinct fives, sixes and so forth. | |
| From: Gottlob Frege (Grundlagen der Arithmetik (Foundations) [1884], §41) | |
| A reaction: A blow to my enthusiasm for Michael Resnik's account of maths as patterns. See, for example, Ideas 6296 and 6301. We are clearly set up to spot patterns long before we arrive at the abstract concepts of numbers. We see the same number in two patterns. |