display all the ideas for this combination of texts
28 ideas
| 13874 | Numbers seem to be objects because they exactly fit the inference patterns for identities [Frege] |
| Full Idea: The most important consideration for numbers being objects is that they sustain the patterns of inference demanded by the reflexivity, transitivity and symmetry of identity. | |
| From: Gottlob Frege (Grundlagen der Arithmetik (Foundations) [1884]), quoted by Crispin Wright - Frege's Concept of Numbers as Objects 1.iii | |
| A reaction: But then if I say that the 'whereabouts of Jack' is identical to the 'whereabouts of Jill', that would seem to make whereaboutses into objects. |
| 13875 | Frege's platonism proposes that objects are what singular terms refer to [Frege, by Wright,C] |
| Full Idea: The basis of Frege's platonism is the thesis that objects are what singular terms, in the ordinary intuitive sense of 'singular term', refer to. | |
| 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 claim strikes me as very bizarre, and is at the root of all the daft aspects of twentieth century linguistic philosophy. See Bob Hale on singular terms, who defends the Fregean view against obvious problems like 'for THE SAKE of the children'. |
| 7731 | How can numbers be external (one pair of boots is two boots), or subjective (and so relative)? [Frege, by Weiner] |
| Full Idea: If the number one is a property of external things, how can one pair of boots be the same as two boots? ...but if the number one is subjective, then the number a thing has for me need not be the same number the object has for you. | |
| From: report of Gottlob Frege (Grundlagen der Arithmetik (Foundations) [1884]) by Joan Weiner - Frege Ch.4 | |
| A reaction: This nicely captures the initial dilemma over the nature of numbers. It is the commonest dilemma in all of philosophy, struggling between subjective and objective accounts of things. Hence Putnam's nice definition of philosophy (Idea 2352). |
| 7737 | Identities refer to objects, so numbers must be objects [Frege, by Weiner] |
| Full Idea: Identity statements are about objects. If we can say that 1 is identical (or not) to 0, then 1 must be an object. | |
| From: report of Gottlob Frege (Grundlagen der Arithmetik (Foundations) [1884]) by Joan Weiner - Frege Ch.4 | |
| A reaction: This seems to point to Platonism about numbers, but maybe we can accept it as being about physical objects. If numbers are essentially patterns, then identity is hypothetical one-to-one identity between sets of objects. |
| 8635 | Numbers are not physical, and not ideas - they are objective and non-sensible [Frege] |
| Full Idea: Number is neither spatial and physical, like Mill's pile of pebbles, nor yet subjective like ideas, but non-sensible and objective. | |
| From: Gottlob Frege (Grundlagen der Arithmetik (Foundations) [1884], §27) | |
| A reaction: This doesn't require commitment to full-blown universals, nor to a dualist world of mind. The thinking of the brain moves far away from the areas of sensation, and the brain's capacity for truth is its capacity for objectivity. |
| 8652 | Numbers are objects, because they can take the definite article, and can't be plurals [Frege] |
| Full Idea: Individual numbers are objects, as is indicated by the use of the definite article in expressions like 'the number two', and by the impossibility of speaking of ones, twos, etc. in the plural. | |
| From: Gottlob Frege (Grundlagen der Arithmetik (Foundations) [1884], §68 n) | |
| A reaction: Hm. The beginnings of linguistic philosophy, with all its problems. It is well known that 'for the sake of the children' doesn't make an ontological commitment to 'sakes'. The children might 'enter in threes', but the second half is a good point. |
| 17816 | Frege's logicism aimed at removing the reliance of arithmetic on intuition [Frege, by Yourgrau] |
| Full Idea: In reducing arithmetic to logic Frege was precisely trying to show the independence of this study from any peculiarly mathematical intuitions. | |
| From: report of Gottlob Frege (Grundlagen der Arithmetik (Foundations) [1884]) by Palle Yourgrau - Sets, Aggregates and Numbers 'Two' |
| 8633 | There is no physical difference between two boots and one pair of boots [Frege] |
| Full Idea: One pair of boots may be the same visible and tangible phenomenon as two boots. This is a difference in number to which no physical difference corresponds; for 'two' and 'one pair' are by no means the same thing. | |
| From: Gottlob Frege (Grundlagen der Arithmetik (Foundations) [1884], §25) | |
| A reaction: He is attacking Mill. Those of us who are seeking an empirical account of arithmetic have certainly got to face up to this example. Not insurmountable, I think. |
| 9951 | It appears that numbers are adjectives, but they don't apply to a single object [Frege, by George/Velleman] |
| Full Idea: Numbers as adjectives appear to attribute a property - but to what? Superficially it seems to be to the objects themselves, as it makes sense to say that a plague is 'deadly', but not that it is 'ten'. | |
| From: report of Gottlob Frege (Grundlagen der Arithmetik (Foundations) [1884]) by A.George / D.J.Velleman - Philosophies of Mathematics Ch.2 | |
| A reaction: Surely they could be adjectival if they were properties of groups? Groups can be 'numerous', or 'more than a hundred', or 'too many for this taxi'. |
| 9952 | Numerical adjectives are of the same second-level type as the existential quantifier [Frege, by George/Velleman] |
| Full Idea: A numerical adjective forms part of a predicate of second-level, needing supplementation from the first level (F). So the second-level predicate is of the same type as the existential quantifier, and can be called a 'numerical quantifier'. | |
| From: report of Gottlob Frege (Grundlagen der Arithmetik (Foundations) [1884]) by A.George / D.J.Velleman - Philosophies of Mathematics Ch.2 | |
| A reaction: This seems like a highly plausible account of how numbers work in language, but it leaves you wondering what the ontological status of a quantifier is. I presume platonic heaven is not full of elite entities called quantifiers, marshalling the others. |
| 11031 | 'Jupiter has many moons' won't read as 'The number of Jupiter's moons equals the number many' [Rumfitt on Frege] |
| Full Idea: 'Jupiter has four moons' is semantically and syntactically on all fours with 'Jupiter has many moons'. But it would be brave to construe the latter proposition as a transformation of 'The number of Jupiter's moons is identical with the number many'. | |
| From: comment on Gottlob Frege (Grundlagen der Arithmetik (Foundations) [1884]) by Ian Rumfitt - Concepts and Counting p.49 | |
| A reaction: I take this to be an important insight. Number words are continuous with (are in the same category as) words for general numerical quantity, such as 'just a few' or 'many' or 'rather a lot'. Numbers are part of normal language. |
| 8637 | The number 'one' can't be a property, if any object can be viewed as one or not one [Frege] |
| Full Idea: How can it make sense to ascribe the property 'one' to any object whatever, when every object, according as to how we look at it, can be either one or not one? | |
| From: Gottlob Frege (Grundlagen der Arithmetik (Foundations) [1884], §30) | |
| A reaction: This remark seems to point to numbers being highly subjective, but the interest of Frege is that he then makes out a case for numbers being totally objective, despite being entirely non-physical in nature. How do they do that? |
| 9999 | For science, we can translate adjectival numbers into noun form [Frege] |
| Full Idea: We want a concept of number usable for science; we should not, therefore, be deterred by everyday language using numbers in attributive constructions. The proposition 'Jupiter has four moons' can be converted to 'the number of Jupiter's moons is four'. | |
| From: Gottlob Frege (Grundlagen der Arithmetik (Foundations) [1884], §57) | |
| A reaction: Critics are quick to point out that this could work the other way (noun-to-adjective), so Frege hasn't got an argument here, only an escape route. How about the verb version ('the moons of Jupiter four'), or the adverb ('J's moons behave fourly')? |
| 7739 | Arithmetic is analytic [Frege, by Weiner] |
| Full Idea: Frege's project was to show that arithmetic is analytic. | |
| From: report of Gottlob Frege (Grundlagen der Arithmetik (Foundations) [1884]) by Joan Weiner - Frege Ch.7 | |
| A reaction: This particularly opposes Kant (e.g. Idea 5525). My favoured view (which may have few friends) is that arithmetic is a set of facts about the necessary pattern relationships within any possible physical world. That will make it synthetic. |
| 9945 | Logicism shows that no empirical truths are needed to justify arithmetic [Frege, by George/Velleman] |
| Full Idea: Frege claims that his logicist project directly shows that no empirical truths about the natural world need be employed in the justification of arithmetic (nor need any truths that are apprehended through some kind of intuition). | |
| From: report of Gottlob Frege (Grundlagen der Arithmetik (Foundations) [1884]) by A.George / D.J.Velleman - Philosophies of Mathematics Ch.2 | |
| A reaction: This simple way of putting it creates a sticking-point for me. It occurs to me that the best description of arithmetic is that it 'models' the natural world. If a beautiful system failed to count objects, it wouldn't be accepted as 'arithmetic'. |
| 8782 | Frege offered a Platonist version of logicism, committed to cardinal and real numbers [Frege, by Hale/Wright] |
| Full Idea: Since Frege's defence of his thesis that the laws of arithmetic are analytic depended upon a realm of independently existing objects - the finite cardinal numbers and the real numbers - his view amounted to a Platonist version of logicism. | |
| From: report of Gottlob Frege (Grundlagen der Arithmetik (Foundations) [1884]) by B Hale / C Wright - Logicism in the 21st Century 1 | |
| A reaction: Nice to have this spelled out. Along with Gödel, Frege is the most distinguished Platonist since the great man. Frege has lots of modern fans, but I would have thought that this makes his position a non-starter. Alternatives are needed. |
| 13608 | Mathematics has no special axioms of its own, but follows from principles of logic (with definitions) [Frege, by Bostock] |
| Full Idea: Frege's logicism is the theory that mathematics has no special axioms of its own, but follows just from the principles of logic themselves, when augmented with suitable definitions. | |
| From: report of Gottlob Frege (Grundlagen der Arithmetik (Foundations) [1884]) by David Bostock - Intermediate Logic 5.1 | |
| A reaction: Thus logicism is opposed to the Dedekind-Peano axioms, which are not logic, but are specific to mathematics. Hence modern logicists try to derive the Peano Axioms from logical axioms. Logicism rests on logical truths, not inference rules. |
| 16905 | Arithmetic must be based on logic, because of its total generality [Frege, by Jeshion] |
| Full Idea: For Frege, that arithmetic is essentially general, governing (applying to) everything, entails that its ultimate building blocks are purely logical. | |
| From: report of Gottlob Frege (Grundlagen der Arithmetik (Foundations) [1884]) by Robin Jeshion - Frege's Notion of Self-Evidence 2 | |
| A reaction: Put like that, it doesn't sound very persuasive. If any truth is totally general, then it must be purely logical? |
| 5658 | Numbers are definable in terms of mapping items which fall under concepts [Frege, by Scruton] |
| Full Idea: Frege defines numbers in terms of 'equinumerosity', which says two concepts are equinumerous if the items falling under one of them can be placed in one-to-one correspondence with the items falling under the other. | |
| From: report of Gottlob Frege (Grundlagen der Arithmetik (Foundations) [1884]) by Roger Scruton - Short History of Modern Philosophy Ch.17 | |
| A reaction: This doesn't sound quite enough. What is the difference between three and four? The extensions of items generate separate sets, but why does one follow the other, and how do you count the items to get the one-to-one correspondence? |
| 8655 | Arithmetic is analytic and a priori, and thus it is part of logic [Frege] |
| Full Idea: It is probable that the laws of arithmetic are analytic and consequently a priori; arithmetic thus becomes simply a development of logic, and every proposition of arithmetic a law of logic, albeit a derivative one. | |
| From: Gottlob Frege (Grundlagen der Arithmetik (Foundations) [1884], §87) | |
| A reaction: I'm not sure about 'thus', without more explication. Empiricists loved this, because it placed arithmetic firmly among Hume's 'relations of ideas', thus avoiding the difficulties Mill encountered trying to explain arithmetic through piles of pebbles. |
| 10831 | Frege only managed to prove that arithmetic was analytic with a logic that included set-theory [Quine on Frege] |
| Full Idea: Frege claimed to have proved that the truths of arithmetic are analytic, but the logic capable of encompassing this reduction was logic inclusive of set theory. | |
| From: comment on Gottlob Frege (Grundlagen der Arithmetik (Foundations) [1884]) by Willard Quine - Philosophy of Logic Ch.5 |
| 13864 | Frege's platonism and logicism are in conflict, if logic must dictates an infinity of objects [Wright,C on Frege] |
| Full Idea: Frege's platonism seems to be in some tension with logicism: for the thought is unprepossessing that logic should dictate the existence of infinitely many objects of some kind. | |
| From: comment on Gottlob Frege (Grundlagen der Arithmetik (Foundations) [1884]) by Crispin Wright - Frege's Concept of Numbers as Objects Intro | |
| A reaction: Obviously Frege didn't think this, but then the crux seems to be that Frege believed that there was a multitude of logical truths awaiting discovery, while modern logic just seems to be about the logical consequences of things. |
| 10033 | Why should the existence of pure logic entail the existence of objects? [George/Velleman on Frege] |
| Full Idea: If a distinguishing features of logic is its complete generality, focusing on truth in general, why should the existence of logic entail the existence of infinitely many objects? ..How can it be completely general if it has ontological commitments? | |
| From: comment on Gottlob Frege (Grundlagen der Arithmetik (Foundations) [1884]) by A.George / D.J.Velleman - Philosophies of Mathematics Ch.2 | |
| A reaction: This strikes me as simple and devastating. It depends how you conceive logic, but I only conceive it as the formalised rules of successful reasoning. I can't comprehend the claim that without certain objects, reasoning would be impossible. |
| 10010 | Frege's belief in logicism and in numerical objects seem uncomfortable together [Hodes on Frege] |
| Full Idea: Frege's views on arithmetic centred on two central theses, that mathematics is really logic, and that it is about distinctively mathematical sorts of objects, such as cardinal numbers. These theses seem uncomfortable passengers in a single boat. | |
| From: comment on Gottlob Frege (Grundlagen der Arithmetik (Foundations) [1884]) by Harold Hodes - Logicism and Ontological Commits. of Arithmetic | |
| A reaction: This question pinpoints precisely my unease about Frege. I take logic to be the rules for successful reasoning, so I don't see how they can have ontological implications. It is very extreme platonism to say that right reasoning requires logical objects. |
| 9631 | Formalism fails to recognise types of symbols, and also meta-games [Frege, by Brown,JR] |
| Full Idea: Early formalism (Thomae etc) was crushed by Frege: first, mathematics must be about classes of symbols (abstract types), not the symbols themselves (the tokens); second, games may be meaningless, but meta-games are not. | |
| From: report of Gottlob Frege (Grundlagen der Arithmetik (Foundations) [1884]) by James Robert Brown - Philosophy of Mathematics Ch.5 | |
| A reaction: Brown goes on to show how Hilbert revived the formalist project. A really austere formalist view of mathematics clearly seems to be missing something basic, either in physical nature, or in the world of ideas. |
| 9875 | Frege was completing Bolzano's work, of expelling intuition from number theory and analysis [Frege, by Dummett] |
| Full Idea: Frege was completing Bolzano's work, of expelling intuition from number theory and analysis (while leaving it its due place in geometry). | |
| From: report of Gottlob Frege (Grundlagen der Arithmetik (Foundations) [1884]) by Michael Dummett - Frege philosophy of mathematics Ch.18 | |
| A reaction: It was Kant who had placed the emphasis on intuition. Frege eventually thought arithmetic might be geometric, and so intuition had to triumph after all. |
| 8642 | Abstraction from things produces concepts, and numbers are in the concepts [Frege] |
| Full Idea: What we actually get by means of abstraction from things is the concept, and in this we then discover the number. | |
| From: Gottlob Frege (Grundlagen der Arithmetik (Foundations) [1884], §47) | |
| A reaction: And how do we 'discover' it, if not by a process of further abstraction? The concept of the moon (see Idea 8641) no more contains the number one than the actual moon does |
| 8621 | Mental states are irrelevant to mathematics, because they are vague and fluctuating [Frege] |
| Full Idea: Sensations and mental pictures, formed from the amalgamated traces of earlier sense-impressions, are absolutely no concern of arithmetic; they are characteristically fluctuating and indefinite, in contrast to the concepts and objects of mathematics. | |
| From: Gottlob Frege (Grundlagen der Arithmetik (Foundations) [1884], Intro) | |
| A reaction: Sounds very like Plato's distinction between the worlds of opinion and knowledge (Ideas 1170 and 2133). This view is fine amidst the implicit dualism of all nineteenth century thought, but how does abstract mathematics link to the soggy brain? |