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| 13876 | The syntactic category is primary, and the ontological category is derivative |
| Full Idea: For Frege it is the syntactic category which is primary, the ontological one derivative. | |||
| From: report of Gottlob Frege (Grundlagen der Arithmetik (Foundations) [1884]) by Crispin Wright - Frege's Concept of Numbers as Objects 1.iii | |||
| A reaction: I take the recent revival of metaphysics to be a rebellion against precisely this thought. Ontology disappeared for a hundred years into a hopeless miasma of linguistic complexity. Language is cludge, but the world isn't. |
| 8415 | Never lose sight of the distinction between concept and object |
| Full Idea: Never lose sight of the distinction between concept and object. | |||
| From: Gottlob Frege (Grundlagen der Arithmetik (Foundations) [1884], Intro p.x) | |||
| A reaction: Along with 8414 and 7732, we have the three axioms of modern analytical philosophy. Russell uses this distinction from Frege to attack Berkeley's idealism (see Idea 1103). The idea is strong in causal theories of reference. We realists love it. |
| 9841 | Frege was the first to give linguistic answers to non-linguistic questions |
| Full Idea: Frege was the first philosopher to ask a non-linguistic question, and return a linguistic answer. | |||
| From: report of Gottlob Frege (Grundlagen der Arithmetik (Foundations) [1884]) by Michael Dummett - Frege philosophy of mathematics Ch.10 | |||
| A reaction: This is both heroic and infuriating. It is like erecting a road block in front of a beautiful valley. You say 'Is there a God?' and I reply 'Let us consider the semantics of that sentence'. |
| 9840 | Frege initiated linguistic philosophy, studying number through the sense of sentences |
| Full Idea: §62 of Frege's 'Grundlagen' is arguably the most pregnant philosophical paragraph ever written; ..it is the very first example of what has become known as the 'linguistic turn' in philosophy. His enquiry into numbers focuses on the sense of sentences. | |||
| From: report of Gottlob Frege (Grundlagen der Arithmetik (Foundations) [1884], §62) by Michael Dummett - Frege philosophy of mathematics | |||
| A reaction: Dummett is a great fan of this, possibly the last great fan. It is undeniable that Frege has found one way to get at the problem, but I doubt if it is the only way. |
| 15948 | Frege developed formal systems to avoid unnoticed assumptions |
| Full Idea: Frege developed a formal system to make sure that he hadn't employed unnoticed assumptions about arithmetic. | |||
| From: report of Gottlob Frege (Grundlagen der Arithmetik (Foundations) [1884]) by Shaughan Lavine - Understanding the Infinite VIII.2 | |||
| A reaction: It is interesting that Frege seems to have had far more influence on analytic philosophy than he ever had on mathematics. |
| 10804 | Thoughts have a natural order, to which human thinking is drawn |
| Full Idea: Burge has argued that Frege's rationalism runs very deep. Frege holds that there is a natural order of thoughts to which human thinking is naturally drawn. | |||
| From: report of Gottlob Frege (Grundlagen der Arithmetik (Foundations) [1884]) by Stephen Yablo - Carving Content at the Joints § 8 | |||
| A reaction: [Yablo cites Burge 1984,1992,1998] What an intriguing idea. I always start from empiricist beginnings, but some aspects of rationalism just sieze you by the throat. |
| 9832 | Frege sees no 'intersubjective' category, between objective and subjective |
| Full Idea: Frege left no place for a category of the intersubjective, intermediate between the wholly objective and the radically subjective. | |||
| From: comment on Gottlob Frege (Grundlagen der Arithmetik (Foundations) [1884]) by Michael Dummett - Frege philosophy of mathematics Ch.7 | |||
| A reaction: Interesting. More sophisticated accounts of language (with the Private Language Argument as background) hold out possibilities of objectivity arising from an articulate community. See Idea 95. |
| 8414 | Keep the psychological and subjective separate from the logical and objective |
| Full Idea: Always separate sharply the psychological from the logical, the subjective from the objective. | |||
| From: Gottlob Frege (Grundlagen der Arithmetik (Foundations) [1884], Intro p.x) | |||
| A reaction: This (with Ideas 7732 and 8415) is said to be the foundation of modern analytical philosophy. It contrasts with Husserl's 'Logical Investigations', which are the foundations of phenomenology. I think it is time someone challenged Frege here. |
| 9844 | Originally Frege liked contextual definitions, but later preferred them fully explicit |
| Full Idea: In his middle period, Frege became hostile to contextual definitions, and any definition other than an explicit one, ..but at the time of the 'Grundlagen' he conceived of his context principle as licensing contextual definitions. | |||
| From: report of Gottlob Frege (Grundlagen der Arithmetik (Foundations) [1884]) by Michael Dummett - Frege philosophy of mathematics Ch.11 | |||
| A reaction: His context principle says words only have a meaning in a context. Intuitively, I would say that there is no correct answer to how something should be defined. Totally circularity is hopeless, but presuppositions just weaken a definition. |
| 9822 | Nothing should be defined in terms of that to which it is conceptually prior |
| Full Idea: Frege appeals to a general principle that nothing should be defined in terms of that to which it is conceptually prior. | |||
| From: report of Gottlob Frege (Grundlagen der Arithmetik (Foundations) [1884], §64) by Michael Dummett - Frege philosophy of mathematics Ch.3 | |||
| A reaction: The point is that the terms of the definition would depend on the thing being defined. But of all the elusive concepts, that of 'conceptual priority' is one of the slipperiest. An example is the question of precedence between 'parallel' and 'direction'. |
| 17495 | Proof aims to remove doubts, but also to show the interdependence of truths |
| Full Idea: Proof has as its goal not only to raise the truth of a proposition above all doubts, but additionally to provide insight into the interdependence of truths. | |||
| From: Gottlob Frege (Grundlagen der Arithmetik (Foundations) [1884], §02) | |||
| A reaction: This is a major idea in Frege's thinking, and a reason why he is the father of modern metaphysics as well as the father of modern logic. You study the framework of truths by studying the logic that connects them. |
| 8632 | You can't transfer external properties unchanged to apply to ideas |
| Full Idea: It would be remarkable if a property abstracted from external things could be transferred without any change of sense to events, to ideas and to concepts, like speaking of 'blue ideas' or 'salty concepts'. | |||
| From: Gottlob Frege (Grundlagen der Arithmetik (Foundations) [1884], §24) | |||
| A reaction: Since those phrases make perfectly good metaphorical sense, I presume the Frege was a fairly literal sort of chap. Is this the earliest emergence of the idea of a category mistake? |
| 13881 | We need to grasp not number-objects, but the states of affairs which make number statements true |
| Full Idea: For Frege (as opposed to Gödel) the epistemological aim is not to relate to the objects which are the subject-matter of number theory, but to relate to the states of affairs that make for the truth of number-theoretic statements. | |||
| From: report of Gottlob Frege (Grundlagen der Arithmetik (Foundations) [1884]) by Crispin Wright - Frege's Concept of Numbers as Objects 1.v | |||
| A reaction: I am beginning to see that this is a key issue in modern philosophy, of whether we build our metaphysics on the things of the world or on the truths about the world. I vote for the things, because the other way slides into anti-realism. |
| 9154 | Frege agreed with Euclid that the axioms of logic and mathematics are known through self-evidence |
| Full Idea: Frege maintained a sophisticated version of the Euclidean position that knowledge of the axioms and theorems of logic, geometry, and arithmetic rests on the self-evidence of the axioms, definitions, and rules of inference. | |||
| From: report of Gottlob Frege (Grundlagen der Arithmetik (Foundations) [1884]) by Tyler Burge - Frege on Apriority Intro | |||
| A reaction: I am inclined to agree that they are indeed self-evident, but not in a purely a priori way. They are self-evident general facts about how reality is and how (it seems) that it must be. It seems to me closer to a perception than an insight. |
| 9157 | The null set is only defensible if it is the extension of an empty concept |
| Full Idea: Frege regarded the null set as an indefensible entity from the point of view of iterative set theory. It collects nothing. He thought a null entity (a null extension) is derivable only as the extension of an empty concept. | |||
| From: report of Gottlob Frege (Grundlagen der Arithmetik (Foundations) [1884]) by Tyler Burge - Frege on Apriority II | |||
| A reaction: Frege is right, if you like sets. Othewise all the other sets are going to be defined simply by their extension, and the empty set has to be defined in a different way, which looks like appalling theory. Empty concepts bother me though! |
| 9835 | It is because a concept can be empty that there is such a thing as the empty class |
| Full Idea: Since he thought of classes as extensions of concepts, ...it is because a concept can be empty that there is such a thing as the empty class. | |||
| From: report of Gottlob Frege (Grundlagen der Arithmetik (Foundations) [1884]) by Michael Dummett - Frege philosophy of mathematics Ch.8 | |||
| A reaction: Frege was already up against the awaiting Russell Paradox, but this view also seems to imply that there are many empty classes, since the absences of sandwiches would be different from the absence of heroism. |
| 9854 | We can introduce new objects, as equivalence classes of objects already known |
| Full Idea: We can introduce a new type of object from the obtaining of some equivalence relation between objects of some already known kind, by identifying the new objects as equivalence classes of the old ones under that equivalence relation. | |||
| From: report of Gottlob Frege (Grundlagen der Arithmetik (Foundations) [1884]) by Michael Dummett - Frege philosophy of mathematics Ch.14 | |||
| A reaction: Some accounts of abstraction merely describe the concept, but this is a rival to the traditional pyschological abstractionism that Frege attacked so vigorously. Should we take a platonist or constructivist view of the new objects? |
| 9883 | Frege introduced the standard device, of defining logical objects with equivalence classes |
| Full Idea: Frege decided that all logical objects, or at least all those needed for mathematics, could be defined by logical abstraction, except the classes needed for such definitions. ..This definition by equivalence classes has been adopted as a standard device. | |||
| From: report of Gottlob Frege (Grundlagen der Arithmetik (Foundations) [1884], §64-68) by Michael Dummett - Frege philosophy of mathematics | |||
| A reaction: This means if we are to understand modern abstraction (instead of the psychological method of ignoring selected properties of objects), we must understand the presuppositions needed for a definition by equivalence. |
| 18104 | Frege, unlike Russell, has infinite individuals because numbers are individuals |
| Full Idea: Frege was able to prove that there are infinitely many individuals by taking the numbers themselves to be individuals, but this course was not open to Russell. | |||
| From: report of Gottlob Frege (Grundlagen der Arithmetik (Foundations) [1884]) by David Bostock - Philosophy of Mathematics 5.2 |
| 9834 | A class is, for Frege, the extension of a concept |
| Full Idea: A class is, for Frege, the extension of a concept. | |||
| From: report of Gottlob Frege (Grundlagen der Arithmetik (Foundations) [1884]) by Michael Dummett - Frege philosophy of mathematics Ch.8 | |||
| A reaction: This simple idea was the source of all his troubles, because there are concepts which can't have an extension, because of contradiction. ...And yet all intuition says Frege is right.. |
| 8645 | Convert "Jupiter has four moons" into "the number of Jupiter's moons is four" |
| Full Idea: The proposition "Jupiter has four moons" can be converted into "the number of Jupiter's moons is four". | |||
| From: Gottlob Frege (Grundlagen der Arithmetik (Foundations) [1884], §57) | |||
| A reaction: This seems to be the beginning of the modern exploration of the whole idea of logical form. It is one thing to find a logical forms which suits your current thesis (here, that numbers are not adjectival), but another to prove that it is the right form. |
| 16891 | Despite Gödel, Frege's epistemic ordering of all the truths is still plausible |
| Full Idea: Gödel undermined Frege's assumption that all but the basic truths are provable in a system, but insofar as one conceives of proof informally as an epistemic ordering among truths, one can see his vision as worth developing. | |||
| From: report of Gottlob Frege (Grundlagen der Arithmetik (Foundations) [1884]) by Tyler Burge - Frege on Apriority (with ps) 1 | |||
| A reaction: [compressed] This 'epistemic ordering' fits my thesis of seeing the world through our explanations of it. |
| 16906 | The primitive simples of arithmetic are the essence, determining the subject, and its boundaries |
| Full Idea: The primitive truths contain the core of arithmetic because their constituents are simples which define the essential boundaries of the subject. …The primitive truths are the most general ones, containing the basic, essence determining elements. | |||
| From: report of Gottlob Frege (Grundlagen der Arithmetik (Foundations) [1884]) by Robin Jeshion - Frege's Notion of Self-Evidence 2 | |||
| A reaction: This presents Frege as explicable in essentialist terms, as identifying the core of an abstract discipline, from which the rest of it is generated. Jeshion says 'simples are the essence'. |
| 14236 | Each horse doesn't fall under the concept 'horse that draws the carriage', because all four are needed |
| Full Idea: Frege says the number four is assigned to the concept 'horse that draws the Kaiser's carriage', but the four horses that drew the carriage did so together, not separately. No horses, not four, fall under the Fregean concept. | |||
| From: comment on Gottlob Frege (Grundlagen der Arithmetik (Foundations) [1884], §46) by Oliver,A/Smiley,T - What are Sets and What are they For? Intro | |||
| A reaction: They say that Frege stumbles because he is blind to irreducibly plural predicates. |
| 22294 | We can show that a concept is consistent by producing something which falls under it |
| Full Idea: We can only establish that a concept is free from contradiction by first producing something that falls under it. | |||
| From: Gottlob Frege (Grundlagen der Arithmetik (Foundations) [1884], §095), quoted by Michael Potter - The Rise of Analytic Philosophy 1879-1930 19 'Exist' | |||
| A reaction: Potter quotes this as an example of proof by modelling. If it has one model then it must be consistent. Then we ask whether all the models are or are not consistent with one another. Circular squares fail the test. |
| 17624 | To understand axioms you must grasp their logical power and priority |
| Full Idea: Understanding the axioms depends not only on understanding Frege's elucidatory remarks about the interpretation of his symbols, but also on understanding their logical structure - their power to entail other truths, and their reason-giving priority. | |||
| From: report of Gottlob Frege (Grundlagen der Arithmetik (Foundations) [1884], 4) by Tyler Burge - Frege on Knowing the Foundations 4 | |||
| A reaction: This is a distinctively Burgean spin put on what Frege has to say about axioms, but I like it, and it seems well enough supported in Frege's writings (e.g. 1914). |
| 8640 | We cannot define numbers from the idea of a series, because numbers must precede that |
| Full Idea: We cannot define number by the generalized concept of a series. Positions in the series cannot be the basis on which we distinguish the objects, since they must already have been distinguished somehow or other, for us to arrange them in a series. | |||
| From: Gottlob Frege (Grundlagen der Arithmetik (Foundations) [1884], §42) | |||
| A reaction: You can arrange things in a line without the use of numbers. You need prior mastery of counting, though, to say where an item comes in the line. And yet... why shouldn't you define counting by the use of some original primitive line? Numbers map onto it. |
| 9838 | Treating 0 as a number avoids antinomies involving treating 'nobody' as a person |
| Full Idea: Frege's point was that by treating 0 as a number, we run into none of the antinomies that result from treating 'never' as the name of a time, or 'nobody' as the name of a person. | |||
| From: report of Gottlob Frege (Grundlagen der Arithmetik (Foundations) [1884]) by Michael Dummett - Frege philosophy of mathematics Ch.8 | |||
| A reaction: I don't think that is a good enough reason. Daft problems like that are solved by settling the underlying proposition or logical form (of a sentence containing 'nobody') before one begins to reason. Other antinomies arise with zero. |
| 9564 | For Frege 'concept' and 'extension' are primitive, but 'zero' and 'successor' are defined |
| Full Idea: In Frege's system 'concept' and 'extension of a concept' are primitive notions; whereas 'zero' and 'successor' are defined. | |||
| From: report of Gottlob Frege (Grundlagen der Arithmetik (Foundations) [1884]) by Charles Chihara - A Structural Account of Mathematics 7.5 | |||
| A reaction: This is in contrast to the earlier Peano Postulates for arithmetic, which treat 'zero' and 'successor' as primitive. Interesting, given that Frege is famous for being a platonist. |
| 10551 | If objects exist because they fall under a concept, 0 is the object under which no objects fall |
| Full Idea: On Frege's approach (of accepting abstract objects if they fall under a concept) the existence of the number 0, from which the series of numbers starts, is of course guaranteed by the citation of a concept under which nothing falls. | |||
| From: report of Gottlob Frege (Grundlagen der Arithmetik (Foundations) [1884]) by Michael Dummett - Frege Philosophy of Language (2nd ed) Ch.14 | |||
| A reaction: Frege cites the set of all non-self-identical objects, but he could have cited the set of circular squares. Given his Russell Paradox problems, this whole claim is thrown in doubt. Actually doesn't Frege's view make 0 impossible? Am I missing something? |
| 8653 | Nought is the number belonging to the concept 'not identical with itself' |
| Full Idea: I define nought as the Number which belongs to the concept 'not identical with itself'. ...I choose this definition as it can be proved on purely logical grounds. | |||
| From: Gottlob Frege (Grundlagen der Arithmetik (Foundations) [1884], §74) | |||
| A reaction: An important part of Frege's logicist programme, along with his use of Hume's Principle (Idea 8649). He needed a prior definition of 'Number' (in §68). Clever, but intuitively a rather weird idea of zero. It is more of an example than a definition. |
| 8636 | We can say 'a and b are F' if F is 'wise', but not if it is 'one' |
| Full Idea: We combine 'Solon was wise' and 'Thales was wise' into 'Solon and Thales were wise', but we can't say 'Solon and Thales were one', which implies that 'one' is not a property in the same way 'wise' is. | |||
| From: Gottlob Frege (Grundlagen der Arithmetik (Foundations) [1884], §29) | |||
| A reaction: Maybe 'one' is still a property, but of a different sort. However, Frege builds up a very persuasive case that just because numbers function as adjectives it does not follow that they are properties. See Idea 8637. |
| 8654 | One is the Number which belongs to the concept "identical with 0" |
| Full Idea: One is the Number which belongs to the concept "identical with 0". | |||
| From: Gottlob Frege (Grundlagen der Arithmetik (Foundations) [1884], §77) | |||
| A reaction: This follows from Idea 8653, which defined zero. Zero is the number of a non-existent set, and one is how many sets you have when you have only got zero. Very clever. |
| 8641 | You can abstract concepts from the moon, but the number one is not among them |
| Full Idea: What are we supposed to abstract from to get from the moon to the number 1? We do get certain concepts, such as satellite, but 1 is not to be met with. In the case of 0 we have no objects at all. ..The essence of number must work for 0 and 1. | |||
| From: Gottlob Frege (Grundlagen der Arithmetik (Foundations) [1884], §44) | |||
| A reaction: Note that Frege seems to be conceding psychological abstraction for most other concepts. But why can't you abstract from your abstractions, to reach high-level abstractions? And why should numbers not emerge at those higher levels? |
| 9989 | Units can be equal without being identical |
| Full Idea: The fact that units are equal does not mean that they are identical. The units can be equal just in the sense that once can be substituted for any other without altering the name assigned, i.e. the number. | |||
| From: comment on Gottlob Frege (Grundlagen der Arithmetik (Foundations) [1884], §54) by William W. Tait - Frege versus Cantor and Dedekind XI | |||
| A reaction: [this is in reference to Thomae 1880] Presumably this might mean that units have type-identity, rather than token-dentity. 'This' unit might be a token, but 'a' unit would be a type. I am extremely reluctant to ditch the old concept of a unit. |
| 17429 | Frege says only concepts which isolate and avoid arbitrary division can give units |
| Full Idea: It is Frege's view that only concepts which satisfy isolation and non-arbitrary division can play the role of dividing up what falls under them into countable units. | |||
| From: report of Gottlob Frege (Grundlagen der Arithmetik (Foundations) [1884], §54) by Kathrin Koslicki - Isolation and Non-arbitrary Division 2.1 | |||
| A reaction: Compare Idea 17429. If I count out a 'team of players', I need this unit concept to get what a 'player' is, but then need the 'team' concept to do the counting. Number doesn't attach to the unit concept. |
| 17438 | Our concepts decide what is countable, as in seeing the leaves of the tree, or the foliage |
| Full Idea: For Frege, the distinction between what we count and what we do not count is drawn by our concepts. ...We can describe the very same external phenomena either as the leaves of a tree or its foliage. | |||
| From: report of Gottlob Frege (Grundlagen der Arithmetik (Foundations) [1884]) by Kathrin Koslicki - Isolation and Non-arbitrary Division 3 | |||
| A reaction: Hm. We can't obey 'count the foliage', but we all know that foliage is countable stuff, where water isn't. Nature has a say here - it isn't just a matter of our concepts. |
| 17426 | A concept creating a unit must isolate and unify what falls under it |
| Full Idea: Only a concept which isolates what falls under it in a definite manner, and which does not permit any arbitrary division of it into parts, can be a unit relative to finite Number. | |||
| From: Gottlob Frege (Grundlagen der Arithmetik (Foundations) [1884], §54), quoted by Kathrin Koslicki - Isolation and Non-arbitrary Division 1 | |||
| A reaction: This is the key modern proposal for the basis of counting, by trying to get at the sort of concept which will turn something into a 'unit'. The concept must isolate and unify. Why should just one concept do that each time? |
| 17428 | Frege says counting is determining what number belongs to a given concept |
| Full Idea: Roughly, Frege's picture of counting is this. When we count something, we determine what number belongs to a given concept. | |||
| From: report of Gottlob Frege (Grundlagen der Arithmetik (Foundations) [1884], §54) by Kathrin Koslicki - Isolation and Non-arbitrary Division 2.1 | |||
| A reaction: If the concept were 'herd of sheep' that would need a context before there could be a fixed number. You can count until you get bored, like counting stars to get to sleep. 'Count off 20 sheep' has the number before the counting starts. |
| 17427 | Frege's 'isolation' could be absence of overlap, or drawing conceptual boundaries |
| Full Idea: Frege's proposal can be isolation as discreteness, i.e. absence of overlap, between the objects counted; and isolation as drawing of conceptual boundaries. | |||
| From: report of Gottlob Frege (Grundlagen der Arithmetik (Foundations) [1884]) by Kathrin Koslicki - Isolation and Non-arbitrary Division 1 |
| 17437 | Non-arbitrary division means that what falls under the concept cannot be divided into more of the same |
| Full Idea: Non-arbitrary division concerns the internal structure of the things falling under a concept. Its point is to ensure that we cannot go on dividing these units arbitrarily and still expect to find more things of the same kind. | |||
| From: report of Gottlob Frege (Grundlagen der Arithmetik (Foundations) [1884]) by Kathrin Koslicki - Isolation and Non-arbitrary Division 2.3 | |||
| A reaction: Counting something red is given as an example. This seems to define mass-terms, or stuff. |
| 15916 | Frege's one-to-one correspondence replaces well-ordering, because infinities can't be counted |
| Full Idea: Frege assumed that since infinite collections cannot be counted, he needed a theory of number that is independent of counting. He therefore took one-to-one correspondence to be basic, not well-orderings. Hence cardinals are basic, not ordinals. | |||
| From: report of Gottlob Frege (Grundlagen der Arithmetik (Foundations) [1884]) by Shaughan Lavine - Understanding the Infinite III.4 |
| 10034 | The number of natural numbers is not a natural number |
| Full Idea: Frege shows that the number of natural numbers is not identical to any natural number. This is because, while no natural number is identical to its successor, the number of natural numbers is. | |||
| From: report of Gottlob Frege (Grundlagen der Arithmetik (Foundations) [1884]) by A.George / D.J.Velleman - Philosophies of Mathematics Ch.2 | |||
| A reaction: Frege is notorious for the lack of respect shown in his writings for the great Cantor, and this seems to have blocked him from a more sophisticated account of infinity, but this idea seems a nice one. |
| 16883 | Arithmetical statements can't be axioms, because they are provable |
| 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. |
| 10625 | Frege had a motive to treat numbers as objects, but not a justification |
| 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 |
| 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 |
| 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 |
| 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. |
| 17636 | A cardinal number may be defined as a class of similar classes |
| 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 |
| 9953 | Numbers are more than just 'second-level concepts', since existence is also one |
| 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 |
| 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 |
| 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 |
| 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? |
| 10029 | Numbers need to be objects, to define the extension of the concept of each successor to n |
| 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' |
| 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 |
| 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 |
| 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 |
| 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 |
| 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 |
| 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 |
| 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 |
| 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 |
| 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 |
| 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 |
| 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. |
| 17460 | A statement of number contains a predication about a concept |
| 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. |
| 17820 | If you can subdivide objects many ways for counting, you can do that to set-elements too |
| 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. |
| 16890 | Frege's problem is explaining the particularity of numbers by general laws |
| 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 |
| 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' |
| 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 |
| 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) |
| 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 |
| 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? |
| 9956 | 'The number of Fs' is the extension (a collection of first-level concepts) of the concept 'equinumerous with F' |
| 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 |
| 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 |
| 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 |
| 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 |
| 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 |
| 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? |
| 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) |
| 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. |
| 9046 | Our definition will not tell us whether or not Julius Caesar is a number |
| 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? |
| 13889 | Fregean numbers are numbers, and not 'Caesar', because they correlate 1-1 |
| 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 |
| 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 |
| 16896 | If numbers can be derived from logic, then set theory is superfluous |
| 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 |
| 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. |
| 13874 | Numbers seem to be objects because they exactly fit the inference patterns for identities |
| 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 |
| 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)? |
| 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 |
| 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 |
| 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 |
| 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 |
| 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 |
| 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 |
| 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 |
| 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' |
| 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 |
| 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 |
| 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')? |
| 9945 | Logicism shows that no empirical truths are needed to justify arithmetic |
| 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'. |
| 7739 | Arithmetic is analytic |
| 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. |
| 8782 | Frege offered a Platonist version of logicism, committed to cardinal and real numbers |
| 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) |
| 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 |
| 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 |
| 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 |
| 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 |
| 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 |
| 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? |
| 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 |
| 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 |
| 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 |
| 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 |
| 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 |
| 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? |
| 8643 | Affirmation of existence is just denial of zero |
| Full Idea: Affirmation of existence is nothing but denial of the number nought. | |||
| From: Gottlob Frege (Grundlagen der Arithmetik (Foundations) [1884], §53) | |||
| A reaction: Mathematicians - don't you luv 'em. No doubt this is helpful in placing existence within the great network of logical inferences, but his 'nothing but' is laughable. I don't see much existential anguish in the denial of zero. |
| 8911 | If abstracta are non-mental, quarks are abstracta, and yet chess and God's thoughts are mental |
| Full Idea: Frege's identification of the abstract with the realm of non-mental things entails that unobservables such as quarks are abstract. The abstract nature of chess, and the possibility of abstracta in the mind of God, show they can be mind-dependent. | |||
| From: comment on Gottlob Frege (Grundlagen der Arithmetik (Foundations) [1884]) by Gideon Rosen - Abstract Objects 'Way of Neg' | |||
| A reaction: I like the robust question 'if a is said to 'exist', what is it said to be made of?' I consider the views of Frege to have had too much influence in this area, and recognising the role of the mind (psychology!) in abstraction is a start. |
| 8634 | The equator is imaginary, but not fictitious; thought is needed to recognise it |
| Full Idea: We speak of the equator as an imaginary line, but it is not a fictitious line; it is not a creature of thought, the product of a psychological process, but is only recognised or apprehended by thought. | |||
| From: Gottlob Frege (Grundlagen der Arithmetik (Foundations) [1884], §26) | |||
| A reaction: Nice point. The same goes for the apparently very abstract and theoretical concept of a 'circle', because a perfect circle could be imagined in a very specific location, perhaps passing through three specified points. |
| 17443 | Many of us find Frege's claim that truths depend on one another an obscure idea |
| Full Idea: Frege sometimes speaks of 'the dependence of truths upon one another' (1884:§2), but I find such ideas obscure, and suspect I'm not the only one who does. | |||
| From: comment on Gottlob Frege (Grundlagen der Arithmetik (Foundations) [1884], §02) by Richard G. Heck - Cardinality, Counting and Equinumerosity 1 | |||
| A reaction: He refers to Burge 'struggling mightily' with this aspect of Frege's thought. I intend to defend Frege. See his 1914 lectures. I thought this dependence was basic to the whole modern project of doing metaphysics through logic? |
| 17445 | Parallelism is intuitive, so it is more fundamental than sameness of direction |
| Full Idea: Frege says that parallelism is more fundamental than sameness of direction because all geometrical notions must originally be given in intuition. | |||
| From: report of Gottlob Frege (Grundlagen der Arithmetik (Foundations) [1884], §64) by Richard G. Heck - Cardinality, Counting and Equinumerosity 3 | |||
| A reaction: If Frege thinks some truths are more fundamental, this gives an indication of his reasons. But intuition is not a very strong base. |
| 10539 | Frege refers to 'concrete' objects, but they are no different in principle from abstract ones |
| Full Idea: Frege employs the notion of 'concrete' (wirklich, literally 'actual') objects, in arguing that not every object is concrete, but it does not work; abstract objects are just as much objects as concrete ones. | |||
| From: report of Gottlob Frege (Grundlagen der Arithmetik (Foundations) [1884], §26,85) by Michael Dummett - Frege Philosophy of Language (2nd ed) Ch.14 | |||
| A reaction: See Idea 10516 for why Dummett is keen on the distinction. Frege strikes me as being wildly irresponsible about ontology. |
| 17431 | Vagueness is incomplete definition |
| Full Idea: Frege seems to assimilate vagueness to incompleteness of definition. | |||
| From: report of Gottlob Frege (Grundlagen der Arithmetik (Foundations) [1884]) by Kathrin Koslicki - Isolation and Non-arbitrary Division 2.1 |
| 13879 | For Frege, ontological questions are to be settled by reference to syntactic structures |
| Full Idea: For Frege, syntactic categories are prior to ontological ones, and it is by reference to the syntactic structure of true statements that ontological questions are to be understood and settled. | |||
| From: report of Gottlob Frege (Grundlagen der Arithmetik (Foundations) [1884]) by Crispin Wright - Frege's Concept of Numbers as Objects 1.v |
| 10642 | Second-order quantifiers are committed to concepts, as first-order commits to objects |
| Full Idea: Frege claims that second-order quantifiers are committed to concepts, just as singular first-order quantifiers are committed to objects. | |||
| From: report of Gottlob Frege (Grundlagen der Arithmetik (Foundations) [1884]) by Řystein Linnebo - Plural Quantification 5.3 | |||
| A reaction: It increasingly strikes me that Fregeans try to get away with this nonsense by diluting both the notion of a 'concept' and the notion of an 'object'. |
| 10032 | 'Ancestral' relations are derived by iterating back from a given relation |
| Full Idea: Any relation will yield a new relation, called the 'ancestral', which is the iterated relation which leads up to it, as when 'x is the parent of y' can lead us to the relation 'x is an ancestor of y' | |||
| From: report of Gottlob Frege (Grundlagen der Arithmetik (Foundations) [1884], §79) by A.George / D.J.Velleman - Philosophies of Mathematics Ch.2 | |||
| A reaction: This idea is one of Frege's notable discoveries. The ancestral seems to be a generalisation of a given relation. |
| 10606 | Frege treats properties as a kind of function, and maybe a property is its characteristic function |
| Full Idea: Frege urges us to regard properties as just a special kind of function, and in the case of numerical properties he comes close to identifying a property with its characteristic function. | |||
| From: report of Gottlob Frege (Grundlagen der Arithmetik (Foundations) [1884]) by Peter Smith - Intro to Gödel's Theorems 11.3 n 5 | |||
| A reaction: Every now and then really interesting bits of metaphysics pop out of Frege, though it usually needs commentators to show the implications. Does the 'characteristic' imply a teleological view? |
| 8647 | Not all objects are spatial; 4 can still be an object, despite lacking spatial co-ordinates |
| Full Idea: To give spatial co-ordinates for the number four makes no sense; but the only conclusion to be drawn from that is, that 4 is not a spatial object, not that it is not an object at all. Not every object has a place. | |||
| From: Gottlob Frege (Grundlagen der Arithmetik (Foundations) [1884], §61) | |||
| A reaction: This is the modern philosophical concept of an 'object', though I find such talk very peculiar. It sounds like extreme Platonism, though this is usually denied. This is how logicians seem to see the world. |
| 10309 | Frege says singular terms denote objects, numerals are singular terms, so numbers exist |
| Full Idea: Frege's argument for abstract objects is: 1) singular terms in true expressions must denote objects, 2) numerals function as singular terms, 3) there must exist numbers denoted by those expressions. | |||
| From: report of Gottlob Frege (Grundlagen der Arithmetik (Foundations) [1884]) by Bob Hale - Abstract Objects Ch.1 | |||
| A reaction: [compressed] Given that most of the singular term usages can be rephrased adjectively, this strikes me as a weak argument, though Hale pins his whole book on it. |
| 10550 | Frege establishes abstract objects independently from concrete ones, by falling under a concept |
| Full Idea: For Frege it is legitimate, in order to establish the existence of a certain number, to cite a concept under which only abstract objects fall, and in such a way guarantee the existence of the number quite independently of what concrete objects there are. | |||
| From: report of Gottlob Frege (Grundlagen der Arithmetik (Foundations) [1884]) by Michael Dummett - Frege Philosophy of Language (2nd ed) Ch.14 | |||
| A reaction: This approach of Frege's got into trouble with Russell's Paradox, which gave a concept under which nothing could fall. It strikes me as misguided even without that problem. I say abstracta are rooted in the concrete. |
| 8785 | For Frege, objects just are what singular terms refer to |
| Full Idea: In Frege's 'Grundlagen' objects, as distinct from entities of other types (properties, relations, or various functions), just are what (actual or possible) singular terms refer to. | |||
| From: report of Gottlob Frege (Grundlagen der Arithmetik (Foundations) [1884]) by B Hale / C Wright - Logicism in the 21st Century 2 | |||
| A reaction: This seems to be the key claim that results in twentieth century metaphysics being done through analysis of language. The culmination is, of course, a denial of metaphysics, and then an eventual realisation that Frege was wrong. |
| 10278 | Without concepts we would not have any objects |
| Full Idea: Frege is known for the idea that we do not have objects without concepts. Without concepts, there is nothing - no thing - to count. | |||
| From: report of Gottlob Frege (Grundlagen der Arithmetik (Foundations) [1884]) by Stewart Shapiro - Philosophy of Mathematics 8.4 | |||
| A reaction: A very helpful clarification. Thinking about the probable mental life of higher and lower animals, the proposal seems extremely plausible. Dogs have some concepts, slugs have none, so slugs do not exist in a world of objects. I like it. |
| 17432 | Frege's universe comes already divided into objects |
| Full Idea: Frege's universe is one that comes already divided into objects. | |||
| From: report of Gottlob Frege (Grundlagen der Arithmetik (Foundations) [1884]) by Kathrin Koslicki - Isolation and Non-arbitrary Division 2.1 | |||
| A reaction: Nice to have this spelled out. I get frustrated with metaphysics built on logic, with domains of objects, without worry about where all these objects came from. They're axiomatic, it seems. She cites Geach as having a universe of 'goo'. |
| 16022 | The idea of a criterion of identity was introduced by Frege |
| Full Idea: The notion of a criterion of identity was introduced into philosophical terminology in Frege's 'Grundlagen', and was strong emphasised in Wittgenstein's 'Philosophical Investigations'. | |||
| From: report of Gottlob Frege (Grundlagen der Arithmetik (Foundations) [1884]) by Harold Noonan - Identity §4 | |||
| A reaction: For Frege a thing can only have an intrinsic identity if it can participate in an equality relation. For abstract objects (such as directions or numbers) the relation is an equivalence. The general idea is that identical objects must relate in some way. |
| 11100 | Frege's algorithm of identity is the law of putting equals for equals |
| Full Idea: Frege's algorithm of identity is the law of putting equals for equals. | |||
| From: report of Gottlob Frege (Grundlagen der Arithmetik (Foundations) [1884]) by Willard Quine - Identity, Ostension, and Hypostasis 4 | |||
| A reaction: Quine, and most modern philosophers, seem to accept universal substitutivity as a sufficient condition for identity. But you then get the problem of coextensionality (renate/cordate), which can only be solved by introducing modality. |
| 12153 | Geach denies Frege's view, that 'being the same F' splits into being the same and being F |
| Full Idea: Frege's position is that 'being the same F as' splits up into a general relation and an assertion about the referent ('being the same' and 'being an F'). This is what Geach denies, when he says there is no such thing as being 'just the same'. | |||
| From: comment on Gottlob Frege (Grundlagen der Arithmetik (Foundations) [1884]) by John Perry - The Same F I | |||
| A reaction: It looks as if you can take your pick - whether two things are perfectly identical, or whether they are identical in some respect. Get an unambiguous proposition before you begin the discussion. Identify referents, not kinds of identity, says Perry. |
| 9853 | Identity between objects is not a consequence of identity, but part of what 'identity' means |
| Full Idea: Part of Frege's profound new idea of identity is that the criteria for identity of objects of a given kind is not a consequence of the way that kind of object is characterised, but has to be expressly stipulated as part of that characterisation. | |||
| From: report of Gottlob Frege (Grundlagen der Arithmetik (Foundations) [1884]) by Michael Dummett - Frege philosophy of mathematics Ch.13 | |||
| A reaction: This makes identity a relative concept, rather than an instrinsic concept. Does a unique object have an identity? Do properties have intrinsic identity conditions, making them usable to identify two objects. Deep waters. Has Frege muddied them? |
| 17623 | To understand a thought you must understand its logical structure |
| Full Idea: For Frege, coming to a full understanding of logical structure is necessary to full understanding of a thought. And understanding logical structure derives from seeing what structures are most fruitful in accounting for the patterns of inference. | |||
| From: report of Gottlob Frege (Grundlagen der Arithmetik (Foundations) [1884]) by Tyler Burge - Frege on Knowing the Foundations 4 | |||
| A reaction: To me, the notion of finding what is 'fruitful' implies finding the essence of the structure. |
| 9158 | For Frege a priori knowledge derives from general principles, so numbers can't be primitive |
| Full Idea: If one took the numbers as primitive, one would not be deriving their existence and character from general principles- thus controverting Frege's view of the nature of an a priori subject. | |||
| From: Gottlob Frege (Grundlagen der Arithmetik (Foundations) [1884]), quoted by Tyler Burge - Frege on Apriority II | |||
| A reaction: He seems to be in tune with Leibniz on this. His view begs the obvious question of where the general principles come from. I would have thought that relationships between concepts might be known a priori, without principles being involved. |
| 8657 | Mathematicians just accept self-evidence, whether it is logical or intuitive |
| Full Idea: The mathematician rests content if every transition to a fresh judgement is self-evidently correct, without enquiring into the nature of this self-evidence, whether it is logical or intuitive. | |||
| From: Gottlob Frege (Grundlagen der Arithmetik (Foundations) [1884], §90) | |||
| A reaction: Note the suggestion that there are two different sorts of self-evidence. But see Idea 1410. Frege presumably drifted into philosophy because he wasn't happy with this blissful ignorance. |
| 9352 | An a priori truth is one derived from general laws which do not require proof |
| Full Idea: If the proof of a truth can be derived exclusively from general laws, which themselves neither need nor admit of proof, then the truth is a priori. | |||
| From: Gottlob Frege (Grundlagen der Arithmetik (Foundations) [1884], §03) | |||
| A reaction: Presumably the unproved general laws from which the derivation comes are more securely a priori, as are the principles used to make the derivation. As Frege says, he is trying to spell out Kant's view; see Idea 9345. |
| 16889 | A truth is a priori if it can be proved entirely from general unproven laws |
| Full Idea: If it is possible to derive a proof purely from general laws, which themselves neither need nor admit of proof, then the truth is a priori. | |||
| From: Gottlob Frege (Grundlagen der Arithmetik (Foundations) [1884], §03), quoted by Tyler Burge - Frege on Apriority (with ps) 1 | |||
| A reaction: Burge brings out the contrast with Kant, for whom a priori truths are derived from particular facts, not general ones. |
| 2514 | Frege tried to explain synthetic a priori truths by expanding the concept of analyticity |
| Full Idea: Frege challenged synthetic a priori truths by expanding the concept of analyticity, undertaken in order to provide a semantic basis for his logicist explanation of mathematical truth as analytic truth. | |||
| From: report of Gottlob Frege (Grundlagen der Arithmetik (Foundations) [1884]) by Jerrold J. Katz - Realistic Rationalism Int.xx |
| 16900 | Intuitions cannot be communicated |
| Full Idea: Frege makes a notorious claim that what is intuitable is not communicable. | |||
| From: report of Gottlob Frege (Grundlagen der Arithmetik (Foundations) [1884], §26) by Tyler Burge - Frege on Apriority (with ps) 4 |
| 16903 | Justifications show the ordering of truths, and the foundation is what is self-evident |
| Full Idea: Frege thought that the relations of epistemic justification in a science mirrors the natural ordering of truths: in particular, what is self-evident is selbstverstandlich. | |||
| From: report of Gottlob Frege (Grundlagen der Arithmetik (Foundations) [1884], §02) by Robin Jeshion - Frege's Notion of Self-Evidence 1 | |||
| A reaction: I'm not sure that I can accept a 'natural ordering of truths'. Is there a natural ordering of the facts of the world? The most I can see is a direction to causation. Maybe inferences have a direction, but humans intrude on those. |
| 8624 | Induction is merely psychological, with a principle that it can actually establish laws |
| Full Idea: Induction depends on the general proposition that the inductive method can establish the truth of a law, or the probability for it. If we deny this, induction becomes nothing more than a psychological phenomenon. | |||
| From: Gottlob Frege (Grundlagen der Arithmetik (Foundations) [1884], §03 n) | |||
| A reaction: The problem is that we can't seem to 'establish' the requisite proposition, even for probability, since probability is in part subjective. I think induction needs the premiss that nature has underlying uniformity, which we then tease out by observation. |
| 8626 | In science one observation can create high probability, while a thousand might prove nothing |
| Full Idea: The procedure of the sciences, with its objective standards, will at times find a high probability established by a single confirmatory instance, while at others it will dismiss a thousand as almost worthless. | |||
| From: Gottlob Frege (Grundlagen der Arithmetik (Foundations) [1884], §10) | |||
| A reaction: This thought is presumably what pushes theorists away from traditional induction and towards Bayes's Theorem (Idea 2798). The remark is a great difficulty for anyone trying to defend traditional induction. |
| 8648 | Ideas are not spatial, and don't have distances between them |
| Full Idea: Spatial predicates are not applicable to ideas; an idea is neither to the right nor to the left of another idea; we cannot give the distances between ideas in millimetres. | |||
| From: Gottlob Frege (Grundlagen der Arithmetik (Foundations) [1884], §61) | |||
| A reaction: This Fregean thought should be music to the ears of Cartesians, though it does not seem intended as support for dualism. This is the logicians' view of reality, where true inferences are what matter, and brains and souls are irrelevant. |
| 8620 | Thought is the same everywhere, and the laws of thought do not vary |
| Full Idea: Thought is in essentials the same everywhere: it is not true that there are different kinds of laws of thought to suit the different kinds of objects thought about. | |||
| From: Gottlob Frege (Grundlagen der Arithmetik (Foundations) [1884], Intro) | |||
| A reaction: Different kinds of thinker might also be candidates for different laws of thought. I'm unsure of Frege's grounds for this claim; most continental philosophers would probably reject it. |
| 9870 | Early Frege takes the extensions of concepts for granted |
| Full Idea: In the 'Grundlagen' Frege takes the notion of the extension of a concept for granted as unproblematic. | |||
| From: report of Gottlob Frege (Grundlagen der Arithmetik (Foundations) [1884]) by Michael Dummett - Frege philosophy of mathematics Ch.16 | |||
| A reaction: This comfortable notion was undermined by Russell's discovery of a concept which couldn't have an extension. Maybe we could defeat the Russell problem (and return to Frege's common sense) by denying that sets are objects. |
| 7736 | A concept is a non-psychological one-place function asserting something of an object |
| Full Idea: A concept is a one-place function - something that can be asserted of an object - as found in 'Earth is a planet' and 'Venus is a planet'. This notion of concept does not belong to psychology at all. | |||
| From: report of Gottlob Frege (Grundlagen der Arithmetik (Foundations) [1884]) by Joan Weiner - Frege Ch.4 | |||
| A reaction: This doesn't seem to leave room for the concept of the object or substance of which the something is asserted. In 'x is a planet' we need a concept of what x is. But then Frege will reduce the reference to a set of descriptions (i.e. functions). |
| 17430 | Fregean concepts have precise boundaries and universal applicability |
| Full Idea: Both precise boundaries and universal applicability are built into the very notion of a Fregean concept from the outset, while isolation and non-arbitrary division are additional criteria imposed on concepts. | |||
| From: report of Gottlob Frege (Grundlagen der Arithmetik (Foundations) [1884]) by Kathrin Koslicki - Isolation and Non-arbitrary Division 2.1 | |||
| A reaction: The latter two criteria are for concepts which create counting units. |
| 8622 | Psychological accounts of concepts are subjective, and ultimately destroy truth |
| Full Idea: Defining concepts psychologically, in terms of the nature of the human mind, makes everything subjective, and if we follow it through to the end, does away with truth. | |||
| From: Gottlob Frege (Grundlagen der Arithmetik (Foundations) [1884], Intro) | |||
| A reaction: This is the reason for Frege's passionate opposition to psychological approaches to thought. The problem, though, is to give an account in which the fixity of truth connects to the fluctuations of mental life. How does it do that?? |
| 13878 | Concepts are, precisely, the references of predicates |
| Full Idea: For Frege concepts are, precisely, the Bedeutungen of predicates. | |||
| From: report of Gottlob Frege (Grundlagen der Arithmetik (Foundations) [1884]) by Crispin Wright - Frege's Concept of Numbers as Objects 1.iv | |||
| A reaction: On p.17 Wright challenges Frege's right to make that assumption. |
| 8651 | A concept is a possible predicate of a singular judgement |
| Full Idea: A concept is for me that which can be predicate of a singular judgement-content. | |||
| From: Gottlob Frege (Grundlagen der Arithmetik (Foundations) [1884], §66 n) | |||
| A reaction: This seems intuitively odd, given that a predicate could (in principle) be of almost infinite complexity, whereas I would be reluctant to call anything a 'concept' if it couldn't be grasped by a single action of a normal conscious mind. |
| 9846 | Defining 'direction' by parallelism doesn't tell you whether direction is a line |
| Full Idea: The stipulation that the direction of a line a is to be the same as that of a line b just in case a is parallel to b does not determine whether the direction of a line is itself a line or something quite different. | |||
| From: comment on Gottlob Frege (Grundlagen der Arithmetik (Foundations) [1884], §60) by Michael Dummett - Frege philosophy of mathematics Ch.11 | |||
| A reaction: Nice point. Maybe not being able to say exactly what something is is either a symptom of nonsense, and simply a symptom that we are dealing with an abstract concept. If abstractions don't exist, they don't need individuation criteria. |
| 9976 | Frege accepts abstraction to the concept of all sets equipollent to a given one |
| Full Idea: Frege's own conception of abstraction (although he disapproves of the term) is in agreement with the view that abstracting from the particular nature of the elements of M would yield the concept under which fall all sets equipollent to M. | |||
| From: comment on Gottlob Frege (Grundlagen der Arithmetik (Foundations) [1884]) by William W. Tait - Frege versus Cantor and Dedekind III | |||
| A reaction: Nice! This shows how difficult it is to slough off the concept of abstractionism and live with purely logical concepts of it. If we 'construct' a set, then there is a process of creation to be explained; we can't just think of platonic givens. |
| 10803 | Frege himself abstracts away from tone and color |
| Full Idea: Frege himself abstracts away from tone and color. | |||
| From: comment on Gottlob Frege (Grundlagen der Arithmetik (Foundations) [1884]) by Stephen Yablo - Carving Content at the Joints §3 | |||
| A reaction: Gotcha! I have been searching for instances where Frege perpetrates psychological abstraction right in the heart of his theory. No one can avoid it, if they are in the business of trying to formulate new concepts. Reference ignores sense, and vice versa. |
| 9988 | If we abstract 'from' two cats, the units are not black or white, or cats |
| Full Idea: When from a set of two cats, one black and one white, we 'abstract' the number two as a set of pure units, the units are not black and white, respectively, and they are not cats. | |||
| From: comment on Gottlob Frege (Grundlagen der Arithmetik (Foundations) [1884], §34) by William W. Tait - Frege versus Cantor and Dedekind XI | |||
| A reaction: Well said. Frege is contemptuous of this approach, as if we were incapable of thinking of a black cat as anything other than as black or cat, when we can catch cats as 'food', or 'objects', or just plain 'countables'. |
| 9855 | Frege's logical abstaction identifies a common feature as the maximal set of equivalent objects |
| Full Idea: Like psychological abstractionism, Frege's method (which we can call 'logical abstraction') aims at isolating what is in common between the members of any equivalent sets of objects, by identifying the feature with the maximal set of such objects. | |||
| From: report of Gottlob Frege (Grundlagen der Arithmetik (Foundations) [1884]) by Michael Dummett - Frege philosophy of mathematics Ch.14 | |||
| A reaction: [compressed] So Frege's approach to abstraction is a branch of the view that properties are sets. This view, in addition to being vulnerable to Russell's paradox, ignores the causal role of properties, making them all categorical (which is daft). |
| 10802 | Frege's 'parallel' and 'direction' don't have the same content, as we grasp 'parallel' first |
| Full Idea: Frege's discussion of 'direction' borders on incoherent. He claims that the equivalence of lines a and b and their directions being equal have the same content, which leads to the concept of direction, but we grasp the equivalence before the equality. | |||
| From: comment on Gottlob Frege (Grundlagen der Arithmetik (Foundations) [1884]) by Stephen Yablo - Carving Content at the Joints § 1 | |||
| A reaction: [The Frege is in Grundlagen §64] Well said. The notion that we get the full concept of 'direction' from such paltry resources seems very weak. For a start, parallel lines exhibit two directions, not one. |
| 10526 | Fregean abstraction creates concepts which are equivalences between initial items |
| Full Idea: Fregean abstraction rests on initial items, taken to be related by an equivalence relation (e.g. parallelism, or equinumerosity), and then an operation for forming abstraction (e.g. direction or number), with identity related to their equivalence. | |||
| From: report of Gottlob Frege (Grundlagen der Arithmetik (Foundations) [1884]) by Kit Fine - Precis of 'Limits of Abstraction' p.305 | |||
| A reaction: [compressed] This is the best summary I have found of the modern theory of abstraction, as opposed to the nature of the abstracta themselves. A minimum of two items is needed to implement the process. |
| 10525 | Frege put the idea of abstraction on a rigorous footing |
| Full Idea: It was Frege who first showed how the idea of abstraction could be put on a rigorous footing. | |||
| From: report of Gottlob Frege (Grundlagen der Arithmetik (Foundations) [1884]) by Kit Fine - Precis of 'Limits of Abstraction' p.305 | |||
| A reaction: This refers to the crucial landmark in philosophical thought about abstraction. The question is whether Frege had to narrow the concept of abstraction and abstract entities too severely, in order to achieve his rigour. |
| 10556 | We create new abstract concepts by carving up the content in a different way |
| Full Idea: (In creating the concept of direction..) We carve up the content in a way different from the original way, and this yields us a new concept. ...It is a matter of drawing boundary lines that were not previously given. | |||
| From: Gottlob Frege (Grundlagen der Arithmetik (Foundations) [1884], §64) | |||
| A reaction: [second half in §88] 'Recarving' is now the useful shorthand for Frege's way of creating abstract concepts (rather than the old psychological way of ignoring some features of an object). |
| 9882 | You can't simultaneously fix the truth-conditions of a sentence and the domain of its variables |
| Full Idea: Frege's root confusion (over abstraction by identity, and other things) was to believe that he could simultaneously fix the truth-conditions of such statements and the domain over which the individual variables were to range. | |||
| From: comment on Gottlob Frege (Grundlagen der Arithmetik (Foundations) [1884], §64-68) by Michael Dummett - Frege philosophy of mathematics Ch.18 | |||
| A reaction: This strikes me as a wonderfully penetrating criticism, but it also seems to me to threaten Dummett's whole programme of doing ontology through language. If a quantified sentences needs a domain, how do you first decide your domain? |
| 9881 | From basing 'parallel' on identity of direction, Frege got all abstractions from identity statements |
| Full Idea: Having rightly perceived that the fundamental class here was statements of identity between directions, Frege leapt to the conclusion that the basis for introducing new abstract terms consisted of determining the truth-conditions of identity-statements. | |||
| From: report of Gottlob Frege (Grundlagen der Arithmetik (Foundations) [1884], §64-68) by Michael Dummett - Frege philosophy of mathematics Ch.18 | |||
| A reaction: This seems to be the modern view - that abstraction consists of the assertion of an equivalence principle. Dummett criticises Frege here (see Idea 9882). There always seems to be a chicken/egg problem. Why would the identity be asserted? |
| 8646 | Words in isolation seem to have ideas as meanings, but words have meaning in propositions |
| Full Idea: We consider the meanings of words in isolation, which leads us to accept an idea as the meaning, and words with no mental picture appear to have no mental content. But only in a proposition have the words really a meaning. | |||
| From: Gottlob Frege (Grundlagen der Arithmetik (Foundations) [1884], §60) | |||
| A reaction: Frege (later) sees concepts as functions, which need input and output to be understood. It points to the idea that meaning is nothing more than usage. Something, though, is missing. As ever, WHY does something have a particular function? |
| 7732 | Never ask for the meaning of a word in isolation, but only in the context of a proposition |
| Full Idea: Never ask for the meaning of a word in isolation, but only in the context of a proposition. | |||
| From: Gottlob Frege (Grundlagen der Arithmetik (Foundations) [1884], p.x) | |||
| A reaction: [Called the 'Contextual Principle']. But surely the word 'pig' has a known meaning, even if I don't give it a context? A word like 'the' seems to need a context, though. One might demand the context of the proposition as well. |
| 9370 | A statement is analytic if substitution of synonyms can make it a logical truth |
| Full Idea: According to Frege, a statement's analyticity (in my epistemological sense) is to be explained by the fact that it is transformable into a logical truth by the substitution of synonyms for synonyms. | |||
| From: report of Gottlob Frege (Grundlagen der Arithmetik (Foundations) [1884], §03) by Paul Boghossian - Analyticity Reconsidered §I | |||
| A reaction: [He says this interpretation of Frege's semantical notion of analyticity may be controversial] Presumably we see that 'bachelors are unmarried men' is analytic when we start substituting for 'bachelor'. Sounds reasonable. |
| 8743 | Frege considered analyticity to be an epistemic concept |
| Full Idea: Frege held that analyticity is like a priority in being an epistemic concept. | |||
| From: report of Gottlob Frege (Grundlagen der Arithmetik (Foundations) [1884], §03) by Stewart Shapiro - Thinking About Mathematics 5.1 | |||
| A reaction: Kripke very firmly says that this is not so. While a priori is an epistemic concept, analyticity is a semantic concept. I cling on to Kripke's framework, but probably more because it is neat and comfortable than because it is true. |
| 20295 | All analytic truths can become logical truths, by substituting definitions or synonyms |
| Full Idea: Frege appealed to definition, or (if 'meaning' is preserved) synonymy: the non-logical analytic truths can be converted to logical truths by substitution of definitions for defined terms, or synonyms for synonyms. | |||
| From: report of Gottlob Frege (Grundlagen der Arithmetik (Foundations) [1884], §005, 88) by Georges Rey - The Analytic/Synthetic Distinction 1.2 | |||
| A reaction: This is a 'dogma of empiricism' attacked by Quine. It seems rather obvious (with hindsight?) that you can smuggle whatever is required to do the job into your definition. Or assert some slightly dubious synonymy. |
| 2515 | Frege fails to give a concept of analyticity, so he fails to explain synthetic a priori truth that way |
| Full Idea: Frege's approach provides no concept of analyticity (hence Quine's attack), so there is no notion of the analytic a priori under which to bring the metaphysician's synthetic a priori propositions. | |||
| From: comment on Gottlob Frege (Grundlagen der Arithmetik (Foundations) [1884]) by Jerrold J. Katz - Realistic Rationalism Int.xxi | |||
| A reaction: So Frege might have been a logical positivist, if only he had given himself the right tools for the job? |
| 8619 | To learn something, you must know that you don't know |
| Full Idea: The first prerequisite for knowing anything is the knowledge that we do not know. | |||
| From: Gottlob Frege (Grundlagen der Arithmetik (Foundations) [1884], Intro) | |||
| A reaction: This is serious practical advice for teachers. Intelligent people are aware of most philosophical problems, but tongue-tied when asked to discuss them. |
| 8656 | The laws of number are not laws of nature, but are laws of the laws of nature |
| Full Idea: The laws of number are not applicable to external things, and are not laws of nature, but they are applicable to judgements of external things: they are laws of the laws of nature. | |||
| From: Gottlob Frege (Grundlagen der Arithmetik (Foundations) [1884], §87) | |||
| A reaction: We seem to be somewhere between pythagoreanism and 'the mind of God'. I feel fairly strongly that we are looking through the wrong end of the telescope here. The laws of nature 'emerge' from nature, and high-level abstractions emerge with them. |
| 22286 | Existence is not a first-level concept (of God), but a second-level property of concepts |
| Full Idea: For Frege (unlike Kant) existence is a genuine concept, but of the second level, not the first. Since God's perfections are of the first level, existence is not a candidate to be one of them. | |||
| From: report of Gottlob Frege (Grundlagen der Arithmetik (Foundations) [1884], §053) by Michael Potter - The Rise of Analytic Philosophy 1879-1930 09 'App' | |||
| A reaction: That is, God's perfections are of God, but existence is a concept of concepts (that they are instantiated). So existence is a metaconcept. I'm not convinced. If I bake a successful cake, its existence is its most wonderful feature. |
| 8644 | Because existence is a property of concepts the ontological argument for God fails |
| Full Idea: Because existence is a property of concepts the ontological argument for the existence of God breaks down. | |||
| From: Gottlob Frege (Grundlagen der Arithmetik (Foundations) [1884], §53) | |||
| A reaction: The point being that existence (like number) is not a property of actual things. His proposition sounds rather dubious. The concept of unicorns exists quite entertainingly; it is the failure of actual unicorns to exist that is so heartbreaking. |