| 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. |
| 22270 | Frege changed philosophy by extending logic's ability to check the grounds of thinking |
| Full Idea: Frege's 1879 logic transformed philosophy because it greatly expanded logic's reach - what thought can achieve unaided - and hence compelled a re-examination of everything previously said about the grounds of thought when logic gives out. | |
| From: comment on Gottlob Frege (Begriffsschrift [1879]) by Michael Potter - The Rise of Analytic Philosophy 1879-1930 Intro | |
| A reaction: I loved the gloss on logic as 'what thought can achieve unaided'. I largely see logic in terms of what is mechanically computable. |
| 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. |
| 7740 | There exists a realm, beyond objects and ideas, of non-spatio-temporal thoughts |
| Full Idea: There is, in addition to the external world of physical objects and the internal world of ideas, a third realm of non-spatio-temporal objective objects, among which are thoughts. | |
| From: report of Gottlob Frege (The Thought: a Logical Enquiry [1918]) by Joan Weiner - Frege Ch.7 | |
| A reaction: This seems to be Platonism, and, in particular, to give a Platonic existent status to propositions. Personally I believe in propositions, but as glimpses of how our brains actually work, not as mystical objects. |
| 8939 | We should not describe human laws of thought, but how to correctly track truth |
| Full Idea: Frege disagree that logic should merely describe the laws of thought - how people actually did reason. Logic is essentially normative, not descriptive. We want the one logic which successfully tracks the truth. | |
| From: report of Gottlob Frege (Begriffsschrift [1879]) by Jennifer Fisher - On the Philosophy of Logic 1.III | |
| A reaction: This explains Frege's sustained attack on psychologism, and it also explains we he ended up as a platonist about logic - because he wanted its laws to be valid independently of human thinking. A step too far, perhaps. Brains are truth machines. |
| 9821 | A definition need not capture the sense of an expression - just get the reference right |
| Full Idea: Frege expressly denies that a correct definition need capture the sense of the expression it defines: it need only get the reference right. | |
| From: report of Gottlob Frege (Review of Husserl's 'Phil of Arithmetic' [1894]) by Michael Dummett - Frege philosophy of mathematics Ch.3 | |
| A reaction: This might hit up against the renate/cordate problem, of two co-extensive concepts, where the definition gets the extension right, but the intension wrong. |
| 13886 | Later Frege held that definitions must fix a function's value for every possible argument |
| Full Idea: Frege later became fastidious about definitions, and demanded that they must provide for every possible case, and that no function is properly determined unless its value is fixed for every conceivable object as argument. | |
| From: report of Gottlob Frege (Grundgesetze der Arithmetik 2 (Basic Laws) [1903]) by Crispin Wright - Frege's Concept of Numbers as Objects 3.xiv | |
| A reaction: Presumably definitions come in degrees of completeness, but it seems harsh to describe a desire for the perfect definition as 'fastidious', especially if we are talking about mathematics, rather than defining 'happiness'. |
| 16877 | A 'constructive' (as opposed to 'analytic') definition creates a new sign |
| Full Idea: We construct a sense out of its constituents and introduce an entirely new sign to express this sense. This may be called a 'constructive definition', but we prefer to call it a 'definition' tout court. It contrasts with an 'analytic' definition. | |
| From: Gottlob Frege (Logic in Mathematics [1914], p.210) | |
| A reaction: An analytic definition is evidently a deconstruction of a past constructive definition. Fregean definition is a creative activity. |
| 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'. |
| 9845 | We can't define a word by defining an expression containing it, as the remaining parts are a problem |
| Full Idea: Given the reference (bedeutung) of an expression and a part of it, obviously the reference of the remaining part is not always determined. So we may not define a symbol or word by defining an expression in which it occurs, whose remaining parts are known | |
| From: Gottlob Frege (Grundgesetze der Arithmetik 2 (Basic Laws) [1903], §66) | |
| A reaction: Dummett cites this as Frege's rejection of contextual definitions, which he had employed in the Grundlagen. I take it not so much that they are wrong, as that Frege decided to set the bar a bit higher. |
| 11219 | Frege suggested that mathematics should only accept stipulative definitions |
| Full Idea: Frege has defended the austere view that, in mathematics at least, only stipulative definitions should be countenanced. | |
| From: report of Gottlob Frege (Logic in Mathematics [1914]) by Anil Gupta - Definitions 1.3 | |
| A reaction: This sounds intriguingly at odds with Frege's well-known platonism about numbers (as sets of equinumerous sets). It makes sense for other mathematical concepts. |
| 10019 | Only what is logically complex can be defined; what is simple must be pointed to |
| Full Idea: Only what is logically complex can be defined; what is simple can only be pointed to. | |
| From: Gottlob Frege (Grundgesetze der Arithmetik 2 (Basic Laws) [1903], §180), quoted by Harold Hodes - Logicism and Ontological Commits. of Arithmetic p.137 | |
| A reaction: Frege presumably has in mind his treasured abstract objects, such as cardinal numbers. It is hard to see how you could 'point to' anything in the phenomenal world that had atomic simplicity. Hodes calls this a 'desperate Kantian move'. |
| 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. |
| 16878 | We must be clear about every premise and every law used in a proof |
| Full Idea: It is so important, if we are to have a clear insight into what is going on, for us to be able to recognise the premises of every inference which occurs in a proof and the law of inference in accordance with which it takes place. | |
| From: Gottlob Frege (Logic in Mathematics [1914], p.212) | |
| A reaction: Teachers of logic like natural deduction, because it reduces everything to a few clear laws, which can be stated at each step. |
| 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? |
| 19466 | The word 'true' seems to be unique and indefinable |
| Full Idea: It seems likely that the content of the word 'true' is sui generis and indefinable | |
| From: Gottlob Frege (The Thought: a Logical Enquiry [1918], p.327 (60)) | |
| A reaction: This is the view I associate with Davidson, though fans of Axiomatic Truth give up defining it, and just describe how it behaves. Defining it is very elusive, but I don't accept that nothing can be said about the contents of the concept of truth. |
| 8187 | Frege was strongly in favour of taking truth to attach to propositions |
| Full Idea: Frege was strongly in favour of taking truth to attach to propositions, which he called 'thoughts' and regarded as being expressed by sentences. | |
| From: report of Gottlob Frege (On Sense and Reference [1892]) by Michael Dummett - Truth and the Past 1 | |
| A reaction: Sometimes it is necessary to know the time, the place, and the speaker before one can evaluate the truth of a proposition. Not just indexical words, but the indexical aspect of, say, "the team played badly". |
| 22317 | Truth does not admit of more and less |
| Full Idea: What is only half true is untrue. Truth does not admit of more and less. | |
| From: Gottlob Frege (works [1890], CP 353), quoted by Michael Potter - The Rise of Analytic Philosophy 1879-1930 48 'Truth' | |
| A reaction: What about a measurement which is accurate to three decimal places? Maybe being 'close to' the truth is not the same as being 'more' true. The truth about a distance between two points is unknowable? |
| 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. |
| 19465 | There cannot be complete correspondence, because ideas and reality are quite different |
| Full Idea: It is essential that the reality shall be distinct from the idea. But then there can be no complete correspondence, no complete truth. | |
| From: Gottlob Frege (The Thought: a Logical Enquiry [1918], p.327 (60)) | |
| A reaction: He thinks that logic can give a perfect account of truth, or at least the extension of truth, where ordinary language will always fail. I wonder what he would have thought of Tarski's theory? |
| 19468 | The property of truth in 'It is true that I smell violets' adds nothing to 'I smell violets' |
| Full Idea: The sentence 'I smell the scent of violets' has just the same content as 'It is true that I smell the scent of violets'. So it seems that nothing is added to the thought by my ascribing to it the property of truth. | |
| From: Gottlob Frege (The Thought: a Logical Enquiry [1918], p.328 (61)) | |
| A reaction: This idea predates Ramsey's similar proposal, for which, oddly, Ramsey always seems to get the credit. To a logician they may have identical content, but pragmatically they are likely to differ in context. 'True' certainly doesn't add to the thought. |
| 18806 | Frege thought traditional categories had psychological and linguistic impurities |
| Full Idea: Frege rejected the traditional categories as importing psychological and linguistic impurities into logic. | |
| From: report of Gottlob Frege (Function and Concept [1891]) by Ian Rumfitt - The Boundary Stones of Thought 1.2 | |
| A reaction: Resisting such impurities is the main motivation for making logic entirely symbolic, but it doesn't follow that the traditional categories have to be dropped. |
| 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. |
| 9585 | Since every definition is an equation, one cannot define equality itself |
| Full Idea: Since every definition is an equation, one cannot define equality itself. | |
| From: Gottlob Frege (Review of Husserl's 'Phil of Arithmetic' [1894], p.327) | |
| A reaction: This seems a particularly nice instance of the general rule that 'you have to start somewhere'. It is a nice test case for the nature of meaning to ask 'what do you understand when you understand equality?', given that you can't define it. |
| 4971 | I don't use 'subject' and 'predicate' in my way of representing a judgement |
| Full Idea: A distinction of subject and predicate finds no place in my way of representing a judgement. | |
| From: Gottlob Frege (Begriffsschrift [1879], §03) | |
| A reaction: Perhaps this sentence could be taken as the beginning of modern analytical philosophy. The old view doesn't seem to me entirely redundant - merely replaced by a much more detailed analysis of what makes a 'subject' and what makes a 'predicate'. |
| 17745 | For Frege, 'All A's are B's' means that the concept A implies the concept B |
| Full Idea: 'All A's are B's' meant for Frege that the concept A implies the concept B, or that to be A implies also to be B. Moreover this applies to arbitrary x which happens to be A. | |
| From: report of Gottlob Frege (Begriffsschrift [1879]) by Michal Walicki - Introduction to Mathematical Logic History D.2 | |
| A reaction: This seems to hit the renate/cordate problem. If all creatures with hearts also have kidneys, does that mean that being enhearted logically implies being kidneyfied? If all chimps are hairy, is that a logical requirement? Is inclusion implication? |
| 13455 | Frege did not think of himself as working with sets |
| Full Idea: Frege did not think of himself as working with sets. | |
| From: report of Gottlob Frege (works [1890]) by William D. Hart - The Evolution of Logic 1 | |
| A reaction: One can hardly blame him, given that set theory was only just being invented. |
| 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. |
| 16895 | The null set is indefensible, because it collects nothing |
| Full Idea: Frege regarded the null set as an indefensible entity from the point of view of iterative set theory. It collects nothing. | |
| From: report of Gottlob Frege (works [1890]) by Tyler Burge - Frege on Apriority (with ps) 2 | |
| A reaction: The null set defines the possibility that something could be collected. At the very least, it introduces curly brackets into the language. |
| 14238 | A class is an aggregate of objects; if you destroy them, you destroy the class; there is no empty class |
| Full Idea: A class consists of objects; it is an aggregate, a collective unity, of them; if so, it must vanish when these objects vanish. If we burn down all the trees of a wood, we thereby burn down the wood. Thus there can be no empty class. | |
| From: Gottlob Frege (Elucidation of some points in E.Schröder [1895], p.212), quoted by Oliver,A/Smiley,T - What are Sets and What are they For? | |
| A reaction: This rests on Cantor's view of a set as a collection, rather than on Dedekind, which allows null and singleton sets. |
| 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.. |
| 3328 | Frege proposed a realist concept of a set, as the extension of a predicate or concept or function |
| Full Idea: Contrary to Dedekind's anti-realism, Frege proposed a realist definition of a set as the extension of a predicate (or concept, or function). | |
| From: report of Gottlob Frege (works [1890]) by José A. Benardete - Metaphysics: the logical approach Ch.13 |
| 16881 | The laws of logic are boundless, so we want the few whose power contains the others |
| Full Idea: Since in view of the boundless multitude of laws that can be enunciated we cannot list them all, we cannot achieve completeness except by searching out those that, by their power, contain all of them. | |
| From: Gottlob Frege (Begriffsschrift [1879], §13) | |
| A reaction: He refers to these laws in the previous sentence as the 'core'. His talk of 'power' is music to my ears, since it implies a direction of explanation. Burge says the power is that of defining other concepts. |
| 7728 | Frege has a judgement stroke (vertical, asserting or judging) and a content stroke (horizontal, expressing) |
| Full Idea: Frege distinguished between asserting a proposition and expressing it, and he introduced the judgement stroke (a small vertical line, assertion) and the content stroke (a long horizontal line, expression) to represent them. | |
| From: report of Gottlob Frege (Begriffsschrift [1879]) by Joan Weiner - Frege Ch.3 | |
| A reaction: There are also strokes for conditional and denial. |
| 7622 | In 1879 Frege developed second order logic |
| Full Idea: By 1879 Frege had discovered an algorithm, a mechanical proof procedure, that embraces what is today standard 'second order logic'. | |
| From: report of Gottlob Frege (Begriffsschrift [1879]) by Hilary Putnam - Reason, Truth and History Ch.5 | |
| A reaction: Note that Frege did more than introduce quantifiers, and the logic of predicates. |
| 9179 | Frege frequently expressed a contempt for language |
| Full Idea: Frege frequently expressed a contempt for language. | |
| From: report of Gottlob Frege (works [1890], p.228) by Michael Dummett - Frege's Distinction of Sense and Reference p.228 | |
| A reaction: This strikes me as exactly the right attitude for a logician to have. Russell seems to have agreed. Attitudes to vagueness are the test case. Over-ambitious modern logicians dream of dealing with vagueness. Forget it. Stick to your last. |
| 16867 | Logic not only proves things, but also reveals logical relations between them |
| Full Idea: A proof does not only serve to convince us of the truth of what is proved: it also serves to reveal logical relations between truths. Hence we find in Euclid proofs of truths that appear to stand in no need of proof because they are obvious without one. | |
| From: Gottlob Frege (Logic in Mathematics [1914], p.204) | |
| A reaction: This is a key idea in Frege's philosophy, and a reason why he is the founder of modern analytic philosophy, with logic placed at the centre of the subject. I take the value of proofs to be raising questions, more than giving answers. |
| 16863 | Does some mathematical reasoning (such as mathematical induction) not belong to logic? |
| Full Idea: Are there perhaps modes of inference peculiar to mathematics which …do not belong to logic? Here one may point to inference by mathematical induction from n to n+1. | |
| From: Gottlob Frege (Logic in Mathematics [1914], p.203) | |
| A reaction: He replies that it looks as if induction can be reduced to general laws, and those can be reduced to logic. |
| 16862 | The closest subject to logic is mathematics, which does little apart from drawing inferences |
| Full Idea: Mathematics has closer ties with logic than does almost any other discipline; for almost the entire activity of the mathematician consists in drawing inferences. | |
| From: Gottlob Frege (Logic in Mathematics [1914], p.203) | |
| A reaction: The interesting question is who is in charge - the mathematician or the logician? |
| 13473 | Frege thinks there is an independent logical order of the truths, which we must try to discover |
| Full Idea: Frege thinks there is a single right deductive order of the truths. This is not an epistemic order, but a logical order, and it is our job to arrange our beliefs in this order if we can make it out. | |
| From: report of Gottlob Frege (works [1890]) by William D. Hart - The Evolution of Logic 2 | |
| A reaction: Frege's dream rests on the belief that there exists a huge set of logical truths. Pluralism, conventionalism, constructivism etc. about logic would challenge this dream. I think the defence of Frege must rest on Russellian rooting of logic in nature. |
| 7729 | Frege replaced Aristotle's subject/predicate form with function/argument form |
| Full Idea: Frege's regimentation is based on the view of the simplest sort of statement as having, not subject/predicate form (as in Aristotle), but function/argument form. | |
| From: report of Gottlob Frege (Begriffsschrift [1879]) by Joan Weiner - Frege | |
| A reaction: This looks like being a crucial move into the modern world, where one piece of information is taken in and dealt with, as in computer procedures. Have educated people reorganised their minds along Fregean lines? |
| 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. |
| 4975 | A thought can be split in many ways, so that different parts appear as subject or predicate |
| Full Idea: A thought can be split up in many ways, so that now one thing, now another, appears as subject or predicate | |
| From: Gottlob Frege (On Concept and Object [1892], p.199) | |
| A reaction: Thus 'the mouse is in the box', and 'the box contains the mouse'. A simple point, but important when we are trying to distinguish thought from language. |
| 8490 | First-level functions have objects as arguments; second-level functions take functions as arguments |
| Full Idea: Just as functions are fundamentally different from objects, so also functions whose arguments are and must be functions are fundamentally different from functions whose arguments are objects. The latter are first-level, the former second-level, functions. | |
| From: Gottlob Frege (Function and Concept [1891], p.38) | |
| A reaction: In 1884 he called it 'second-order'. This is the standard distinction between first- and second-order logic. The first quantifies over objects, the second over intensional entities such as properties and propositions. |
| 8492 | Relations are functions with two arguments |
| Full Idea: Functions of one argument are concepts; functions of two arguments are relations. | |
| From: Gottlob Frege (Function and Concept [1891], p.39) | |
| A reaction: Nowadays we would say 'two or more'. Another interesting move in the aim of analytic philosophy to reduce the puzzling features of the world to mathematical logic. There is, of course, rather more to some relations than being two-argument functions. |
| 3319 | Frege gives a functional account of predication so that we can dispense with predicates |
| Full Idea: The whole point of Frege's functional account of predication lies in its allowing us to dispense with all properties across the board. | |
| From: report of Gottlob Frege (works [1890]) by José A. Benardete - Metaphysics: the logical approach Ch.9 |
| 6076 | For Frege, predicates are names of functions that map objects onto the True and False |
| Full Idea: For Frege, a predicate does not refer to the objects of which it is true, but to the function that maps these objects onto the True and False; ..a predicate is a name for this function. | |
| From: report of Gottlob Frege (works [1890]) by Colin McGinn - Logical Properties Ch.3 | |
| A reaction: McGinn says this is close to the intuitive sense of a property. Perhaps 'predicates are what make objects the things they are?' |
| 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'. |
| 16865 | 'Theorems' are both proved, and used in proofs |
| Full Idea: Usually a truth is only called a 'theorem' when it has not merely been obtained by inference, but is used in turn as a premise for a number of inferences in the science. ….Proofs use non-theorems, which only occur in that proof. | |
| From: Gottlob Frege (Logic in Mathematics [1914], p.204) |
| 18772 | We can treat designation by a few words as a proper name |
| Full Idea: The designation of a single object can also consist of several words or other signs. For brevity, let every such designation be called a proper name. | |
| From: Gottlob Frege (On Sense and Reference [1892]), quoted by Bernard Linsky - Quantification and Descriptions 1 | |
| A reaction: Frege regards names and descriptions as in the same class. Russell, and then Kripke, had things to say about that. |
| 8447 | In 'Etna is higher than Vesuvius' the whole of Etna, including all the lava, can't be the reference |
| Full Idea: The reference of 'Etna' cannot be Mount Etna itself, because each piece of frozen lava which is part of Mount Etna would then also be part of the thought that Etna is higher than Vesuvius. | |
| From: Gottlob Frege (Letters to Jourdain [1910], p.43) | |
| A reaction: This seems to be a straight challenge to Kripke's baptismal account of reference. I think I side with Kripke. Frege is allergic to psychological accounts, but the mind only has the capacity to think of the aspect of Etna that is relevant. |
| 10424 | A Fregean proper name has a sense determining an object, instead of a concept |
| Full Idea: We could think of a referring expression in Fregean terms as what he calls a proper name (Eigenname): its Sinn (sense) is supposed to determine an object as opposed to a concept as its Bedeutung (referent). | |
| From: report of Gottlob Frege (On Sense and Reference [1892]) by Mark Sainsbury - The Essence of Reference 18.1 | |
| A reaction: The problem would be that the same expression could precisely indicate an object on one occasion, nearly do so on another, and totally fail on a third. |
| 18773 | People may have different senses for 'Aristotle', like 'pupil of Plato' or 'teacher of Alexander' |
| Full Idea: In the case of an actual proper name such as 'Aristotle' opinions as to the sense may differ. It might, for instance, be taken to be the following: the pupil of Plato and teacher of Alexander the Great. | |
| From: Gottlob Frege (On Sense and Reference [1892], note), quoted by Bernard Linsky - Quantification and Descriptions 1 | |
| A reaction: This note is 'notorious', and was a central target for Kripke's critique. Frege says people's senses may vary on this, and thinks the sense of 'Aristotle' can be accurately expressed. |
| 14075 | Proper name in modal contexts refer obliquely, to their usual sense |
| Full Idea: According to Frege, a proper name in a modal context refers obliquely; its reference there is its usual sense. | |
| From: report of Gottlob Frege (On Sense and Reference [1892]) by Allan Gibbard - Contingent Identity V | |
| A reaction: [he cites the fourth page of Frege's 'Sense and Reference'] One can foresee problems with the word 'usual' here. Frege might be offering something better than Kripke does here. |
| 8448 | Any object can have many different names, each with a distinct sense |
| Full Idea: An object can be determined in different ways, and every one of these ways of determining it can give rise to a special name, and these different names then have different senses. | |
| From: Gottlob Frege (Letters to Jourdain [1910], p.44) | |
| A reaction: This seems right. No name is an entirely neutral designator. Imagine asking a death-camp survivor their name, and they give you their prison number. Sense clearly intrudes into names. But picking out the object is what really matters. |
| 4978 | The meaning of a proper name is the designated object |
| Full Idea: The meaning of a proper name is the object itself which we designate by using it. | |
| From: Gottlob Frege (On Sense and Reference [1892], p.30) | |
| A reaction: I can't actually make sense of this. How can a physical object be identical with a meaning? What sort of thing is a 'meaning'? Meanings are just 'in the head', I suspect. |
| 10510 | Frege ascribes reference to incomplete expressions, as well as to singular terms |
| Full Idea: Frege ascribes reference not only to singular terms, but equally to expressions of other kinds (the various kinds of incomplete expressions). | |
| From: report of Gottlob Frege (On Sense and Reference [1892]) by Bob Hale - Abstract Objects Ch.3 Intro | |
| A reaction: The incomplete expressions presumably make reference to concepts. Frege may not seem, therefore, to have a notion of reference as what plugs language into reality - except that he is presumably a platonist about concepts. |
| 18937 | If sentences have a 'sense', empty name sentences can be understood that way |
| Full Idea: Frege's theory of 'sense' showed how sentences with empty names can have meaning and be understood. One just has to grasp the sense of the sentence (the thought expressed), and this is available even in the absence of a referent for the name. | |
| From: report of Gottlob Frege (On Sense and Reference [1892]) by Sarah Sawyer - Empty Names 2 | |
| A reaction: My immediate reaction is that this provides a promising solution to the empty names problem, which certainly never bothered me before I started reading philosophy. Sawyer says co-reference and truth problems remain. |
| 18940 | It is a weakness of natural languages to contain non-denoting names |
| Full Idea: Languages have the fault of containing expressions which fail to designate an object. | |
| From: Gottlob Frege (On Sense and Reference [1892], p.40) | |
| A reaction: Wrong, Frege! This is a strength of natural languages! Names are tools. It isn't a failure of your hammer if you can't find any nails. |
| 18939 | In a logically perfect language every well-formed proper name designates an object |
| Full Idea: A logically perfect language should satisfy the conditions that every expression grammatically well constructed as a proper name out of signs already introduced shall in fact designate an object. | |
| From: Gottlob Frege (On Sense and Reference [1892], p.41) | |
| A reaction: This seems to cramp your powers of reasoning, if you must know the object to use the name ('Jack the Ripper'), and reasoning halts once you deny the object's existence ('Pegasus'), or you don't know if names co-refer ('Hesperus/Phosphorus'). |
| 13733 | Frege considered definite descriptions to be genuine singular terms |
| Full Idea: Frege (1893) considered a definite description to be a genuine singular term (as we do), so that a sentence like 'The present King of France is bald' would have the same logical form as 'Harry Truman is bald'. | |
| From: report of Gottlob Frege (Grundgesetze der Arithmetik 1 (Basic Laws) [1893]) by M Fitting/R Mendelsohn - First-Order Modal Logic | |
| A reaction: The difficulty is what the term refers to, and they embrace a degree of Meinongianism - that is that non-existent objects can still have properties attributed to them, and so can be allowed some sort of 'existence'. |
| 9950 | A quantifier is a second-level predicate (which explains how it contributes to truth-conditions) |
| Full Idea: The contribution of the quantifier to the truth conditions of sentences of which it is a part cannot be adequately explained if it is treated as other than a second-level predicate (for instance, if it is viewed as name). | |
| From: report of Gottlob Frege (Begriffsschrift [1879]) by A.George / D.J.Velleman - Philosophies of Mathematics Ch.2 | |
| A reaction: They suggest that this makes it something like a 'property of properties'. With this account it becomes plausible to think of numbers as quantifiers (since they do, after all, specify quantities). |
| 9991 | For Frege the variable ranges over all objects |
| Full Idea: For Frege the variable ranges over all objects. | |
| From: report of Gottlob Frege (Begriffsschrift [1879]) by William W. Tait - Frege versus Cantor and Dedekind XII | |
| A reaction: The point is that Frege had not yet seen the necessity to define the domain of quantification, and this leads him into various difficulties. |
| 10536 | Frege's domain for variables is all objects, but modern interpretations first fix the domain |
| Full Idea: For Frege there is no need to specify the domain of the individual variables, which is taken as the totality of all objects. This contrasts with the standard notion of an interpretation, which demands that we first fix the domain. | |
| From: comment on Gottlob Frege (Begriffsschrift [1879]) by Michael Dummett - Frege Philosophy of Language (2nd ed) Ch.14 | |
| A reaction: What intrigues me is how domains of quantification shift according to context in ordinary usage, even in mid-sentence. I ought to go through every idea in this database, specifying its domain of quantification. Any volunteers? |
| 9871 | Frege always, and fatally, neglected the domain of quantification |
| Full Idea: Frege persistently neglected the question of the domain of quantification, which proved in the end to be fatal. | |
| From: comment on Gottlob Frege (works [1890]) by Michael Dummett - Frege philosophy of mathematics Ch.16 | |
| A reaction: The 'fatality' refers to Russell's paradox, and the fact that not all concepts have extensions. Common sense now says that this is catastrophic. A domain of quantification is a topic of conversation, which is basic to all language. Cf. Idea 9874. |
| 7730 | Frege introduced quantifiers for generality |
| Full Idea: In order to express generality, Frege introduced quantifier notation. | |
| From: report of Gottlob Frege (Begriffsschrift [1879]) by Joan Weiner - Frege | |
| A reaction: This is the birth of predicate logic, beloved of analytical philosophers (but of no apparent interest to phenomenalists, deconstructionists, existentialists?). Generality is what you get from induction (which is, of course, problematic). |
| 7742 | Frege reduced most quantifiers to 'everything' combined with 'not' |
| Full Idea: Frege treated 'everything' as basic, and suggested ways of recasting propositions containing other quantifiers so that this was the only one remaining. He recast 'something' as 'at least one thing', and defined this in terms of 'everything' and 'not'. | |
| From: report of Gottlob Frege (Begriffsschrift [1879]) by Gregory McCullogh - The Game of the Name 1.6 | |
| A reaction: Extreme parsimony seems highly desirable in logic as well as ontology, but it can lead to frustrations, especially over the crucial question of the existence of things quantified over. See Idea 6068. |
| 9874 | Contradiction arises from Frege's substitutional account of second-order quantification |
| Full Idea: The contradiction in Frege's system is due to the presence of second-order quantification, ..and Frege's explanation of the second-order quantifier, unlike that which he provides for the first-order one, appears to be substitutional rather than objectual. | |
| From: comment on Gottlob Frege (Grundgesetze der Arithmetik 1 (Basic Laws) [1893], §25) by Michael Dummett - Frege philosophy of mathematics Ch.17 | |
| A reaction: In Idea 9871 Dummett adds the further point that Frege lacks a clear notion of the domain of quantification. At this stage I don't fully understand this idea, but it is clearly of significance, so I will return to it. |
| 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. |
| 13824 | Proof theory began with Frege's definition of derivability |
| Full Idea: Frege's formal definition of derivability is perhaps the first investigation in general proof theory. | |
| From: report of Gottlob Frege (Begriffsschrift [1879]) by Dag Prawitz - Gentzen's Analysis of First-Order Proofs 2 n2 | |
| A reaction: In 'On General Proof Theory §1' Prawitz says "proof theory originated with Hilbert" in 1900. Presumably Frege offered a theory, and then Hilbert saw it as a general project. |
| 13609 | Frege produced axioms for logic, though that does not now seem the natural basis for logic |
| Full Idea: Frege's work supplied a set of axioms for logic itself, at least partly because it was a well-known way of presenting the foundations in other disciplines, especially mathematics, but it does not nowadays strike us as natural for logic. | |
| From: report of Gottlob Frege (Begriffsschrift [1879]) by David Kaplan - Dthat 5.1 | |
| A reaction: What Bostock has in mind is the so-called 'natural' deduction systems, which base logic on rules of entailment, rather than on a set of truths. The axiomatic approach uses a set of truths, plus the idea of possible contradictions. |
| 16884 | Basic truths of logic are not proved, but seen as true when they are understood |
| Full Idea: In Frege's view axioms are basic truth, and basic truths do not need proof. Basic truths can be (justifiably) recognised as true by understanding their content. | |
| From: report of Gottlob Frege (works [1890]) by Tyler Burge - Frege on Knowing the Foundations 1 | |
| A reaction: This is the underpinning of the rationalism in Frege's philosophy. |
| 9462 | Frege is intensionalist about reference, as it is determined by sense; identity of objects comes first |
| Full Idea: Intensionalism of reference is owing to Frege (in his otherwise extensionalist philosophy of language). Sense determines reference, so intension determines extension. An object must first satisfy identity requirements, and is thus in a set. | |
| From: report of Gottlob Frege (On Sense and Reference [1892]) by Dale Jacquette - Intro to 'Philosophy of Logic' §4 | |
| A reaction: The notion that identity of objects comes first sounds right - you can't just take objects as basic - they have to be individuated in order to be discussed. |
| 18936 | Frege moved from extensional to intensional semantics when he added the idea of 'sense' |
| Full Idea: Frege moved from an extensional semantic theory (that countenances only linguistic expressions and their referents) to an intensional theory that invokes in addition a notion of sense. | |
| From: report of Gottlob Frege (On Sense and Reference [1892]) by Sarah Sawyer - Empty Names 2 | |
| A reaction: This was because of Frege's famous 'puzzles', such as the morning/evening star. Quine loudly proclaimed himself an 'extensionalist', implying that he had extensional solutions for Frege's Puzzles. |
| 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). |
| 16886 | The truth of an axiom must be independently recognisable |
| Full Idea: It is part of the concept of an axiom that it can be recognised as true independently of other truths. | |
| From: Gottlob Frege (On Euclidean Geometry [1900], 183/168), quoted by Tyler Burge - Frege on Knowing the Foundations 4 | |
| A reaction: Frege thinks the axioms of arithmetic all reside in logic. |
| 16871 | A truth can be an axiom in one system and not in another |
| Full Idea: It is possible for a truth to be an axiom in one system and not in another. | |
| From: Gottlob Frege (Logic in Mathematics [1914], p.205) | |
| A reaction: Frege aspired to one huge single system, so this is a begrudging concession, one which modern thinkers would probably take for granted. |
| 16866 | Tracing inference backwards closes in on a small set of axioms and postulates |
| Full Idea: We can trace the chains of inference backwards, …and the circle of theorems closes in more and more. ..We must eventually come to an end by arriving at truths can cannot be inferred, …which are the axioms and postulates. | |
| From: Gottlob Frege (Logic in Mathematics [1914], p.204) | |
| A reaction: The rival (more modern) view is that that all theorems are equal in status, and axioms are selected for convenience. |
| 16868 | The essence of mathematics is the kernel of primitive truths on which it rests |
| Full Idea: Science must endeavour to make the circle of unprovable primitive truths as small as possible, for the whole of mathematics is contained in this kernel. The essence of mathematics has to be defined by this kernel of truths. | |
| From: Gottlob Frege (Logic in Mathematics [1914], p.204-5) | |
| A reaction: [compressed] I will make use of this thought, by arguing that mathematics may be 'explained' by this kernel. |
| 16870 | Axioms are truths which cannot be doubted, and for which no proof is needed |
| Full Idea: The axioms are theorems, but truths for which no proof can be given in our system, and no proof is needed. It follows from this that there are no false axioms, and we cannot accept a thought as an axiom if we are in doubt about its truth. | |
| From: Gottlob Frege (Logic in Mathematics [1914], p.205) | |
| A reaction: He struggles to be as objective as possible, but has to concede that whether we can 'doubt' the axiom is one of the criteria. |
| 16869 | To create order in mathematics we need a full system, guided by patterns of inference |
| Full Idea: We cannot long remain content with the present fragmentation [of mathematics]. Order can be created only by a system. But to construct a system it is necessary that in any step forward we take we should be aware of the logical inferences involved. | |
| From: Gottlob Frege (Logic in Mathematics [1914], p.205) |
| 9886 | Cardinals say how many, and reals give measurements compared to a unit quantity |
| Full Idea: The cardinals and the reals are completely disjoint domains. The cardinal numbers answer the question 'How many objects of a given kind are there?', but the real numbers are for measurement, saying how large a quantity is compared to a unit quantity. | |
| From: Gottlob Frege (Grundgesetze der Arithmetik 2 (Basic Laws) [1903], §157), quoted by Michael Dummett - Frege philosophy of mathematics Ch.19 | |
| A reaction: We might say that cardinals are digital and reals are analogue. Frege is unusual in totally separating them. They map onto one another, after all. Cardinals look like special cases of reals. Reals are dreams about the gaps between cardinals. |
| 18256 | Quantity is inconceivable without the idea of addition |
| Full Idea: There is so intimate a connection between the concepts of addition and of quantity that one cannot begin to grasp the latter without the former. | |
| From: Gottlob Frege (Rechnungsmethoden (dissertation) [1874], p.2), quoted by Michael Dummett - Frege philosophy of mathematics 22 'Quantit' | |
| A reaction: Frege offers good reasons for making cardinals prior to ordinals, though plenty of people disagree. |
| 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. |
| 18252 | Real numbers are ratios of quantities, such as lengths or masses |
| Full Idea: If 'number' is the referent of a numerical symbol, a real number is the same as a ratio of quantities. ...A length can have to another length the same ratio as a mass to another mass. | |
| From: Gottlob Frege (Grundgesetze der Arithmetik 1 (Basic Laws) [1893], III.1.73), quoted by Michael Dummett - Frege philosophy of mathematics 21 'Frege's' | |
| A reaction: This is part of a critique of Cantor and the Cauchy series approach. Interesting that Frege, who is in the platonist camp, is keen to connect the real numbers with natural phenomena. He is always keen to keep touch with the application of mathematics. |
| 18253 | I wish to go straight from cardinals to reals (as ratios), leaving out the rationals |
| Full Idea: You need a double transition, from cardinal numbes (Anzahlen) to the rational numbers, and from the latter to the real numbers generally. I wish to go straight from the cardinal numbers to the real numbers as ratios of quantities. | |
| From: Gottlob Frege (Letters to Russell [1902], 1903.05.21), quoted by Michael Dummett - Frege philosophy of mathematics 21 'Frege's' | |
| A reaction: Note that Frege's real numbers are not quantities, but ratios of quantities. In this way the same real number can refer to lengths, masses, intensities etc. |
| 9889 | Real numbers are ratios of quantities |
| Full Idea: Frege fixed on construing real numbers as ratios of quantities (in agreement with Newton). | |
| From: report of Gottlob Frege (Grundgesetze der Arithmetik 2 (Basic Laws) [1903]) by Michael Dummett - Frege philosophy of mathematics Ch.20 | |
| A reaction: If 3/4 is the same real number as 6/8, which is the correct ratio? Why doesn't the square root of 9/16 also express it? Why should irrationals be so utterly different from rationals? In what sense are they both 'numbers'? |
| 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 |
| 17446 | Counting rests on one-one correspondence, of numerals to objects |
| Full Idea: Counting rests itself on a one-one correlation, namely of numerals 1 to n and the objects. | |
| From: Gottlob Frege (Review of Husserl's 'Phil of Arithmetic' [1894]), quoted by Richard G. Heck - Cardinality, Counting and Equinumerosity 3 | |
| A reaction: Parsons observes that counting will establish a one-one correspondence, but that doesn't make it the aim of counting, and so Frege hasn't answered Husserl properly. Which of the two is conceptually prior? How do you decide. |
| 9582 | Husserl rests sameness of number on one-one correlation, forgetting the correlation with numbers themselves |
| Full Idea: When Husserl says that sameness of number can be shown by one-one correlation, he forgets that this counting itself rests on a univocal one-one correlation, namely that between the numerals 1 to n and the objects of the set. | |
| From: Gottlob Frege (Review of Husserl's 'Phil of Arithmetic' [1894], p.326) | |
| A reaction: This is the platonist talking. Neo-logicism is attempting to build numbers just from the one-one correlation of objects. |
| 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. |
| 18271 | We can't prove everything, but we can spell out the unproved, so that foundations are clear |
| Full Idea: It cannot be demanded that everything be proved, because that is impossible; but we can require that all propositions used without proof be expressly declared as such, so that we can see distinctly what the whole structure rests upon. | |
| From: Gottlob Frege (Grundgesetze der Arithmetik 1 (Basic Laws) [1893], p.2), quoted by J. Alberto Coffa - The Semantic Tradition from Kant to Carnap 7 'What' |
| 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. |
| 16864 | If principles are provable, they are theorems; if not, they are axioms |
| Full Idea: If the law [of induction] can be proved, it will be included amongst the theorems of mathematics; if it cannot, it will be included amongst the axioms. | |
| From: Gottlob Frege (Logic in Mathematics [1914], p.203) | |
| A reaction: This links Frege with the traditional Euclidean view of axioms. The question, then, is how do we know them, given that we can't prove them. |
| 17855 | It may be possible to define induction in terms of the ancestral relation |
| Full Idea: Frege's account of the ancestral has made it possible, in effect, to define the natural numbers as entities for which induction holds. | |
| From: report of Gottlob Frege (Begriffsschrift [1879]) by Crispin Wright - Frege's Concept of Numbers as Objects 4.xix | |
| A reaction: This is the opposite of the approach in the Peano Axioms, where induction is used to define the natural numbers. |
| 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? |
| 3331 | If '5' is the set of all sets with five members, that may be circular, and you can know a priori if the set has content |
| Full Idea: There is a suspicion that Frege's definition of 5 (as the set of all sets with 5 members) may be infected with circularity, …and how can we be sure on a priori grounds that 4 and 5 are not both empty sets, and hence identical? | |
| From: comment on Gottlob Frege (works [1890]) by José A. Benardete - Metaphysics: the logical approach Ch.14 |
| 9949 | There is the concept, the object falling under it, and the extension (a set, which is also an object) |
| Full Idea: For Frege, the extension of the concept F is an object, as revealed by the fact that we use a name to refer to it. ..We must distinguish the concept, the object that falls under it, and the extension of the concept, which is the set containing the object. | |
| From: report of Gottlob Frege (On Concept and Object [1892]) by A.George / D.J.Velleman - Philosophies of Mathematics Ch.2 | |
| A reaction: This I take to be the key distinction needed if one is to grasp Frege's account of what a number is. When we say that Frege is a platonist about numbers, it is because he is committed to the notion that the extension is an object. |
| 10623 | Frege defined number in terms of extensions of concepts, but needed Basic Law V to explain extensions |
| Full Idea: Frege opts for his famous definition of numbers in terms of extensions of the concept 'equal to the concept F', but he then (in 'Grundgesetze') needs a theory of extensions or classes, which he provided by means of Basic Law V. | |
| From: report of Gottlob Frege (Grundgesetze der Arithmetik 1 (Basic Laws) [1893]) by B Hale / C Wright - Intro to 'The Reason's Proper Study' §1 |
| 9975 | Frege ignored Cantor's warning that a cardinal set is not just a concept-extension |
| Full Idea: Cantor pointed out explicitly to Frege that it is a mistake to take the notion of a set (i.e. of that which has a cardinal number) to simply mean the extension of a concept. ...Frege's later assumption of this was an act of recklessness. | |
| From: comment on Gottlob Frege (Grundgesetze der Arithmetik 1 (Basic Laws) [1893]) by William W. Tait - Frege versus Cantor and Dedekind III | |
| A reaction: ['recklessness' is on p.61] Tait has no sympathy with the image of Frege as an intellectual martyr. Frege had insufficient respect for a great genius. Cantor, crucially, understood infinity much better than Frege. |
| 9586 | In a number-statement, something is predicated of a concept |
| Full Idea: In a number-statement, something is predicated of a concept. | |
| From: Gottlob Frege (Review of Husserl's 'Phil of Arithmetic' [1894], p.328) | |
| A reaction: A succinct statement of Frege's theory of numbers. By my lights that would make numbers at least second-order abstractions. |
| 10553 | A number is a class of classes of the same cardinality |
| Full Idea: For Frege, in 'Grundgesetze', a number is a class of classes of the same cardinality. | |
| From: report of Gottlob Frege (Grundgesetze der Arithmetik 2 (Basic Laws) [1903]) by Michael Dummett - Frege Philosophy of Language (2nd ed) Ch.14 |
| 10020 | Frege's biggest error is in not accounting for the senses of number terms |
| Full Idea: The inconsistency of Grundgesetze was only a minor flaw. Its fundamental flaw was its inability to account for the way in which the senses of number terms are determined. It leaves the reference-magnetic nature of the standard numberer a mystery. | |
| From: comment on Gottlob Frege (Grundgesetze der Arithmetik 2 (Basic Laws) [1903]) by Harold Hodes - Logicism and Ontological Commits. of Arithmetic p.139 | |
| A reaction: A point also made by Hofweber. As a logician, Frege was only concerned with the inferential role of number terms, and he felt he had captured their logical form, but it is when you come to look at numbers in natural language that he seem in trouble. |
| 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. |
| 9580 | Our concepts recognise existing relations, they don't change them |
| Full Idea: The bringing of an object under a concept is merely the recognition of a relation which previously already obtained, [but in the abstractionist view] objects are essentially changed by the process, so that objects brought under a concept become similar. | |
| From: Gottlob Frege (Review of Husserl's 'Phil of Arithmetic' [1894], p.324) | |
| A reaction: Frege's view would have to account for occasional misapplications of concepts, like taking a dolphin to be a fish, or falsely thinking there is someone in the cellar. |
| 9589 | Numbers are not real like the sea, but (crucially) they are still objective |
| Full Idea: The sea is something real and a number is not; but this does not prevent it from being something objective; and that is the important thing. | |
| From: Gottlob Frege (Review of Husserl's 'Phil of Arithmetic' [1894], p.337) | |
| A reaction: This seems a qualification of Frege's platonism. It is why people start talking about abstract items which 'subsist', instead of 'exist'. It shows Frege's motivation in all this, which is to secure logic and maths from the vagaries of psychology. |
| 9831 | Geometry appeals to intuition as the source of its axioms |
| Full Idea: The elements of all geometrical constructions are intuitions, and geometry appeals to intuition as the source of its axioms. | |
| From: Gottlob Frege (Rechnungsmethoden (dissertation) [1874], Ch.6), quoted by Michael Dummett - Frege philosophy of mathematics | |
| A reaction: Very early Frege, but he stuck to this view, while firmly rejecting intuition as a source of arithmetic. Frege would have known well that Euclid's assumption about parallels had been challenged. |
| 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. |
| 9577 | The naďve view of number is that it is like a heap of things, or maybe a property of a heap |
| Full Idea: The most naďve opinion of number is that it is something like a heap in which things are contained. The next most naďve view is the conception of number as the property of a heap, cleansing the objects of their particulars. | |
| From: Gottlob Frege (Review of Husserl's 'Phil of Arithmetic' [1894], p.323) | |
| A reaction: A hundred toothbrushes and a hundred sponges can be seen to contain the same number (by one-to-one mapping), without actually knowing what that number is. There is something numerical in the heap, even if the number is absent. |
| 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')? |
| 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. |
| 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? |
| 16880 | Frege aimed to discover the logical foundations which justify arithmetical judgements |
| Full Idea: Frege saw arithmetical judgements as resting on a foundation of logical principles, and the discovery of this foundation as a discovery of the nature and structure of the justification of arithmetical truths and judgments. | |
| From: report of Gottlob Frege (works [1890]) by Tyler Burge - Frege on Knowing the Foundations Intro | |
| A reaction: Burge's point is that the logic justifies the arithmetic, as well as underpinning it. |
| 8689 | Eventually Frege tried to found arithmetic in geometry instead of in logic |
| Full Idea: After the problem with Russell's paradox, Frege did not publish for fourteen years, and he then tried to re-found arithmetic in Euclidean geometry, rather than in logic. | |
| From: report of Gottlob Frege (works [1890], 3.4) by Michčle Friend - Introducing the Philosophy of Mathematics 3.4 | |
| A reaction: I take it that his new road would have led him to modern Structuralism, so I think he was probably on the right lines. Unfortunately Frege had already done enough for one good lifetime. |
| 8487 | Arithmetic is a development of logic, so arithmetical symbolism must expand into logical symbolism |
| Full Idea: I am of the opinion that arithmetic is a further development of logic, which leads to the requirement that the symbolic language of arithmetic must be expanded into a logical symbolism. | |
| From: Gottlob Frege (Function and Concept [1891], p.30) | |
| A reaction: This may the the one key idea at the heart of modern analytic philosophy (even though logicism may be a total mistake!). Logic and arithmetical foundations become the master of ontology, instead of the servant. The jury is out on the whole enterprise. |
| 18165 | My Basic Law V is a law of pure logic |
| Full Idea: I hold that my Basic Law V is a law of pure logic. | |
| From: Gottlob Frege (Grundgesetze der Arithmetik 1 (Basic Laws) [1893], p.4), quoted by Penelope Maddy - Naturalism in Mathematics I.1 | |
| A reaction: This is, of course, the notorious law which fell foul of Russell's Paradox. It is said to be pure logic, even though it refers to things that are F and things that are G. |
| 18166 | The loss of my Rule V seems to make foundations for arithmetic impossible |
| Full Idea: With the loss of my Rule V, not only the foundations of arithmetic, but also the sole possible foundations of arithmetic, seem to vanish. | |
| From: Gottlob Frege (Letters to Russell [1902], 1902.06.22) | |
| A reaction: Obviously he was stressed, but did he really mean that there could be no foundation for arithmetic, suggesting that the subject might vanish into thin air? |
| 10607 | Frege's logic has a hierarchy of object, property, property-of-property etc. |
| Full Idea: Frege's general logical system involves a type hierarchy, distinguishing objects from properties from properties-of-properties etc., with every item belonging to a determinate level. | |
| From: report of Gottlob Frege (Begriffsschrift [1879]) by Peter Smith - Intro to Gödel's Theorems 14.1 | |
| A reaction: The Theory of Types went on to apply this hierarchy to classes, where Frege's disastrous Basic Law V flattens the hierarchy of classes, putting them on the same level (Smith p.119) |
| 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. |
| 9545 | Late in life Frege abandoned logicism, and saw the source of arithmetic as geometrical |
| Full Idea: Near the end of his life, Frege completely abandoned his logicism, and came to the conclusion that the source of our arithmetical knowledge is what he called 'the Geometrical Source of Knowledge'. | |
| From: report of Gottlob Frege (Sources of Knowledge of Mathematics [1922]) by Charles Chihara - A Structural Account of Mathematics Intro n3 | |
| A reaction: We have, rather crucially, lost touch with the geometrical origins of arithmetic (such as 'square' numbers), which is good news for the practice of mathematics, but probably a disaster for the philosophy of the subject. |
| 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. |
| 9887 | Formalism misunderstands applications, metatheory, and infinity |
| Full Idea: Frege's three main objections to radical formalism are that it cannot account for the application of mathematics, that it confuses a formal theory with its metatheory, and it cannot explain an infinite sequence. | |
| From: report of Gottlob Frege (Grundgesetze der Arithmetik 2 (Basic Laws) [1903], §86-137) by Michael Dummett - Frege philosophy of mathematics | |
| A reaction: The application is because we don't design maths randomly, but to be useful. The third objection might be dealt with by potential infinities (from formal rules). The second objection sounds promising. |
| 8751 | Only applicability raises arithmetic from a game to a science |
| Full Idea: It is applicability alone which elevates arithmetic from a game to the rank of a science. | |
| From: Gottlob Frege (Grundgesetze der Arithmetik 2 (Basic Laws) [1903], §91), quoted by Stewart Shapiro - Thinking About Mathematics 6.1.2 | |
| A reaction: This is the basic objection to Formalism. It invites the question of why it is applicable, which platonists like Frege don't seem to answer (though Plato himself has reality modelled on the Forms). This is why I like structuralism. |
| 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? |
| 11008 | Existence is not a first-order property, but the instantiation of a property |
| Full Idea: When Kant said that existence was not a property, what he meant was, according to Frege, that existence is not a first-order property - it is not a property of individuals but a property of properties, that the property has an instance. | |
| From: report of Gottlob Frege (Begriffsschrift [1879]) by Stephen Read - Thinking About Logic Ch.5 |
| 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. |
| 19470 | Thoughts in the 'third realm' cannot be sensed, and do not need an owner to exist |
| Full Idea: Thoughts are neither things in the external world nor ideas. A third realm must be recognised. Anything in this realm has it in common with ideas that it cannot be perceived by the senses, and does not need an owner to belong with his consciousness. | |
| From: Gottlob Frege (The Thought: a Logical Enquiry [1918], p.337(69)) | |
| A reaction: This important idea is the creed for modern platonists. We don't have to accept Forms, or any particular content, but there is a mode of existence which is distinct from both mental and physical, and is the residence of 'abstracta'. I deny it! |
| 5657 | Frege's logic showed that there is no concept of being |
| Full Idea: Frege's quantificational logic vindicates Kant's insight that existence is not a predicate and leads to fallacies when treated as one; and we might also say, despite Hegel, that there is no concept of being. | |
| From: report of Gottlob Frege (works [1890]) by Roger Scruton - Short History of Modern Philosophy Ch.17 | |
| A reaction: I notice that Colin McGinn has questioned the value of quantificational logic. It is difficult to assert that 'there is no concept of x', if several people have written large books about it. |
| 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. |
| 18899 | Frege takes the existence of horses to be part of their concept |
| Full Idea: Frege regarded the existence of horses as a property of the concept 'horse'. | |
| From: report of Gottlob Frege (Function and Concept [1891]) by Fred Sommers - Intellectual Autobiography 'Realism' |
| 18995 | Frege mistakenly takes existence to be a property of concepts, instead of being about things |
| Full Idea: Frege's theory treats existence as a property, not of things we call existent, but of concepts instantiated by those things. 'Biden exists' says our Biden-concept has instances. That is certainly not how it feels! We speak of the thing, not of concepts. | |
| From: report of Gottlob Frege (On Concept and Object [1892]) by Stephen Yablo - Aboutness 01.4 | |
| A reaction: Yablo's point is that you must ask what the sentence is 'about', and then the truth will refer to those things. Frege gets into a tangle because he thinks remarks using concepts are about the concepts. |
| 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. |
| 9578 | If objects are just presentation, we get increasing abstraction by ignoring their properties |
| Full Idea: If an object is just presentation, we can pay less attention to a property and it disappears. By letting one characteristic after another disappear, we obtain concepts that are increasingly more abstract. | |
| From: Gottlob Frege (Review of Husserl's 'Phil of Arithmetic' [1894], p.324) | |
| A reaction: Frege despises this view. Note there is scope in the despised view for degrees or levels of abstraction, defined in terms of number of properties ignored. Part of Frege's criticism is realist. He retains the object, while Husserl imagines it different. |
| 19471 | A fact is a thought that is true |
| Full Idea: A fact is a thought that is true. | |
| From: Gottlob Frege (The Thought: a Logical Enquiry [1918], p.342(74)) | |
| A reaction: It strikes me as pretty obvious that facts are not thoughts, because they concern the contents of thoughts. You can't discuss facts without the notion of what a thought is 'about'. If I think about my garden, the relevant fact is aspects of my garden. |
| 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? |
| 4028 | Frege allows either too few properties (as extensions) or too many (as predicates) |
| Full Idea: Frege's theory of properties (which he calls 'concepts') yields too few properties, by identifying coextensive properties, and also too many, by letting every predicate express a property. | |
| From: comment on Gottlob Frege (Function and Concept [1891]) by DH Mellor / A Oliver - Introduction to 'Properties' §2 | |
| A reaction: Seems right; one extension may have two properties (have heart/kidneys), two predicates might express the same property. 'Cutting nature at the joints' covers properties as well as objects. |
| 10317 | It is unclear whether Frege included qualities among his abstract objects |
| Full Idea: Expositors of Frege's views have disagreed over whether abstract qualities are to be reckoned among his objects. | |
| From: report of Gottlob Frege (On Concept and Object [1892]) by Bob Hale - Abstract Objects Ch.2.II | |
| A reaction: [he cites Dummett 1973:70-80, and Wright 1983:25-8] There seems to be a danger here of a collision between Fregean verbal approaches to ontological commitment and the traditional views about universals. No wonder they can't decide. |
| 10533 | We can't get a semantics from nouns and predicates referring to the same thing |
| Full Idea: Frege is denying that on a traditional basis we can construct a workable semantics for a language; we can't regard terms like 'wisdom' as standing for the very same thing as the predicate 'x is wise' stands for. | |
| From: report of Gottlob Frege (On Sense and Reference [1892]) by Michael Dummett - Frege Philosophy of Language (2nd ed) Ch.14 | |
| A reaction: This follows from Idea 10532, indicating how to deal with the problem of universals. So predicates refer to concepts, and singular terms to objects. But I see no authoritative way of deciding which is which, given that paraphrases are possible. |
| 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. |
| 18269 | Logical objects are extensions of concepts, or ranges of values of functions |
| Full Idea: How are we to conceive of logical objects? My only answer is, we conceive of them as extensions of concepts or, more generally, as ranges of values of functions ...what other way is there? | |
| From: Gottlob Frege (Letters to Russell [1902], 1902.07.28), quoted by J. Alberto Coffa - The Semantic Tradition from Kant to Carnap 7 epigr | |
| A reaction: This is the clearest statement I have found of what Frege means by an 'object'. But an extension is a collection of things, so an object is a group treated as a unity, which is generally how we understand a 'set'. Hence Quine's ontology. |
| 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. |
| 8489 | The concept 'object' is too simple for analysis; unlike a function, it is an expression with no empty place |
| Full Idea: I regard a regular definition of 'object' as impossible, since it is too simple to admit of logical analysis. Briefly: an object is anything that is not a function, so that an expression for it does not contain any empty place. | |
| From: Gottlob Frege (Function and Concept [1891], p.32) | |
| A reaction: Here is the core of the programme for deriving our ontology from our logic and language, followed through by Russell and Quine. Once we extend objects beyond the physical, it becomes incredibly hard to individuate them. |
| 10535 | Frege's 'objects' are both the referents of proper names, and what predicates are true or false of |
| Full Idea: Frege's notion of an object plays two roles in his semantics. Objects are the referents of proper names, and they are equally what predicates are true and false of. | |
| From: report of Gottlob Frege (On Concept and Object [1892]) by Michael Dummett - Frege Philosophy of Language (2nd ed) Ch.4 | |
| A reaction: Frege is the source of a desperate desire to turn everything into an object (see Idea 8858!), and he has the irritating authority of the man who invented quantificational logic. Nothing but trouble, that man. |
| 9877 | Late Frege saw his non-actual objective objects as exclusively thoughts and senses |
| Full Idea: Earlier, Frege divided objects into subjective, actual objective, and non-actual objective; in the 'Grundgesetze' he emphasised logical objects; but in 'The Thought' the non-actual objects become exclusively thoughts and their constituent senses. | |
| From: report of Gottlob Frege (The Thought: a Logical Enquiry [1918]) by Michael Dummett - Frege philosophy of mathematics Ch.18 | |
| A reaction: Sounds to me like Frege was finally waking up and taking a dose of common sense. The Equator is the standard example of a non-actual objective object. |
| 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'. |
| 9891 | The first demand of logic is of a sharp boundary |
| Full Idea: The first demand of logic is of a sharp boundary. | |
| From: Gottlob Frege (Grundgesetze der Arithmetik 2 (Basic Laws) [1903], §160), quoted by Michael Dummett - Frege philosophy of mathematics Ch.22 | |
| A reaction: Nothing I have read about vagueness has made me doubt Frege's view of this, although precisification might allow you to do logic with vague concepts without having to finally settle where the actual boundaries are. |
| 9388 | Every concept must have a sharp boundary; we cannot allow an indeterminate third case |
| Full Idea: Of any concept, we must require that it have a sharp boundary. Of any object it must hold either that it falls under the concept or it does not. We may not allow a third case in which it is somehow indeterminate whether an object falls under a concept. | |
| From: Gottlob Frege (Logic in Mathematics [1914], p.229), quoted by Ian Rumfitt - The Logic of Boundaryless Concepts p.1 n1 | |
| A reaction: This is the voice of the classical logician, which has echoed by Russell. I'm with them, I think, in the sense that logic can only work with precise concepts. The jury is still out. Maybe we can 'precisify', without achieving total precision. |
| 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. |
| 4893 | Frege was asking how identities could be informative |
| Full Idea: A problem which Frege called to our attention is: how can identities be informative? | |
| From: report of Gottlob Frege (On Sense and Reference [1892]) by John Perry - Knowledge, Possibility and Consciousness §5.2 | |
| A reaction: E.g. (in Russell's example) how is "Scott is the author of 'Waverley'" more informative than "Scott is Scott"? A simple answer might just be that informative identities also tell you of a thing's properties. "The red ball is the heavy ball". |
| 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. |
| 3318 | Frege made identity a logical notion, enshrined above all in the formula 'for all x, x=x' |
| Full Idea: It was Frege who first made identity a logical notion, enshrining it above all in the formula (x) x=x. | |
| From: report of Gottlob Frege (works [1890]) by José A. Benardete - Metaphysics: the logical approach Ch.9 |
| 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. |
| 16885 | To understand a thought, understand its inferential connections to other thoughts |
| Full Idea: Frege famously realised that understanding a thought requires understanding its inferential connections to other thoughts. | |
| From: report of Gottlob Frege (works [1890]) by Tyler Burge - Frege on Knowing the Foundations 1 | |
| A reaction: If true, this is probably our greatest advance in grasping the concept of 'understanding' since Aristotle - but is it true? It is a striking and interesting idea, and central to the importance of Frege in modern analytic philosophy. |
| 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. |
| 16887 | Frege's concept of 'self-evident' makes no reference to minds |
| Full Idea: Frege's terms that translate 'self-evident' usually make no explicit reference to actual minds. | |
| From: report of Gottlob Frege (works [1890]) by Tyler Burge - Frege on Knowing the Foundations 4 | |
| A reaction: This follows the distinction in Aquinas, between things that are intrinsically self-evident, and things that are self-evident to particular people. God, presumably, knows all of the former. |
| 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. |
| 16894 | An apriori truth is grounded in generality, which is universal quantification |
| Full Idea: Generality for Frege is simply universal quantification; what makes a truth apriori is that its ultimate grounds are universally quantified. | |
| From: report of Gottlob Frege (works [1890]) by Tyler Burge - Frege on Apriority (with ps) 2 |
| 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. |
| 11052 | Psychological logic can't distinguish justification from causes of a belief |
| Full Idea: With the psychological conception of logic we lose the distinction between the grounds that justify a conviction and the causes that actually produce it. | |
| From: Gottlob Frege (Logic [1897] [1897]) | |
| A reaction: Thus Frege kicked the causal theory of justification well into touch long before it had even been properly formulated. That is not to say that there is no psychological aspect to logic, because there is. |
| 16882 | The building blocks contain the whole contents of a discipline |
| Full Idea: The ultimate building blocks of a discipline contain, as it were in a nutshell, its whole contents. | |
| From: Gottlob Frege (works [1890]), quoted by Tyler Burge - Frege on Knowing the Foundations 1 | |
| A reaction: [Burge gives a reference] I would describe this nutshell as the 'essence' of the subject, and it fits Aristotle's concept of an essence perfectly. Does it fit biology or sociology, in the way it might fit maths or logic? Think of DNA or cells in biology. |
| 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. |
| 9581 | Many people have the same thought, which is the component, not the private presentation |
| Full Idea: The same thought can be grasped by many people. The components of a thought, and even more so the things themselves, must be distinguished from the presentations which in the soul accompany the grasping of a thought. | |
| From: Gottlob Frege (Review of Husserl's 'Phil of Arithmetic' [1894], p.325) | |
| A reaction: This is the basic realisation, also found in Russell, of how so much confusion has crept into philosophy, in Berkeley, for example. Frege starts down the road which leads to the externalist view of content. |
| 19469 | We grasp thoughts (thinking), decide they are true (judgement), and manifest the judgement (assertion) |
| Full Idea: We distinguish the grasp of a thought, which is 'thinking', from the acknowledgement of the truth of a thought, which is the act of 'judgement', from the manifestation of this judgement, which is an 'assertion'. | |
| From: Gottlob Frege (The Thought: a Logical Enquiry [1918], p.329 (62)) |
| 8162 | Thoughts have their own realm of reality - 'sense' (as opposed to the realm of 'reference') |
| Full Idea: For Frege, thoughts belong to a special realm of reality, which he called the 'realm of sense' and distinguished from the 'realm of reference'. | |
| From: report of Gottlob Frege (The Thought: a Logical Enquiry [1918]) by Michael Dummett - Thought and Reality 1 | |
| A reaction: A thought is, for Frege, a proposition. There is a halfway Platonism possible here, where the 'realm' for such things exists, but within that realm the objects might be conventional, or some such. Real possible worlds containing fictions! |
| 9818 | A thought is distinguished from other things by a capacity to be true or false |
| Full Idea: On Frege's view, what distinguishes thoughts from everything else is that they may meaningfully be called 'true' and 'false'. | |
| From: report of Gottlob Frege (The Thought: a Logical Enquiry [1918]) by Michael Dummett - Frege philosophy of mathematics Ch.2 | |
| A reaction: A lot of thinking is imagistic, and while the image may or may not truly picture the world, we tend to think that the truth or otherwise of daydreaming is simply irrelevant. Does Frege take all thought to be propositional? |
| 18265 | We don't judge by combining subject and concept; we get a concept by splitting up a judgement |
| Full Idea: Instead of putting a judgement together out of an individual as subject and an already previously formed concept as predicate, we do the opposite and arrive at a concept by splitting up the content of possible judgement. | |
| From: Gottlob Frege (Boole calculus and the Concept script [1881], p.17) | |
| A reaction: This is behind holistic views of sentences, and hence of whole languages, and behind Quine's rejection of 'properties' inferred from the predicates in judgements. |
| 16379 | Thoughts about myself are understood one way to me, and another when communicated |
| Full Idea: When Dr Lauben thinks he has been wounded, ..only Dr Lauben can grasp thoughts determined in this way. But he cannot communicate a thought which only he can grasp. To say 'I have been wounded' he must use 'I' in a sense graspable by others. | |
| From: Gottlob Frege (The Thought: a Logical Enquiry [1918]), quoted by François Recanati - Mental Files 16.1 | |
| A reaction: [compressed] This seems to be the first, and very influential, attempt to explain the unusual and revealing semantics of indexicals. It seems to be the ultimate source of 2-D semantics, by introducing two modes of meaning for one term. |
| 16876 | We need definitions to cram retrievable sense into a signed receptacle |
| Full Idea: If we need such signs, we also need definitions so that we can cram this sense into the receptacle and also take it out again. | |
| From: Gottlob Frege (Logic in Mathematics [1914], p.209) | |
| A reaction: Has anyone noticed that Frege is the originator of the idea of the mental file? Has anyone noticed the role that definition plays in his account? |
| 16875 | We use signs to mark receptacles for complex senses |
| Full Idea: We often need to use a sign with which we associate a very complex sense. Such a sign seems a receptacle for the sense, so that we can carry it with us, while being always aware that we can open this receptacle should we need what it contains. | |
| From: Gottlob Frege (Logic in Mathematics [1914], p.209) | |
| A reaction: This exactly the concept of a mental file, which I enthusiastically endorse. Frege even talks of 'opening the receptacle'. For Frege a definition (which he has been discussing) is the assigment of a label (the 'definiendum') to the file (the 'definiens'). |
| 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. |
| 9947 | Concepts are the ontological counterparts of predicative expressions |
| Full Idea: Concepts, for Frege, are the ontological counterparts of predicative expressions. | |
| From: report of Gottlob Frege (Function and Concept [1891]) by A.George / D.J.Velleman - Philosophies of Mathematics Ch.2 | |
| A reaction: That sounds awfully like what many philosophers call 'universals'. Frege, as a platonist (at least about numbers), I would take to be in sympathy with that. At least we can say that concepts seem to be properties. |
| 10319 | An assertion about the concept 'horse' must indirectly speak of an object |
| Full Idea: Frege had a notorious difficulty over the concept 'horse', when he suggests that if we wish to assert something about a concept, we are obliged to proceed indirectly by speaking of an object that represents it. | |
| From: report of Gottlob Frege (Function and Concept [1891], Ch.2.II) by Bob Hale - Abstract Objects | |
| A reaction: This sounds like the thin end of a wedge. The great champion of objects is forced to accept them here as a façon de parler, when elsewhere they have ontological status. |
| 8488 | A concept is a function whose value is always a truth-value |
| Full Idea: A concept in logic is closely connected with what we call a function. Indeed, we may say at once: a concept is a function whose value is always a truth-value. ..I give the name 'function' to what is meant by the 'unsaturated' part. | |
| From: Gottlob Frege (Function and Concept [1891], p.30) | |
| A reaction: So a function becomes a concept when the variable takes a value. Problems arise when the value is vague, or the truth-value is indeterminable. |
| 18752 | 'The concept "horse"' denotes a concept, yet seems also to denote an object |
| Full Idea: The phrase 'the concept "horse"' can be the subject of a sentence, and ought to denote an object. But it clearly denotes the concept "horse". Yet Fregean concepts are said to be 'incomplete' objects, which led to confusion. | |
| From: report of Gottlob Frege (On Sense and Reference [1892]) by Vann McGee - Logical Consequence 4 | |
| A reaction: This is the notorious 'concept "horse"' problem, which was bad news for Frege's idea of a concept. |
| 9839 | Frege equated the concepts under which an object falls with its properties |
| Full Idea: Frege equated the concepts under which an object falls with its properties. | |
| From: report of Gottlob Frege (On Concept and Object [1892], p.201) by Michael Dummett - Frege philosophy of mathematics Ch.8 | |
| A reaction: I take this to be false, as objects can fall under far more concepts than they have properties. I don't even think 'being a pencil' is a property of pencils, never mind 'being my favourite pencil', or 'not being Alexander the Great'. |
| 9190 | A concept is a function mapping objects onto truth-values, if they fall under the concept |
| Full Idea: In later Frege, a concept could be taken as a particular case of a function, mapping every object on to one of the truth-values (T or F), according as to whether, as we should ordinarily say, that object fell under the concept or not. | |
| From: report of Gottlob Frege (Grundgesetze der Arithmetik 1 (Basic Laws) [1893]) by Michael Dummett - The Philosophy of Mathematics 3.5 | |
| A reaction: As so often in these attempts at explanation, this sounds circular. You can't decide whether an object truly falls under a concept, if you haven't already got the concept. His troubles all arise (I say) because he scorns abstractionist accounts. |
| 13665 | Frege took the study of concepts to be part of logic |
| Full Idea: Frege took the study of concepts and their extensions to be within logic. | |
| From: report of Gottlob Frege (Grundgesetze der Arithmetik 1 (Basic Laws) [1893]) by Stewart Shapiro - Foundations without Foundationalism 7.1 | |
| A reaction: This is part of the plan to make logic a universal language (see Idea 13664). I disagree with this, and with the general logicist view of the position of logic. The logical approach thins concepts out. See Deleuze/Guattari's horror at this. |
| 9948 | Unlike objects, concepts are inherently incomplete |
| Full Idea: For Frege, concepts differ from objects in being inherently incomplete in nature. | |
| From: report of Gottlob Frege (Function and Concept [1891]) by A.George / D.J.Velleman - Philosophies of Mathematics Ch.2 | |
| A reaction: This is because they are 'unsaturated', needing a quantified variable to complete the sentence. This could be a pointer towards Quine's view of properties, as simply an intrinsic feature of predication about objects, with no separate identity. |
| 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. |
| 4973 | As I understand it, a concept is the meaning of a grammatical predicate |
| Full Idea: As I understand it, a concept is the meaning of a grammatical predicate. | |
| From: Gottlob Frege (On Concept and Object [1892], p.193) | |
| A reaction: All the ills of twentieth century philosophy reside here, because it makes a concept an entirely linguistic thing, so that animals can't have concepts, and language is cut off from reality, leading to relativism, pragmatism, and other nonsense. |
| 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'. |
| 9579 | Disregarding properties of two cats still leaves different objects, but what is now the difference? |
| Full Idea: If from a black cat and a white cat we disregard colour, then posture, then location, ..we finally derive something which is completely without restrictions on content; but what is derived from the objects does differ, although it is not easy to say how. | |
| From: Gottlob Frege (Review of Husserl's 'Phil of Arithmetic' [1894], p.324) | |
| A reaction: This is a key objection to abstractionism for Frege - we are counting two cats, not two substrata of essential catness, or whatever. But what makes a cat countable? (Key question!) It isn't its colour, or posture or location. |
| 9587 | How do you find the right level of inattention; you eliminate too many or too few characteristics |
| Full Idea: Inattention is a very strong lye which must not be too concentrated, or it dissolves everything (such as the connection between the objects), but must not be too weak, to produce sufficient change. Personally I cannot find the proper dilution. | |
| From: Gottlob Frege (Review of Husserl's 'Phil of Arithmetic' [1894], p.330) | |
| A reaction: We may sympathise with the lack of precision here (frustrating for a logician), but it is not difficult to say of a baseball defence 'just concentrate on the relations, and ignore the individuals who implement it'. You retain basic baseball skills. |
| 9890 | The modern account of real numbers detaches a ratio from its geometrical origins |
| Full Idea: From geometry we retain the interpretation of a real number as a ratio of quantities or measurement-number; but in more recent times we detach it from geometrical quantities, and from all particular types of quantity. | |
| From: Gottlob Frege (Grundgesetze der Arithmetik 2 (Basic Laws) [1903], §159), quoted by Michael Dummett - Frege philosophy of mathematics | |
| A reaction: Dummett glosses the 'recent' version as by Cantor and Dedekind in 1872. This use of 'detach' seems to me startlingly like the sort of psychological abstractionism which Frege was so desperate to avoid. |
| 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? |
| 5816 | Frege said concepts were abstract entities, not mental entities |
| Full Idea: Frege, rebelling against 'psychologism', identified concepts (and hence 'intensions' or meanings) with abstract entities rather than mental entities. | |
| From: report of Gottlob Frege (works [1890]) by Hilary Putnam - Meaning and Reference p.119 | |
| A reaction: This, of course, assumes that 'abstract' entities and 'mental' entities are quite distinct things. A concept is presumably a mental item which has content, and the word 'concept' is simply ambiguous, between the container and the contents. |
| 9588 | Number-abstraction somehow makes things identical without changing them! |
| Full Idea: Number-abstraction simply has the wonderful and very fruitful property of making things absolutely the same as one another without altering them. Something like this is possible only in the psychological wash-tub. | |
| From: Gottlob Frege (Review of Husserl's 'Phil of Arithmetic' [1894], p.332) | |
| A reaction: Frege can be awfully sarcastic. I don't really see his difficulty. For mathematics we only need to know what is countable about an object - we don't need to know how many hairs there are on the cat, only that it has identity. |
| 11846 | If we abstract the difference between two houses, they don't become the same house |
| Full Idea: If abstracting from the difference between my house and my neighbour's, I were to regard both houses as mine, the defect of the abstraction would soon be made clear. It may, though, be possible to obtain a concept by means of abstraction... | |
| From: Gottlob Frege (Grundgesetze der Arithmetik 2 (Basic Laws) [1903], §99) | |
| A reaction: Note the important concession at the end, which shows Frege could never deny the abstraction process, despite all the modern protests by Geach and Dummett that he totally rejected it. |
| 9167 | Frege felt that meanings must be public, so they are abstractions rather than mental entities |
| Full Idea: Frege felt that meanings are public property, and identified concepts (and hence 'intensions' or meanings) with abstract entities rather than mental entities. | |
| From: report of Gottlob Frege (On Concept and Object [1892]) by Hilary Putnam - Meaning and Reference p.150 | |
| A reaction: This is the germ of Wittgenstein's private language argument. I am inclined to feel that Frege approached language strictly as a logician, and didn't really care that he got himself into implausible platonist ontological commitments. |
| 9583 | Psychological logicians are concerned with sense of words, but mathematicians study the reference |
| Full Idea: The psychological logicians are concerned with the sense of the words and with the presentations, which they do not distinguish from the sense; but the mathematicians are concerned with the matter itself, with the reference of the words. | |
| From: Gottlob Frege (Review of Husserl's 'Phil of Arithmetic' [1894], p.326) | |
| A reaction: This is helpful for showing the point of his sense/reference distinction; it is part of his campaign against psychologism, by showing that there is a non-psychological component to language - the reference, where it meets the public world. |
| 9584 | Identity baffles psychologists, since A and B must be presented differently to identify them |
| Full Idea: The relation of sameness remains puzzling to a psychological logician. They cannot say 'A is the same as B', because that requires distinguishing A from B, so that these would have to be different presentations. | |
| From: Gottlob Frege (Review of Husserl's 'Phil of Arithmetic' [1894], p.327) | |
| A reaction: This is why Frege needed the concept of reference, so that identity could be outside the mind (as in Hesperus = Phosophorus). Think about an electron; now think about a different electron. |
| 7307 | A thought is not psychological, but a condition of the world that makes a sentence true |
| Full Idea: For Frege, a thought is not something psychological or subjective; rather, it is objective in the sense that it specifies some condition in the world the obtaining of which is necessary and sufficient for the truth of the sentence that expresses it. | |
| From: report of Gottlob Frege (works [1890]) by Alexander Miller - Philosophy of Language 2.2 | |
| A reaction: It is worth emphasising Russell's anti-Berkeley point about 'ideas', that the idea is in the mind, but its contents are in the world. Since the contents are what matter, this endorses Frege, and also points towards modern externalism. |
| 4980 | The meaning (reference) of a sentence is its truth value - the circumstance of it being true or false |
| Full Idea: We are driven into accepting the truth-value of a sentence as constituting what it means (refers to). By the truth-value I understand the circumstance that it is true or false. | |
| From: Gottlob Frege (On Sense and Reference [1892], p.34) | |
| A reaction: Sounds bizarre, but Black's translation doesn't help. The notion of what the whole sentence refers to (rather than its sense) is a very theoretical notion. 'All true sentences refer to the truth' sounds harmless enough. |
| 22318 | Frege failed to show when two sets of truth-conditions are equivalent |
| Full Idea: Frege's account suffered from a lack of precision about when two sets of truth-conditions should count as equivalent. (Wittgenstein aimed to rectify this defect). | |
| From: report of Gottlob Frege (On Sense and Reference [1892]) by Michael Potter - The Rise of Analytic Philosophy 1879-1930 50 Intro |
| 16879 | A sign won't gain sense just from being used in sentences with familiar components |
| Full Idea: No sense accrues to a sign by the mere fact that it is used in one or more sentences, the other constituents of which are known. | |
| From: Gottlob Frege (Logic in Mathematics [1914], p.213) | |
| A reaction: Music to my ears. I've never grasped how meaning could be grasped entirely through use. |
| 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. |
| 8446 | We understand new propositions by constructing their sense from the words |
| Full Idea: The possibility of our understanding propositions which we have never heard before rests on the fact that we construct the sense of a proposition out of parts that correspond to words. | |
| From: Gottlob Frege (Letters to Jourdain [1910], p.43) | |
| A reaction: This is the classic statement of the principle of compositionality, which seems to me so obviously correct that I cannot understand anyone opposing it. Which comes first, the thought or the word, may be a futile debate. |
| 9180 | Holism says all language use is also a change in the rules of language |
| Full Idea: Frege thought of a language as a game played with fixed rules, there being all the difference in the world between a move in the game and an alteration of the rules; but, if holism is correct, every move in the game changes the rules. | |
| From: report of Gottlob Frege (On Sense and Reference [1892]) by Michael Dummett - Frege's Distinction of Sense and Reference p.248 | |
| A reaction: Rules do shift over time, so there must be some mechanism for that - the rules can't sit in sacrosanct isolation. People play games with the language itself, as well as using it to play other games. |
| 4981 | The reference of a word should be understood as part of the reference of the sentence |
| Full Idea: I have transferred the relation between the parts and the whole of the sentence to its reference, by calling the reference of the word part of the reference of the sentence, if the word itself is part of the sentence. | |
| From: Gottlob Frege (On Sense and Reference [1892], p.35) | |
| A reaction: Since Frege says the reference of a true sentence is simply to truth, words have reference insofar as they make contributions to attempts at stating truths. |
| 15597 | Frege's Puzzle: from different semantics we infer different reference for two names with the same reference |
| Full Idea: Frege's Puzzle: If two sentences convey different information, they have different semantic roles, so the names 'Cicero' and 'Tully' are semantically different, in which case they are referentially different - but they are not referentially different. | |
| From: report of Gottlob Frege (On Sense and Reference [1892]) by Kit Fine - Semantic Relationism 2.A | |
| A reaction: [this is my summary of Fine's summary] Given the paradox, the question is which of these premisses should be challenged. Fregeans reject their being referentially different. Referentialists reject the different semantic roles. |
| 17002 | Frege's 'sense' is ambiguous, between the meaning of a designator, and how it fixes reference |
| Full Idea: Frege should be criticised for using the term 'sense' in two senses. He takes the sense of a designator to be its meaning; and he also takes it to be the way its reference is determined. …They correspond to two ordinary uses of 'definition'. | |
| From: comment on Gottlob Frege (On Sense and Reference [1892]) by Saul A. Kripke - Naming and Necessity lectures Lecture 1 | |
| A reaction: Stalnaker quotes this, but seems unconvinced that Frege is guilty. If the 'meaning' largely consists of a way of determining a reference, Frege would be in the clear. |
| 18778 | Every descriptive name has a sense, but may not have a reference |
| Full Idea: It may perhaps be granted that every grammatically well-formed expression representing a proper name always has a sense. But this is not to say that to this sense there also corresponds a reference. | |
| From: Gottlob Frege (On Sense and Reference [1892]), quoted by Bernard Linsky - Quantification and Descriptions 3.1 | |
| A reaction: Presumably this concerns fictional names such as 'Pegasus'. It seems to be good simple evidence for the distinction between sense and reference. |
| 7805 | Frege started as anti-realist, but the sense/reference distinction led him to realism |
| Full Idea: In the Grundlagen of 1884 Frege was an anti-realist, but in Grundgesetze of 1893 he is a realist, who has profited by his interim discovery of the sense/reference distinction. | |
| From: report of Gottlob Frege (On Sense and Reference [1892]) by José A. Benardete - Logic and Ontology | |
| A reaction: This is the germ of the new realist philosophy which seems to be growing out of Kripke and co's causal theory of reference. The very notion of reference is realist (hence Russell's realism). |
| 4976 | The meaning (reference) of 'evening star' is the same as that of 'morning star', but not the sense |
| Full Idea: The meaning (reference) of 'evening star' is the same as that of 'morning star', but not the sense. | |
| From: Gottlob Frege (On Sense and Reference [1892], p.27) | |
| A reaction: Max Black translates 'bedeutung' as 'meaning', but nowadays everyone calls it 'reference'. This is Frege's crucial distinction, which greatly clarified analytical philosophy. Nevertheless, is it a sharp distinction? E.g. referring to a fictional name? |
| 4977 | In maths, there are phrases with a clear sense, but no actual reference |
| Full Idea: The expression 'the least rapidly convergent series' has a sense but demonstrably there is no reference, since a less rapidly convergent series (for any given series) can always be found. | |
| From: Gottlob Frege (On Sense and Reference [1892], p.28) | |
| A reaction: A nice example. 'The second Kennedy assassin' has a clear meaning, but does it have a reference? The meaning 'points at' a possible reference. We yet discover an identity. |
| 4979 | We are driven from sense to reference by our desire for truth |
| Full Idea: The striving for truth drives us always to advance from the sense to the thing meant (the reference). | |
| From: Gottlob Frege (On Sense and Reference [1892], p.33) | |
| A reaction: As in, we want to know the reference of 'the person who shot Kennedy'. I always perk up if truth is mentioned in a discussion of language, because it reminds us of the point of the whole thing. In 'Is he the best man?' I have the reference, not the truth. |
| 8449 | Senses can't be subjective, because propositions would be private, and disagreement impossible |
| Full Idea: If the sense of a name was subjective, then the proposition and the thought would be subjective; the thought one man connects with this proposition would be different from that of another man. One man could not then contradict another. | |
| From: Gottlob Frege (Letters to Jourdain [1910], p.44) | |
| A reaction: This is an implicit argument for the identity of 'proposition' and 'thought'. This argument resembles Plato's argument for universals (Idea 223). See also Kant on existence as a predicate (Idea 4475). But people do misunderstand one another. |
| 15155 | Expressions always give ways of thinking of referents, rather than the referents themselves |
| Full Idea: For Frege, expressions always contribute ways of thinking of their referents, rather than the referents themselves, to the thoughts expressed by sentences. | |
| From: report of Gottlob Frege (On Sense and Reference [1892]) by Scott Soames - Philosophy of Language 1.16 | |
| A reaction: I have some sympathy for Frege. It always strikes me as daft to think that if I say 'my dustbin is empty', the dustbin becomes 'part' of my sentence. Sentences don't contain large plastic objects. |
| 4972 | I may regard a thought about Phosphorus as true, and the same thought about Hesperus as false |
| Full Idea: From sameness of meaning there does not follow sameness of thought expressed. A fact about the Morning Star may express something different from a fact about the Evening Star, as someone may regard one as true and the other false. | |
| From: Gottlob Frege (Function and Concept [1891], p.14) | |
| A reaction: This all gets clearer if we distinguish internalist and externalist theories of content. Why take sides on this? Why not just ask 'what is in the speaker's head?', 'what does the sentence mean in the community?', and 'what is the corresponding situation?' |
| 22280 | Frege's account was top-down and decompositional, not bottom-up and compositional |
| Full Idea: Frege's account was top-down, not bottom-up: he aimed to decompose and discern function-argument structure in already existing sentences, not to explain how those sentences acquired their meanings in the first place. | |
| From: report of Gottlob Frege (Begriffsschrift [1879]) by Michael Potter - The Rise of Analytic Philosophy 1879-1930 03 'Func' | |
| A reaction: This goes with the holistic account of meaning, which leads to Quine's gavagai and Kuhn's obfuscation of science. I recommend compositionality for everthing. |
| 7309 | Frege's 'sense' is the strict and literal meaning, stripped of tone |
| Full Idea: Frege held that "and" and "but" have the same 'sense' but different 'tones' (note: they have the same truth tables); the sense of an expression is what a sentence strictly and literally means, stripped of its tone. | |
| From: report of Gottlob Frege (works [1890]) by Alexander Miller - Philosophy of Language 2.6 | |
| A reaction: It seems important when studying Frege to remember what has been stripped out. In "he is a genius and he plays football", if you substitute 'but' for 'and', the new version says (literally?) something very distinctive about football. |
| 7312 | 'Sense' solves the problems of bearerless names, substitution in beliefs, and informativeness |
| Full Idea: Frege's introduction of 'sense' was motivated by the desire to solve three problems: the problem of bearerless names, the problem of substitution in belief contexts, and the problem of informativeness. | |
| From: report of Gottlob Frege (works [1890]) by Alexander Miller - Philosophy of Language 2.9 | |
| A reaction: A proposal which solves three problems sounds pretty good! These three problems can be used to test the counter-proposals of Russell and Kripke. |
| 11126 | 'Sense' gives meaning to non-referring names, and to two expressions for one referent |
| Full Idea: Frege notes that an expression without a referent ('Pegasus') needn't lack a meaning, since it still has a sense, and the same referent (Eric Blair) can be associated with different expressions (George Orwell) because they convey different senses. | |
| From: report of Gottlob Frege (On Sense and Reference [1892]) by E Margolis/S Laurence - Concepts 1.3 | |
| A reaction: A nice neat summary of the value of Frege's introduction of the sense/reference distinction, which seems to me to be virtually undeniable (a rare event in modern philosophy). |
| 8164 | Frege was the first to construct a plausible theory of meaning |
| Full Idea: Frege was the first to construct a plausible theory of meaning, that is, a theory of how a human language functions. | |
| From: report of Gottlob Frege (On Sense and Reference [1892]) by Michael Dummett - Thought and Reality 1 | |
| A reaction: Presumably Frege had an advantage because he was the first to distinguish sense from reference, and hence to identify the subject-matter of the theory. Essentially Frege's theory is that of truth-conditions. |
| 9817 | Earlier Frege focuses on content itself; later he became interested in understanding content |
| Full Idea: Earlier Frege was interested solely in the content of our statements, not in our grasp of that content. His notion of 'sense' from 1891 onwards, has to do with understanding; the sense of an expression is something we grasp. | |
| From: report of Gottlob Frege (On Sense and Reference [1892]) by Michael Dummett - Frege philosophy of mathematics Ch.2 | |
| A reaction: The important point must be that the later theory depends on the earlier, so we can hardly give theories of understanding, if we don't have a view about what it is that is understood. |
| 8171 | Frege divided the meaning of a sentence into sense, force and tone |
| Full Idea: Frege distinguished three components of the meaning of a sentence: sense, force and tone; he used no single term for 'linguistic meaning' in general. ...The sense is only what bears on the truth or falsity of what the sentence expresses. | |
| From: report of Gottlob Frege (On Sense and Reference [1892]) by Michael Dummett - Thought and Reality 3 | |
| A reaction: Modern theories of meaning seem to assume that there is one item called 'meaning' which needs to be explained, but presumably this is 'strict and literal meaning', leaving the rest to pragmatics. |
| 4954 | Frege uses 'sense' to mean both a designator's meaning, and the way its reference is determined |
| Full Idea: Frege should be criticised for using the term 'sense' in two senses. For he takes the sense of a designator to be its meaning; and he also takes it to be the way its reference is determined. | |
| From: comment on Gottlob Frege (On Sense and Reference [1892]) by Saul A. Kripke - Naming and Necessity lectures Lecture 1 | |
| A reaction: This criticism doesn't surprise me, as heroic pioneers like Frege seem to have been extremely unclear about what they were claiming. Kripke has helped, but we still need some great mind to step in and sort out the mess. |
| 7304 | Frege explained meaning as sense, semantic value, reference, force and tone |
| Full Idea: Frege analysed the intuitive notion of meaning in terms of the notions of sense, semantic value, reference, force and tone. | |
| From: report of Gottlob Frege (On Sense and Reference [1892], Pref) by Alexander Miller - Philosophy of Language Pref | |
| A reaction: This suggests that there are two approaches to the explanation of meaning: either a simple identity with some other mental fact, or an analysis (as here) into a range of components. I remain open-minded on that. |
| 4974 | For all the multiplicity of languages, mankind has a common stock of thoughts |
| Full Idea: For all the multiplicity of languages, mankind has a common stock of thoughts. | |
| From: Gottlob Frege (On Concept and Object [1892], p.196n) | |
| A reaction: Given the acknowledgement here that two very different sentences in different languages can express the same thought, he should recognise that at least some aspects of a thought are non-linguistic. |
| 16873 | Thoughts are not subjective or psychological, because some thoughts are the same for us all |
| Full Idea: A thought is not something subjective, is not the product of any form of mental activity; for the thought that we have in Pythagoras's theorem is the same for everybody. | |
| From: Gottlob Frege (Logic in Mathematics [1914], p.206) | |
| A reaction: When such thoughts are treated as if the have objective (platonic) existence, I become bewildered. I take a thought (or proposition) to be entirely psychological, but that doesn't stop two people from having the same thought. |
| 16872 | A thought is the sense expressed by a sentence, and is what we prove |
| Full Idea: The sentence is of value to us because of the sense that we grasp in it, which is recognisably the same in a translation. I call this sense the thought. What we prove is not a sentence, but a thought. | |
| From: Gottlob Frege (Logic in Mathematics [1914], p.206) | |
| A reaction: The 'sense' is presumably the German 'sinn', and a 'thought' in Frege is what we normally call a 'proposition'. So the sense of a sentence is a proposition, and logic proves propositions. I'm happy with that. |
| 19467 | A 'thought' is something for which the question of truth can arise; thoughts are senses of sentences |
| Full Idea: I call a 'thought' something for which the question of truth can arise at all. ...So I can say: thoughts are senses of sentences, without wishing to assert that the sense of every sentence is a thought. | |
| From: Gottlob Frege (The Thought: a Logical Enquiry [1918], p.327-8 (61)) | |
| A reaction: This builds on his distinction between sense and reference. The reference of every truth sentence is just 'the true', and the sense is the proposition. The concept of a proposition seems indispensable to logic, I would say. |
| 16874 | The parts of a thought map onto the parts of a sentence |
| Full Idea: A sentence is generally a complex sign, so the thought expressed by it is complex too: in fact it is put together in such a way that parts of a thought correspond to parts of the sentence. | |
| From: Gottlob Frege (Logic in Mathematics [1914], p.207) | |
| A reaction: This is the compositional view of propositions, as opposed to the holistic view. |
| 19472 | A sentence is only a thought if it is complete, and has a time-specification |
| Full Idea: Only a sentence with the time-specification filled out, a sentence complete in every respect, expresses a thought. | |
| From: Gottlob Frege (The Thought: a Logical Enquiry [1918], p.343(76)) | |
| A reaction: I take the 'every respect' to include the avoidance of ambiguity, and some sort of perspicacious reference for the terms. I wish philosophers would focus on the thoughts in their subject, and not nit-pick about the sentences. Does he mean 'utterances'? |
| 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. |
| 7725 | 'P or not-p' seems to be analytic, but does not fit Kant's account, lacking clear subject or predicate |
| Full Idea: 'It is raining or it is not raining' appears to true because of the general principle 'p or not-p', so it is analytic; but this does not fit Kant's idea of an analytic truth, because it is not obvious that it has a subject concept or a predicate concept. | |
| From: report of Gottlob Frege (works [1890]) by Joan Weiner - Frege Ch.2 | |
| A reaction: The general progress of logic seems to be a widening out to embrace problem sentences. However, see Idea 7315 for the next problem that arises with analyticity. All this culminates in Quine's attack (e.g. Idea 1624). |
| 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. |
| 7316 | Analytic truths are those that can be demonstrated using only logic and definitions |
| Full Idea: Frege (according to Quine) characterises analytic truths as those that can be demonstrated or proved using only logical laws and definitions as premises. | |
| From: report of Gottlob Frege (works [1890]) by Alexander Miller - Philosophy of Language 4.2 | |
| A reaction: This is the big shift away from the Kantian version (predicate contained in the subject) towards a modern version, perhaps fixed by a truth table giving true for all values. |
| 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. |
| 3307 | Frege put forward an ontological argument for the existence of numbers |
| Full Idea: Frege put forward an ontological argument for the existence of numbers. | |
| From: report of Gottlob Frege (works [1890]) by José A. Benardete - Metaphysics: the logical approach Ch.4 |
| 7741 | The predicate 'exists' is actually a natural language expression for a quantifier |
| Full Idea: On Frege's logical analysis, the predicate 'exists' is actually a natural language expression for a quantifier. | |
| From: report of Gottlob Frege (Begriffsschrift [1879]) by Joan Weiner - Frege Ch.8 | |
| A reaction: However see Idea 6067, for McGinn's alternative view of quantifiers. In the normal conventions of predicate logic it may be that existence is treated as a quantifier, but that is not the same as saying that existence just IS a quantifier. |
| 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. |
| 8491 | The Ontological Argument fallaciously treats existence as a first-level concept |
| Full Idea: The ontological proof of God's existence suffers from the fallacy of treating existence as a first-level concept. | |
| From: Gottlob Frege (Function and Concept [1891], p.38 n) | |
| A reaction: [See Idea 8490 for first- and second-order functions] This is usually summarised as the idea that existence is a quantifier rather than a predicate. |