display all the ideas for this combination of philosophers
50 ideas
| 7728 | Frege has a judgement stroke (vertical, asserting or judging) and a content stroke (horizontal, expressing) [Frege, by Weiner] |
| 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. |
| 16881 | The laws of logic are boundless, so we want the few whose power contains the others [Frege] |
| 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. |
| 7622 | In 1879 Frege developed second order logic [Frege, by Putnam] |
| 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 [Frege, by Dummett] |
| 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 [Frege] |
| 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? [Frege] |
| 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 [Frege] |
| 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 [Frege, by Hart,WD] |
| 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 [Frege, by Weiner] |
| 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" [Frege] |
| 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 [Frege] |
| 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 [Frege] |
| 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 [Frege] |
| 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 [Frege, by Benardete,JA] |
| 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 [Frege, by McGinn] |
| 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 [Frege, by Burge] |
| 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 [Frege, by Jeshion] |
| 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 [Frege] |
| 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 [Frege] |
| 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 [Frege] |
| 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 [Frege, by Sainsbury] |
| 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' [Frege] |
| 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. |
| 8448 | Any object can have many different names, each with a distinct sense [Frege] |
| 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. |
| 14075 | Proper name in modal contexts refer obliquely, to their usual sense [Frege, by Gibbard] |
| 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. |
| 4978 | The meaning of a proper name is the designated object [Frege] |
| 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 [Frege, by Hale] |
| 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 [Frege, by Sawyer] |
| 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 [Frege] |
| 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 [Frege] |
| 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 [Frege, by Fitting/Mendelsohn] |
| 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) [Frege, by George/Velleman] |
| 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 [Frege, by Tait] |
| 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 [Dummett on Frege] |
| 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 [Dummett on Frege] |
| 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 [Frege, by Weiner] |
| 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' [Frege, by McCullogh] |
| 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 [Dummett on Frege] |
| 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 [Oliver/Smiley on Frege] |
| 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 [Frege, by Prawitz] |
| 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 [Frege, by Kaplan] |
| 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 [Frege, by Burge] |
| 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 [Frege, by Jacquette] |
| 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' [Frege, by Sawyer] |
| 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 [Frege] |
| 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. |
| 16886 | The truth of an axiom must be independently recognisable [Frege] |
| 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. |
| 17624 | To understand axioms you must grasp their logical power and priority [Frege, by Burge] |
| 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). |
| 16866 | Tracing inference backwards closes in on a small set of axioms and postulates [Frege] |
| 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 [Frege] |
| 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. |
| 16871 | A truth can be an axiom in one system and not in another [Frege] |
| 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. |
| 16870 | Axioms are truths which cannot be doubted, and for which no proof is needed [Frege] |
| 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. |