14 ideas
| 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'. |
| 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. |
| 18252 | Real numbers are ratios of quantities, such as lengths or masses [Frege] |
| 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. |
| 18271 | We can't prove everything, but we can spell out the unproved, so that foundations are clear [Frege] |
| 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' |
| 10623 | Frege defined number in terms of extensions of concepts, but needed Basic Law V to explain extensions [Frege, by Hale/Wright] |
| 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 [Tait on Frege] |
| 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. |
| 18165 | My Basic Law V is a law of pure logic [Frege] |
| 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. |
| 14296 | Dispositions are physical states of mechanism; when known, these replace the old disposition term [Quine] |
| Full Idea: Each disposition, in my view, is a physical state or mechanism. ...In some cases nowadays we understand the physical details and set them forth explicitly in terms of the arrangement and interaction of small bodies. This replaces the old disposition. | |
| From: Willard Quine (The Roots of Reference [1990], p.11), quoted by Stephen Mumford - Dispositions 01.3 | |
| A reaction: A challenge to the dispositions and powers view of nature, one which rests on the 'categorical' structural properties, rather than the 'hypothetical' dispositions. But can we define a mechanism without mentioning its powers? |
| 9190 | A concept is a function mapping objects onto truth-values, if they fall under the concept [Frege, by Dummett] |
| 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 [Frege, by Shapiro] |
| 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. |
| 9086 | The idea of abstract objects is not ontological; it comes from the epistemological idea of abstraction [Plantinga] |
| Full Idea: The notion of an abstract object comes from the notion of abstraction; it is in origin an epistemological rather than an ontological category. | |
| From: Alvin Plantinga (Why Propositions cannot be concrete [1993], p.232) | |
| A reaction: Etymology doesn't prove anything. However, if you define abstract objects as not existing in space or time, you must recognise that this may only be because that is how humans imaginatively created them in the first place. |
| 9087 | Theists may see abstract objects as really divine thoughts [Plantinga] |
| Full Idea: Theists may find attractive a view popular among medieval philosophers from Augustine on: that abstract objects are really divine thoughts. More exactly, propositions are divine thoughts, properties divine concepts, and sets divine collections. | |
| From: Alvin Plantinga (Why Propositions cannot be concrete [1993], p.233) | |
| A reaction: Hm. I pass this on because we should be aware that there is a theological history to discussions of abstract objects, and some people have vested interests in keeping them outside of the natural world. Aren't properties natural? Does God gerrymander sets? |
| 9085 | If propositions are concrete they don't have to exist, and so they can't be necessary truths [Plantinga] |
| Full Idea: Someone who believes propositions are concrete cannot agree that some propositions are necessary. For propositions are contingent beings, and could have failed to exist. But if they fail to exist, then they fail to be true. | |
| From: Alvin Plantinga (Why Propositions cannot be concrete [1993], p.230) | |
| A reaction: [compressed] He implies the actual existence of an infinity of trivial, boring or ridiculous necessary truths. I suspect that he is just confusing a thought with its content. Or we might just treat necessary propositions as hypothetical. |
| 9084 | Propositions can't just be in brains, because 'there are no human beings' might be true [Plantinga] |
| Full Idea: If propositions are brain inscriptions, then if there had been no human beings there would have been no propositions. But then 'there are no human beings' would have been true, so there would have been at least one truth (and thus one proposition). | |
| From: Alvin Plantinga (Why Propositions cannot be concrete [1993], p.229) | |
| A reaction: This would make 'there are no x's' true for any value of x apart from actual objects, which implies an infinity of propositions. Does Plantinga really believe that these all exist? He may be confusing propositions with facts. |