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13886
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Later Frege held that definitions must fix a function's value for every possible argument [Frege, by Wright,C]
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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.
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From:
report of Gottlob Frege (Grundgesetze der Arithmetik 2 (Basic Laws) [1903]) by Crispin Wright - Frege's Concept of Numbers as Objects 3.xiv
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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'.
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9845
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We can't define a word by defining an expression containing it, as the remaining parts are a problem [Frege]
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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
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From:
Gottlob Frege (Grundgesetze der Arithmetik 2 (Basic Laws) [1903], §66)
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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.
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9886
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Cardinals say how many, and reals give measurements compared to a unit quantity [Frege]
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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.
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From:
Gottlob Frege (Grundgesetze der Arithmetik 2 (Basic Laws) [1903], §157), quoted by Michael Dummett - Frege philosophy of mathematics Ch.19
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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.
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9887
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Formalism misunderstands applications, metatheory, and infinity [Frege, by Dummett]
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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.
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From:
report of Gottlob Frege (Grundgesetze der Arithmetik 2 (Basic Laws) [1903], §86-137) by Michael Dummett - Frege philosophy of mathematics
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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.
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11846
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If we abstract the difference between two houses, they don't become the same house [Frege]
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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...
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From:
Gottlob Frege (Grundgesetze der Arithmetik 2 (Basic Laws) [1903], §99)
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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.
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23684
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Morality gives everyone reasons to act, irrespective of their desires [Foot, by Hacker-Wright]
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Full Idea:
In her early work she also defends moral rationalism, which is the idea that morality gives reasons for action to everyone, even those who lack the desire to do what is right.
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From:
report of Philippa Foot (Moral Beliefs [1959]) by John Hacker-Wright - Philippa Foot's Moral Thought Intro
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A reaction:
Evidently a rejection of the Humean view that only a desire can motivate action, including moral action. There is an ongoing debate about whether reasons can cause anything, or motivate anything. I think the contents of reasons pull us towards action.
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23690
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We all have reason to cultivate the virtues, even when we lack the desire [Foot, by Hacker-Wright]
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Full Idea:
Foot advocates the view that anyone has reason to cultivate the virtues, even if they lack the desire to do so at a given moment.
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From:
report of Philippa Foot (Moral Beliefs [1959], Pt II) by John Hacker-Wright - Philippa Foot's Moral Thought 2 'Concepts'
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A reaction:
The view which she soon abandoned, but then returned to later. It specifically repudiates the view of Hume, that only desires can motivate. I'm unsure, because the concept of 'reason' strikes me as too imprecise. She sees self-interest as a reason.
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22379
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The meaning of 'good' and other evaluations must include the object to which they attach [Foot]
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Full Idea:
There is no describing the evaluative meaning of 'good', evaluation, commending, or anything of the sort, without fixing the object to which they are supposed to be attached.
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From:
Philippa Foot (Moral Beliefs [1959], p.112)
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A reaction:
I go further, and say that a specification of the feature(s) of the object that produce the value must also be available (if requested). 'That's a good car, but I've no idea why' makes no sense. 'Apparently that's a good car', if other people know why.
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