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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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14193
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'Substance theorists' take modal properties as primitive, without structure, just falling under a sortal [Paul,LA]
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Full Idea:
Some deep essentialists resist the need to explain the structure under de re modal properties, taking them as primitive. One version (which we can call 'substance theory') takes them to fall under a sortal concept, with no further explanation.
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From:
L.A. Paul (In Defense of Essentialism [2006], §1)
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A reaction:
A very helpful identification of what Wiggins stands for, and why I disagree with him. The whole point of essences is to provide a notion that fits in with sciences, which means they must have an explanatory role, which needs structures.
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14195
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If an object's sort determines its properties, we need to ask what determines its sort [Paul,LA]
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Full Idea:
If the substance essentialist holds that the sort an object belongs to determines its de re modal properties (rather than the other way round), then he needs to give an (ontological, not conceptual) explanation of what determines an object's sort.
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From:
L.A. Paul (In Defense of Essentialism [2006], §1)
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A reaction:
See Idea 14193 for 'substance essentialism'. I find it quite incredible that anyone could think that a thing's sort could determine its properties, rather than the other way round. Even if sortals are conventional, they are not arbitrary.
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14196
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Substance essentialism says an object is multiple, as falling under various different sortals [Paul,LA]
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Full Idea:
The explanation of material constitution given by substance essentialism is that there are multiple objects. A person is essentially human-shaped (falling under the human sort), while their hunk of tissue is accidentally human-shaped (as tissue).
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From:
L.A. Paul (In Defense of Essentialism [2006], §1)
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A reaction:
At this point sortal essentialism begins to look crazy. Persons are dubious examples (with sneaky dualism involved). A bronze statue is essentially harder to dent than a clay one, because of its bronze. If you remake it of clay, it isn't the same statue.
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14190
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Deep essentialist objects have intrinsic properties that fix their nature; the shallow version makes it contextual [Paul,LA]
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Full Idea:
Essentialism says that objects have their properties essentially. 'Deep' essentialists take the (nontrivial) essential properties of an object to determine its nature. 'Shallow' essentialists substitute context-dependent truths for the independent ones.
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From:
L.A. Paul (In Defense of Essentialism [2006], Intro)
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A reaction:
If the deep essence determines a things nature, we should not need to say 'nontrivial'. This is my bete noire, the confusion of essential properties with necessary ones, where necessary properties (or predicates, at least) can indeed be trivial.
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14189
|
'Modal realists' believe in many concrete worlds, 'actualists' in just this world, 'ersatzists' in abstract other worlds [Paul,LA]
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Full Idea:
A 'modal realist' believes that there are many concrete worlds, while the 'actualist' believes in only one concrete world, the actual world. The 'ersatzist' is an actualist who takes nonactual possible worlds and their contents to be abstracta.
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From:
L.A. Paul (In Defense of Essentialism [2006], Intro)
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A reaction:
My view is something like that modal realism is wrong, and actualism is right, and possible worlds (if they really are that useful) are convenient abstract fictions, constructed (if we have any sense) out of the real possibilities in the actual world.
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11846
|
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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