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.
|
5880
|
Xenocrates held that the soul had no form or substance, but was number [Xenocrates, by Cicero]
|
|
Full Idea:
Xenocrates denied that the soul had form or any substance, but said that it was number, and the power of number, as had been held by Pythagoras long before, was the highest in nature.
|
|
From:
report of Xenocrates (fragments/reports [c.327 BCE]) by M. Tullius Cicero - Tusculan Disputations I.x.20
|
|
A reaction:
This shows how strong the Pythagorean influence was in the Academy. This is not totally stupid. Dawkins holds that the essence of DNA is information, which can be expressed mathematically. Xenocrates was a functionalist.
|
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.
|
14080
|
Are causal descriptions part of the causal theory of reference, or are they just metasemantic? [Kaplan, by Schaffer,J]
|
|
Full Idea:
Kaplan notes that the causal theory of reference can be understood in two quite different ways, as part of the semantics (involving descriptions of causal processes), or as metasemantics, explaining why a term has the referent it does.
|
|
From:
report of David Kaplan (Dthat [1970]) by Jonathan Schaffer - Deflationary Metaontology of Thomasson 1
|
|
A reaction:
[Kaplan 'Afterthought' 1989] The theory tends to be labelled as 'direct' rather than as 'causal' these days, but causal chains are still at the heart of the story (even if more diffused socially). Nice question. Kaplan takes the meta- version as orthodox.
|