18253
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I wish to go straight from cardinals to reals (as ratios), leaving out the rationals [Frege]
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Full Idea:
You need a double transition, from cardinal numbes (Anzahlen) to the rational numbers, and from the latter to the real numbers generally. I wish to go straight from the cardinal numbers to the real numbers as ratios of quantities.
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From:
Gottlob Frege (Letters to Russell [1902], 1903.05.21), quoted by Michael Dummett - Frege philosophy of mathematics 21 'Frege's'
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A reaction:
Note that Frege's real numbers are not quantities, but ratios of quantities. In this way the same real number can refer to lengths, masses, intensities etc.
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18269
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Logical objects are extensions of concepts, or ranges of values of functions [Frege]
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Full Idea:
How are we to conceive of logical objects? My only answer is, we conceive of them as extensions of concepts or, more generally, as ranges of values of functions ...what other way is there?
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From:
Gottlob Frege (Letters to Russell [1902], 1902.07.28), quoted by J. Alberto Coffa - The Semantic Tradition from Kant to Carnap 7 epigr
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A reaction:
This is the clearest statement I have found of what Frege means by an 'object'. But an extension is a collection of things, so an object is a group treated as a unity, which is generally how we understand a 'set'. Hence Quine's ontology.
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19554
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Denying closure is denying we know P when we know P and Q, which is absurd in simple cases [Hawthorne]
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Full Idea:
How could we know that P and Q but not be in a position to know that P (as deniers of closure must say)? If my glass is full of wine, we know 'g is full of wine, and not full of non-wine'. How can we deny that we know it is not full of non-wine?
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From:
John Hawthorne (The Case for Closure [2005], 2)
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A reaction:
Hawthorne merely raises this doubt. Dretske is concerned with heavyweight implications, but how do you accept lightweight implications like this one, and then suddenly reject them when they become too heavy? [see p.49]
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15282
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Facts should be deducible from the theory and initial conditions, and prefer the simpler theory [Osiander, by Harré/Madden]
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Full Idea:
The two positivist criteria for a scientific theory are that the facts must be deducible from the theory together with initial conditions, and if there is more than one theory the simplest must be chosen.
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From:
report of Andreas Osiander (Preface to 'De Revolutionibus' [1543]) by Harré,R./Madden,E.H. - Causal Powers 7.I
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A reaction:
Harré and Madden cite this as a famous early statement of positivism. It seems to combine Hempel and Lewis very concisely. Wrong, of course. It does not, though, appear to mention 'laws'.
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