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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.
Gist of Idea
Real numbers are ratios of quantities, such as lengths or masses
Source
Gottlob Frege (Grundgesetze der Arithmetik 1 (Basic Laws) [1893], III.1.73), quoted by Michael Dummett - Frege philosophy of mathematics 21 'Frege's'
Book Ref
Dummett,Michael: 'Frege: philosophy of mathematics' [Duckworth 1991], p.269
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.
| 13733 | Frege considered definite descriptions to be genuine singular terms [Frege, by Fitting/Mendelsohn] |
| 10623 | Frege defined number in terms of extensions of concepts, but needed Basic Law V to explain extensions [Frege, by Hale/Wright] |
| 9975 | Frege ignored Cantor's warning that a cardinal set is not just a concept-extension [Tait on Frege] |
| 9190 | A concept is a function mapping objects onto truth-values, if they fall under the concept [Frege, by Dummett] |
| 13665 | Frege took the study of concepts to be part of logic [Frege, by Shapiro] |
| 9874 | Contradiction arises from Frege's substitutional account of second-order quantification [Dummett on Frege] |
| 18252 | Real numbers are ratios of quantities, such as lengths or masses [Frege] |
| 18271 | We can't prove everything, but we can spell out the unproved, so that foundations are clear [Frege] |
| 18165 | My Basic Law V is a law of pure logic [Frege] |