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Full Idea
From geometry we retain the interpretation of a real number as a ratio of quantities or measurement-number; but in more recent times we detach it from geometrical quantities, and from all particular types of quantity.
Gist of Idea
The modern account of real numbers detaches a ratio from its geometrical origins
Source
Gottlob Frege (Grundgesetze der Arithmetik 2 (Basic Laws) [1903], §159), quoted by Michael Dummett - Frege philosophy of mathematics
Book Ref
Dummett,Michael: 'Frege: philosophy of mathematics' [Duckworth 1991], p.262
A Reaction
Dummett glosses the 'recent' version as by Cantor and Dedekind in 1872. This use of 'detach' seems to me startlingly like the sort of psychological abstractionism which Frege was so desperate to avoid.
| 13886 | Later Frege held that definitions must fix a function's value for every possible argument [Frege, by Wright,C] |
| 9889 | Real numbers are ratios of quantities [Frege, by Dummett] |
| 10553 | A number is a class of classes of the same cardinality [Frege, by Dummett] |
| 10020 | Frege's biggest error is in not accounting for the senses of number terms [Hodes on Frege] |
| 9886 | Cardinals say how many, and reals give measurements compared to a unit quantity [Frege] |
| 9890 | The modern account of real numbers detaches a ratio from its geometrical origins [Frege] |
| 9891 | The first demand of logic is of a sharp boundary [Frege] |
| 10019 | Only what is logically complex can be defined; what is simple must be pointed to [Frege] |
| 9845 | We can't define a word by defining an expression containing it, as the remaining parts are a problem [Frege] |
| 9887 | Formalism misunderstands applications, metatheory, and infinity [Frege, by Dummett] |
| 8751 | Only applicability raises arithmetic from a game to a science [Frege] |
| 11846 | If we abstract the difference between two houses, they don't become the same house [Frege] |