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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.
Gist of Idea
Cardinals say how many, and reals give measurements compared to a unit quantity
Source
Gottlob Frege (Grundgesetze der Arithmetik 2 (Basic Laws) [1903], §157), quoted by Michael Dummett - Frege philosophy of mathematics Ch.19
Book Ref
Dummett,Michael: 'Frege: philosophy of mathematics' [Duckworth 1991], p.246
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.
| 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] |