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Full Idea
Frege's three main objections to radical formalism are that it cannot account for the application of mathematics, that it confuses a formal theory with its metatheory, and it cannot explain an infinite sequence.
Gist of Idea
Formalism misunderstands applications, metatheory, and infinity
Source
report of Gottlob Frege (Grundgesetze der Arithmetik 2 (Basic Laws) [1903], §86-137) by Michael Dummett - Frege philosophy of mathematics
Book Ref
Dummett,Michael: 'Frege: philosophy of mathematics' [Duckworth 1991], p.252
A Reaction
The application is because we don't design maths randomly, but to be useful. The third objection might be dealt with by potential infinities (from formal rules). The second objection sounds promising.
Related Idea
Idea 6425 Formalism can't apply numbers to reality, so it is an evasion [Russell]
| 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] |