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Full Idea
The elements of all geometrical constructions are intuitions, and geometry appeals to intuition as the source of its axioms.
Gist of Idea
Geometry appeals to intuition as the source of its axioms
Source
Gottlob Frege (Rechnungsmethoden (dissertation) [1874], Ch.6), quoted by Michael Dummett - Frege philosophy of mathematics
Book Ref
Dummett,Michael: 'Frege: philosophy of mathematics' [Duckworth 1991], p.68
A Reaction
Very early Frege, but he stuck to this view, while firmly rejecting intuition as a source of arithmetic. Frege would have known well that Euclid's assumption about parallels had been challenged.
| 12421 | Kant's intuitions struggle to judge relevance, impossibility and exactness [Kitcher on Kant] |
| 16910 | Mathematics can only start from an a priori intuition which is not empirical but pure [Kant] |
| 16917 | All necessary mathematical judgements are based on intuitions of space and time [Kant] |
| 9830 | Bolzano began the elimination of intuition, by proving something which seemed obvious [Bolzano, by Dummett] |
| 17816 | Frege's logicism aimed at removing the reliance of arithmetic on intuition [Frege, by Yourgrau] |
| 9831 | Geometry appeals to intuition as the source of its axioms [Frege] |
| 12420 | If mathematics comes through intuition, that is either inexplicable, or too subjective [Kitcher] |
| 12393 | Intuition is no basis for securing a priori knowledge, because it is fallible [Kitcher] |
| 18061 | Mathematical intuition is not the type platonism needs [Kitcher] |
| 10244 | Intuition is an outright hindrance to five-dimensional geometry [Shapiro] |
| 8756 | Intuition doesn't support much mathematics, and we should question its reliability [Maddy, by Shapiro] |