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Full Idea
Even if spatial intuition provides a little help in the heuristics of four-dimensional geometry, intuition is an outright hindrance for five-dimensional geometry and beyond.
Gist of Idea
Intuition is an outright hindrance to five-dimensional geometry
Source
Stewart Shapiro (Philosophy of Mathematics [1997], 5.2)
Book Ref
Shapiro,Stewart: 'Philosophy of Mathematics:structure and ontology' [OUP 1997], p.150
A Reaction
One might respond by saying 'so much the worse for five-dimensional geometry'. One could hardly abolish the subject, though, so the point must be taken.
| 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] |