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Full Idea
Geometers by no means assume that there are lines without width or surfaces without depth. They only think it is possible to consider the length without paying attention to the width. We can measure the length of a path without its width.
Gist of Idea
No one denies that a line has width, but we can just attend to its length
Source
Arnauld / Nicole (Logic (Port-Royal Art of Thinking) [1662], I.5)
Book Ref
Arnauld,A/Nicole,P: 'Logic, or the Art of Thinking (Port-Royal)', ed/tr. Buroker,J.V. [CUP 1996], p.38
A Reaction
A nice example which makes the point indubitable. The modern 'rigorous' account of abstraction that starts with Frege seems to require more than one object, in order to derive abstractions like direction or number. Path widths are not comparatives.
| 10502 | We can rise by degrees through abstraction, with higher levels representing more things [Arnauld,A/Nicole,P] |
| 10500 | No one denies that a line has width, but we can just attend to its length [Arnauld,A/Nicole,P] |
| 10499 | We know by abstraction because we only understand composite things a part at a time [Arnauld,A/Nicole,P] |
| 10501 | A triangle diagram is about all triangles, if some features are ignored [Arnauld,A/Nicole,P] |
| 16784 | Forms make things distinct and explain the properties, by pure form, or arrangement of parts [Arnauld,A/Nicole,P] |
| 18258 | We can only know the exterior world via our ideas [Arnauld,A/Nicole,P] |