display all the ideas for this combination of texts
5 ideas
| 17185 | Mathematics deals with the essences and properties of forms [Spinoza] |
| Full Idea: Mathematics does not deal with ends, but with the essences and properties of forms (figures), …and has placed before us another rule of truth. | |
| From: Baruch de Spinoza (The Ethics [1675], IApp) | |
| A reaction: Just what I need - a nice clear assertion of essentialism in mathematics. Many say maths is all necessary, so essence is irrelevant, but I say explanations occur in mathematics, and that points to essentialism. |
| 17222 | The sum of its angles follows from a triangle's nature [Spinoza] |
| Full Idea: It follows from the nature of a triangle that its three angles are equal to two right angles. | |
| From: Baruch de Spinoza (The Ethics [1675], IV Pr 57) | |
| A reaction: This is the essentialist view of mathematics, which I take to be connected to explanation, which I take to be connected to the direction of explanation. |
| 17197 | The idea of a triangle involves truths about it, so those are part of its essence [Spinoza] |
| Full Idea: The idea of the triangle must involve the affirmation that its three angles are equal to two right angles. Therefore this affirmation pertains to the essence of the idea of a triangle. | |
| From: Baruch de Spinoza (The Ethics [1675], II Pr 49) | |
| A reaction: This seems to say that the essence is what is inescapable when you think of something. Does that mean that brandy is part of the essence of Napoleon? (Presumably not) Spinoza is ignoring the direction of explanation here. |
| 18256 | Quantity is inconceivable without the idea of addition [Frege] |
| Full Idea: There is so intimate a connection between the concepts of addition and of quantity that one cannot begin to grasp the latter without the former. | |
| From: Gottlob Frege (Rechnungsmethoden (dissertation) [1874], p.2), quoted by Michael Dummett - Frege philosophy of mathematics 22 'Quantit' | |
| A reaction: Frege offers good reasons for making cardinals prior to ordinals, though plenty of people disagree. |
| 9831 | Geometry appeals to intuition as the source of its axioms [Frege] |
| Full Idea: The elements of all geometrical constructions are intuitions, and geometry appeals to intuition as the source of its axioms. | |
| From: Gottlob Frege (Rechnungsmethoden (dissertation) [1874], Ch.6), quoted by Michael Dummett - Frege philosophy of mathematics | |
| 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. |