display all the ideas for this combination of texts
6 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. |
| 10718 | A natural number is a property of sets [Maddy, by Oliver] |
| Full Idea: Maddy takes a natural number to be a certain property of sui generis sets, the property of having a certain number of members. | |
| From: report of Penelope Maddy (Realism in Mathematics [1990], 3 §2) by Alex Oliver - The Metaphysics of Properties | |
| A reaction: [I believe Maddy has shifted since then] Presumably this will make room for zero and infinities as natural numbers. Personally I want my natural numbers to count things. |
| 8756 | Intuition doesn't support much mathematics, and we should question its reliability [Maddy, by Shapiro] |
| Full Idea: Maddy says that intuition alone does not support very much mathematics; more importantly, a naturalist cannot accept intuition at face value, but must ask why we are justified in relying on intuition. | |
| From: report of Penelope Maddy (Realism in Mathematics [1990]) by Stewart Shapiro - Thinking About Mathematics 8.3 | |
| A reaction: It depends what you mean by 'intuition', but I identify with her second objection, that every faculty must ultimately be subject to criticism, which seems to point to a fairly rationalist view of things. |
| 17733 | We know mind-independent mathematical truths through sets, which rest on experience [Maddy, by Jenkins] |
| Full Idea: Maddy proposes that we can know (some) mind-independent mathematical truths through knowing about sets, and that we can obtain knowledge of sets through experience. | |
| From: report of Penelope Maddy (Realism in Mathematics [1990]) by Carrie Jenkins - Grounding Concepts 6.5 | |
| A reaction: Maddy has since backed off from this, and now tries to merely defend 'objectivity' about sets (2011:114). My amateurish view is that she is overrating the importance of sets, which merely model mathematics. Look at category theory. |