8623 | Proof reveals the interdependence of truths, as well as showing their certainty [Euclid, by Frege] |
Full Idea: Euclid gives proofs of many things which anyone would concede to him without question. ...The aim of proof is not merely to place the truth of a proposition beyond doubt, but also to afford us insight into the dependence of truths upon one another. | |
From: report of Euclid (Elements of Geometry [c.290 BCE]) by Gottlob Frege - Grundlagen der Arithmetik (Foundations) §02 | |
A reaction: This connects nicely with Shoemaker's view of analysis (Idea 8559), which I will adopt as my general view. I've always thought of philosophy as the aspiration to wisdom through the cartography of concepts. |
1885 | Proof moves from agreed premises to a non-evident inference [Sext.Empiricus] |
Full Idea: Dogmatists define proof as "an argument which, by means of agreed premises, reveals by way of deduction a nonevident inference". | |
From: Sextus Empiricus (Outlines of Pyrrhonism [c.180], II.135) |
8627 | Leibniz is inclined to regard all truths as provable [Leibniz, by Frege] |
Full Idea: Leibniz has an inclination to regard all truths as provable. | |
From: report of Gottfried Leibniz (works [1690]) by Gottlob Frege - Grundlagen der Arithmetik (Foundations) §15 | |
A reaction: Leibniz sounds like the epitome of Enlightenment optimism about the powers of reason. Could God prove every truth? It's a nice thought. |
17495 | Proof aims to remove doubts, but also to show the interdependence of truths [Frege] |
Full Idea: Proof has as its goal not only to raise the truth of a proposition above all doubts, but additionally to provide insight into the interdependence of truths. | |
From: Gottlob Frege (Grundlagen der Arithmetik (Foundations) [1884], §02) | |
A reaction: This is a major idea in Frege's thinking, and a reason why he is the father of modern metaphysics as well as the father of modern logic. You study the framework of truths by studying the logic that connects them. |
16878 | We must be clear about every premise and every law used in a proof [Frege] |
Full Idea: It is so important, if we are to have a clear insight into what is going on, for us to be able to recognise the premises of every inference which occurs in a proof and the law of inference in accordance with which it takes place. | |
From: Gottlob Frege (Logic in Mathematics [1914], p.212) | |
A reaction: Teachers of logic like natural deduction, because it reduces everything to a few clear laws, which can be stated at each step. |
2898 | Anything which must first be proved is of little value [Nietzsche] |
Full Idea: What has first to have itself proved is of little value. | |
From: Friedrich Nietzsche (Twilight of the Idols [1889], 1.05) |
19067 | A successful proof requires recognition of truth at every step [Dummett] |
Full Idea: For a demonstration to be cogent it is necessary that the passage from step to step involve a recognition of truth at each line. | |
From: Michael Dummett (The Justification of Deduction [1973], p.313) | |
A reaction: Dummett cited Quine (esp. 1970) as having an almost entirely syntactic view of logic. Rumfitt points out that logic can move validly from one falsehood to another. Even a 'proof' might detour into falsehood, but it would not be a 'canonical' proof! |
17773 | Proof shows that it is true, but also why it must be true [Mayberry] |
Full Idea: When you have proved something you know not only that it is true, but why it must be true. | |
From: John Mayberry (What Required for Foundation for Maths? [1994], p.405-2) | |
A reaction: Note the word 'must'. Presumably both the grounding and the necessitation of the truth are revealed. |