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2. Reason / E. Argument / 6. Conclusive Proof

[securely establishing a result by precise defined steps]

8 ideas
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.
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)
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.
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.
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.
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)
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!
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.