Combining Texts

All the ideas for 'fragments/reports', 'Elements of Geometry' and 'Rationality and Virtue'

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17 ideas

1. Philosophy / D. Nature of Philosophy / 3. Philosophy Defined
Even pointing a finger should only be done for a reason [Epictetus]
     Full Idea: Philosophy says it is not right even to stretch out a finger without some reason.
     From: Epictetus (fragments/reports [c.57], 15)
     A reaction: The key point here is that philosophy concerns action, an idea on which Epictetus is very keen. He rather despise theory. This idea perfectly sums up the concept of the wholly rational life (which no rational person would actually want to live!).
2. Reason / E. Argument / 6. Conclusive Proof
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.
4. Formal Logic / C. Predicate Calculus PC / 2. Tools of Predicate Calculus / c. Derivations rules of PC
If you pick an arbitrary triangle, things proved of it are true of all triangles [Euclid, by Lemmon]
     Full Idea: Euclid begins proofs about all triangles with 'let ABC be a triangle', but ABC is not a proper name. It names an arbitrarily selected triangle, and if that has a property, then we can conclude that all triangles have the property.
     From: report of Euclid (Elements of Geometry [c.290 BCE]) by E.J. Lemmon - Beginning Logic 3.2
     A reaction: Lemmon adds the proviso that there must be no hidden assumptions about the triangle we have selected. You must generalise the properties too. Pick a triangle, any triangle, say one with three angles of 60 degrees; now generalise from it.
6. Mathematics / A. Nature of Mathematics / 2. Geometry
Euclid's geometry is synthetic, but Descartes produced an analytic version of it [Euclid, by Resnik]
     Full Idea: Euclid's geometry is a synthetic geometry; Descartes supplied an analytic version of Euclid's geometry, and we now have analytic versions of the early non-Euclidean geometries.
     From: report of Euclid (Elements of Geometry [c.290 BCE]) by Michael D. Resnik - Maths as a Science of Patterns One.4
     A reaction: I take it that the original Euclidean axioms were observations about the nature of space, but Descartes turned them into a set of pure interlocking definitions which could still function if space ceased to exist.
6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / b. Types of number
An assumption that there is a largest prime leads to a contradiction [Euclid, by Brown,JR]
     Full Idea: Assume a largest prime, then multiply the primes together and add one. The new number isn't prime, because we assumed a largest prime; but it can't be divided by a prime, because the remainder is one. So only a larger prime could divide it. Contradiction.
     From: report of Euclid (Elements of Geometry [c.290 BCE]) by James Robert Brown - Philosophy of Mathematics Ch.1
     A reaction: Not only a very elegant mathematical argument, but a model for how much modern logic proceeds, by assuming that the proposition is false, and then deducing a contradiction from it.
6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / m. One
A unit is that according to which each existing thing is said to be one [Euclid]
     Full Idea: A unit is that according to which each existing thing is said to be one.
     From: Euclid (Elements of Geometry [c.290 BCE], 7 Def 1)
     A reaction: See Frege's 'Grundlagen' §29-44 for a sustained critique of this. Frege is good, but there must be something right about the Euclid idea. If I count stone, paper and scissors as three, each must first qualify to be counted as one. Psychology creeps in.
6. Mathematics / A. Nature of Mathematics / 5. The Infinite / a. The Infinite
Postulate 2 says a line can be extended continuously [Euclid, by Shapiro]
     Full Idea: Euclid's Postulate 2 says the geometer can 'produce a finite straight line continuously in a straight line'.
     From: report of Euclid (Elements of Geometry [c.290 BCE]) by Stewart Shapiro - Thinking About Mathematics 4.2
     A reaction: The point being that this takes infinity for granted, especially if you start counting how many points there are on the line. The Einstein idea that it might eventually come round and hit you on the back of the head would have charmed Euclid.
6. Mathematics / B. Foundations for Mathematics / 3. Axioms for Geometry
Euclid relied on obvious properties in diagrams, as well as on his axioms [Potter on Euclid]
     Full Idea: Euclid's axioms were insufficient to derive all the theorems of geometry: at various points in his proofs he appealed to properties that are obvious from the diagrams but do not follow from the stated axioms.
     From: comment on Euclid (Elements of Geometry [c.290 BCE]) by Michael Potter - The Rise of Analytic Philosophy 1879-1930 03 'aim'
     A reaction: I suppose if the axioms of a system are based on self-evidence, this would licence an appeal to self-evidence elsewhere in the system. Only pedants insist on writing down what is obvious to everyone!
Euclid's parallel postulate defines unique non-intersecting parallel lines [Euclid, by Friend]
     Full Idea: Euclid's fifth 'parallel' postulate says if there is an infinite straight line and a point, then there is only one straight line through the point which won't intersect the first line. This axiom is independent of Euclid's first four (agreed) axioms.
     From: report of Euclid (Elements of Geometry [c.290 BCE]) by Michèle Friend - Introducing the Philosophy of Mathematics 2.2
     A reaction: This postulate was challenged in the nineteenth century, which was a major landmark in the development of modern relativist views of knowledge.
Euclid needs a principle of continuity, saying some lines must intersect [Shapiro on Euclid]
     Full Idea: Euclid gives no principle of continuity, which would sanction an inference that if a line goes from the outside of a circle to the inside of circle, then it must intersect the circle at some point.
     From: comment on Euclid (Elements of Geometry [c.290 BCE]) by Stewart Shapiro - Philosophy of Mathematics 6.1 n2
     A reaction: Cantor and Dedekind began to contemplate discontinuous lines.
Euclid says we can 'join' two points, but Hilbert says the straight line 'exists' [Euclid, by Bernays]
     Full Idea: Euclid postulates: One can join two points by a straight line; Hilbert states the axiom: Given any two points, there exists a straight line on which both are situated.
     From: report of Euclid (Elements of Geometry [c.290 BCE]) by Paul Bernays - On Platonism in Mathematics p.259
Modern geometries only accept various parts of the Euclid propositions [Russell on Euclid]
     Full Idea: In descriptive geometry the first 26 propositions of Euclid hold. In projective geometry the 1st, 7th, 16th and 17th require modification (as a straight line is not a closed series). Those after 26 depend on the postulate of parallels, so aren't assumed.
     From: comment on Euclid (Elements of Geometry [c.290 BCE]) by Bertrand Russell - The Principles of Mathematics §388
6. Mathematics / B. Foundations for Mathematics / 5. Definitions of Number / b. Greek arithmetic
Euclid's common notions or axioms are what we must have if we are to learn anything at all [Euclid, by Roochnik]
     Full Idea: The best known example of Euclid's 'common notions' is "If equals are subtracted from equals the remainders are equal". These can be called axioms, and are what "the man who is to learn anything whatever must have".
     From: report of Euclid (Elements of Geometry [c.290 BCE], 72a17) by David Roochnik - The Tragedy of Reason p.149
20. Action / C. Motives for Action / 3. Acting on Reason / a. Practical reason
Possessing the virtue of justice disposes a person to good practical rationality [Foot]
     Full Idea: If justice is a virtue it must make action good by disposing its possessor to goodness in practical rationality; the latter consisting of the right recognition of reasons, and corresponding action.
     From: Philippa Foot (Rationality and Virtue [1994], p.174)
     A reaction: This somewhat inverts Aristotle, who says the possessing of good practical reason is the key to acquiring the virtues. Foot suggests that possessing the virtue promotes the practical rationality. Someone can be sensible without being virtuous.
22. Metaethics / A. Ethics Foundations / 2. Source of Ethics / h. Expressivism
Calling a knife or farmer or speech or root good does not involve attitudes or feelings [Foot]
     Full Idea: No one thinks that calling a knife a good knife, a farmer a good farmer, a speech a good speech, a root a good root, necessarily expresses or even involves an attitude or feeling towards it.
     From: Philippa Foot (Rationality and Virtue [1994], p.163)
     A reaction: This is the Aristotelian idea (which I favour) that good derives from function. In such a case it seems obvious that it has nothing to do with expressing emotions.
23. Ethics / C. Virtue Theory / 1. Virtue Theory / b. Basis of virtue
The essential thing is the 'needs' of plants and animals, and their operative parts [Foot]
     Full Idea: The key notion is the concept of need, …as when we say what a plant or animal of a certain species needs to have, …and what its operative features, such roots, leaves, hearts and lungs, need to do.
     From: Philippa Foot (Rationality and Virtue [1994], p.164)
     A reaction: Good. That takes it away from the idea of a function, which could be possessed by an inanimate machine (even though that still entails success and failure). Strictly, we need oxygen, but the goodness resides in the lungs.
23. Ethics / C. Virtue Theory / 3. Virtues / c. Justice
Observing justice is necessary to humans, like hunting to wolves or dancing to bees [Foot]
     Full Idea: The teaching and observing of the rules of justice is as necessary a part of the life of human beings as hunting together in packs with a leader is a necessary part of the lives of wolves, or dancing part of the life of the dancing bee.
     From: Philippa Foot (Rationality and Virtue [1994], p.168)
     A reaction: So why are some men unjust? All wolves hunt, and all appropriate bees dance. A few men even thrive on injustice.