28 ideas
| 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. |
| 13907 | 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. |
| 6297 | 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. |
| 9603 | 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. |
| 9894 | 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. |
| 8738 | 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. |
| 22278 | 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! |
| 8673 | 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. |
| 10250 | 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. |
| 10302 | 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 |
| 14157 | 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 |
| 1600 | 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 |
| 7518 | If folk psychology gives a network of causal laws, that fits neatly with functionalism [Churchland,PM] |
| Full Idea: The portrait of folk psychology as a network of causal laws dovetailed neatly with the emerging philosophy of mind called functionalism. | |
| From: Paul M. Churchland (Folk Psychology [1996], II) | |
| A reaction: And from the lower levels functionalism is supported by the notion that the brain is modular. Note the word 'laws'; this implies an underlying precision in folk psychology, which is then easily attacked. Maybe the network is too complex for simple laws. |
| 7519 | Many mental phenomena are totally unexplained by folk psychology [Churchland,PM] |
| Full Idea: Folk psychology fails utterly to explain a considerable variety of central psychological phenomena: mental illness, sleep, creativity, memory, intelligence differences, and many forms of learning, to cite just a few. | |
| From: Paul M. Churchland (Folk Psychology [1996], III) | |
| A reaction: If folk psychology is a theory, it will have been developed to predict behaviour, rather than as a full-blown psychological map. The odd thing is that some people seem to be very bad at folk psychology. |
| 7520 | Folk psychology never makes any progress, and is marginalised by modern science [Churchland,PM] |
| Full Idea: Folk psychology has not progressed significantly in the last 2500 years; if anything, it has been steadily in retreat during this period; it does not integrate with modern science, and its emerging wallflower status bodes ill for its future. | |
| From: Paul M. Churchland (Folk Psychology [1996], III) | |
| A reaction: [compressed] However, while shares in alchemy and astrology have totally collapsed, folk psychology shows not the slightest sign of going away, and it is unclear how it ever could. See Idea 3177. |
| 20712 | God is 'eternal' either by being non-temporal, or by enduring forever [Davies,B] |
| Full Idea: Saying 'God is eternal' means either that God is non-temporal or timeless, or that God has no beginning and no end. The first ('classical') view is found in Anselm, Augustine, Boethius, Aquinas, Calvin and Descartes. | |
| From: Brian Davies (Introduction to the Philosophy of Religion [1982], 8 'Meaning') | |
| A reaction: A God who is outside of time but performs actions is a bit of a puzzle. It seems that Augustine started the idea of a timeless God. |
| 20701 | Can God be good, if he has not maximised goodness? [Davies,B] |
| Full Idea: We may wonder whether God can be good since he has not produced more moral goodness than he has. We may wonder whether God is guilty by neglect. | |
| From: Brian Davies (Introduction to the Philosophy of Religion [1982], 3 'Freedom') | |
| A reaction: The orthodox response is that we cannot possibly know what the maximum of moral goodness would look like, so we can't make this judgement. Atheists say that God fails by human standards, which are not particularly high. |
| 20702 | The goodness of God may be a higher form than the goodness of moral agents [Davies,B] |
| Full Idea: If we can know that God exists and if God's goodness is not moral goodness, then moral goodness is not the highest form of goodness we know. There is the goodness of God to be reckoned with. | |
| From: Brian Davies (Introduction to the Philosophy of Religion [1982], 3 'Goodness') | |
| A reaction: This idea is to counter the charge that God fails to meet human standards for an ideal moral agent. But it sounds hand-wavy, since we presumably cannot comprehend the sort of goodness that is postulated here. |
| 20703 | How could God have obligations? What law could possibly impose them? [Davies,B] |
| Full Idea: We have good reason for resisting the suggestion that God has any duties or obligations. …What can oblige God in relation to his creatures? Could there be a law saying God has such obligations? Where does such a law come from? | |
| From: Brian Davies (Introduction to the Philosophy of Religion [1982], 3 'Goodness') | |
| A reaction: Plato can answer this question. Greek gods are not so supreme that nothing could put them under an obligation, but 'God' has to be supreme in every respect. |
| 20694 | 'Natural theology' aims to prove God to anyone (not just believers) by reason or argument [Davies,B] |
| Full Idea: 'Natural theology' is the attempt to show that belief in God's existence can be defended with reference to reason or argument which ought to be acceptable to anyone, not simply to those who believe in God's existence. | |
| From: Brian Davies (Introduction to the Philosophy of Religion [1982], 1 'Other') | |
| A reaction: I assume by 'reason or argument' he primarily means evidence (plus the ontological argument). He cites Karl Barth as objecting to the assumption of natural theology (preferring revelation). Presumably Kierkegaard offers a rival view too. |
| 20706 | A distinct cause of the universe can't be material (which would be part of the universe) [Davies,B] |
| Full Idea: If the universe was caused to come into being, it presumably could not have been caused to do so by anything material. For a material object would be part of the universe, and we are now asking for a cause distinct from the universe. | |
| From: Brian Davies (Introduction to the Philosophy of Religion [1982], 5 'God') | |
| A reaction: We're out of our depth here. We only have two modes of existence to offer, material and spiritual, and 'spiritual' means little more than non-material. |
| 20707 | The universe exhibits design either in its sense of purpose, or in its regularity [Davies,B] |
| Full Idea: The design argument offers two lines: the first states that the universe displays design in the sense of purpose; the second that it displays design in the sense of regularity. | |
| From: Brian Davies (Introduction to the Philosophy of Religion [1982], 6 'Versions') | |
| A reaction: I would have thought that you would infer the purpose from the regularity. How could you see purpose in a totally chaotic universe? |
| 20708 | If God is an orderly being, he cannot be the explanation of order [Davies,B] |
| Full Idea: If God is an instance of something orderly, how can he serve to account for the order of orderly things? | |
| From: Brian Davies (Introduction to the Philosophy of Religion [1982], 6 'b Has') | |
| A reaction: You can at least explain the tidiness of a house by the tidiness of its owner, but obviously that won't explain the phenomenon of tidiness. |
| 20710 | Maybe an abnormal state of mind is needed to experience God? [Davies,B] |
| Full Idea: Might it not be possible that experience of God requires an unusual state or psychological abnormality, just as an aerial view of Paris requires that one be in the unusual state of being abnormally elevated? | |
| From: Brian Davies (Introduction to the Philosophy of Religion [1982], 7 'Are the') | |
| A reaction: That would make sense if it were analogous to great mathematical or musical ability, but it sounds more like ouija boards in darkened rooms. Talent has a wonderful output, but people in mystical states don't return with proofs. |
| 20711 | A believer can experience the world as infused with God [Davies,B] |
| Full Idea: Maybe someone who believes in God can be regarded as experiencing everything as something behind which God lies. Believers see the world as a world in which God is present. | |
| From: Brian Davies (Introduction to the Philosophy of Religion [1982], 7 'Experiencing') | |
| A reaction: [Attributed to John Hick] This would count as supporting evidence for God, perhaps, if seeing reality as infused with God produces a consistent and plausible picture. But seeing reality as infused with other things might pass the same test. |
| 20709 | The experiences of God are inconsistent, not universal, and untestable [Davies,B] |
| Full Idea: A proclaimed experience of God must be rejected because a) there is no agreed test that it is such an experience, b) some people experience God's absence, and c) there is no uniformity of testimony about the experience. | |
| From: Brian Davies (Introduction to the Philosophy of Religion [1982], 7 'Objections') | |
| A reaction: [compressed] I'm not sure that absence of an experience is experience of an absence. Compare it with experiencing the greatness of Beethoven's Ninth. |
| 20697 | One does not need a full understanding of God in order to speak of God [Davies,B] |
| Full Idea: In order to speak meaningfully about God, it is not necessary that one should understand exactly the import of one's statements about him. | |
| From: Brian Davies (Introduction to the Philosophy of Religion [1982], 2 'Sayng') | |
| A reaction: Perfectly reasonable. To insist that all discussion of a thing requires exact understanding of the thing is ridiculous. Equally, though, to discuss God while denying all understanding of God is just as ridiculous. |
| 20699 | Paradise would not contain some virtues, such as courage [Davies,B] |
| Full Idea: There are virtues (such as courage) that would not be present in a paradise. | |
| From: Brian Davies (Introduction to the Philosophy of Religion [1982], 3 'Evil') | |
| A reaction: Part of a suggestion that morality would be entirely inapplicable in paradise, and so we need dangers etc in the world. |