26 ideas
| 23367 | 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!). |
| 19342 | Reason avoids multiplying hypotheses or principles [Leibniz] |
| Full Idea: Reason requires that we avoid multiplying hypotheses or principles, in somewhat the same way that the simplest system is always preferred in astronomy. | |
| From: Gottfried Leibniz (Discourse on Metaphysics [1686], 5) | |
| A reaction: He offers this principle without mentioning Ockham, as if it were self-evident. |
| 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 |
| 12711 | The immediate cause of movements is more real [than geometry] [Leibniz] |
| Full Idea: The force or proximate cause of these changes [of position] is something more real, and there is sufficient basis to attribute it to one body more than to another. | |
| From: Gottfried Leibniz (Discourse on Metaphysics [1686], §18), quoted by Daniel Garber - Leibniz:Body,Substance,Monad 3 | |
| A reaction: The force is said to be 'more real' than geometry. Leibniz seems to have embraced fairly physical powers in the period 1678-1698, and then seen them as more and more like spirits. |
| 19349 | The complete notion of a substance implies all of its predicates or attributes [Leibniz] |
| Full Idea: The nature of an individual substance or of a complete being is to have a notion so complete that it is sufficient to contain and to allow us to deduce from it all the predicates of the subject to which this notion is attributed. | |
| From: Gottfried Leibniz (Discourse on Metaphysics [1686], §8) | |
| A reaction: This is the unusual Leibnizian view of such things, which he takes to extremes. I think it depends on whether you are talking of predicates, or of real intrinsic properties. I don't see how what happens to a substance can be contained in the subject. |
| 7558 | Substances mirror God or the universe, each from its own viewpoint [Leibniz] |
| Full Idea: Each substance is like a whole world, and like a mirror of God, or indeed of the whole universe, which each one expresses in its own fashion. | |
| From: Gottfried Leibniz (Discourse on Metaphysics [1686]), quoted by Nicholas Jolley - Leibniz Intro | |
| A reaction: Leibniz isn't a pantheist, so he does not identify God with the universe, so it is a bit revealing that substance could reflect either one or the other, and he doesn't seem to care which. In the end, for all the sophistication, he just made it up. |
| 16761 | Forms are of no value in physics, but are indispensable in metaphysics [Leibniz] |
| Full Idea: The consideration of forms serves no purpose in the details of physics and must not be used to explain particular phenomena. …but their misuse must not lead us to reject something which is so useful to metaphysics. | |
| From: Gottfried Leibniz (Discourse on Metaphysics [1686], 10), quoted by Robert Pasnau - Metaphysical Themes 1274-1671 24.5 | |
| A reaction: This is a key test for the question of whether metaphysics is separate from science (as Leibniz and Pasnau think), or whether there is a continuum. Is 'substantial form' an illuminating way to undestand modern physics? |
| 13088 | Subjects include predicates, so full understanding of subjects reveals all the predicates [Leibniz] |
| Full Idea: The subject-term must always include the predicate-term, in such a way that the man who understood the notion of the subject perfectly would also judge that the predicate belongs to it. | |
| From: Gottfried Leibniz (Discourse on Metaphysics [1686], §8) | |
| A reaction: Sounds as if every sentence is analytic, but he doesn't mean that. He does, oddly, mean that if we fully understand the name 'Alexander', we understand his complete history, which is a bit silly, I'm afraid. Even God doesn't learn things just from names. |
| 13085 | Leibniz is some form of haecceitist [Leibniz, by Cover/O'Leary-Hawthorne] |
| Full Idea: Some form of haecceitism is central to the Leibnizian metaphysic. | |
| From: report of Gottfried Leibniz (Discourse on Metaphysics [1686], §8) by Cover,J/O'Leary-Hawthorne,J - Substance and Individuation in Leibniz 5.2.1 | |
| A reaction: That is, there is some inner hallmark that individuates each thing (though they don't mean the Duns Scotus idea of a haecceity which has no qualities apart from the capacity to individuate). Leibniz thinks essences individuate. |
| 5024 | Knowledge doesn't just come from the senses; we know the self, substance, identity, being etc. [Leibniz] |
| Full Idea: It is always false to say that all our notions come from the so-called external senses, for the notion I have of myself and of my thoughts, and consequently of being, substance, action, identity, and many others, come from an internal experience. | |
| From: Gottfried Leibniz (Discourse on Metaphysics [1686], §27) | |
| A reaction: Of course, an empiricist like Hume would not deny this, as he bases his views on 'experience' (including anger, for example), not just 'sense experience'. But Hume, famously, said he has no experience of a Self, so can't get started on Leibniz's journey. |
| 5027 | If a person's memories became totally those of the King of China, he would be the King of China [Leibniz] |
| Full Idea: If someone were suddenly to become the King of China, forgetting what he has been, as if born anew, is this not as if he were annihilated, and a King of China created in his place at the same moment? | |
| From: Gottfried Leibniz (Discourse on Metaphysics [1686], §34) | |
| A reaction: Strikingly, this clearly endorse the view of the empiricist Locke. It is a view about the continuity of the self, not its essence, but Descartes must have turned in his grave when he read this. When this 'King of China' introspects his self, what is it? |
| 5023 | Future contingent events are certain, because God foresees them, but that doesn't make them necessary [Leibniz] |
| Full Idea: We must distinguish between what is certain and what is necessary; everyone agrees that future contingents are certain, since God foresees them, but it is not thereby admitted that they are necessary. | |
| From: Gottfried Leibniz (Discourse on Metaphysics [1686], §13) | |
| A reaction: An interesting point, since there is presumably a difference between God foreseeing that future squares will have four corners, and His foreseeing the next war. It seems to me, though, that 'certainty' is bad enough news for free will, without necessity. |
| 2119 | People argue for God's free will, but it isn't needed if God acts in perfection following supreme reason [Leibniz] |
| Full Idea: People try to safeguard God's freedom, as though it were not freedom of the highest sort to act in perfection following sovereign reason. | |
| From: Gottfried Leibniz (Discourse on Metaphysics [1686], §03) |
| 5025 | Mind and body can't influence one another, but God wouldn't intervene in the daily routine [Leibniz] |
| Full Idea: It is inconceivable that mind and body should have any influence on one another, and it is unreasonable simply to have recourse to the extraordinary operation of the universal cause in a matter which is ordinary and particular. | |
| From: Gottfried Leibniz (Discourse on Metaphysics [1686], §33) | |
| A reaction: Leibniz was the ultimate intellectual contortionist! Here he is rejecting Cartesian interactionism, and also Malebranche's Occasionalism (God bridges the gap), in order to prepare for his own (daft) theory of what is now called Parallelism. |
| 5026 | Animals lack morality because they lack self-reflection [Leibniz] |
| Full Idea: It is for lack of reflection on themselves that beasts have no moral qualities. | |
| From: Gottfried Leibniz (Discourse on Metaphysics [1686], §34) | |
| A reaction: Interesting, but I think this is false. I would say animals do have a sense of their self, because that is the most basic feature of any mind, but what they lack is second-order thought, that is, ability to reflect on and judge their own beliefs and acts. |