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 |
| 7720 | Two things can only resemble one another in some respect, and that may reintroduce a universal [Lowe] |
| Full Idea: A problem for resemblance nominalism is that in saying that two particulars 'resemble' one another, it is necessary to specify in what respect they do so (e.g. colour, shape, size), and this threatens to reintroduce what appears to be talk of universals. | |
| From: E.J. Lowe (Locke on Human Understanding [1995], Ch.7) | |
| A reaction: We see resemblance between faces instantly, long before we can specify the 'respects' of the resemblance. This supports the Humean hard-wired view of resemblance, rather than some appeal to Platonic universals. |
| 7712 | On substances, Leibniz emphasises unity, Spinoza independence, Locke relations to qualities [Lowe] |
| Full Idea: Later philosophers emphasised different strands of Aristotle's concept of substances: Leibniz (in his theory of monads) emphasised their unity; Spinoza emphasised their ontological independence; Locke emphasised their role in relation to qualities. | |
| From: E.J. Lowe (Locke on Human Understanding [1995], Ch.4) | |
| A reaction: Note that this Aristotelian idea had not been jettisoned in the late seventeenth century, unlike other Aristotelianisms. I think it is only with the success of atomism in chemistry that the idea of substance is forced to recede. |
| 19712 | Maybe there is plain 'animal' knowledge, and clearly justified 'reflective' knowledge [Vahid] |
| Full Idea: There is a distinction between 'animal knowledge' (which requires only apt belief), and 'reflective knowledge' (requiring both apt and justified belief). | |
| From: Hamid Vahid (Externalism/Internalism [2011], 5) | |
| A reaction: [He cites Sosa 1991] My inclination (Idea 19711) was to think of knowledge as a continuum (possibly with a contextual component), and this distinction doesn't change my view, though it makes the point. |
| 7710 | Perception is a mode of belief-acquisition, and does not involve sensation [Lowe] |
| Full Idea: According to one school of thought, perception is simply a mode of belief-acquisition,and there is no reason to suppose that any element of sensation is literally involved in perception. | |
| From: E.J. Lowe (Locke on Human Understanding [1995], Ch.3) | |
| A reaction: Blindsight would be an obvious supporting case for this view. I think this point is crucial in understanding what is wrong with Jackson's 'knowledge argument' (involving Mary, see Idea 7377). Sensation gives knowledge, so it can't be knowledge. |
| 7711 | Science requires a causal theory - perception of an object must be an experience caused by the object [Lowe] |
| Full Idea: Only a causal theory of perception will respect the facts of physiology and physics ...meaning a theory which maintains that for a subject to perceive a physical object the subject should enjoy some appropriate perceptual experience caused by the object. | |
| From: E.J. Lowe (Locke on Human Understanding [1995], Ch.3) | |
| A reaction: If I hallucinate an object, then presumably I am not allowed to say that I 'perceive' it, but that seems to make the causal theory an idle tautology. If we are in virtual reality then there aren't any objects. |
| 19703 | Epistemic is normally marked out from moral or pragmatic justifications by its truth-goal [Vahid] |
| Full Idea: It is widely believed that epistemic justification is distinct from other species of justification such as moral or pragmatic justification in that it is intended to serve the so-called 'truth-goal'. | |
| From: Hamid Vahid (Externalism/Internalism [2011], 1) | |
| A reaction: Kvanvig explicitly argues against this view. He broadens the aims, but it strikes me that other aims are all intertwined with truth in some way, so I find this idea quite plausible. |
| 19705 | 'Mentalist' internalism seems to miss the main point, if it might not involve an agent's access [Vahid] |
| Full Idea: Since mentalism remains neutral on whether mental states need be accessible to an agent ...it does not seem to do justice to the intuitions that drive paradigm internalist positions. | |
| From: Hamid Vahid (Externalism/Internalism [2011], 2 A) | |
| A reaction: The rival view is 'access internalism', which implies that you can act on and take responsibility for your knowledge, because you are aware of its grounding. If animals know things, that might fit the mentalist picture better. |
| 19706 | Strong access internalism needs actual awareness; weak versions need possibility of access [Vahid] |
| Full Idea: A strong form of 'access internalism' is when an agent is required to be actually aware of the conditions that constitute justification; a weaker version loosens the accessibility condition, requiring only the ability to access the justification. | |
| From: Hamid Vahid (Externalism/Internalism [2011], 2 B) | |
| A reaction: The super strong version implies that you probably only know one thing at a time, so it must be nonsense. The weaker version has grey areas. I remember roughly the justification, but not the details. The justification is in my diary. Etc. |
| 19707 | Maybe we need access to our justification, and also to know why it justifies [Vahid] |
| Full Idea: Access internalism may also have a truth-conducive conception of justification, where one should not only know what one's reasons are, but also why one's beliefs are probable on one's reasons. | |
| From: Hamid Vahid (Externalism/Internalism [2011], 2 B) | |
| A reaction: [he cites Bonjour 1985] Sounds reasonable. It would seem odd if you had clear access to the reason, but didn't understand it, because you had just learned it by rote. |
| 19709 | Internalism in epistemology over-emphasises deliberation about beliefs [Vahid] |
| Full Idea: The internalist approach in epistemology seems to suggest an over-inellectualized and deliberative picture of our belief-forming activities. | |
| From: Hamid Vahid (Externalism/Internalism [2011], 2.2 B) | |
| A reaction: This strikes me as confused. The question is not how do I arrive at my beliefs but what justifies my believing them, and what justifies the beliefs in themselves? My head is full of daft beliefs produced by TV advertising. |
| 19704 | Externalism may imply that identical mental states might go with different justifications [Vahid] |
| Full Idea: According to the 'mentalist' version of internalism, an externalist is someone who maintains that two people can be in the same present mental states while one has a justified belief and the other does not. | |
| From: Hamid Vahid (Externalism/Internalism [2011], 2 A) | |
| A reaction: It seems an unlikely coincidence, that we have identical mental states, but your is (say) reliably created but mine isn't. Nevertheless this does seem to be an implication of externalism, though not a definition of it. |
| 19710 | With a counterfactual account of the causal theory, we get knowledge as tracking or sensitive to truth [Vahid] |
| Full Idea: The causal theory of justification was soon replaced by Nozick's construal of knowledge as counterfactually sensitive to its truth value (that is, it tracks truth). A counterfactual theory of causation connects this to the causal theory. | |
| From: Hamid Vahid (Externalism/Internalism [2011], 3) | |
| A reaction: This is presented as an externalist theory, close to the causal theory (and prior to the reliability theory). But how could you be 'sensitive' to a changing truth if the justification was all external? Externally supported beliefs seem ossified. |
| 19711 | Externalism makes the acquisition of knowledge too easy? [Vahid] |
| Full Idea: Internalists say that externalism is inadequate because it makes the obtaining of knowledge and justified beliefs too easy | |
| From: Hamid Vahid (Externalism/Internalism [2011], 4) | |
| A reaction: This looks like a key issue in epistemology. Do children and animals have lots of knowledge, which they soak up unthinkingly, or do only thinking adults really 'know' things? Why not have degrees of knowledge? |
| 7714 | Personal identity is a problem across time (diachronic) and at an instant (synchronic) [Lowe] |
| Full Idea: There is the question of the identity of a person over or across time ('diachronic' personal identity), and there is also the question of what makes for personal identity at a time ('synchronic' personal identity). | |
| From: E.J. Lowe (Locke on Human Understanding [1995], Ch.5) | |
| A reaction: This seems to me to be the first and most important distinction in the philosophy of personal identity, and they regularly get run together. Locke, for example, has an account of synchronic identity, which is often ignored. It applies to objects too. |
| 7715 | Mentalese isn't a language, because it isn't conventional, or a means of public communication [Lowe] |
| Full Idea: 'Mentalese' would be neither conventional nor a means of public communication so that even to call it a language is seriously misleading. | |
| From: E.J. Lowe (Locke on Human Understanding [1995], Ch.7) | |
| A reaction: It is, however, supposed to contain symbolic representations which are then used as tokens for computation, so it seems close to a language, if (for example) symbolic logic or mathematics were accepted as languages. But who understands it? |
| 7722 | If meaning is mental pictures, explain "the cat (or dog!) is NOT on the mat" [Lowe] |
| Full Idea: If meaning is a private mental picture, what does 'the cat is NOT on the mat' mean, and how does it differ from 'the dog is not on the mat?'. | |
| From: E.J. Lowe (Locke on Human Understanding [1995], Ch.7) | |
| A reaction: Not insurmountable. We picture an empty mat, combined with a cat (or whatever) located somewhere else. A mental 'picture' of something shouldn't be contrued as a single image in a neat black frame. |