25 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. |
| 8820 | Rules of reasoning precede the concept of truth, and they are what characterize it [Pollock] |
| Full Idea: Rather than truth being fundamental and rules for reasoning being derived from it, the rules for reasoning come first and truth is characterized by the rules for reasoning about truth. | |
| From: John L. Pollock (Epistemic Norms [1986], 'Cog.Mach') | |
| A reaction: This nicely disturbs our complacency about such things. There is plenty of reasoning in Homer, but I bet there is no talk of 'truth'. Pontius Pilate seems to have been a pioneer (Idea 8821). Do the truth tables define or describe logical terms? |
| 8819 | We need the concept of truth for defeasible reasoning [Pollock] |
| Full Idea: It might be wondered why we even have a concept of truth. The answer is that this concept is required for defeasible reasoning. | |
| From: John L. Pollock (Epistemic Norms [1986], 'Cog.Mach') | |
| A reaction: His point is that we must be able to think critically about our beliefs ('is p true?') if we are to have any knowledge at all. An excellent point. Give that man a teddy bear. |
| 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. |
| 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 |
| 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. |
| 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 |
| 8822 | Statements about necessities need not be necessarily true [Pollock] |
| Full Idea: True statements about the necessary properties of things need not be necessarily true. The well-known example is that the number of planets (9) is necessarily an odd number. The necessity is de re, but not de dicto. | |
| From: John L. Pollock (Epistemic Norms [1986], 'Nat.Internal') | |
| A reaction: This would be a matter of the scope (the placing of the brackets) of the 'necessarily' operator in a formula. The quick course in modal logic should eradicate errors of this kind in your budding philosopher. |
| 8818 | Defeasible reasoning requires us to be able to think about our thoughts [Pollock] |
| Full Idea: Defeasible reasoning requires us to be able to think about our thoughts. | |
| From: John L. Pollock (Epistemic Norms [1986], 'Cog.Mach') | |
| A reaction: This is why I do not think animals 'know' anything, though they seem to have lots of true beliefs about their immediate situation. |
| 8811 | What we want to know is - when is it all right to believe something? [Pollock] |
| Full Idea: When we ask whether a belief is justified, we want to know whether it is all right to believe it. The question we must ask is 'when is it permissible (epistemically) to believe P?'. | |
| From: John L. Pollock (Epistemic Norms [1986], 'Ep.Norms') | |
| A reaction: Nice to see someone trying to get the question clear. The question clearly points to the fact that there must at least be some sort of social aspect to criteria of justification. I can't cheerfully follow my intuitions if everyone else laughs at them. |
| 8817 | Logical entailments are not always reasons for beliefs, because they may be irrelevant [Pollock] |
| Full Idea: Epistemologists have noted that logical entailments do not always constitute reasons. P may entail Q without the connection between P and Q being at all obvious. | |
| From: John L. Pollock (Epistemic Norms [1986], 'Ref.of Extern') | |
| A reaction: Graham Priest and others try to develop 'relevance logic' to deal with this. This would deny the peculiar classical claim that everything is entailed by a falsehood. A belief looks promising if it entails lots of truths about the world. |
| 8814 | Epistemic norms are internalised procedural rules for reasoning [Pollock] |
| Full Idea: Epistemic norms are to be understood in terms of procedural knowledge involving internalized rules for reasoning. | |
| From: John L. Pollock (Epistemic Norms [1986], 'How regulate?') | |
| A reaction: He offers analogies with bicycly riding, but the simple fact that something is internalized doesn't make it a norm. Some mention of truth is needed, equivalent to 'don't crash the bike'. |
| 8823 | Reasons are always for beliefs, but a perceptual state is a reason without itself being a belief [Pollock] |
| Full Idea: When one makes a perceptual judgement on the basis of a perceptual state, I want to say that the perceptual state itself is one's reason. ..Reason are always reasons for beliefs, but the reasons themselves need not be beliefs. | |
| From: John L. Pollock (Epistemic Norms [1986], 'Dir.Realism') | |
| A reaction: A crucial issue. I think I prefer the view of Davidson, in Ideas 8801 and 8804. Three options: a pure perception counts as a reason, or perceptions involve some conceptual content, or you only acquire a reason when a proposition is formulated. |
| 8813 | If we have to appeal explicitly to epistemic norms, that will produce an infinite regress [Pollock] |
| Full Idea: If we had to make explicit appeal to epistemic norms for justification (the 'intellectualist model') we would find ourselves in an infinite regress. The norms, their existence and their application would themselves have to be justified. | |
| From: John L. Pollock (Epistemic Norms [1986], 'How regulate?') | |
| A reaction: This is counter to the 'space of reasons' picture, where everything is rationally assessed. There are regresses for both reasons and for experiences, when they are offered as justifications. |
| 8812 | Norm Externalism says norms must be internal, but their selection is partly external [Pollock] |
| Full Idea: Norm Externalism acknowledges that the content of our epistemic norms must be internalist, but employs external considerations in the selection of the norms themselves. | |
| From: John L. Pollock (Epistemic Norms [1986], 'Ep.Norms') | |
| A reaction: It can't be right that you just set your own norms, so this must contain some truth. Equally, even the most hardened externalist can't deny that what goes on in the head of the person concerned must have some relevance. |
| 8816 | Externalists tend to take a third-person point of view of epistemology [Pollock] |
| Full Idea: Externalists tend to take a third-person point of view in discussing epistemology. | |
| From: John L. Pollock (Epistemic Norms [1986], 'Ref.of Extern') | |
| A reaction: Pollock's point, quite reasonably, is that the first-person aspect must precede any objective assessment of whether someone knows. External facts, such as unpublicised information, can undermine high quality internal justification. |
| 8815 | Belief externalism is false, because external considerations cannot be internalized for actual use [Pollock] |
| Full Idea: External considerations of reliability could not be internalized. Consequently, it is in principle impossible for us to actually employ externalist norms. I take this to be a conclusive refutation of belief externalism. | |
| From: John L. Pollock (Epistemic Norms [1986], 'Ref.of Extern') | |
| A reaction: Not so fast. He earlier rejected the 'intellectualist model' (Idea 8813), so he doesn't think norms have to be fully conscious and open to criticism. So they could be innate, or the result of indoctrination (sorry, teaching), or just forgotten. |
| 1513 | The Egyptians were the first to say the soul is immortal and reincarnated [Herodotus] |
| Full Idea: The Egyptians were the first to claim that the soul of a human being is immortal, and that each time the body dies the soul enters another creature just as it is being born. | |
| From: Herodotus (The Histories [c.435 BCE], 2.123.2) |