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. |
| 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 |
| 5901 | Is 'productive of happiness' the definition of 'right', or the cause of it? [Ross on Bentham] |
| Full Idea: Bentham has not made up his mind whether he thinks that 'right' means 'productive of the general happiness', or that being productive of the general happiness is what makes acts right (and he would have thought the difference unimportant). | |
| From: comment on Jeremy Bentham (Intro to Principles of Morals and Legislation [1789]) by W. David Ross - The Right and the Good §I | |
| A reaction: The issue is whether rightness exists as a concept separate from happiness. I take it Bentham would vote for the first reading, as he has no interest in what is right, apart from increasing happiness. |
| 5934 | Of Bentham's 'dimensions' of pleasure, only intensity and duration matter [Ross on Bentham] |
| Full Idea: Most of Bentham's 'dimensions' of pleasure refer to further pleasures, or are irrelevant to the pleasure; we are left with intensity and duration as the characteristics on which depend the value of a pleasure qua pleasure, and there is nothing to add. | |
| From: comment on Jeremy Bentham (Intro to Principles of Morals and Legislation [1789]) by W. David Ross - The Right and the Good §VI | |
| A reaction: I agree. When Bentham produces his list he seems to be trying to add a bogus enrichment to what is really a rather crude and basic notion of the aim of life. Your simple hedonist appears to only desire high intensity and long duration. |
| 5271 | Prejudice apart, push-pin has equal value with music and poetry [Bentham] |
| Full Idea: Prejudice apart, the game of push-pin is of equal value with the arts and science of music and poetry. | |
| From: Jeremy Bentham (Constitutional Code I [1827], p.139), quoted by J.R. Dinwiddy - Bentham p.114 | |
| A reaction: Mill quoted this with implied outrage, but Bentham was attacking public subsidies to the arts when he said it. It is a basic idea in the debate on pleasure - that pleasures are only distinguished by their intensity, not some other value. |
| 3777 | Pleasure and pain control all human desires and duties [Bentham] |
| Full Idea: Nature has placed mankind under the governance of two sovereign masters, pain and pleasure. It is for them alone to point out what we ought to do, as well as to determine what we shall do. | |
| From: Jeremy Bentham (Intro to Principles of Morals and Legislation [1789], I.1) | |
| A reaction: Ridiculous. Both halves are false. We pursue things for other reasons, and to deny this makes his idea a tautology. Deep ecology has nothing to do with human pleasure or pain. |
| 3554 | Bentham thinks happiness is feeling good, but why use morality to achieve that? [Annas on Bentham] |
| Full Idea: It is easy to fall into Bentham's mindless assumption that happiness must be a specific state of feeling good about something, but it is mysterious why anyone would think morality a good strategy for achieving this. | |
| From: comment on Jeremy Bentham (Intro to Principles of Morals and Legislation [1789]) by Julia Annas - The Morality of Happiness 2.7 |
| 3781 | The value of pleasures and pains is their force [Bentham] |
| Full Idea: It behoves the legislator to understand the force of pleasures and pains, which is their value. | |
| From: Jeremy Bentham (Intro to Principles of Morals and Legislation [1789], IV.1) |
| 20977 | Natural rights are nonsense, and unspecified natural rights is nonsense on stilts [Bentham] |
| Full Idea: Natural rights is simple nonsense: natural and imprescriptible rights, rhetorical nonsense — nonsense upon stilts. | |
| From: Jeremy Bentham (Anarchical Fallacies: on the Declaration of Rights [1796]) | |
| A reaction: If you want your opinion to be remembered, express it memorably! I take natural rights to be the basic principles and values which are obvious to almost everyone when they come for formulate legal rights (which are the only true rights). |
| 3778 | The community's interest is a sum of individual interests [Bentham] |
| Full Idea: The interest of the community is the sum of the interests of the several members who compose it. | |
| From: Jeremy Bentham (Intro to Principles of Morals and Legislation [1789], I.4) |
| 21003 | Only laws can produce real rights; rights from 'law of nature' are imaginary [Bentham] |
| Full Idea: Right, the substantive right, is the child of law; from real laws come real rights; but from imaginary laws, from 'law of nature' can come only imaginary rights. | |
| From: Jeremy Bentham (Anarchical Fallacies: on the Declaration of Rights [1796], II.523), quoted by Amartya Sen - The Idea of Justice 17 'Ethics' | |
| A reaction: I am coming to agree with this. What are called 'natural rights' are just self-evident good reasons why someone should be allowed a right. A right can, of course, come from an informal agreement. The question is: why award that particular legal right? |
| 7810 | The 'Eumenides' of Aeschylus shows blood feuds replaced by law [Aeschylus, by Grayling] |
| Full Idea: The 'Eumenides' of Aeschylus tells how the old rule of revenge and blood feud was replaced by a due process of law before a civil jury. | |
| From: report of Aeschylus (The Eumenides [c.458 BCE]) by A.C. Grayling - What is Good? Ch.2 | |
| A reaction: Compare Idea 1659, where this revolution is attributed to Protagoras (a little later than Aeschylus). I take the rule of law and of society to be above all the rule of reason, because the aim is calm objectivity instead of emotion. |
| 20280 | Large mature animals are more rational than babies. But all that really matters is - can they suffer? [Bentham] |
| Full Idea: A full-grown horse or dog is beyond comparison a more rational animal than an infant of a day, or even a month, old. But suppose they be otherwise, what would it avail? The question is not, Can they reason? nor Can they talk? but, Can they suffer? | |
| From: Jeremy Bentham (Intro to Principles of Morals and Legislation [1789], XVIII 1 n), quoted by Peter Singer - Practical Ethics 03 | |
| A reaction: This is certainly an inspiring, and even shocking question, which never seems to have been so directly asked before in the entire history of European thought. Awesome. |
| 3779 | Unnatural, when it means anything, means infrequent [Bentham] |
| Full Idea: Unnatural, when it means anything, means unfrequent. | |
| From: Jeremy Bentham (Intro to Principles of Morals and Legislation [1789], II.14 n8.9) |
| 3780 | We must judge a thing morally to know if it conforms to God's will [Bentham] |
| Full Idea: It is necessary to know first whether a thing is right in order to know from thence whether it be conformable to the will of God. | |
| From: Jeremy Bentham (Intro to Principles of Morals and Legislation [1789], II.18) |