27 ideas
| 14519 | It is a great good to show reverence for a wise man [Epicurus] |
| Full Idea: To show reverence for a wise man is itself a great good for him who reveres [the wise man]. | |
| From: Epicurus (Principle Doctrines ('Kuriai Doxai') (frags) [c.290 BCE], 32) | |
| A reaction: It is characteristic of Epicurus to move up a level in his thinking, and not merely respect wisdom, but ask after the value of his own respect. Compare Idea 14517. Nice. |
| 14518 | In the study of philosophy, pleasure and knowledge arrive simultaneously [Epicurus] |
| Full Idea: In philosophy the pleasure accompanies the knowledge. For the enjoyment does not come after the learning but the learning and the enjoyment are simultaneous. | |
| From: Epicurus (Principle Doctrines ('Kuriai Doxai') (frags) [c.290 BCE], 27) |
| 15924 | Predicative definitions are acceptable in mathematics if they distinguish objects, rather than creating them? [Zermelo, by Lavine] |
| Full Idea: On Zermelo's view, predicative definitions are not only indispensable to mathematics, but they are unobjectionable since they do not create the objects they define, but merely distinguish them from other objects. | |
| From: report of Ernst Zermelo (Investigations in the Foundations of Set Theory I [1908]) by Shaughan Lavine - Understanding the Infinite V.1 | |
| A reaction: This seems to have an underlying platonism, that there are hitherto undefined 'objects' lying around awaiting the honour of being defined. Hm. |
| 17608 | We take set theory as given, and retain everything valuable, while avoiding contradictions [Zermelo] |
| Full Idea: Starting from set theory as it is historically given ...we must, on the one hand, restrict these principles sufficiently to exclude as contradiction and, on the other, take them sufficiently wide to retain all that is valuable in this theory. | |
| From: Ernst Zermelo (Investigations in the Foundations of Set Theory I [1908], Intro) | |
| A reaction: Maddy calls this the one-step-back-from-disaster rule of thumb. Zermelo explicitly mentions the 'Russell antinomy' that blocked Frege's approach to sets. |
| 17607 | Set theory investigates number, order and function, showing logical foundations for mathematics [Zermelo] |
| Full Idea: Set theory is that branch whose task is to investigate mathematically the fundamental notions 'number', 'order', and 'function', taking them in their pristine, simple form, and to develop thereby the logical foundations of all of arithmetic and analysis. | |
| From: Ernst Zermelo (Investigations in the Foundations of Set Theory I [1908], Intro) | |
| A reaction: At this point Zermelo seems to be a logicist. Right from the start set theory was meant to be foundational to mathematics, and not just a study of the logic of collections. |
| 10870 | ZFC: Existence, Extension, Specification, Pairing, Unions, Powers, Infinity, Choice [Zermelo, by Clegg] |
| Full Idea: Zermelo-Fraenkel axioms: Existence (at least one set); Extension (same elements, same set); Specification (a condition creates a new set); Pairing (two sets make a set); Unions; Powers (all subsets make a set); Infinity (set of successors); Choice | |
| From: report of Ernst Zermelo (Investigations in the Foundations of Set Theory I [1908]) by Brian Clegg - Infinity: Quest to Think the Unthinkable Ch.15 |
| 13012 | Zermelo published his axioms in 1908, to secure a controversial proof [Zermelo, by Maddy] |
| Full Idea: Zermelo proposed his listed of assumptions (including the controversial Axiom of Choice) in 1908, in order to secure his controversial proof of Cantor's claim that ' we can always bring any well-defined set into the form of a well-ordered set'. | |
| From: report of Ernst Zermelo (Investigations in the Foundations of Set Theory I [1908]) by Penelope Maddy - Believing the Axioms I §1 | |
| A reaction: This is interesting because it sometimes looks as if axiom systems are just a way of tidying things up. Presumably it is essential to get people to accept the axioms in their own right, the 'old-fashioned' approach that they be self-evident. |
| 17609 | Set theory can be reduced to a few definitions and seven independent axioms [Zermelo] |
| Full Idea: I intend to show how the entire theory created by Cantor and Dedekind can be reduced to a few definitions and seven principles, or axioms, which appear to be mutually independent. | |
| From: Ernst Zermelo (Investigations in the Foundations of Set Theory I [1908], Intro) | |
| A reaction: The number of axioms crept up to nine or ten in subsequent years. The point of axioms is maximum reduction and independence from one another. He says nothing about self-evidence (though Boolos claimed a degree of that). |
| 13017 | Zermelo introduced Pairing in 1930, and it seems fairly obvious [Zermelo, by Maddy] |
| Full Idea: Zermelo's Pairing Axiom superseded (in 1930) his original 1908 Axiom of Elementary Sets. Like Union, its only justification seems to rest on 'limitations of size' and on the 'iterative conception'. | |
| From: report of Ernst Zermelo (Investigations in the Foundations of Set Theory I [1908]) by Penelope Maddy - Believing the Axioms I §1.3 | |
| A reaction: Maddy says of this and Union, that they seem fairly obvious, but that their justification is of prime importance, if we are to understand what the axioms should be. |
| 13015 | Zermelo used Foundation to block paradox, but then decided that only Separation was needed [Zermelo, by Maddy] |
| Full Idea: Zermelo used a weak form of the Axiom of Foundation to block Russell's paradox in 1906, but in 1908 felt that the form of his Separation Axiom was enough by itself, and left the earlier axiom off his published list. | |
| From: report of Ernst Zermelo (Investigations in the Foundations of Set Theory I [1908]) by Penelope Maddy - Believing the Axioms I §1.2 | |
| A reaction: Foundation turns out to be fairly controversial. Barwise actually proposes Anti-Foundation as an axiom. Foundation seems to be the rock upon which the iterative view of sets is built. Foundation blocks infinite descending chains of sets, and circularity. |
| 13020 | The Axiom of Separation requires set generation up to one step back from contradiction [Zermelo, by Maddy] |
| Full Idea: The most characteristic Zermelo axiom is Separation, guided by a new rule of thumb: 'one step back from disaster' - principles of set generation should be as strong as possible short of contradiction. | |
| From: report of Ernst Zermelo (Investigations in the Foundations of Set Theory I [1908]) by Penelope Maddy - Believing the Axioms I §1.4 | |
| A reaction: Why is there an underlying assumption that we must have as many sets as possible? We are then tempted to abolish axioms like Foundation, so that we can have even more sets! |
| 13486 | Not every predicate has an extension, but Separation picks the members that satisfy a predicate [Zermelo, by Hart,WD] |
| Full Idea: Zermelo assumes that not every predicate has an extension but rather that given a set we may separate out from it those of its members satisfying the predicate. This is called 'separation' (Aussonderung). | |
| From: report of Ernst Zermelo (Investigations in the Foundations of Set Theory I [1908]) by William D. Hart - The Evolution of Logic 3 |
| 13487 | In ZF, the Burali-Forti Paradox proves that there is no set of all ordinals [Zermelo, by Hart,WD] |
| Full Idea: In Zermelo's set theory, the Burali-Forti Paradox becomes a proof that there is no set of all ordinals (so 'is an ordinal' has no extension). | |
| From: report of Ernst Zermelo (Investigations in the Foundations of Set Theory I [1908]) by William D. Hart - The Evolution of Logic 3 |
| 18178 | For Zermelo the successor of n is {n} (rather than n U {n}) [Zermelo, by Maddy] |
| Full Idea: For Zermelo the successor of n is {n} (rather than Von Neumann's successor, which is n U {n}). | |
| From: report of Ernst Zermelo (Investigations in the Foundations of Set Theory I [1908]) by Penelope Maddy - Naturalism in Mathematics I.2 n8 | |
| A reaction: I could ask some naive questions about the comparison of these two, but I am too shy about revealing my ignorance. |
| 13027 | Zermelo believed, and Von Neumann seemed to confirm, that numbers are sets [Zermelo, by Maddy] |
| Full Idea: Zermelo was a reductionist, and believed that theorems purportedly about numbers (cardinal or ordinal) are really about sets, and since Von Neumann's definitions of ordinals and cardinals as sets, this has become common doctrine. | |
| From: report of Ernst Zermelo (Investigations in the Foundations of Set Theory I [1908]) by Penelope Maddy - Believing the Axioms I §1.8 | |
| A reaction: Frege has a more sophisticated take on this approach. It may just be an updating of the Greek idea that arithmetic is about treating many things as a unit. A set bestows an identity on a group, and that is all that is needed. |
| 9627 | Different versions of set theory result in different underlying structures for numbers [Zermelo, by Brown,JR] |
| Full Idea: In Zermelo's set-theoretic definition of number, 2 is a member of 3, but not a member of 4; in Von Neumann's definition every number is a member of every larger number. This means they have two different structures. | |
| From: report of Ernst Zermelo (Investigations in the Foundations of Set Theory I [1908]) by James Robert Brown - Philosophy of Mathematics Ch. 4 | |
| A reaction: This refers back to the dilemma highlighted by Benacerraf, which was supposed to be the motivation for structuralism. My intuition says that the best answer is that they are both wrong. In a pattern, the nodes aren't 'members' of one another. |
| 14524 | Bodies are combinations of shape, size, resistance and weight [Epicurus] |
| Full Idea: Epicurus said that body was conceived as an aggregate of shape and size and resistance and weight. | |
| From: Epicurus (Principle Doctrines ('Kuriai Doxai') (frags) [c.290 BCE]) | |
| A reaction: [Source Sextus 'Adversus Mathematicos' 10.257] Note that this is how we 'conceive' them. They might be intrinsically different, except that Epicurus is pretty much a phenomenalist. |
| 14521 | If everything is by necessity, then even denials of necessity are by necessity [Epicurus] |
| Full Idea: He who claims that everything occurs by necessity has no complaint against him who claims that everything does not occur by necessity. For he makes the very claim in question by necessity. | |
| From: Epicurus (Principle Doctrines ('Kuriai Doxai') (frags) [c.290 BCE], 40) |
| 14522 | What happens to me if I obtain all my desires, and what if I fail? [Epicurus] |
| Full Idea: One should bring this question to bear on all one's desires: what will happen to me if what is sought by desire is achieved, and what will happen if it is not? | |
| From: Epicurus (Principle Doctrines ('Kuriai Doxai') (frags) [c.290 BCE], 71) | |
| A reaction: Yet another example of Epicurus moving up a level in his thinking about ethical issues, as in Idea 14517 and Idea 14519. The mark of a true philosopher. This seems to be a key idea for wisdom - to think further ahead than merely what you desire. |
| 3563 | Pleasure and virtue entail one another [Epicurus] |
| Full Idea: It is not possible to live pleasantly without living intelligently and finely and justly, nor to live intelligently and finely and justly without living pleasantly. | |
| From: Epicurus (Principle Doctrines ('Kuriai Doxai') (frags) [c.290 BCE], 5), quoted by Julia Annas - The Morality of Happiness Ch.16 | |
| A reaction: A person with all these virtues might still suffer from depression. And I don't see why having limited intelligence should stop someone from living pleasantly. Just be warm-hearted. |
| 3560 | Justice is merely a contract about not harming or being harmed [Epicurus] |
| Full Idea: There is no such things as justice in itself; in people's relations with one another in any place and at any time it is a contract about not harming or being harmed. | |
| From: Epicurus (Principle Doctrines ('Kuriai Doxai') (frags) [c.290 BCE], 33), quoted by Julia Annas - The Morality of Happiness 13.2 |
| 14517 | We value our own character, whatever it is, and we should respect the characters of others [Epicurus] |
| Full Idea: We value our characters as our own personal possessions, whether they are good and envied by men or not. We must regard our neighbours' characters thus too, if they are respectable. | |
| From: Epicurus (Principle Doctrines ('Kuriai Doxai') (frags) [c.290 BCE], 15) | |
| A reaction: I like this because it introduces a metaethical dimension to the whole problem of virtue. We should value our own character - so should we try to improve it? Should we improve so much as to become unrecognisable? |
| 14513 | Justice is a pledge of mutual protection [Epicurus] |
| Full Idea: The justice of nature is a pledge of reciprocal usefulness, neither to harm one another nor to be harmed. | |
| From: Epicurus (Principle Doctrines ('Kuriai Doxai') (frags) [c.290 BCE], 31) | |
| A reaction: Notice that justice is not just reciprocal usefulness, but a 'pledge' to that effect. This implies a metaethical value of trust and honesty in keeping the pledge. Is it better to live by the pledge, or to be always spontaneously useful? |
| 14515 | A law is not just if it is not useful in mutual associations [Epicurus] |
| Full Idea: If someone passes a law and it does not turn out to be in accord with what is useful in mutual associations, this no longer possesses the nature of justice. | |
| From: Epicurus (Principle Doctrines ('Kuriai Doxai') (frags) [c.290 BCE], 37) |
| 14520 | It is small-minded to find many good reasons for suicide [Epicurus] |
| Full Idea: He is utterly small-minded for whom there are many plausible reasons for committing suicide. | |
| From: Epicurus (Principle Doctrines ('Kuriai Doxai') (frags) [c.290 BCE], 38) | |
| A reaction: It is a pity that the insult of 'small-minded' has slipped out of philosophy. The Greeks use it all the time, and know exactly what it means. We all recognise small-mindedness, and it is a great (and subtle) vice. |
| 1748 | Archelaus was the first person to say that the universe is boundless [Archelaus, by Diog. Laertius] |
| Full Idea: Archelaus was the first person to say that the universe is boundless. | |
| From: report of Archelaus (fragments/reports [c.450 BCE]) by Diogenes Laertius - Lives of Eminent Philosophers 02.Ar.3 |
| 5989 | Archelaus said life began in a primeval slime [Archelaus, by Schofield] |
| Full Idea: Archelaus wrote that life on Earth began in a primeval slime. | |
| From: report of Archelaus (fragments/reports [c.450 BCE]) by Malcolm Schofield - Archelaus | |
| A reaction: This sounds like a fairly clearcut assertion of the production of life by evolution. Darwin's contribution was to propose the mechanism for achieving it. We should honour the name of Archelaus for this idea. |