116 ideas
| 8014 | Resolve a complex into simple elements, then reconstruct the complex by using them [Hobbes, by MacIntyre] |
| 224 | When questions are doubtful we should concentrate not on objects but on ideas of the intellect [Plato] |
| 232 | Opposites are as unlike as possible [Plato] |
| 8937 | Plato's 'Parmenides' is the greatest artistic achievement of the ancient dialectic [Hegel on Plato] |
| 8721 | An 'impredicative' definition seems circular, because it uses the term being defined [Friend] |
| 8680 | Classical definitions attempt to refer, but intuitionist/constructivist definitions actually create objects [Friend] |
| 3678 | Reductio ad absurdum proves an idea by showing that its denial produces contradiction [Friend] |
| 8705 | Anti-realists see truth as our servant, and epistemically contrained [Friend] |
| 8713 | In classical/realist logic the connectives are defined by truth-tables [Friend] |
| 8708 | Double negation elimination is not valid in intuitionist logic [Friend] |
| 8694 | Free logic was developed for fictional or non-existent objects [Friend] |
| 8665 | A 'proper subset' of A contains only members of A, but not all of them [Friend] |
| 8672 | A 'powerset' is all the subsets of a set [Friend] |
| 8677 | Set theory makes a minimum ontological claim, that the empty set exists [Friend] |
| 8666 | Infinite sets correspond one-to-one with a subset [Friend] |
| 8682 | Major set theories differ in their axioms, and also over the additional axioms of choice and infinity [Friend] |
| 8709 | The law of excluded middle is syntactic; it just says A or not-A, not whether they are true or false [Friend] |
| 8711 | Intuitionists read the universal quantifier as "we have a procedure for checking every..." [Friend] |
| 13986 | Plato found antinomies in ideas, Kant in space and time, and Bradley in relations [Plato, by Ryle] |
| 14150 | Plato's 'Parmenides' is perhaps the best collection of antinomies ever made [Russell on Plato] |
| 8675 | Paradoxes can be solved by talking more loosely of 'classes' instead of 'sets' [Friend] |
| 8674 | The Burali-Forti paradox asks whether the set of all ordinals is itself an ordinal [Friend] |
| 8667 | The 'integers' are the positive and negative natural numbers, plus zero [Friend] |
| 8668 | The 'rational' numbers are those representable as fractions [Friend] |
| 8670 | A number is 'irrational' if it cannot be represented as a fraction [Friend] |
| 8661 | The natural numbers are primitive, and the ordinals are up one level of abstraction [Friend] |
| 8664 | Cardinal numbers answer 'how many?', with the order being irrelevant [Friend] |
| 8671 | The 'real' numbers (rationals and irrationals combined) is the Continuum, which has no gaps [Friend] |
| 8663 | Raising omega to successive powers of omega reveal an infinity of infinities [Friend] |
| 8662 | The first limit ordinal is omega (greater, but without predecessor), and the second is twice-omega [Friend] |
| 8669 | Between any two rational numbers there is an infinite number of rational numbers [Friend] |
| 8676 | Is mathematics based on sets, types, categories, models or topology? [Friend] |
| 8678 | Most mathematical theories can be translated into the language of set theory [Friend] |
| 8701 | The number 8 in isolation from the other numbers is of no interest [Friend] |
| 8702 | In structuralism the number 8 is not quite the same in different structures, only equivalent [Friend] |
| 8699 | Are structures 'ante rem' (before reality), or are they 'in re' (grounded in physics)? [Friend] |
| 8696 | Structuralist says maths concerns concepts about base objects, not base objects themselves [Friend] |
| 8695 | Structuralism focuses on relations, predicates and functions, with objects being inessential [Friend] |
| 8700 | 'In re' structuralism says that the process of abstraction is pattern-spotting [Friend] |
| 16150 | One is, so numbers exist, so endless numbers exist, and each one must partake of being [Plato] |
| 8681 | The big problem for platonists is epistemic: how do we perceive, intuit, know or detect mathematical facts? [Friend] |
| 8712 | Mathematics should be treated as true whenever it is indispensable to our best physical theory [Friend] |
| 8716 | Formalism is unconstrained, so cannot indicate importance, or directions for research [Friend] |
| 8706 | Constructivism rejects too much mathematics [Friend] |
| 8707 | Intuitionists typically retain bivalence but reject the law of excluded middle [Friend] |
| 229 | The one was and is and will be and was becoming and is becoming and will become [Plato] |
| 21821 | Plato's Parmenides has a three-part theory, of Primal One, a One-Many, and a One-and-Many [Plato, by Plotinus] |
| 221 | Absolute ideas, such as the Good and the Beautiful, cannot be known by us [Plato] |
| 7559 | Every part of the universe is body, and non-body is not part of it [Hobbes] |
| 223 | If you deny that each thing always stays the same, you destroy the possibility of discussion [Plato] |
| 227 | You must always mean the same thing when you utter the same name [Plato] |
| 228 | Greatness and smallness must exist, to be opposed to one another, and come into being in things [Plato] |
| 211 | If admirable things have Forms, maybe everything else does as well [Plato] |
| 219 | If absolute ideas existed in us, they would cease to be absolute [Plato] |
| 16151 | Plato moves from Forms to a theory of genera and principles in his later work [Plato, by Frede,M] |
| 210 | It would be absurd to think there were abstract Forms for vile things like hair, mud and dirt [Plato] |
| 220 | The concept of a master includes the concept of a slave [Plato] |
| 212 | The whole idea of each Form must be found in each thing which participates in it [Plato] |
| 216 | If things are made alike by participating in something, that thing will be the absolute idea [Plato] |
| 215 | If things partake of ideas, this implies either that everything thinks, or that everything actually is thought [Plato] |
| 213 | Each idea is in all its participants at once, just as daytime is a unity but in many separate places at once [Plato] |
| 218 | Participation is not by means of similarity, so we are looking for some other method of participation [Plato] |
| 214 | If absolute greatness and great things are seen as the same, another thing appears which makes them seem great [Plato] |
| 217 | Nothing can be like an absolute idea, because a third idea intervenes to make them alike (leading to a regress) [Plato] |
| 8704 | Structuralists call a mathematical 'object' simply a 'place in a structure' [Friend] |
| 15851 | Parts must belong to a created thing with a distinct form [Plato] |
| 15846 | In Parmenides, if composition is identity, a whole is nothing more than its parts [Plato, by Harte,V] |
| 15849 | Plato says only a one has parts, and a many does not [Plato, by Harte,V] |
| 15850 | Anything which has parts must be one thing, and parts are of a one, not of a many [Plato] |
| 13259 | It seems that the One must be composed of parts, which contradicts its being one [Plato] |
| 15847 | Two things relate either as same or different, or part of a whole, or the whole of the part [Plato] |
| 2356 | Appearance and reality can be separated by mirrors and echoes [Hobbes] |
| 2357 | Dreams must be false because they seem absurd, but dreams don't see waking as absurd [Hobbes] |
| 2358 | Freedom is absence of opposition to action; the idea of 'free will' is absurd [Hobbes] |
| 6214 | Liberty and necessity are consistent, as when water freely flows, by necessity [Hobbes] |
| 8685 | Studying biology presumes the laws of chemistry, and it could never contradict them [Friend] |
| 23987 | The 'simple passions' are appetite, desire, love, aversion, hate, joy, and grief [Hobbes, by Goldie] |
| 8688 | Concepts can be presented extensionally (as objects) or intensionally (as a characterization) [Friend] |
| 2362 | The will is just the last appetite before action [Hobbes] |
| 2363 | Reason is usually general, but deliberation is of particulars [Hobbes] |
| 2360 | 'Good' is just what we desire, and 'Evil' what we hate [Hobbes] |
| 2368 | Men's natural desires are no sin, and neither are their actions, until law makes it so [Hobbes] |
| 2359 | Desire and love are the same, but in the desire the object is absent, and in love it is present [Hobbes] |
| 2370 | All voluntary acts aim at some good for the doer [Hobbes] |
| 2371 | A contract is a mutual transfer of rights [Hobbes] |
| 2372 | The person who performs first in a contract is said to 'merit' the return, and is owed it [Hobbes] |
| 8015 | Hobbes wants a contract to found morality, but shared values are needed to make a contract [MacIntyre on Hobbes] |
| 5337 | For Hobbes the Golden Rule concerns not doing things, whereas Jesus encourages active love [Hobbes, by Flanagan] |
| 2374 | In the violent state of nature, the merest suspicion is enough to justify breaking a contract [Hobbes] |
| 2375 | Suspicion will not destroy a contract, if there is a common power to enforce it [Hobbes] |
| 8016 | Fear of sanctions is the only motive for acceptance of authority that Hobbes can think of [MacIntyre on Hobbes] |
| 2377 | No one who admitted to not keeping contracts could ever be accepted as a citizen [Hobbes] |
| 2379 | If there is a good reason for breaking a contract, the same reason should have stopped the making of it [Hobbes] |
| 2373 | The first performer in a contract is handing himself over to an enemy [Hobbes] |
| 2382 | Someone who keeps all his contracts when others are breaking them is making himself a prey to others [Hobbes] |
| 2383 | Virtues are a means to peaceful, sociable and comfortable living [Hobbes] |
| 2376 | Injustice is the failure to keep a contract, and justice is the constant will to give what is owed [Hobbes] |
| 2367 | In time of war the life of man is solitary, poor, nasty, brutish and short [Hobbes] |
| 19764 | Hobbes attributed to savages the passions which arise in a law-bound society [Hobbes, by Rousseau] |
| 20566 | Hobbes says the people voluntarily give up their sovereignty, in a contract with a ruler [Hobbes, by Oksala] |
| 2366 | There is not enough difference between people for one to claim more benefit than another [Hobbes] |
| 20485 | Hobbes says people are roughly equal; Locke says there is no right to impose inequality [Hobbes, by Wolff,J] |
| 2369 | If we seek peace and defend ourselves, we must compromise on our rights [Hobbes] |
| 20484 | We should obey the laws of nature, provided other people are also obeying them [Hobbes, by Wolff,J] |
| 7573 | The legal positivism of Hobbes said law is just formal or procedural [Hobbes, by Jolley] |
| 2380 | Punishment should only be for reform or deterrence [Hobbes] |
| 2361 | If fear of unknown powers is legal it is religion, if it is illegal it is superstition [Hobbes] |
| 222 | Only a great person can understand the essence of things, and an even greater person can teach it [Plato] |
| 225 | The unlimited has no shape and is endless [Plato] |
| 231 | Everything partakes of the One in some way [Plato] |
| 2062 | The only movement possible for the One is in space or in alteration [Plato] |
| 233 | Some things do not partake of the One [Plato] |
| 2364 | Causation is only observation of similar events following each other, with nothing visible in between [Hobbes] |
| 234 | We couldn't discuss the non-existence of the One without knowledge of it [Plato] |
| 2365 | Religion is built on ignorance and misinterpretation of what is unknown or frightening [Hobbes] |
| 2378 | Belief in an afterlife is based on poorly founded gossip [Hobbes] |