94 ideas
| 18274 | Analysis complicates a statement, but only as far as the complexity of its meaning [Wittgenstein] |
| 23917 | Contrary statements can both be reasonable, if they are meant in two different ways [Aristotle] |
| 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] |
| 16908 | We can dispense with self-evidence, if language itself prevents logical mistakes [Jeshion on Wittgenstein] |
| 8709 | The law of excluded middle is syntactic; it just says A or not-A, not whether they are true or false [Friend] |
| 18276 | A statement's logical form derives entirely from its constituents [Wittgenstein] |
| 6563 | 'And' and 'not' are non-referring terms, which do not represent anything [Wittgenstein, by Fogelin] |
| 8711 | Intuitionists read the universal quantifier as "we have a procedure for checking every..." [Friend] |
| 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] |
| 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] |
| 23472 | The sense of propositions relies on the world's basic logical structure [Wittgenstein] |
| 5869 | The thesis of the Form of the Good (or of anything else) is verbal and vacuous [Aristotle] |
| 8704 | Structuralists call a mathematical 'object' simply a 'place in a structure' [Friend] |
| 22505 | The two right angles of a triangle necessitate that a quadrilateral has four [Aristotle] |
| 22513 | Knowing is having knowledge; understanding is using knowledge [Aristotle] |
| 23500 | My main problem is the order of the world, and whether it is knowable a priori [Wittgenstein] |
| 23906 | Courage from spirit is natural and unconquerable, as seen in the young [Aristotle] |
| 20204 | Whether the mind has parts is irrelevant, since it obviously has distinct capacities [Aristotle] |
| 22323 | The philosophical I is the metaphysical subject, the limit - not a part of the world [Wittgenstein] |
| 22506 | A man is the cause of what is within his power, and what he causes is in his power [Aristotle] |
| 22504 | Only a human being can be a starting point for an action [Aristotle] |
| 8685 | Studying biology presumes the laws of chemistry, and it could never contradict them [Friend] |
| 22510 | Some emotional states are too strong for human nature [Aristotle] |
| 23913 | Nearly all the good and bad states of character are concerned with feelings [Aristotle] |
| 8688 | Concepts can be presented extensionally (as objects) or intensionally (as a characterization) [Friend] |
| 23481 | Propositions assemble a world experimentally, like the model of a road accident [Wittgenstein] |
| 23918 | Akrasia is the clash of two feelings - goodness and pleasure [Aristotle] |
| 22515 | Choice results when deliberation brings together an opinion with an inclination [Aristotle] |
| 22508 | Unlike in inanimate things, in animate things actions have more than one starting point [Aristotle] |
| 22514 | The deliberative part of the soul discerns explanatory causes [Aristotle] |
| 22507 | An action is voluntary when it is accompanied by thought of some kind [Aristotle] |
| 23319 | We are responsible if our actions reflect our motivation [Aristotle, by Frede,M] |
| 22512 | Acts are voluntary if done knowingly, by the agent, and in his power to avoid it [Aristotle] |
| 22509 | What is natural for us is either there at birth, or appears by normal processes [Aristotle] |
| 18230 | No one would choose life just for activities not done for their own sake [Aristotle] |
| 23909 | Wearing a shoe is its intrinsic use, and selling it (as a shoe) is its coincidental use [Aristotle] |
| 5870 | Everything seeks, not a single good, but its own separate good [Aristotle] |
| 5877 | We judge people from their deeds because we cannot see their choices (which matter more) [Aristotle] |
| 5868 | Horses, birds and fish are not happy, lacking a divine aspect to their natures [Aristotle] |
| 5865 | Happiness involves three things, of which the greatest is either wisdom, virtue, or pleasure [Aristotle] |
| 5876 | Virtue is different from continence [Aristotle] |
| 5872 | Excellence is the best state of anything (like a cloak) which has an employment or function [Aristotle] |
| 5874 | Character virtues (such as courage) are of the non-rational part, which follows the rational part [Aristotle] |
| 22516 | Character is shown by what is or is not enjoyed, and virtue chooses the mean among them [Aristotle] |
| 22517 | We judge character not by their actions, but by their reasons for actions [Aristotle] |
| 5875 | Character (éthos) is developed from habit (ethos) [Aristotle] |
| 23914 | People sometimes exhibit both extremes together, but the mean is contrary to both of them [Aristotle] |
| 23911 | Possessors of a virtue tend to despise what reason shows to be its opposite [Aristotle] |
| 23910 | Greatness of soul produces all the virtues - and vice versa [Aristotle] |
| 23908 | If someone just looks at or listens to beautiful things, they would not be thought intemperate [Aristotle] |
| 23905 | Courage follows reason, which tells us to choose what is noble [Aristotle] |
| 23912 | Honour depends on what it is for, and whether it is bestowed by worthy people [Aristotle] |
| 5871 | Goods in the soul are more worthy than those outside it, as everybody wants them [Aristotle] |
| 23920 | Decent people can be friends with base people [Aristotle] |
| 23919 | Friendship cannot be immediate; it takes time, and needs testing [Aristotle] |
| 23915 | The main function of politics is to produce friendship [Aristotle] |
| 23916 | The best cure for mutual injustice is friendship [Aristotle] |
| 4678 | Absolute prohibitions are the essence of ethics, and suicide is the most obvious example [Wittgenstein] |
| 5866 | It is folly not to order one's life around some end [Aristotle] |
| 5878 | Eyes could be used for a natural purpose, or for unnatural seeing, or for a non-seeing activity [Aristotle] |
| 5873 | Each thing's function is its end [Aristotle] |