133 ideas
| 13773 | For the truth you need Prodicus's fifty-drachma course, not his one-drachma course [Socrates] |
| 7421 | A philosopher is one who cares about what other people care about [Socrates, by Foucault] |
| 1649 | Socrates opened philosophy to all, but Plato confined moral enquiry to a tiny elite [Vlastos on Socrates] |
| 5842 | Philosophical discussion involves dividing subject-matter into categories [Socrates, by Xenophon] |
| 648 | Socrates began the quest for something universal with his definitions, but he didn't make them separate [Socrates, by Aristotle] |
| 10237 | Coherence is a primitive, intuitive notion, not reduced to something formal [Shapiro] |
| 164 | It is legitimate to play the devil's advocate [Socrates] |
| 1647 | In Socratic dialogue you must say what you believe, so unasserted premises are not debated [Vlastos on Socrates] |
| 115 | Socrates was pleased if his mistakes were proved wrong [Socrates] |
| 22099 | The method of Socrates shows the student is discovering the truth within himself [Socrates, by Carlisle] |
| 5844 | Socrates always proceeded in argument by general agreement at each stage [Socrates, by Xenophon] |
| 11389 | Socrates sought essences, which are the basis of formal logic [Socrates, by Aristotle] |
| 10204 | An 'implicit definition' gives a direct description of the relations of an entity [Shapiro] |
| 639 | Socrates developed definitions as the basis of syllogisms, and also inductive arguments [Socrates, by Aristotle] |
| 10206 | Modal operators are usually treated as quantifiers [Shapiro] |
| 10208 | Axiom of Choice: some function has a value for every set in a given set [Shapiro] |
| 10252 | The Axiom of Choice seems to license an infinite amount of choosing [Shapiro] |
| 10207 | Anti-realists reject set theory [Shapiro] |
| 10259 | The two standard explanations of consequence are semantic (in models) and deductive [Shapiro] |
| 10257 | Intuitionism only sanctions modus ponens if all three components are proved [Shapiro] |
| 10253 | Either logic determines objects, or objects determine logic, or they are separate [Shapiro] |
| 10251 | The law of excluded middle might be seen as a principle of omniscience [Shapiro] |
| 10212 | Classical connectives differ from their ordinary language counterparts; '∧' is timeless, unlike 'and' [Shapiro] |
| 10209 | A function is just an arbitrary correspondence between collections [Shapiro] |
| 10268 | Maybe plural quantifiers should be understood in terms of classes or sets [Shapiro] |
| 10235 | A sentence is 'satisfiable' if it has a model [Shapiro] |
| 10240 | Model theory deals with relations, reference and extensions [Shapiro] |
| 10239 | The central notion of model theory is the relation of 'satisfaction' [Shapiro] |
| 10214 | Theory ontology is never complete, but is only determined 'up to isomorphism' [Shapiro] |
| 10238 | The set-theoretical hierarchy contains as many isomorphism types as possible [Shapiro] |
| 10234 | Any theory with an infinite model has a model of every infinite cardinality [Shapiro] |
| 10201 | Virtually all of mathematics can be modeled in set theory [Shapiro] |
| 10213 | Real numbers are thought of as either Cauchy sequences or Dedekind cuts [Shapiro] |
| 18243 | Understanding the real-number structure is knowing usage of the axiomatic language of analysis [Shapiro] |
| 18245 | Cuts are made by the smallest upper or largest lower number, some of them not rational [Shapiro] |
| 10236 | There is no grounding for mathematics that is more secure than mathematics [Shapiro] |
| 10256 | For intuitionists, proof is inherently informal [Shapiro] |
| 10202 | Natural numbers just need an initial object, successors, and an induction principle [Shapiro] |
| 10205 | Mathematics originally concerned the continuous (geometry) and the discrete (arithmetic) [Shapiro] |
| 10222 | Mathematical foundations may not be sets; categories are a popular rival [Shapiro] |
| 10218 | Baseball positions and chess pieces depend entirely on context [Shapiro] |
| 10224 | The even numbers have the natural-number structure, with 6 playing the role of 3 [Shapiro] |
| 10228 | Could infinite structures be apprehended by pattern recognition? [Shapiro] |
| 10230 | The 4-pattern is the structure common to all collections of four objects [Shapiro] |
| 10249 | The main mathematical structures are algebraic, ordered, and topological [Shapiro] |
| 10273 | Some structures are exemplified by both abstract and concrete [Shapiro] |
| 10276 | Mathematical structures are defined by axioms, or in set theory [Shapiro] |
| 10270 | The main versions of structuralism are all definitionally equivalent [Shapiro] |
| 10221 | Is there is no more to structures than the systems that exemplify them? [Shapiro] |
| 10248 | Number statements are generalizations about number sequences, and are bound variables [Shapiro] |
| 10220 | Because one structure exemplifies several systems, a structure is a one-over-many [Shapiro] |
| 10223 | There is no 'structure of all structures', just as there is no set of all sets [Shapiro] |
| 8703 | Shapiro's structuralism says model theory (comparing structures) is the essence of mathematics [Shapiro, by Friend] |
| 10274 | Does someone using small numbers really need to know the infinite structure of arithmetic? [Shapiro] |
| 10200 | We distinguish realism 'in ontology' (for objects), and 'in truth-value' (for being either true or false) [Shapiro] |
| 10210 | If mathematical objects are accepted, then a number of standard principles will follow [Shapiro] |
| 10215 | Platonists claim we can state the essence of a number without reference to the others [Shapiro] |
| 10233 | Platonism must accept that the Peano Axioms could all be false [Shapiro] |
| 10244 | Intuition is an outright hindrance to five-dimensional geometry [Shapiro] |
| 10280 | A stone is a position in some pattern, and can be viewed as an object, or as a location [Shapiro] |
| 10254 | Can the ideal constructor also destroy objects? [Shapiro] |
| 10255 | Presumably nothing can block a possible dynamic operation? [Shapiro] |
| 10279 | Can we discover whether a deck is fifty-two cards, or a person is time-slices or molecules? [Shapiro] |
| 10227 | The abstract/concrete boundary now seems blurred, and would need a defence [Shapiro] |
| 10226 | Mathematicians regard arithmetic as concrete, and group theory as abstract [Shapiro] |
| 10262 | Fictionalism eschews the abstract, but it still needs the possible (without model theory) [Shapiro] |
| 10277 | Structuralism blurs the distinction between mathematical and ordinary objects [Shapiro] |
| 1652 | Socrates did not consider universals or definitions as having separate existence, but Plato made Forms of them [Socrates, by Aristotle] |
| 10272 | The notion of 'object' is at least partially structural and mathematical [Shapiro] |
| 10275 | A blurry border is still a border [Shapiro] |
| 10258 | Logical modalities may be acceptable, because they are reducible to satisfaction in models [Shapiro] |
| 10266 | Why does the 'myth' of possible worlds produce correct modal logic? [Shapiro] |
| 10203 | We apprehend small, finite mathematical structures by abstraction from patterns [Shapiro] |
| 1650 | For Socrates our soul, though hard to define, is our self [Vlastos on Socrates] |
| 23252 | Socrates first proposed that we are run by mind or reason [Socrates, by Frede,M] |
| 10229 | Simple types can be apprehended through their tokens, via abstraction [Shapiro] |
| 10217 | We can apprehend structures by focusing on or ignoring features of patterns [Shapiro] |
| 9554 | We can focus on relations between objects (like baseballers), ignoring their other features [Shapiro] |
| 10231 | Abstract objects might come by abstraction over an equivalence class of base entities [Shapiro] |
| 199 | The common belief is that people can know the best without acting on it [Socrates] |
| 195 | No one willingly commits an evil or base act [Socrates] |
| 1653 | Socrates did not accept the tripartite soul (which permits akrasia) [Vlastos on Socrates] |
| 5843 | People do what they think they should do, and only ever do what they think they should do [Socrates, by Xenophon] |
| 5253 | Socrates was shocked by the idea of akrasia, but observation shows that it happens [Aristotle on Socrates] |
| 5839 | For Socrates, wisdom and prudence were the same thing [Socrates, by Xenophon] |
| 5867 | For Socrates, virtues are forms of knowledge, so knowing justice produces justice [Socrates, by Aristotle] |
| 5069 | Socrates was the first to base ethics upon reason, and use reason to explain it [Taylor,R on Socrates] |
| 5836 | All human virtues are increased by study and practice [Socrates, by Xenophon] |
| 5840 | The wise perform good actions, and people fail to be good without wisdom [Socrates, by Xenophon] |
| 23590 | Criminal responsibility can be fully assigned to each member of a group [Walzer] |
| 23578 | Double Effect needs a double intention - to achieve the good, and minimise the evil [Walzer] |
| 185 | Socrates despised good looks [Socrates, by Plato] |
| 23564 | Deep ethical theory is very controversial, but we have to live with higher ethical practice [Walzer] |
| 5070 | Socrates conservatively assumed that Athenian conventions were natural and true [Taylor,R on Socrates] |
| 5838 | A well-made dung basket is fine, and a badly-made gold shield is base, because of function [Socrates, by Xenophon] |
| 5837 | Things are both good and fine by the same standard [Socrates, by Xenophon] |
| 3017 | The only good is knowledge, and the only evil is ignorance [Socrates, by Diog. Laertius] |
| 1646 | Socrates was the first to put 'eudaimonia' at the centre of ethics [Socrates, by Vlastos] |
| 1663 | By 'areté' Socrates means just what we mean by moral virtue [Vlastos on Socrates] |
| 4323 | Socrates is torn between intellectual virtue, which is united and teachable, and natural virtue, which isn't [PG on Socrates] |
| 8003 | Socrates agrees that virtue is teachable, but then denies that there are teachers [Socrates, by MacIntyre] |
| 126 | We should ask what sort of people we want to be [Socrates] |
| 4111 | Socrates believed that basically there is only one virtue, the power of right judgement [Socrates, by Williams,B] |
| 7808 | Socrates made the civic values of justice and friendship paramount [Socrates, by Grayling] |
| 23907 | Courage is scientific knowledge [Socrates, by Aristotle] |
| 7585 | Socrates emphasises that the knower is an existing individual, with existence his main task [Socrates, by Kierkegaard] |
| 5841 | Obedience to the law gives the best life, and success in war [Socrates, by Xenophon] |
| 23568 | If whole states possess rights, there can be social relations between states [Walzer] |
| 1661 | Socrates was the first to grasp that a cruelty is not justified by another cruelty [Vlastos on Socrates] |
| 23571 | States can rightly pre-empt real and serious threats [Walzer] |
| 23572 | Just wars are self-defence, or a rightful intercession in another's troubles [Walzer] |
| 23581 | The aim of reprisals is to enforce the rules of war [Walzer] |
| 23582 | Reprisal is defensible, as an alternative to war [Walzer] |
| 23588 | With nuclear weapons we have a permanent supreme emergency (which is unstable) [Walzer] |
| 23580 | States need not endure attacks passively, and successful reprisals are legitimate [Walzer] |
| 23567 | Even non-violent intrusive acts between states count as aggression, if they justify resistance [Walzer] |
| 23570 | The only good reason for fighting is in defence of rights [Walzer] |
| 23587 | Nuclear bombs are not for normal war; they undermine the 'just war', with a new morality [Walzer] |
| 23573 | For moral reasons, a just war must be a limited war [Walzer] |
| 23577 | Napoleon said 'I don't care about the deaths of a million men' [Walzer] |
| 23593 | Jus ad bellum and Jus in bello are independent; unjust wars can be fought in a just way [Walzer] |
| 23574 | The duties and moral status of loyal and obedient soldiers is the same in defence and aggression [Walzer] |
| 23575 | We can't blame soldiers for anything they do which clearly promotes victory [Walzer] |
| 23584 | Rejecting Combatant Equality allows just soldiers to be harsher, even to the extreme [Walzer] |
| 23589 | Kidnapped sailors and volunteers have different obligations to the passengers [Walzer] |
| 23614 | Even aggressor soldiers are not criminals, so they have equal rights with their opponents [Walzer] |
| 23579 | Soldiers will only protect civilians if they feel safe from them [Walzer] |
| 23586 | What matters in war is unacceptable targets, not unacceptable weapons [Walzer] |
| 23591 | If the oppressor is cruel, nonviolence is either surrender, or a mere gesture [Walzer] |
| 23592 | We can only lead war towards peace if we firmly enforce the rules of war [Walzer] |
| 5846 | A lover using force is a villain, but a seducer is much worse, because he corrupts character [Socrates, by Xenophon] |
| 1657 | Socrates holds that right reason entails virtue, and this must also apply to the gods [Vlastos on Socrates] |
| 1662 | A new concept of God as unswerving goodness emerges from Socrates' commitment to virtue [Vlastos on Socrates] |