122 ideas
13773 | For the truth you need Prodicus's fifty-drachma course, not his one-drachma course [Socrates] |
343 | The unexamined life is not worth living for men [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] |
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] |
8623 | Proof reveals the interdependence of truths, as well as showing their certainty [Euclid, by Frege] |
639 | Socrates developed definitions as the basis of syllogisms, and also inductive arguments [Socrates, by Aristotle] |
13907 | If you pick an arbitrary triangle, things proved of it are true of all triangles [Euclid, by Lemmon] |
18194 | 'Forcing' can produce new models of ZFC from old models [Maddy] |
18195 | A Large Cardinal Axiom would assert ever-increasing stages in the hierarchy [Maddy] |
13011 | New axioms are being sought, to determine the size of the continuum [Maddy] |
13013 | The Axiom of Extensionality seems to be analytic [Maddy] |
13014 | Extensional sets are clearer, simpler, unique and expressive [Maddy] |
13021 | The Axiom of Infinity states Cantor's breakthrough that launched modern mathematics [Maddy] |
13022 | Infinite sets are essential for giving an account of the real numbers [Maddy] |
18191 | Axiom of Infinity: completed infinite collections can be treated mathematically [Maddy] |
13023 | The Power Set Axiom is needed for, and supported by, accounts of the continuum [Maddy] |
18193 | The Axiom of Foundation says every set exists at a level in the set hierarchy [Maddy] |
13024 | Efforts to prove the Axiom of Choice have failed [Maddy] |
13025 | Modern views say the Choice set exists, even if it can't be constructed [Maddy] |
13026 | A large array of theorems depend on the Axiom of Choice [Maddy] |
17610 | The Axiom of Choice paradoxically allows decomposing a sphere into two identical spheres [Maddy] |
18169 | Axiom of Reducibility: propositional functions are extensionally predicative [Maddy] |
13019 | The Iterative Conception says everything appears at a stage, derived from the preceding appearances [Maddy] |
13018 | Limitation of Size is a vague intuition that over-large sets may generate paradoxes [Maddy] |
17824 | The master science is physical objects divided into sets [Maddy] |
8755 | Maddy replaces pure sets with just objects and perceived sets of objects [Maddy, by Shapiro] |
10594 | Henkin semantics is more plausible for plural logic than for second-order logic [Maddy] |
17620 | Critics of if-thenism say that not all starting points, even consistent ones, are worth studying [Maddy] |
18168 | 'Propositional functions' are propositions with a variable as subject or predicate [Maddy] |
17605 | Hilbert's geometry and Dedekind's real numbers were role models for axiomatization [Maddy] |
17625 | If two mathematical themes coincide, that suggest a single deep truth [Maddy] |
6297 | Euclid's geometry is synthetic, but Descartes produced an analytic version of it [Euclid, by Resnik] |
9603 | An assumption that there is a largest prime leads to a contradiction [Euclid, by Brown,JR] |
9894 | A unit is that according to which each existing thing is said to be one [Euclid] |
8738 | Postulate 2 says a line can be extended continuously [Euclid, by Shapiro] |
18190 | Completed infinities resulted from giving foundations to calculus [Maddy] |
18171 | Cantor and Dedekind brought completed infinities into mathematics [Maddy] |
17615 | Every infinite set of reals is either countable or of the same size as the full set of reals [Maddy] |
18172 | Infinity has degrees, and large cardinals are the heart of set theory [Maddy] |
18175 | For any cardinal there is always a larger one (so there is no set of all sets) [Maddy] |
18196 | An 'inaccessible' cardinal cannot be reached by union sets or power sets [Maddy] |
18187 | Theorems about limits could only be proved once the real numbers were understood [Maddy] |
22278 | Euclid relied on obvious properties in diagrams, as well as on his axioms [Potter on Euclid] |
8673 | Euclid's parallel postulate defines unique non-intersecting parallel lines [Euclid, by Friend] |
10250 | Euclid needs a principle of continuity, saying some lines must intersect [Shapiro on Euclid] |
10302 | Euclid says we can 'join' two points, but Hilbert says the straight line 'exists' [Euclid, by Bernays] |
14157 | Modern geometries only accept various parts of the Euclid propositions [Russell on Euclid] |
1600 | Euclid's common notions or axioms are what we must have if we are to learn anything at all [Euclid, by Roochnik] |
18182 | The extension of concepts is not important to me [Maddy] |
18177 | In the ZFC hierarchy it is impossible to form Frege's set of all three-element sets [Maddy] |
18164 | Frege solves the Caesar problem by explicitly defining each number [Maddy] |
17825 | Set theory (unlike the Peano postulates) can explain why multiplication is commutative [Maddy] |
17826 | Standardly, numbers are said to be sets, which is neat ontology and epistemology [Maddy] |
17828 | Numbers are properties of sets, just as lengths are properties of physical objects [Maddy] |
10718 | A natural number is a property of sets [Maddy, by Oliver] |
18184 | Making set theory foundational to mathematics leads to very fruitful axioms [Maddy] |
18185 | Unified set theory gives a final court of appeal for mathematics [Maddy] |
18183 | Set theory brings mathematics into one arena, where interrelations become clearer [Maddy] |
18186 | Identifying geometric points with real numbers revealed the power of set theory [Maddy] |
18188 | The line of rationals has gaps, but set theory provided an ordered continuum [Maddy] |
17618 | Set-theory tracks the contours of mathematical depth and fruitfulness [Maddy] |
18163 | Mathematics rests on the logic of proofs, and on the set theoretic axioms [Maddy] |
17830 | Number theory doesn't 'reduce' to set theory, because sets have number properties [Maddy] |
17827 | Sets exist where their elements are, but numbers are more like universals [Maddy] |
17823 | If mathematical objects exist, how can we know them, and which objects are they? [Maddy] |
8756 | Intuition doesn't support much mathematics, and we should question its reliability [Maddy, by Shapiro] |
17733 | We know mind-independent mathematical truths through sets, which rest on experience [Maddy, by Jenkins] |
18204 | Scientists posit as few entities as possible, but set theorist posit as many as possible [Maddy] |
18207 | Maybe applications of continuum mathematics are all idealisations [Maddy] |
17614 | The connection of arithmetic to perception has been idealised away in modern infinitary mathematics [Maddy] |
17829 | Number words are unusual as adjectives; we don't say 'is five', and numbers always come first [Maddy] |
18167 | We can get arithmetic directly from HP; Law V was used to get HP from the definition of number [Maddy] |
18205 | The theoretical indispensability of atoms did not at first convince scientists that they were real [Maddy] |
1652 | Socrates did not consider universals or definitions as having separate existence, but Plato made Forms of them [Socrates, by Aristotle] |
18206 | Science idealises the earth's surface, the oceans, continuities, and liquids [Maddy] |
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] |
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] |
185 | Socrates despised good looks [Socrates, by Plato] |
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] |
344 | If death is like a night of dreamless sleep, such nights are very pleasant [Socrates] |
339 | Men fear death as a great evil when it may be a great blessing [Socrates] |
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] |
2 | We should not even harm someone who harms us [Socrates] |
1663 | By 'areté' Socrates means just what we mean by moral virtue [Vlastos on Socrates] |
345 | A good man cannot be harmed, either in life or in death [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] |
346 | One ought not to return a wrong or injury to any person, whatever the provocation [Socrates] |
23907 | Courage is scientific knowledge [Socrates, by Aristotle] |
341 | Wealth is good if it is accompanied by virtue [Socrates] |
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] |
347 | Will I stand up against the law, simply because I have been unjustly judged? [Socrates] |
1661 | Socrates was the first to grasp that a cruelty is not justified by another cruelty [Vlastos on Socrates] |
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] |
338 | Socrates is accused of denying the gods, saying sun is stone and moon is earth [Socrates, by Plato] |