141 ideas
| 2136 | Philosophers become as divine and orderly as possible, by studying divinity and order [Plato] |
| 291 | Don't assume that wisdom is the automatic consequence of old age [Plato] |
| 23767 | The winds of the discussion should decide its destination [Plato] |
| 23682 | It would be absurd to be precise about the small things, but only vague about the big things [Plato] |
| 2151 | Dialectic is the only method of inquiry which uproots the things which it takes for granted [Plato] |
| 2154 | The ability to take an overview is the distinguishing mark of a dialectician [Plato] |
| 4011 | For Plato, rationality is a vision of and love of a cosmic rational order [Plato, by Taylor,C] |
| 2093 | You must never go against what you actually believe [Plato] |
| 2130 | People often merely practice eristic instead of dialectic, because they don't analyse the subject-matter [Plato] |
| 18137 | Impredicative definitions are wrong, because they change the set that is being defined? [Bostock] |
| 2145 | In mathematics certain things have to be accepted without further explanation [Plato] |
| 18122 | Classical interdefinitions of logical constants and quantifiers is impossible in intuitionism [Bostock] |
| 18114 | There is no single agreed structure for set theory [Bostock] |
| 18107 | A 'proper class' cannot be a member of anything [Bostock] |
| 18115 | We could add axioms to make sets either as small or as large as possible [Bostock] |
| 18139 | The Axiom of Choice relies on reference to sets that we are unable to describe [Bostock] |
| 18105 | Replacement enforces a 'limitation of size' test for the existence of sets [Bostock] |
| 18108 | First-order logic is not decidable: there is no test of whether any formula is valid [Bostock] |
| 18109 | The completeness of first-order logic implies its compactness [Bostock] |
| 18123 | Substitutional quantification is just standard if all objects in the domain have a name [Bostock] |
| 18120 | The Deduction Theorem is what licenses a system of natural deduction [Bostock] |
| 18125 | Berry's Paradox considers the meaning of 'The least number not named by this name' [Bostock] |
| 8726 | Geometry can lead the mind upwards to truth and philosophy [Plato] |
| 18101 | Each addition changes the ordinality but not the cardinality, prior to aleph-1 [Bostock] |
| 18100 | ω + 1 is a new ordinal, but its cardinality is unchanged [Bostock] |
| 18102 | A cardinal is the earliest ordinal that has that number of predecessors [Bostock] |
| 18106 | Aleph-1 is the first ordinal that exceeds aleph-0 [Bostock] |
| 18095 | Instead of by cuts or series convergence, real numbers could be defined by axioms [Bostock] |
| 18099 | The number of reals is the number of subsets of the natural numbers [Bostock] |
| 18093 | For Eudoxus cuts in rationals are unique, but not every cut makes a real number [Bostock] |
| 18110 | Infinitesimals are not actually contradictory, because they can be non-standard real numbers [Bostock] |
| 18156 | Modern axioms of geometry do not need the real numbers [Bostock] |
| 18097 | The Peano Axioms describe a unique structure [Bostock] |
| 18148 | Hume's Principle is a definition with existential claims, and won't explain numbers [Bostock] |
| 18145 | Many things will satisfy Hume's Principle, so there are many interpretations of it [Bostock] |
| 18149 | There are many criteria for the identity of numbers [Bostock] |
| 18143 | Frege makes numbers sets to solve the Caesar problem, but maybe Caesar is a set! [Bostock] |
| 18116 | Numbers can't be positions, if nothing decides what position a given number has [Bostock] |
| 18117 | Structuralism falsely assumes relations to other numbers are numbers' only properties [Bostock] |
| 9863 | We aim for elevated discussion of pure numbers, not attaching them to physical objects [Plato] |
| 9864 | In pure numbers, all ones are equal, with no internal parts [Plato] |
| 8727 | Geometry is not an activity, but the study of unchanging knowledge [Plato] |
| 18141 | Nominalism about mathematics is either reductionist, or fictionalist [Bostock] |
| 18157 | Nominalism as based on application of numbers is no good, because there are too many applications [Bostock] |
| 18150 | Actual measurement could never require the precision of the real numbers [Bostock] |
| 9861 | The same thing is both one and an unlimited number at the same time [Plato] |
| 18158 | Ordinals are mainly used adjectively, as in 'the first', 'the second'... [Bostock] |
| 18127 | Simple type theory has 'levels', but ramified type theory has 'orders' [Bostock] |
| 18144 | Neo-logicists agree that HP introduces number, but also claim that it suffices for the job [Bostock] |
| 18147 | Neo-logicists meet the Caesar problem by saying Hume's Principle is unique to number [Bostock] |
| 18146 | If Hume's Principle is the whole story, that implies structuralism [Bostock] |
| 18129 | Many crucial logicist definitions are in fact impredicative [Bostock] |
| 18111 | Treating numbers as objects doesn't seem like logic, since arithmetic fixes their totality [Bostock] |
| 18159 | Higher cardinalities in sets are just fairy stories [Bostock] |
| 18155 | A fairy tale may give predictions, but only a true theory can give explanations [Bostock] |
| 18140 | The best version of conceptualism is predicativism [Bostock] |
| 18138 | Conceptualism fails to grasp mathematical properties, infinity, and objective truth values [Bostock] |
| 18131 | If abstracta only exist if they are expressible, there can only be denumerably many of them [Bostock] |
| 18134 | Predicativism makes theories of huge cardinals impossible [Bostock] |
| 18135 | If mathematics rests on science, predicativism may be the best approach [Bostock] |
| 18136 | If we can only think of what we can describe, predicativism may be implied [Bostock] |
| 18133 | The usual definitions of identity and of natural numbers are impredicative [Bostock] |
| 18132 | The predicativity restriction makes a difference with the real numbers [Bostock] |
| 9862 | To become rational, philosophers must rise from becoming into being [Plato] |
| 21818 | Being depends on the Good, which is not itself being, but superior to being [Plato] |
| 2061 | The best things (gods, healthy bodies, good souls) are least liable to change [Plato] |
| 6562 | Plato's reality has unchanging Parmenidean forms, and Heraclitean flux [Plato, by Fogelin] |
| 2142 | The plurality of beautiful things must belong to a single class, because they have a single particular character [Plato] |
| 5094 | Plato's Forms are said to have no location in space [Plato, by Aristotle] |
| 12043 | Forms are not universals, as they don't cover every general term [Plato, by Annas] |
| 2159 | Craftsmen making furniture refer to the form, but no one manufactures the form of furniture [Plato] |
| 17 | A Form applies to a set of particular things with the same name [Plato] |
| 12122 | Plato mistakenly thought forms were totally abstracted away from matter [Bacon on Plato] |
| 5574 | Plato's Forms not only do not come from the senses, but they are beyond possibility of sensing [Plato, by Kant] |
| 2133 | Knowledge must be of the permanent unchanging nature of things [Plato] |
| 2162 | If theory and practice conflict, the best part of the mind accepts theory, so the other part is of lower grade [Plato] |
| 2140 | True belief without knowledge is like blind people on the right road [Plato] |
| 2096 | Is the function of the mind management, authority and planning - or is it one's whole way of life? [Plato] |
| 6009 | Psychic conflict is clear if appetite is close to the body and reason fairly separate [Plato, by Modrak] |
| 6041 | There is a third element to the mind - spirit - lying between reason and appetite [Plato] |
| 2127 | The mind has parts, because we have inner conflicts [Plato] |
| 1737 | The soul seems to have an infinity of parts [Aristotle on Plato] |
| 5945 | The 'Republic' is a great work of rhetorical theory [Lawson-Tancred on Plato] |
| 18121 | In logic a proposition means the same when it is and when it is not asserted [Bostock] |
| 23316 | For Plato and Aristotle there is no will; there is only rational desire for what is seen as good [Plato, by Frede,M] |
| 16 | We avoid evil either through a natural aversion, or because we have acquired knowledge [Plato] |
| 16565 | Without the surface decoration, poetry shows only appearances and nothing of what is real [Plato] |
| 2160 | Representation is two steps removed from the truth [Plato] |
| 2163 | Artists should be excluded from a law-abiding community, because they destroy the rational mind [Plato] |
| 2135 | Truth is closely related to proportion [Plato] |
| 2141 | I suggest that we forget about trying to define goodness itself for the time being [Plato] |
| 1869 | The good cannot be expressed in words, but imprints itself upon the soul [Plato, by Celsus] |
| 4115 | Plato found that he could only enforce rational moral justification by creating an authoritarian society [Williams,B on Plato] |
| 4547 | Plato measured the degree of reality by the degree of value [Nietzsche on Plato] |
| 2094 | A thing's function is what it alone can do, or what it does better than other things [Plato] |
| 2095 | If something has a function then it has a state of being good [Plato] |
| 2129 | Goodness is mental health, badness is mental sickness [Plato] |
| 12 | If we were invisible, would the just man become like the unjust? [Plato] |
| 2168 | Clever criminals do well at first, but not in the long run [Plato] |
| 2137 | The main aim is to understand goodness, which gives everything its value and advantage [Plato] |
| 2139 | Every person, and every activity, aims at the good [Plato] |
| 2143 | Good has the same role in the world of knowledge as the sun has in the physical world [Plato] |
| 2147 | The sight of goodness leads to all that is fine and true and right [Plato] |
| 4007 | For Plato we abandon honour and pleasure once we see the Good [Plato, by Taylor,C] |
| 2144 | Goodness makes truth and knowledge possible [Plato] |
| 2164 | Bad is always destructive, where good preserves and benefits [Plato] |
| 2138 | Pleasure is commonly thought to be the good, though the more ingenious prefer knowledge [Plato] |
| 2070 | Even people who think pleasure is the good admit that there are bad pleasures [Plato] |
| 2157 | Nice smells are intensive, have no preceding pain, and no bad after-effect [Plato] |
| 2134 | Philosophers are concerned with totally non-physical pleasures [Plato] |
| 2156 | There are three types of pleasure, for reason, for spirit and for appetite [Plato] |
| 2158 | Pleasure-seekers desperately seek illusory satisfaction, like filling a leaky vessel [Plato] |
| 2123 | Excessive pleasure deranges people, making the other virtues impossible [Plato] |
| 2166 | We should behave well even if invisible, for the health of the mind [Plato] |
| 2097 | Isn't it better to have a reputation for goodness than to actually be good? [Plato] |
| 19946 | Morality is a compromise, showing restraint, to avoid suffering wrong without compensation [Plato] |
| 5 | Justice is merely the interests of the stronger party [Plato] |
| 7 | Surely you don't return a borrowed weapon to a mad friend? [Plato] |
| 8 | Is right just the interests of the powerful? [Plato] |
| 15 | Sin first, then sacrifice to the gods from the proceeds [Plato] |
| 5944 | For Plato, virtue is its own reward [Lawson-Tancred on Plato] |
| 2155 | True goodness requires mental unity and harmony [Plato] |
| 2126 | A good community necessarily has wisdom, courage, self-discipline and morality [Plato] |
| 23562 | If the parts of our soul do their correct work, we will be just people, and will act justly [Plato] |
| 2092 | Simonides said morality is helping one's friends and harming one's enemies [Plato] |
| 293 | Being unafraid (perhaps through ignorance) and being brave are two different things [Plato] |
| 19889 | People need society because the individual has too many needs [Plato] |
| 19890 | All exchanges in a community are for mutual benefit [Plato] |
| 10 | After a taste of mutual harm, men make a legal contract to avoid it [Plato] |
| 23561 | People doing their jobs properly is the fourth cardinal virtue for a city [Plato] |
| 2149 | Reluctant rulers make a better and more unified administration [Plato] |
| 2132 | Only rule by philosophers of integrity can keep a community healthy [Plato] |
| 2131 | Is there anything better for a community than to produce excellent people? [Plato] |
| 2152 | Dialectic is the highest and most important part of the curriculum [Plato] |
| 2148 | To gain knowledge, turn away from the world of change, and focus on true goodness [Plato] |
| 2153 | Compulsory intellectual work never remains in the mind [Plato] |
| 2630 | If Plato's God is immaterial, he will lack consciousness, wisdom, pleasure and movement, which are essential to him [Cicero on Plato] |
| 14 | If the gods are non-existent or indifferent, why bother to deceive them? [Plato] |
| 2165 | Something is unlikely to be immortal if it is imperfectly made from diverse parts [Plato] |
| 13 | Is the supreme reward for virtue to be drunk for eternity? [Plato] |
| 2120 | God is responsible for the good things, but we must look elsewhere for the cause of the bad things [Plato] |