76 ideas
| 162 | Can we understand an individual soul without knowing the soul in general? [Plato] |
| 160 | The highest ability in man is the ability to discuss unity and plurality in the nature of things [Plato] |
| 166 | A speaker should be able to divide a subject, right down to the limits of divisibility [Plato] |
| 18137 | Impredicative definitions are wrong, because they change the set that is being defined? [Bostock] |
| 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] |
| 18109 | The completeness of first-order logic implies its compactness [Bostock] |
| 18108 | First-order logic is not decidable: there is no test of whether any formula is valid [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] |
| 18100 | ω + 1 is a new ordinal, but its cardinality is unchanged [Bostock] |
| 18101 | Each addition changes the ordinality but not the cardinality, prior to aleph-1 [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] |
| 18149 | There are many criteria for the identity of numbers [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] |
| 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] |
| 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] |
| 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] |
| 18111 | Treating numbers as objects doesn't seem like logic, since arithmetic fixes their totality [Bostock] |
| 18129 | Many crucial logicist definitions are in fact impredicative [Bostock] |
| 18146 | If Hume's Principle is the whole story, that implies structuralism [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] |
| 18133 | The usual definitions of identity and of natural numbers are impredicative [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] |
| 18132 | The predicativity restriction makes a difference with the real numbers [Bostock] |
| 7953 | Reasoning needs to cut nature accurately at the joints [Plato] |
| 16121 | I revere anyone who can discern a single thing that encompasses many things [Plato] |
| 153 | It takes a person to understand, by using universals, and by using reason to create a unity out of sense-impressions [Plato] |
| 154 | We would have an overpowering love of knowledge if we had a pure idea of it - as with the other Forms [Plato] |
| 151 | True knowledge is of the reality behind sense experience [Plato] |
| 165 | If the apparent facts strongly conflict with probability, it is in everyone's interests to suppress the facts [Plato] |
| 9296 | The soul is self-motion [Plato] |
| 7906 | When the Buddha reached the highest level of insight, he could detect no self in the world [Ashvaghosha] |
| 23997 | Plato saw emotions and appetites as wild horses, in need of taming [Plato, by Goldie] |
| 158 | An excellent speech seems to imply a knowledge of the truth in the mind of the speaker [Plato] |
| 159 | Only a good philosopher can be a good speaker [Plato] |
| 5946 | 'Phaedrus' pioneers the notion of philosophical rhetoric [Lawson-Tancred on Plato] |
| 18121 | In logic a proposition means the same when it is and when it is not asserted [Bostock] |
| 155 | Beauty is the clearest and most lovely of the Forms [Plato] |
| 143 | The two ruling human principles are the natural desire for pleasure, and an acquired love of virtue [Plato] |
| 157 | Most pleasure is release from pain, and is therefore not worthwhile [Plato] |
| 144 | Reason impels us towards excellence, which teaches us self-control [Plato] |
| 156 | Bad people are never really friends with one another [Plato] |
| 148 | If the prime origin is destroyed, it will not come into being again out of anything [Plato] |
| 152 | The mind of God is fully satisfied and happy with a vision of reality and truth [Plato] |
| 150 | We cannot conceive of God, so we have to think of Him as an immortal version of ourselves [Plato] |
| 149 | There isn't a single reason for positing the existence of immortal beings [Plato] |
| 7904 | The first stage of trance is calm amidst applied and discursive thinking [Ashvaghosha] |
| 7905 | The Buddha sought ultimate reality and the final goal of existence in his meditations [Ashvaghosha] |
| 146 | Soul is always in motion, so it must be self-moving and immortal [Plato] |