66 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] |
| 15945 | Second-order set theory just adds a version of Replacement that quantifies over functions [Lavine] |
| 15914 | An 'upper bound' is the greatest member of a subset; there may be several of these, so there is a 'least' one [Lavine] |
| 15921 | Collections of things can't be too big, but collections by a rule seem unlimited in size [Lavine] |
| 15937 | Those who reject infinite collections also want to reject the Axiom of Choice [Lavine] |
| 15936 | The Power Set is just the collection of functions from one collection to another [Lavine] |
| 15899 | Replacement was immediately accepted, despite having very few implications [Lavine] |
| 15930 | Foundation says descending chains are of finite length, blocking circularity, or ungrounded sets [Lavine] |
| 15920 | Pure collections of things obey Choice, but collections defined by a rule may not [Lavine] |
| 15898 | The controversy was not about the Axiom of Choice, but about functions as arbitrary, or given by rules [Lavine] |
| 15919 | The 'logical' notion of class has some kind of definition or rule to characterise the class [Lavine] |
| 15900 | The iterative conception of set wasn't suggested until 1947 [Lavine] |
| 15931 | The iterative conception needs the Axiom of Infinity, to show how far we can iterate [Lavine] |
| 15932 | The iterative conception doesn't unify the axioms, and has had little impact on mathematical proofs [Lavine] |
| 15933 | Limitation of Size: if it's the same size as a set, it's a set; it uses Replacement [Lavine] |
| 15913 | A collection is 'well-ordered' if there is a least element, and all of its successors can be identified [Lavine] |
| 15926 | Second-order logic presupposes a set of relations already fixed by the first-order domain [Lavine] |
| 15934 | Mathematical proof by contradiction needs the law of excluded middle [Lavine] |
| 15907 | Mathematics is nowadays (thanks to set theory) regarded as the study of structure, not of quantity [Lavine] |
| 15942 | Every rational number, unlike every natural number, is divisible by some other number [Lavine] |
| 15922 | For the real numbers to form a set, we need the Continuum Hypothesis to be true [Lavine] |
| 18250 | Cauchy gave a necessary condition for the convergence of a sequence [Lavine] |
| 15904 | The two sides of the Cut are, roughly, the bounding commensurable ratios [Lavine] |
| 15912 | Counting results in well-ordering, and well-ordering makes counting possible [Lavine] |
| 15947 | The infinite is extrapolation from the experience of indefinitely large size [Lavine] |
| 15949 | The theory of infinity must rest on our inability to distinguish between very large sizes [Lavine] |
| 15940 | The intuitionist endorses only the potential infinite [Lavine] |
| 15909 | 'Aleph-0' is cardinality of the naturals, 'aleph-1' the next cardinal, 'aleph-ω' the ω-th cardinal [Lavine] |
| 15915 | Ordinals are basic to Cantor's transfinite, to count the sets [Lavine] |
| 15917 | Paradox: the class of all ordinals is well-ordered, so must have an ordinal as type - giving a bigger ordinal [Lavine] |
| 15918 | Paradox: there is no largest cardinal, but the class of everything seems to be the largest [Lavine] |
| 15929 | Set theory will found all of mathematics - except for the notion of proof [Lavine] |
| 15935 | Modern mathematics works up to isomorphism, and doesn't care what things 'really are' [Lavine] |
| 15928 | Intuitionism rejects set-theory to found mathematics [Lavine] |
| 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] |
| 7647 | The imagination alone perceives all objects; it is the soul, playing all its roles [La Mettrie] |
| 7645 | When falling asleep, the soul becomes paralysed and weak, just like the body [La Mettrie] |
| 23225 | The soul's faculties depend on the brain, and are simply the brain's organisation [La Mettrie] |
| 7652 | Man is a machine, and there exists only one substance, diversely modified [La Mettrie] |
| 23997 | Plato saw emotions and appetites as wild horses, in need of taming [Plato, by Goldie] |
| 7650 | All thought is feeling, and rationality is the sensitive soul contemplating reasoning [La Mettrie] |
| 7651 | With wonderful new machines being made, a speaking machine no longer seems impossible [La Mettrie] |
| 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] |
| 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] |
| 7648 | The sun and rain weren't made for us; they sometimes burn us, or spoil our seeds [La Mettrie] |
| 148 | If the prime origin is destroyed, it will not come into being again out of anything [Plato] |
| 7646 | There is no abrupt transition from man to animal; only language has opened a gap [La Mettrie] |
| 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] |
| 146 | Soul is always in motion, so it must be self-moving and immortal [Plato] |
| 7649 | There is no clear idea of the soul, which should only refer to our thinking part [La Mettrie] |