73 ideas
| 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] |
| 15949 | The theory of infinity must rest on our inability to distinguish between very large sizes [Lavine] |
| 15947 | The infinite is extrapolation from the experience of indefinitely large size [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] |
| 16588 | I prefer a lack of form to mean non-existence, than to think of some quasi-existence [Augustine] |
| 22979 | Three main questions seem to be whether a thing is, what it is, and what sort it is [Augustine] |
| 3912 | I must exist in order to be mistaken, so that even if I am mistaken, I can't be wrong about my own existence [Augustine] |
| 22167 | Our images of bodies are not produced by the bodies, but by our own minds [Augustine, by Aquinas] |
| 22117 | Our minds grasp reality by direct illumination (rather than abstraction from experience) [Augustine, by Matthews] |
| 22981 | Mind and memory are the same, as shown in 'bear it in mind' or 'it slipped from mind' [Augustine] |
| 22980 | Memory contains innumerable principles of maths, as well as past sense experiences [Augustine] |
| 22983 | We would avoid remembering sorrow or fear if that triggered the emotions afresh [Augustine] |
| 22977 | I can distinguish different smells even when I am not experiencing them [Augustine] |
| 22982 | Why does joy in my mind make me happy, but joy in my memory doesn't? [Augustine] |
| 17499 | Theoretical models can represent, by mapping onto the data-models [Portides] |
| 17498 | In the 'received view' models are formal; the 'semantic view' emphasises representation [Portides, by PG] |
| 17501 | Representational success in models depends on success of their explanations [Portides] |
| 17502 | The best model of the atomic nucleus is the one which explains the most results [Portides] |
| 17496 | 'Model' belongs in a family of concepts, with representation, idealisation and abstraction [Portides] |
| 17497 | Models are theory-driven, or phenomenological (more empirical and specific) [Portides] |
| 17500 | General theories may be too abstract to actually explain the mechanisms [Portides] |
| 22978 | Memory is so vast that I cannot recognise it as part of my mind [Augustine] |
| 22984 | Without memory I could not even speak of myself [Augustine] |
| 5982 | If the future does not exist, how can prophets see it? [Augustine] |
| 6683 | The contact of spirit and body is utterly amazing, and incomprehensible [Augustine] |
| 22976 | Memories are preserved separately, according to category [Augustine] |
| 22118 | Augustine created the modern concept of the will [Augustine, by Matthews] |
| 4348 | Love, and do what you will [Augustine] |
| 7821 | Pagans produced three hundred definitions of the highest good [Augustine, by Grayling] |
| 22985 | Everyone wants happiness [Augustine] |
| 22119 | Augustine said (unusually) that 'ought' does not imply 'can' [Augustine, by Matthews] |
| 5984 | Maybe time is an extension of the mind [Augustine] |
| 22888 | To be aware of time it can only exist in the mind, as memory or anticipation [Augustine, by Bardon] |
| 5980 | How can ten days ahead be a short time, if it doesn't exist? [Augustine] |
| 5979 | If the past is no longer, and the future is not yet, how can they exist? [Augustine] |
| 5981 | The whole of the current year is not present, so how can it exist? [Augustine] |
| 5978 | I know what time is, until someone asks me to explain it [Augustine] |
| 5983 | I disagree with the idea that time is nothing but cosmic movement [Augustine] |
| 5977 | Heaven and earth must be created, because they are subject to change [Augustine] |
| 22887 | If God existed before creation, why would a perfect being desire to change things? [Augustine, by Bardon] |
| 5976 | If God is outside time in eternity, can He hear prayers? [Augustine] |
| 16702 | All things are in the present time to God [Augustine] |
| 22116 | Augustine identified Donatism, Pelagianism and Manicheism as the main heresies [Augustine, by Matthews] |
| 19338 | Augustine said evil does not really exist, and evil is a limitation in goodness [Augustine, by Perkins] |