136 ideas
| 5082 | Reason grasps generalities, while the senses grasp particulars [Aristotle] |
| 9955 | Contextual definitions replace a complete sentence containing the expression [George/Velleman] |
| 10031 | Impredicative definitions quantify over the thing being defined [George/Velleman] |
| 10098 | The 'power set' of A is all the subsets of A [George/Velleman] |
| 10101 |
Cartesian Product A x B: the set of all ordered pairs |
| 10099 | The 'ordered pair' <a, b>, for two sets a and b, is the set {{a, b},{a}} [George/Velleman] |
| 10103 | Grouping by property is common in mathematics, usually using equivalence [George/Velleman] |
| 10104 | 'Equivalence' is a reflexive, symmetric and transitive relation; 'same first letter' partitions English words [George/Velleman] |
| 10096 | Even the elements of sets in ZFC are sets, resting on the pure empty set [George/Velleman] |
| 10097 | Axiom of Extensionality: for all sets x and y, if x and y have the same elements then x = y [George/Velleman] |
| 10100 | Axiom of Pairing: for all sets x and y, there is a set z containing just x and y [George/Velleman] |
| 17900 | The Axiom of Reducibility made impredicative definitions possible [George/Velleman] |
| 10109 | ZFC can prove that there is no set corresponding to the concept 'set' [George/Velleman] |
| 10108 | As a reduction of arithmetic, set theory is not fully general, and so not logical [George/Velleman] |
| 13270 | Are a part and whole one or many? Either way, what is the cause? [Aristotle] |
| 10111 | Asserting Excluded Middle is a hallmark of realism about the natural world [George/Velleman] |
| 10129 | A 'model' is a meaning-assignment which makes all the axioms true [George/Velleman] |
| 10105 | Differences between isomorphic structures seem unimportant [George/Velleman] |
| 10119 | Consistency is a purely syntactic property, unlike the semantic property of soundness [George/Velleman] |
| 10126 | A 'consistent' theory cannot contain both a sentence and its negation [George/Velleman] |
| 10120 | Soundness is a semantic property, unlike the purely syntactic property of consistency [George/Velleman] |
| 10127 | A 'complete' theory contains either any sentence or its negation [George/Velleman] |
| 9790 | Geometry studies naturally occurring lines, but not as they occur in nature [Aristotle] |
| 10106 | Rational numbers give answers to division problems with integers [George/Velleman] |
| 10102 | The integers are answers to subtraction problems involving natural numbers [George/Velleman] |
| 22962 | Two is the least number, but there is no least magnitude, because it is always divisible [Aristotle] |
| 10107 | Real numbers provide answers to square root problems [George/Velleman] |
| 9946 | Logicists say mathematics is applicable because it is totally general [George/Velleman] |
| 18090 | Without infinity time has limits, magnitudes are indivisible, and numbers come to an end [Aristotle] |
| 22929 | Aristotle's infinity is a property of the counting process, that it has no natural limit [Aristotle, by Le Poidevin] |
| 10125 | The classical mathematician believes the real numbers form an actual set [George/Velleman] |
| 22930 | Lengths do not contain infinite parts; parts are created by acts of division [Aristotle, by Le Poidevin] |
| 18833 | A continuous line cannot be composed of indivisible points [Aristotle] |
| 17899 | Second-order induction is stronger as it covers all concepts, not just first-order definable ones [George/Velleman] |
| 10128 | The Incompleteness proofs use arithmetic to talk about formal arithmetic [George/Velleman] |
| 17902 | A successor is the union of a set with its singleton [George/Velleman] |
| 10133 | Frege's Theorem shows the Peano Postulates can be derived from Hume's Principle [George/Velleman] |
| 10130 | Set theory can prove the Peano Postulates [George/Velleman] |
| 10089 | Talk of 'abstract entities' is more a label for the problem than a solution to it [George/Velleman] |
| 9974 | Ten sheep and ten dogs are the same numerically, but it is not the same ten [Aristotle] |
| 10131 | If mathematics is not about particulars, observing particulars must be irrelevant [George/Velleman] |
| 10092 | In the unramified theory of types, the types are objects, then sets of objects, sets of sets etc. [George/Velleman] |
| 10094 | The theory of types seems to rule out harmless sets as well as paradoxical ones. [George/Velleman] |
| 10095 | Type theory has only finitely many items at each level, which is a problem for mathematics [George/Velleman] |
| 17901 | Type theory prohibits (oddly) a set containing an individual and a set of individuals [George/Velleman] |
| 10114 | Bounded quantification is originally finitary, as conjunctions and disjunctions [George/Velleman] |
| 10134 | Much infinite mathematics can still be justified finitely [George/Velleman] |
| 10123 | The intuitionists are the idealists of mathematics [George/Velleman] |
| 10124 | Gödel's First Theorem suggests there are truths which are independent of proof [George/Velleman] |
| 5105 | The incommensurability of the diagonal always exists, and so it is not in time [Aristotle] |
| 16115 | Change is the implied actuality of that which exists potentially [Aristotle] |
| 22960 | The sophists thought a man in the Lyceum is different from that man in the marketplace [Aristotle] |
| 17262 | Aristotle's formal and material 'becauses' [aitiai] arguably involve grounding [Aristotle, by Correia/Schnieder] |
| 16656 | The separation from here to there is not the same as the separation from there to here [Aristotle] |
| 16644 | The features of a thing (whether quality or quantity) are inseparable from their subjects [Aristotle] |
| 5117 | Heavy and light are defined by their tendency to move down or up [Aristotle] |
| 17041 | Natural objects include animals and their parts, plants, and the simple elements [Aristotle] |
| 16172 | Substance is not predicated of anything - but it still has something underlying it, that originates it [Aristotle] |
| 16623 | We only infer underlying natures by analogy, observing bronze of a statue, or wood of a bed [Aristotle] |
| 16174 | A nature is related to a substance as shapeless matter is to something which has a shape [Aristotle] |
| 17043 | Form, not matter, is a thing's nature, because it is actual, rather than potential [Aristotle] |
| 16970 | A thing's form and purpose are often the same, and form can be the initiator of change too [Aristotle] |
| 16104 | Unity of the form is just unity of the definition [Aristotle] |
| 11255 | In feature-generation the matter (such as bronze) endures, but in generation it doesn't [Aristotle, by Politis] |
| 9071 | We first sense whole entities, and then move to particular parts of it [Aristotle] |
| 16791 | There is no whole except for the parts [Aristotle] |
| 16972 | The four explanations are the main aspects of a thing's nature [Aristotle, by Moravcsik] |
| 5084 | A thing's nature is what causes its changes and stability [Aristotle] |
| 16173 | Coming to be is by shape-change, addition, subtraction, composition or alteration [Aristotle] |
| 17042 | Natural things are their own source of stability through change [Aristotle] |
| 16691 | A day, or the games, has one thing after another, actually and potentially occurring [Aristotle] |
| 16574 | Coming-to-be may be from nothing in a qualified way, as arising from an absence [Aristotle] |
| 11254 | Matter is potentiality [Aristotle, by Politis] |
| 13110 | Intrinsic cause is prior to coincidence, so nature and intelligence are primary causes, chance secondary [Aristotle] |
| 13106 | Maybe there is no pure chance; a man's choices cause his chance meetings [Aristotle] |
| 13108 | Chance is a coincidental cause among events involving purpose and choice [Aristotle] |
| 8331 | To know something we need understanding, which is grasp of the primary cause [Aristotle] |
| 5080 | We know a thing if we grasp its first causes, principles and basic elements [Aristotle] |
| 16969 | Science refers the question Why? to four causes/explanations: matter, form, source, purpose [Aristotle] |
| 11250 | Four Explanations: the essence and form; the matter; the source; and the end [Aristotle, by Politis] |
| 12045 | Aristotle's four 'causes' are four items which figure in basic explanations of nature [Aristotle, by Annas] |
| 16968 | There are as many causes/explanations as there are different types of why-question [Aristotle] |
| 13109 | Chance is inexplicable, because we can only explain what happens always or usually [Aristotle] |
| 10110 | Corresponding to every concept there is a class (some of them sets) [George/Velleman] |
| 9789 | You can't abstract natural properties to make Forms - objects and attributes are defined together [Aristotle] |
| 9788 | Mathematicians study what is conceptually separable, and doesn't lead to error [Aristotle] |
| 5107 | Predicates are substance, quality, place, relation, quantity and action or affection [Aristotle] |
| 20042 | We assign the cause of someone's walking when we say why they are doing it [Aristotle] |
| 5110 | Goodness is when a thing (such as a circle) is complete, and conforms with its nature [Aristotle] |
| 5111 | All moral virtue is concerned with bodily pleasure and pain [Aristotle] |
| 7903 | The six perfections are giving, morality, patience, vigour, meditation, and wisdom [Nagarjuna] |
| 5092 | Nature is a principle of change, so we must understand change first [Aristotle] |
| 5113 | Nothing natural is disorderly, because nature is responsible for all order [Aristotle] |
| 5085 | 'Nature' refers to two things - form and matter [Aristotle] |
| 5089 | Nature has purpose, and aims at what is better. Is it coincidence that crops grow when it rains? [Aristotle] |
| 5086 | The nature of a thing is its end and purpose [Aristotle] |
| 5087 | A thing's purpose is ambiguous, and from one point of view we ourselves are ends [Aristotle] |
| 5091 | Teeth and crops are predictable, so they cannot be mere chance, but must have a purpose [Aristotle] |
| 5108 | Is ceasing-to-be unnatural if it happens by force, and natural otherwise? [Aristotle] |
| 5093 | Continuity depends on infinity, because the continuous is infinitely divisible [Aristotle] |
| 5095 | The heavens seem to be infinite, because we cannot imagine their end [Aristotle] |
| 16762 | Matter desires form, as female desires male, and ugliness desires beauty [Aristotle] |
| 17464 | When Aristotle's elements compound they are stable, so why would they ever separate? [Weisberg/Needham/Hendry on Aristotle] |
| 11252 | The 'form' of a thing explains why the matter constitutes that particular thing [Aristotle, by Politis] |
| 11253 | A 'material' cause/explanation is the form of whatever is the source [Aristotle, by Politis] |
| 13107 | Causes produce a few things in their own right, and innumerable things coincidentally [Aristotle] |
| 8332 | The four causes are the material, the form, the source, and the end [Aristotle] |
| 9787 | Scientists must know the essential attributes of the things they study [Aristotle] |
| 20063 | Motion fulfils potentiality [Aristotle] |
| 5114 | If movement can arise within an animal, why can't it also arise in the universe? [Aristotle] |
| 5116 | When there is unnatural movement (e.g. fire going downwards) the cause is obvious [Aristotle] |
| 5099 | The universe as a whole is not anywhere [Aristotle] |
| 5097 | If everything has a place, this causes an infinite regress, because each place must have place [Aristotle] |
| 5098 | Place is not shape, or matter, or extension between limits; it is the limits of a body [Aristotle] |
| 20920 | If there were many cosmoses, each would have its own time, giving many times [Aristotle] |
| 5106 | Would there be time if there were no mind? [Aristotle] |
| 22967 | It is unclear whether time depends on the existence of soul [Aristotle] |
| 8590 | Time does not exist without change [Aristotle] |
| 22965 | Time measures rest, as well as change [Aristotle] |
| 22885 | For Aristotle time is not a process but a means for measuring processes [Aristotle, by Bardon] |
| 22959 | Time is not change, but the number we associate with change [Aristotle] |
| 22964 | Change only exists in time through its being temporally measure [Aristotle] |
| 5104 | Time is an aspect of change [Aristotle] |
| 22956 | How can time exist, when it is composed of what has ceased to be and is yet to be? [Aristotle] |
| 5102 | If all of time has either ceased to exist, or has not yet happened, maybe time does not exist [Aristotle] |
| 5103 | Time is not change, but requires change in our minds to be noticed [Aristotle] |
| 22961 | The present moment is obviously a necessary feature of time [Aristotle] |
| 22916 | Unlike time, change goes at different rates, and is usually localised [Aristotle, by Le Poidevin] |
| 16693 | Time has parts, but the now is not one of them, and time is not composed of nows [Aristotle] |
| 22958 | Nows can't be linked together, any more than points on a line [Aristotle] |
| 22968 | Circular motion is the most obvious measure of time, and especially the celestial sphere [Aristotle] |
| 22963 | We measure change by time, and time by change, as they are interdefined [Aristotle] |
| 22966 | The present moment is a link (of past to future), and also a limit (of past and of future) [Aristotle] |
| 22957 | We can't tell whether the changing present moment is one thing, or a succession of things [Aristotle] |
| 5083 | Do things come to be from what is, or from what is not? Both seem problematical. [Aristotle] |
| 5119 | The source of all movement must be indivisible and have no magnitude [Aristotle] |