136 ideas
5082 | Reason grasps generalities, while the senses grasp particulars [Aristotle] |
8721 | An 'impredicative' definition seems circular, because it uses the term being defined [Friend] |
8680 | Classical definitions attempt to refer, but intuitionist/constructivist definitions actually create objects [Friend] |
3678 | Reductio ad absurdum proves an idea by showing that its denial produces contradiction [Friend] |
8705 | Anti-realists see truth as our servant, and epistemically contrained [Friend] |
8713 | In classical/realist logic the connectives are defined by truth-tables [Friend] |
8708 | Double negation elimination is not valid in intuitionist logic [Friend] |
8694 | Free logic was developed for fictional or non-existent objects [Friend] |
8665 | A 'proper subset' of A contains only members of A, but not all of them [Friend] |
8672 | A 'powerset' is all the subsets of a set [Friend] |
8677 | Set theory makes a minimum ontological claim, that the empty set exists [Friend] |
8666 | Infinite sets correspond one-to-one with a subset [Friend] |
8682 | Major set theories differ in their axioms, and also over the additional axioms of choice and infinity [Friend] |
13270 | Are a part and whole one or many? Either way, what is the cause? [Aristotle] |
8709 | The law of excluded middle is syntactic; it just says A or not-A, not whether they are true or false [Friend] |
8711 | Intuitionists read the universal quantifier as "we have a procedure for checking every..." [Friend] |
8675 | Paradoxes can be solved by talking more loosely of 'classes' instead of 'sets' [Friend] |
8674 | The Burali-Forti paradox asks whether the set of all ordinals is itself an ordinal [Friend] |
9790 | Geometry studies naturally occurring lines, but not as they occur in nature [Aristotle] |
8667 | The 'integers' are the positive and negative natural numbers, plus zero [Friend] |
8668 | The 'rational' numbers are those representable as fractions [Friend] |
8670 | A number is 'irrational' if it cannot be represented as a fraction [Friend] |
8661 | The natural numbers are primitive, and the ordinals are up one level of abstraction [Friend] |
8664 | Cardinal numbers answer 'how many?', with the order being irrelevant [Friend] |
8671 | The 'real' numbers (rationals and irrationals combined) is the Continuum, which has no gaps [Friend] |
22962 | Two is the least number, but there is no least magnitude, because it is always divisible [Aristotle] |
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] |
8663 | Raising omega to successive powers of omega reveal an infinity of infinities [Friend] |
8662 | The first limit ordinal is omega (greater, but without predecessor), and the second is twice-omega [Friend] |
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] |
8669 | Between any two rational numbers there is an infinite number of rational numbers [Friend] |
8676 | Is mathematics based on sets, types, categories, models or topology? [Friend] |
8678 | Most mathematical theories can be translated into the language of set theory [Friend] |
8701 | The number 8 in isolation from the other numbers is of no interest [Friend] |
8702 | In structuralism the number 8 is not quite the same in different structures, only equivalent [Friend] |
8699 | Are structures 'ante rem' (before reality), or are they 'in re' (grounded in physics)? [Friend] |
8696 | Structuralist says maths concerns concepts about base objects, not base objects themselves [Friend] |
8695 | Structuralism focuses on relations, predicates and functions, with objects being inessential [Friend] |
8700 | 'In re' structuralism says that the process of abstraction is pattern-spotting [Friend] |
8681 | The big problem for platonists is epistemic: how do we perceive, intuit, know or detect mathematical facts? [Friend] |
9974 | Ten sheep and ten dogs are the same numerically, but it is not the same ten [Aristotle] |
8712 | Mathematics should be treated as true whenever it is indispensable to our best physical theory [Friend] |
8716 | Formalism is unconstrained, so cannot indicate importance, or directions for research [Friend] |
8706 | Constructivism rejects too much mathematics [Friend] |
8707 | Intuitionists typically retain bivalence but reject the law of excluded middle [Friend] |
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] |
8704 | Structuralists call a mathematical 'object' simply a 'place in a structure' [Friend] |
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] |
3913 | Maybe imagination is the source of a priori justification [Casullo] |
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] |
8685 | Studying biology presumes the laws of chemistry, and it could never contradict them [Friend] |
8688 | Concepts can be presented extensionally (as objects) or intensionally (as a characterization) [Friend] |
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] |
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] |