168 ideas
5082 | Reason grasps generalities, while the senses grasp particulars [Aristotle] |
10237 | Coherence is a primitive, intuitive notion, not reduced to something formal [Shapiro] |
10204 | An 'implicit definition' gives a direct description of the relations of an entity [Shapiro] |
19125 | If we define truth, we can eliminate it [Halbach/Leigh] |
19128 | If a language cannot name all objects, then satisfaction must be used, instead of unary truth [Halbach/Leigh] |
19120 | Semantic theories need a powerful metalanguage, typically including set theory [Halbach/Leigh] |
19127 | The T-sentences are deductively weak, and also not deductively conservative [Halbach/Leigh] |
19124 | A natural theory of truth plays the role of reflection principles, establishing arithmetic's soundness [Halbach/Leigh] |
19126 | If deflationary truth is not explanatory, truth axioms should be 'conservative', proving nothing new [Halbach/Leigh] |
19129 | The FS axioms use classical logical, but are not fully consistent [Halbach/Leigh] |
19130 | KF is formulated in classical logic, but describes non-classical truth, which allows truth-value gluts [Halbach/Leigh] |
10206 | Modal operators are usually treated as quantifiers [Shapiro] |
10208 | Axiom of Choice: some function has a value for every set in a given set [Shapiro] |
10252 | The Axiom of Choice seems to license an infinite amount of choosing [Shapiro] |
10207 | Anti-realists reject set theory [Shapiro] |
13270 | Are a part and whole one or many? Either way, what is the cause? [Aristotle] |
10259 | The two standard explanations of consequence are semantic (in models) and deductive [Shapiro] |
10257 | Intuitionism only sanctions modus ponens if all three components are proved [Shapiro] |
10253 | Either logic determines objects, or objects determine logic, or they are separate [Shapiro] |
10251 | The law of excluded middle might be seen as a principle of omniscience [Shapiro] |
10212 | Classical connectives differ from their ordinary language counterparts; '∧' is timeless, unlike 'and' [Shapiro] |
10209 | A function is just an arbitrary correspondence between collections [Shapiro] |
10268 | Maybe plural quantifiers should be understood in terms of classes or sets [Shapiro] |
10235 | A sentence is 'satisfiable' if it has a model [Shapiro] |
10240 | Model theory deals with relations, reference and extensions [Shapiro] |
10239 | The central notion of model theory is the relation of 'satisfaction' [Shapiro] |
10214 | Theory ontology is never complete, but is only determined 'up to isomorphism' [Shapiro] |
10238 | The set-theoretical hierarchy contains as many isomorphism types as possible [Shapiro] |
10234 | Any theory with an infinite model has a model of every infinite cardinality [Shapiro] |
10201 | Virtually all of mathematics can be modeled in set theory [Shapiro] |
9790 | Geometry studies naturally occurring lines, but not as they occur in nature [Aristotle] |
22962 | Two is the least number, but there is no least magnitude, because it is always divisible [Aristotle] |
10213 | Real numbers are thought of as either Cauchy sequences or Dedekind cuts [Shapiro] |
18243 | Understanding the real-number structure is knowing usage of the axiomatic language of analysis [Shapiro] |
18245 | Cuts are made by the smallest upper or largest lower number, some of them not rational [Shapiro] |
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] |
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] |
10236 | There is no grounding for mathematics that is more secure than mathematics [Shapiro] |
10256 | For intuitionists, proof is inherently informal [Shapiro] |
10202 | Natural numbers just need an initial object, successors, and an induction principle [Shapiro] |
10205 | Mathematics originally concerned the continuous (geometry) and the discrete (arithmetic) [Shapiro] |
10222 | Mathematical foundations may not be sets; categories are a popular rival [Shapiro] |
10218 | Baseball positions and chess pieces depend entirely on context [Shapiro] |
10224 | The even numbers have the natural-number structure, with 6 playing the role of 3 [Shapiro] |
10228 | Could infinite structures be apprehended by pattern recognition? [Shapiro] |
10230 | The 4-pattern is the structure common to all collections of four objects [Shapiro] |
10249 | The main mathematical structures are algebraic, ordered, and topological [Shapiro] |
10273 | Some structures are exemplified by both abstract and concrete [Shapiro] |
10276 | Mathematical structures are defined by axioms, or in set theory [Shapiro] |
10270 | The main versions of structuralism are all definitionally equivalent [Shapiro] |
10221 | Is there is no more to structures than the systems that exemplify them? [Shapiro] |
10248 | Number statements are generalizations about number sequences, and are bound variables [Shapiro] |
10220 | Because one structure exemplifies several systems, a structure is a one-over-many [Shapiro] |
10223 | There is no 'structure of all structures', just as there is no set of all sets [Shapiro] |
8703 | Shapiro's structuralism says model theory (comparing structures) is the essence of mathematics [Shapiro, by Friend] |
10274 | Does someone using small numbers really need to know the infinite structure of arithmetic? [Shapiro] |
10200 | We distinguish realism 'in ontology' (for objects), and 'in truth-value' (for being either true or false) [Shapiro] |
10210 | If mathematical objects are accepted, then a number of standard principles will follow [Shapiro] |
10215 | Platonists claim we can state the essence of a number without reference to the others [Shapiro] |
10233 | Platonism must accept that the Peano Axioms could all be false [Shapiro] |
10244 | Intuition is an outright hindrance to five-dimensional geometry [Shapiro] |
10280 | A stone is a position in some pattern, and can be viewed as an object, or as a location [Shapiro] |
9974 | Ten sheep and ten dogs are the same numerically, but it is not the same ten [Aristotle] |
10254 | Can the ideal constructor also destroy objects? [Shapiro] |
10255 | Presumably nothing can block a possible dynamic operation? [Shapiro] |
10279 | Can we discover whether a deck is fifty-two cards, or a person is time-slices or molecules? [Shapiro] |
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] |
10227 | The abstract/concrete boundary now seems blurred, and would need a defence [Shapiro] |
10226 | Mathematicians regard arithmetic as concrete, and group theory as abstract [Shapiro] |
10262 | Fictionalism eschews the abstract, but it still needs the possible (without model theory) [Shapiro] |
10277 | Structuralism blurs the distinction between mathematical and ordinary objects [Shapiro] |
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] |
19121 | We can reduce properties to true formulas [Halbach/Leigh] |
5117 | Heavy and light are defined by their tendency to move down or up [Aristotle] |
19122 | Nominalists can reduce theories of properties or sets to harmless axiomatic truth theories [Halbach/Leigh] |
10272 | The notion of 'object' is at least partially structural and mathematical [Shapiro] |
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] |
10275 | A blurry border is still a border [Shapiro] |
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] |
16791 | There is no whole except for the parts [Aristotle] |
9071 | We first sense whole entities, and then move to particular parts of it [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] |
10258 | Logical modalities may be acceptable, because they are reducible to satisfaction in models [Shapiro] |
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] |
10266 | Why does the 'myth' of possible worlds produce correct modal logic? [Shapiro] |
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] |
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] |
16969 | Science refers the question Why? to four causes/explanations: matter, form, source, purpose [Aristotle] |
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] |
10203 | We apprehend small, finite mathematical structures by abstraction from patterns [Shapiro] |
10229 | Simple types can be apprehended through their tokens, via abstraction [Shapiro] |
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] |
10217 | We can apprehend structures by focusing on or ignoring features of patterns [Shapiro] |
9554 | We can focus on relations between objects (like baseballers), ignoring their other features [Shapiro] |
10231 | Abstract objects might come by abstraction over an equivalence class of base entities [Shapiro] |
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] |
5085 | 'Nature' refers to two things - form and matter [Aristotle] |
5113 | Nothing natural is disorderly, because nature is responsible for all order [Aristotle] |
5092 | Nature is a principle of change, so we must understand change first [Aristotle] |
5089 | Nature has purpose, and aims at what is better. Is it coincidence that crops grow when it rains? [Aristotle] |
5091 | Teeth and crops are predictable, so they cannot be mere chance, but must have a purpose [Aristotle] |
5087 | A thing's purpose is ambiguous, and from one point of view we ourselves are ends [Aristotle] |
5086 | The nature of a thing is its end and 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] |
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] |
20063 | Motion fulfils potentiality [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] |
22967 | It is unclear whether time depends on the existence of soul [Aristotle] |
5106 | Would there be time if there were no mind? [Aristotle] |
22885 | For Aristotle time is not a process but a means for measuring processes [Aristotle, by Bardon] |
8590 | Time does not exist without change [Aristotle] |
5104 | Time is an aspect of change [Aristotle] |
22965 | Time measures rest, as well as change [Aristotle] |
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] |
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] |
22957 | We can't tell whether the changing present moment is one thing, or a succession of things [Aristotle] |
22966 | The present moment is a link (of past to future), and also a limit (of past and of future) [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] |