90 ideas
10237 | Coherence is a primitive, intuitive notion, not reduced to something formal [Shapiro] |
23247 | The need to act produces consciousness, and practical reason is the root of all reason [Fichte] |
23232 | Sufficient reason makes the transition from the particular to the general [Fichte] |
10204 | An 'implicit definition' gives a direct description of the relations of an entity [Shapiro] |
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] |
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] |
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] |
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] |
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] |
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] |
23227 | Each object has a precise number of properties, each to a precise degree [Fichte] |
23228 | The principle of activity and generation is found in a self-moving basic force [Fichte] |
10272 | The notion of 'object' is at least partially structural and mathematical [Shapiro] |
10275 | A blurry border is still a border [Shapiro] |
10258 | Logical modalities may be acceptable, because they are reducible to satisfaction in models [Shapiro] |
10266 | Why does the 'myth' of possible worlds produce correct modal logic? [Shapiro] |
23241 | I am myself, but not the external object; so I only sense myself, and not the object [Fichte] |
21966 | Self-consciousness is the basis of knowledge, and knowing something is knowing myself [Fichte] |
21967 | There is nothing to say about anything which is outside my consciousness [Fichte] |
21969 | Awareness of reality comes from the free activity of consciousness [Fichte] |
23231 | I immediately know myself, and anything beyond that is an inference [Fichte] |
23246 | Faith is not knowledge; it is a decision of the will [Fichte] |
23245 | Knowledge can't be its own foundation; there has to be regress of higher and higher authorities [Fichte] |
23242 | Consciousness has two parts, passively receiving sensation, and actively causing productions [Fichte] |
23240 | We can't know by sight or hearing without realising that we are doing so [Fichte] |
10203 | We apprehend small, finite mathematical structures by abstraction from patterns [Shapiro] |
23243 | Consciousness of external things is always accompanied by an unnoticed consciousness of self [Fichte] |
23244 | Forming purposes is absolutely free, and produces something from nothing [Fichte] |
23237 | The capacity for freedom is above the laws of nature, with its own power of purpose and will [Fichte] |
23235 | I want independent control of the fundamental cause of my decisions [Fichte] |
23230 | Nature contains a fundamental force of thought [Fichte] |
10229 | Simple types can be apprehended through their tokens, via abstraction [Shapiro] |
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] |
23233 | The will is awareness of one of our inner natural forces [Fichte] |
23234 | I cannot change the nature which has been determined for me [Fichte] |
23239 | The self is, apart from outward behaviour, a drive in your nature [Fichte] |
23238 | If life lacks love it becomes destruction [Fichte] |
23236 | Freedom means making yourself become true to your essential nature [Fichte] |
13304 | Learned men gain more in one day than others do in a lifetime [Posidonius] |
23229 | Nature is wholly interconnected, and the tiniest change affects everything [Fichte] |
20820 | Time is an interval of motion, or the measure of speed [Posidonius, by Stobaeus] |