107 ideas
13786 | Wisdom is called 'beautiful', because it performs fine works [Plato] |
13780 | Good people are no different from wise ones [Plato] |
10237 | Coherence is a primitive, intuitive notion, not reduced to something formal [Shapiro] |
13778 | A dialectician is someone who knows how to ask and to answer questions [Plato] |
10204 | An 'implicit definition' gives a direct description of the relations of an entity [Shapiro] |
13776 | Truths say of what is that it is, falsehoods say of what is that it is not [Plato] |
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] |
13777 | A name is a sort of tool [Plato] |
13790 | A name-giver might misname something, then force other names to conform to it [Plato] |
13791 | Things must be known before they are named, so it can't be the names that give us knowledge [Plato] |
13789 | Anyone who knows a thing's name also knows the thing [Plato] |
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] |
10239 | The central notion of model theory is the relation of 'satisfaction' [Shapiro] |
10240 | Model theory deals with relations, reference and extensions [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] |
2063 | How can beauty have identity if it changes? [Plato] |
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] |
13775 | We only succeed in cutting if we use appropriate tools, not if we approach it randomly [Plato] |
14308 | We can bring dispositions into existence, as in creating an identifier [Dennett, by Mumford] |
10272 | The notion of 'object' is at least partially structural and mathematical [Shapiro] |
13787 | Doesn't each thing have an essence, just as it has other qualities? [Plato] |
10275 | A blurry border is still a border [Shapiro] |
13774 | Things don't have every attribute, and essence isn't private, so each thing has an essence [Plato] |
7384 | Words are fixed by being attached to similarity clusters, without mention of 'essences' [Dennett] |
13772 | Is the being or essence of each thing private to each person? [Plato] |
13788 | If we made a perfect duplicate of Cratylus, there would be two Cratyluses [Plato] |
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] |
7374 | Light wavelengths entering the eye are only indirectly related to object colours [Dennett] |
13792 | There can't be any knowledge if things are constantly changing [Plato] |
7369 | Brains are essentially anticipation machines [Dennett] |
13781 | Soul causes the body to live, and gives it power to breathe and to be revitalized [Plato] |
7393 | We can't draw a clear line between conscious and unconscious [Dennett] |
7367 | Perhaps the brain doesn't 'fill in' gaps in consciousness if no one is looking. [Dennett] |
7394 | Conscious events can only be explained in terms of unconscious events [Dennett] |
7391 | We can know a lot of what it is like to be a bat, and nothing important is unknown [Dennett] |
7387 | "Qualia" can be replaced by complex dispositional brain states [Dennett] |
7376 | We can't assume that dispositions will remain normal when qualia have been inverted [Dennett] |
7372 | In peripheral vision we see objects without their details, so blindsight is not that special [Dennett] |
7373 | Blindsight subjects glean very paltry information [Dennett] |
10203 | We apprehend small, finite mathematical structures by abstraction from patterns [Shapiro] |
7385 | People accept blurred boundaries in many things, but insist self is All or Nothing [Dennett] |
7383 | The psychological self is an abstraction, not a thing in the brain [Dennett] |
7386 | Selves are not soul-pearls, but artefacts of social processes [Dennett] |
7381 | We tell stories about ourselves, to protect, control and define who we are [Dennett] |
7382 | We spin narratives about ourselves, and the audience posits a centre of gravity for them [Dennett] |
7370 | The brain is controlled by shifting coalitions, guided by good purposeful habits [Dennett] |
7379 | If an epiphenomenon has no physical effects, it has to be undetectable [Dennett] |
7365 | Dualism wallows in mystery, and to accept it is to give up [Dennett] |
7371 | All functionalism is 'homuncular', of one grain size or another [Dennett] |
7380 | Visual experience is composed of neural activity, which we find pleasing [Dennett] |
7366 | It is arbitrary to say which moment of brain processing is conscious [Dennett] |
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] |
13785 | 'Arete' signifies lack of complexity and a free-flowing soul [Plato] |
7368 | Originally there were no reasons, purposes or functions; since there were no interests, there were only causes [Dennett] |
13779 | The natural offspring of a lion is called a 'lion' (but what about the offspring of a king?) [Plato] |
13783 | Even the gods love play [Plato] |