118 ideas
| 2512 | Philosophy is a battle against the bewitchment of our intelligence by means of language [Wittgenstein] |
| 4148 | What is your aim in philosophy? - To show the fly the way out of the fly-bottle [Wittgenstein] |
| 22490 | Bring words back from metaphysics to everyday use [Wittgenstein] |
| 10237 | Coherence is a primitive, intuitive notion, not reduced to something formal [Shapiro] |
| 6566 | The problem is to explain the role of contradiction in social life [Wittgenstein] |
| 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] |
| 18743 | Wittgenstein says we want the grammar of problems, not their first-order logical structure [Wittgenstein, by Horsten/Pettigrew] |
| 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] |
| 4139 | Naming is a preparation for description [Wittgenstein] |
| 4946 | A name is not determined by a description, but by a cluster or family [Wittgenstein, by Kripke] |
| 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] |
| 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] |
| 10272 | The notion of 'object' is at least partially structural and mathematical [Shapiro] |
| 10275 | A blurry border is still a border [Shapiro] |
| 15106 | Essence is expressed by grammar [Wittgenstein] |
| 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] |
| 6600 | The belief that fire burns is like the fear that it burns [Wittgenstein] |
| 4153 | Are sense-data the material of which the universe is made? [Wittgenstein] |
| 6501 | As sense-data are necessarily private, they are attacked by Wittgenstein's objections [Wittgenstein, by Robinson,H] |
| 11079 | How do I decide when to accept or obey an intuition? [Wittgenstein] |
| 4160 | One can mistrust one's own senses, but not one's own beliefs [Wittgenstein] |
| 19273 | I don't have the opinion that people have minds; I just treat them as such [Wittgenstein] |
| 5663 | It is irresponsible to generalise from my own case of pain to other people's [Wittgenstein] |
| 19272 | To imagine another's pain by my own, I must imagine a pain I don't feel, by one I do feel [Wittgenstein] |
| 4161 | If a lion could talk, we could not understand him [Wittgenstein] |
| 7392 | If a lion could talk, it would be nothing like other lions [Dennett on Wittgenstein] |
| 10203 | We apprehend small, finite mathematical structures by abstraction from patterns [Shapiro] |
| 5676 | To say that I 'know' I am in pain means nothing more than that I AM in pain [Wittgenstein] |
| 4154 | Why are we not aware of the huge gap between mind and brain in ordinary life? [Wittgenstein] |
| 4158 | An 'inner process' stands in need of outward criteria [Wittgenstein] |
| 6165 | Every course of action can either accord or conflict with a rule, so there is no accord or conflict [Wittgenstein] |
| 4143 | One cannot obey a rule 'privately', because that is a practice, not the same as thinking one is obeying [Wittgenstein] |
| 7092 | If individuals can't tell if they are following a rule, how does a community do it? [Grayling on Wittgenstein] |
| 4138 | Is white simple, or does it consist of the colours of the rainbow? [Wittgenstein] |
| 7055 | Externalist accounts of mental content begin in Wittgenstein [Wittgenstein, by Heil] |
| 12576 | Possessing a concept is knowing how to go on [Wittgenstein, by Peacocke] |
| 4157 | Concepts direct our interests and investigations, and express those interests [Wittgenstein] |
| 12606 | Man learns the concept of the past by remembering [Wittgenstein] |
| 4141 | Various games have a 'family resemblance', as their similarities overlap and criss-cross [Wittgenstein] |
| 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] |
| 23450 | Wittgenstein rejected his earlier view that the form of language is the form of the world [Wittgenstein, by Morris,M] |
| 4150 | Asking about verification is only one way of asking about the meaning of a proposition [Wittgenstein] |
| 6567 | For Wittgenstein, words are defined by their use, just as chess pieces are [Wittgenstein, by Fogelin] |
| 6169 | We do not achieve meaning and understanding in our heads, but in the world [Wittgenstein, by Rowlands] |
| 4155 | We all seem able to see quite clearly how sentences represent things when we use them [Wittgenstein] |
| 4137 | In the majority of cases the meaning of a word is its use in the language [Wittgenstein] |
| 4142 | To understand a sentence means to understand a language [Wittgenstein] |
| 4149 | We don't have 'meanings' in our minds in addition to verbal expressions [Wittgenstein] |
| 4156 | Make the following experiment: say "It's cold here" and mean "It's warm here" [Wittgenstein] |
| 4145 | How do words refer to sensations? [Wittgenstein] |
| 4140 | The standard metre in Paris is neither one metre long nor not one metre long [Wittgenstein] |
| 6166 | Was Wittgenstein's problem between individual and community, or between occasions for an individual? [Rowlands on Wittgenstein] |
| 7875 | If a brilliant child invented a name for a private sensation, it couldn't communicate it [Wittgenstein] |
| 4146 | We cannot doublecheck mental images for correctness (or confirm news with many copies of the paper) [Wittgenstein] |
| 4147 | If we only named pain by our own case, it would be like naming beetles by looking in a private box [Wittgenstein] |
| 5659 | If the reference is private, that is incompatible with the sense being public [Wittgenstein, by Scruton] |
| 4152 | Getting from perceptions to words cannot be a private matter; the rules need an institution of use [Wittgenstein] |
| 4136 | To imagine a language means to imagine a form of life [Wittgenstein] |
| 4144 | Common human behaviour enables us to interpret an unknown language [Wittgenstein] |
| 11049 | To communicate, language needs agreement in judgment as well as definition [Wittgenstein] |
| 6658 | What is left over if I subtract my arm going up from my raising my arm? [Wittgenstein] |
| 22489 | 'Good' is an attributive adjective like 'large', not predicative like 'red' [Geach, by Foot] |
| 4151 | Grammar tells what kind of object anything is - and theology is a kind of grammar [Wittgenstein] |
| 4159 | The human body is the best picture of the human soul [Wittgenstein] |