115 ideas
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] |
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] |
8729 | Intuitionists deny excluded middle, because it is committed to transcendent truth or objects [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] |
16357 | Mental files are the counterparts of singular terms [Recanati] |
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] |
10238 | The set-theoretical hierarchy contains as many isomorphism types as possible [Shapiro] |
10214 | Theory ontology is never complete, but is only determined 'up to isomorphism' [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] |
8763 | The number 3 is presumably identical as a natural, an integer, a rational, a real, and complex [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] |
18249 | Cauchy gave a formal definition of a converging sequence. [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] |
8764 | Categories are the best foundation for 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] |
8762 | Two definitions of 3 in terms of sets disagree over whether 1 is a member of 3 [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] |
8760 | Numbers do not exist independently; the essence of a number is its relations to other numbers [Shapiro] |
8761 | A 'system' is related objects; a 'pattern' or 'structure' abstracts the pure relations from them [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] |
8744 | Logicism seems to be a non-starter if (as is widely held) logic has no ontology of its own [Shapiro] |
8749 | Term Formalism says mathematics is just about symbols - but real numbers have no names [Shapiro] |
8750 | Game Formalism is just a matter of rules, like chess - but then why is it useful in science? [Shapiro] |
8752 | Deductivism says mathematics is logical consequences of uninterpreted axioms [Shapiro] |
10254 | Can the ideal constructor also destroy objects? [Shapiro] |
10255 | Presumably nothing can block a possible dynamic operation? [Shapiro] |
8753 | Critics resent the way intuitionism cripples mathematics, but it allows new important distinctions [Shapiro] |
8731 | Conceptualist are just realists or idealist or nominalists, depending on their view of concepts [Shapiro] |
8730 | 'Impredicative' definitions refer to the thing being described [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] |
16360 | Identity statements are informative if they link separate mental files [Recanati] |
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] |
16374 | There is a continuum from acquaintance to description in knowledge, depending on the link [Recanati] |
8725 | Rationalism tries to apply mathematical methodology to all of knowledge [Shapiro] |
10203 | We apprehend small, finite mathematical structures by abstraction from patterns [Shapiro] |
16354 | Indexicality is closely related to singularity, exploiting our direct relations with things [Recanati] |
18409 | Indexicals apply to singular thought, and mental files have essentially indexical features [Recanati] |
16371 | Files can be confused, if two files correctly have a single name, or one file has two names [Recanati] |
16373 | Encylopedic files have further epistemic links, beyond the basic one [Recanati] |
16375 | Singular thoughts need a mental file, and an acquaintance relation from file to object [Recanati] |
16377 | Expected acquaintance can create a thought-vehicle file, but without singular content [Recanati] |
16378 | An 'indexed' file marks a file which simulates the mental file of some other person [Recanati] |
16361 | A mental file treats all of its contents as concerning one object [Recanati] |
16387 | Reference by mental files is Millian, in emphasising acquaintance, rather than satisfaction [Recanati] |
16358 | The reference of a file is fixed by what it relates to, not the information it contains [Recanati] |
16367 | There are transient 'demonstrative' files, habitual 'recognitional' files, cumulative 'encyclopedic' files [Recanati] |
16368 | Files are hierarchical: proto-files, then first-order, then higher-order encyclopedic [Recanati] |
16370 | A file has a 'nucleus' through its relation to the object, and a 'periphery' of links to other files [Recanati] |
16381 | The content of thought is what is required to understand it (which involves hearers) [Recanati] |
16365 | Mental files are individual concepts (thought constituents) [Recanati] |
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] |
16356 | There may be two types of reference in language and thought: descriptive and direct [Recanati] |
16393 | In super-direct reference, the referent serves as its own vehicle of reference [Recanati] |
16386 | Direct reference is strong Millian (just a tag) or weak Kaplanian (allowing descriptions as well) [Recanati] |
16372 | Sense determines reference says same sense/same reference; new reference means new sense [Recanati] |
16388 | We need sense as well as reference, but in a non-descriptive form, and mental files do that [Recanati] |
16359 | Sense is a mental file (not its contents); similar files for Cicero and Tully are two senses [Recanati] |
16352 | A rigid definite description can be attributive, not referential: 'the actual F, whoever he is….' [Recanati] |
16348 | Descriptivism says we mentally relate to objects through their properties [Recanati] |
16384 | Definite descriptions reveal either a predicate (attributive use) or the file it belongs in (referential) [Recanati] |
16355 | Problems with descriptivism are reference by perception, by communications and by indexicals [Recanati] |
16353 | Singularity cannot be described, and it needs actual world relations [Recanati] |
16382 | Fregean modes of presentation can be understood as mental files [Recanati] |
16389 | If two people think 'I am tired', they think the same thing, and they think different things [Recanati] |
16364 | Indexical don't refer; only their tokens do [Recanati] |
16363 | Indexicals (like mental files) determine their reference relationally, not by satisfaction [Recanati] |
16350 | Two-D semantics is said to help descriptivism of reference deal with singular objects [Recanati] |
16351 | In 2-D semantics, reference is determined, then singularity by the truth of a predication [Recanati] |
16380 | Russellian propositions are better than Fregean thoughts, by being constant through communication [Recanati] |