95 ideas
6947 | Metaphysics does not rest on facts, but on what we are inclined to believe [Peirce] |
4739 | In "if and only if" (iff), "if" expresses the sufficient condition, and "only if" the necessary condition [Engel] |
6937 | Reason aims to discover the unknown by thinking about the known [Peirce] |
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] |
4737 | Are truth-bearers propositions, or ideas/beliefs, or sentences/utterances? [Engel] |
4750 | The redundancy theory gets rid of facts, for 'it is a fact that p' just means 'p' [Engel] |
4744 | We can't explain the corresponding structure of the world except by referring to our thoughts [Engel] |
4738 | The coherence theory says truth is an internal relationship between groups of truth-bearers [Engel] |
4745 | Any coherent set of beliefs can be made more coherent by adding some false beliefs [Engel] |
4755 | Deflationism cannot explain why we hold beliefs for reasons [Engel] |
4753 | Deflationism seems to block philosophers' main occupation, asking metatheoretical questions [Engel] |
4751 | Maybe there is no more to be said about 'true' than there is about the function of 'and' in logic [Engel] |
10206 | Modal operators are usually treated as quantifiers [Shapiro] |
10252 | The Axiom of Choice seems to license an infinite amount of choosing [Shapiro] |
10208 | Axiom of Choice: some function has a value for every set in a given set [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] |
4752 | Deflationism must reduce bivalence ('p is true or false') to excluded middle ('p or not-p') [Engel] |
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] |
10239 | The central notion of model theory is the relation of 'satisfaction' [Shapiro] |
10240 | Model theory deals with relations, reference and extensions [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] |
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] |
21492 | Realism is basic to the scientific method [Peirce] |
6949 | If someone doubted reality, they would not actually feel dissatisfaction [Peirce] |
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] |
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] |
4762 | The Humean theory of motivation is that beliefs may be motivators as well as desires [Engel] |
6940 | The feeling of belief shows a habit which will determine our actions [Peirce] |
6943 | A mere question does not stimulate a struggle for belief; there must be a real doubt [Peirce] |
6941 | We are entirely satisfied with a firm belief, even if it is false [Peirce] |
6942 | We want true beliefs, but obviously we think our beliefs are true [Peirce] |
4754 | Our beliefs are meant to fit the world (i.e. be true), where we want the world to fit our desires [Engel] |
4763 | 'Evidentialists' say, and 'voluntarists' deny, that we only believe on the basis of evidence [Engel] |
4746 | Pragmatism is better understood as a theory of belief than as a theory of truth [Engel] |
6598 | We need our beliefs to be determined by some external inhuman permanency [Peirce] |
6944 | Demonstration does not rest on first principles of reason or sensation, but on freedom from actual doubt [Peirce] |
6948 | Doubts should be satisfied by some external permanency upon which thinking has no effect [Peirce] |
4764 | We cannot directly control our beliefs, but we can control the causes of our involuntary beliefs [Engel] |
6945 | Once doubt ceases, there is no point in continuing to argue [Peirce] |
10203 | We apprehend small, finite mathematical structures by abstraction from patterns [Shapiro] |
4759 | Mental states as functions are second-order properties, realised by first-order physical properties [Engel] |
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] |
6939 | What is true of one piece of copper is true of another (unlike brass) [Peirce] |
6938 | Natural selection might well fill an animal's mind with pleasing thoughts rather than true ones [Peirce] |
6946 | If death is annihilation, belief in heaven is a cheap pleasure with no disappointment [Peirce] |