113 ideas
19608 | Wisdom is just the last gasp of a dying civilization [Cioran] |
19624 | Intelligence only fully flourishes at the end of a historical period [Cioran] |
19599 | Ideas are neutral, but people fill them with passion and weakness [Cioran] |
19631 | The history of ideas (and deeds) occurs in a meaningless environment [Cioran] |
19645 | Some thinkers would have been just as dynamic, no matter when they had lived [Cioran] |
19629 | A nation gives expression to its sum of values, and is then exhausted [Cioran] |
19618 | I abandoned philosophy because it didn't acknowledge melancholy and human weakness [Cioran] |
19621 | Originality in philosophy is just the invention of terms [Cioran] |
19607 | The mind is superficial, only concerned with the arrangement of events, not their significance [Cioran] |
19638 | Metaphysics is a universalisation of physical anguish [Cioran] |
19620 | Great systems of philosophy are just brilliant tautologies [Cioran] |
10237 | Coherence is a primitive, intuitive notion, not reduced to something formal [Shapiro] |
19630 | No great idea ever emerged from a dialogue [Cioran] |
10204 | An 'implicit definition' gives a direct description of the relations of an entity [Shapiro] |
19636 | Truth is just an error insufficiently experienced [Cioran] |
19642 | Eventually every 'truth' is guaranteed by the police [Cioran] |
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] |
19632 | An axiom has no more authority than a frenzy [Cioran] |
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] |
6409 | The 'simple theory of types' distinguishes levels among properties [Ramsey, by Grayling] |
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] |
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] |
3212 | Beliefs are maps by which we steer [Ramsey] |
19626 | Our instincts had to be blunted and diminished, to make way for consciousness! [Cioran] |
10203 | We apprehend small, finite mathematical structures by abstraction from patterns [Shapiro] |
19633 | We use concepts to master our fears; saying 'death' releases us from confronting it [Cioran] |
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] |
19615 | I want to suppress in myself the normal reasons people have for action [Cioran] |
19628 | At a civilisation's peak values are all that matters, and people unconsciously live by them [Cioran] |
19646 | Values don't accumulate; they are ruthlessly replaced [Cioran] |
19614 | Lovers are hateful, apart from their hovering awareness of death [Cioran] |
19634 | Man is never himself; he always aims at less than life, or more than life [Cioran] |
19619 | To live authentically, we must see that philosophy is totally useless [Cioran] |
19622 | The pointlessness of our motives and irrelevance of our gestures reveals our vacuity [Cioran] |
19617 | Evidence suggests that humans do not have a purpose [Cioran] |
19612 | The universe is dirty and fragile, as if a scandal in nothingness had produced its matter [Cioran] |
19604 | Unlike other creatures, mankind seems lost in nature [Cioran] |
19606 | We can only live because our imagination and memory are poor [Cioran] |
19601 | Life is now more dreaded than death [Cioran] |
19640 | No one is brave enough to say they don't want to do anything; we despise such a view [Cioran] |
19641 | If you lack beliefs, boredom is your martyrdom [Cioran] |
19602 | You are stuck in the past if you don't know boredom [Cioran] |
19644 | History is the bloody rejection of boredom [Cioran] |
19613 | It is pointless to refuse or accept the social order; we must endure it like the weather [Cioran] |
19627 | Opportunists can save a nation, and heroes can ruin it [Cioran] |
19625 | The ideal is to impose a religion by force, and then live in doubt about its beliefs [Cioran] |
19605 | Despite endless suggestions, no one has found a goal for history [Cioran] |
19637 | History is wonderfully devoid of meaning [Cioran] |
19610 | Religions see suicide as insubordination [Cioran] |
19611 | No one has ever found a good argument against suicide [Cioran] |
19609 | If you have not contemplated suicide, you are a miserable worm [Cioran] |
19639 | We all need sexual secrets! [Cioran] |
19603 | Why is God so boring, and why does God resemble humanity so little? [Cioran] |
19616 | As the perfect wisdom of detachment, philosophy offers no rivals to Taoism [Cioran] |
19600 | When man abandons religion, he then follows new fake gods and mythologies [Cioran] |
19643 | A religion needs to motivate killings, and cannot tolerate rivals [Cioran] |
19623 | Circles of hell are ridiculous; all that matters is to be there [Cioran] |