363 ideas
| 11300 | Agathon: good [PG] |
| 11301 | Aisthesis: perception, sensation, consciousness [PG] |
| 11302 | Aitia / aition: cause, explanation [PG] |
| 11303 | Akrasia: lack of control, weakness of will [PG] |
| 11304 | Aletheia: truth [PG] |
| 11305 | Anamnesis: recollection, remembrance [PG] |
| 11306 | Ananke: necessity [PG] |
| 11307 | Antikeimenon: object [PG] |
| 11375 | Apatheia: unemotional [PG] |
| 11308 | Apeiron: the unlimited, indefinite [PG] |
| 11376 | Aphairesis: taking away, abstraction [PG] |
| 11309 | Apodeixis: demonstration [PG] |
| 11310 | Aporia: puzzle, question, anomaly [PG] |
| 11311 | Arche: first principle, the basic [PG] |
| 11312 | Arete: virtue, excellence [PG] |
| 11313 | Chronismos: separation [PG] |
| 11314 | Diairesis: division [PG] |
| 11315 | Dialectic: dialectic, discussion [PG] |
| 11316 | Dianoia: intellection [cf. Noesis] [PG] |
| 11317 | Diaphora: difference [PG] |
| 11318 | Dikaiosune: moral goodness, justice [PG] |
| 11319 | Doxa: opinion, belief [PG] |
| 11320 | Dunamis: faculty, potentiality, capacity [PG] |
| 11321 | Eidos: form, idea [PG] |
| 11322 | Elenchos: elenchus, interrogation [PG] |
| 11323 | Empeiron: experience [PG] |
| 11324 | Energeia: employment, actuality, power? [PG] |
| 11325 | Enkrateia: control [PG] |
| 11326 | Entelecheia: entelechy, having an end [PG] |
| 11327 | Epagoge: induction, explanation [PG] |
| 11328 | Episteme: knowledge, understanding [PG] |
| 11329 | Epithumia: appetite [PG] |
| 11330 | Ergon: function [PG] |
| 11331 | Eristic: polemic, disputation [PG] |
| 11332 | Eros: love [PG] |
| 11333 | Eudaimonia: flourishing, happiness, fulfilment [PG] |
| 11334 | Genos: type, genus [PG] |
| 11335 | Hexis: state, habit [PG] |
| 11336 | Horismos: definition [PG] |
| 11337 | Hule: matter [PG] |
| 11338 | Hupokeimenon: subject, underlying thing [cf. Tode ti] [PG] |
| 11339 | Kalos / kalon: beauty, fineness, nobility [PG] |
| 11340 | Kath' hauto: in virtue of itself, essentially [PG] |
| 11341 | Kinesis: movement, process [PG] |
| 11342 | Kosmos: order, universe [PG] |
| 11343 | Logos: reason, account, word [PG] |
| 11344 | Meson: the mean [PG] |
| 11345 | Metechein: partaking, sharing [PG] |
| 11377 | Mimesis: imitation, fine art [PG] |
| 11346 | Morphe: form [PG] |
| 11347 | Noesis: intellection, rational thought [cf. Dianoia] [PG] |
| 11348 | Nomos: convention, law, custom [PG] |
| 11349 | Nous: intuition, intellect, understanding [PG] |
| 11350 | Orexis: desire [PG] |
| 11351 | Ousia: substance, (primary) being, [see 'Prote ousia'] [PG] |
| 11352 | Pathos: emotion, affection, property [PG] |
| 11353 | Phantasia: imagination [PG] |
| 11354 | Philia: friendship [PG] |
| 11355 | Philosophia: philosophy, love of wisdom [PG] |
| 11356 | Phronesis: prudence, practical reason, common sense [PG] |
| 11357 | Physis: nature [PG] |
| 11358 | Praxis: action, activity [PG] |
| 11359 | Prote ousia: primary being [PG] |
| 11360 | Psuche: mind, soul, life [PG] |
| 11361 | Sophia: wisdom [PG] |
| 11362 | Sophrosune: moderation, self-control [PG] |
| 11363 | Stoicheia: elements [PG] |
| 11364 | Sullogismos: deduction, syllogism [PG] |
| 11365 | Techne: skill, practical knowledge [PG] |
| 11366 | Telos: purpose, end [PG] |
| 11367 | Theoria: contemplation [PG] |
| 11368 | Theos: god [PG] |
| 11369 | Ti esti: what-something-is, essence [PG] |
| 11370 | Timoria: vengeance, punishment [PG] |
| 11371 | To ti en einai: essence, what-it-is-to-be [PG] |
| 11372 | To ti estin: essence [PG] |
| 11373 | Tode ti: this-such, subject of predication [cf. hupokeimenon] [PG] |
| 11461 | 323 (roughly): Euclid wrote 'Elements', summarising all of geometry [PG] |
| 11390 | 1000 (roughly): Upanishads written (in Sanskrit); religious and philosophical texts [PG] |
| 11391 | 750 (roughly): the Book of Genesis written by Hebrew writers [PG] |
| 11392 | 586: eclipse of the sun on the coast of modern Turkey was predicted by Thales of Miletus [PG] |
| 11395 | 570: Anaximander flourished in Miletus [PG] |
| 11396 | 563: the Buddha born in northern India [PG] |
| 11398 | 540: Lao Tzu wrote 'Tao Te Ching', the basis of Taoism [PG] |
| 11400 | 529: Pythagoras created his secretive community at Croton in Sicily [PG] |
| 11403 | 500: Heraclitus flourishes at Ephesus, in modern Turkey [PG] |
| 11404 | 496: Confucius travels widely, persuading rulers to be more moral [PG] |
| 11408 | 472: Empedocles persuades his city (Acragas in Sicily) to become a democracy [PG] |
| 11412 | 450 (roughly): Parmenides and Zeno visit Athens from Italy [PG] |
| 11414 | 445: Protagoras helps write laws for the new colony of Thurii [PG] |
| 11417 | 436 (roughly): Anaxagoras is tried for impiety, and expelled from Athens [PG] |
| 11535 | 170 (roughly): Marcus Aurelius wrote his private stoic meditations [PG] |
| 11537 | -200 (roughly): Sextus Empiricus wrote a series of books on scepticism [PG] |
| 11541 | 263: Porphyry began to study with Plotinus in Rome [PG] |
| 11545 | 310: Christianity became the official religion of the Roman empire [PG] |
| 11549 | 387: Ambrose converts Augustine to Christianity [PG] |
| 11555 | 523: Boethius imprisoned at Pavia, and begins to write [PG] |
| 11557 | 529: the emperor Justinian closes all the philosophy schools in Athens [PG] |
| 11421 | 427: Gorgias visited Athens as ambassador for Leontini [PG] |
| 11425 | 399: Socrates executed (with Plato absent through ill health) [PG] |
| 11432 | 387 (roughly): Plato returned to Athens, and founded the Academy [PG] |
| 11433 | 387 (roughly): Aristippus the Elder founder a hedonist school at Cyrene [PG] |
| 11440 | 367: the teenaged Aristotle came to study at the Academy [PG] |
| 11443 | 360 (roughly): Diogenes of Sinope lives in a barrel in central Athens [PG] |
| 11445 | 347: death of Plato [PG] |
| 11454 | 343: Aristotle becomes tutor to 13 year old Alexander (the Great) [PG] |
| 11456 | 335: Arisotle founded his school at the Lyceum in Athens [PG] |
| 11459 | 330 (roughly): Chuang Tzu wrote his Taoist book [PG] |
| 11465 | 322: Aristotle retired to Chalcis, and died there [PG] |
| 11468 | 307 (roughly): Epicurus founded his school at the Garden in Athens [PG] |
| 11470 | 301 (roughly): Zeno of Citium founded Stoicism at the Stoa Poikile in Athens [PG] |
| 11483 | 261: Cleanthes replaced Zeno as head of the Stoa [PG] |
| 11486 | 229 (roughly): Chrysippus replaced Cleanthes has head of the Stoa [PG] |
| 11492 | 157 (roughly): Carneades became head of the Academy [PG] |
| 11509 | 85: most philosophical activity moves to Alexandria [PG] |
| 11513 | 78: Cicero visited the stoic school on Rhodes [PG] |
| 11516 | 60 (roughly): Lucretius wrote his Latin poem on epicureanism [PG] |
| 11528 | 65: Seneca forced to commit suicide by Nero [PG] |
| 11531 | 80: the discourses of the stoic Epictetus are written down [PG] |
| 11564 | 1090: Anselm publishes his proof of the existence of God [PG] |
| 11558 | 622 (roughly): Mohammed writes the Koran [PG] |
| 11559 | 642: Arabs close the philosophy schools in Alexandria [PG] |
| 11560 | 910 (roughly): Al-Farabi wrote Arabic commentaries on Aristotle [PG] |
| 11562 | 1015 (roughly): Ibn Sina (Avicenna) writes a book on Aristotle [PG] |
| 11566 | 1115: Abelard is the chief logic teacher in Paris [PG] |
| 11573 | 1166: Ibn Rushd (Averroes) wrote extensive commentaries on Aristotle [PG] |
| 11581 | 1266: Aquinas began writing 'Summa Theologica' [PG] |
| 11586 | 1280: after his death, the teaching of Aquinas becomes official Dominican doctrine [PG] |
| 11591 | 1328: William of Ockham decides the Pope is a heretic, and moves to Munich [PG] |
| 17916 | 1347: the Church persecutes philosophical heresies [PG] |
| 11593 | 1470: Marsilio Ficino founds a Platonic Academy in Florence [PG] |
| 11596 | 1513: Machiavelli wrote 'The Prince' [PG] |
| 11599 | 1543: Copernicus publishes his heliocentric view of the solar system [PG] |
| 11601 | 1580: Montaigne publishes his essays [PG] |
| 11607 | 1600: Giordano Bruno was burned at the stake in Rome [PG] |
| 11613 | 1619: Descartes's famous day of meditation inside a stove [PG] |
| 11614 | 1620: Bacon publishes 'Novum Organum' [PG] |
| 11619 | 1633: Galileo convicted of heresy by the Inquisition [PG] |
| 11623 | 1641: Descartes publishes his 'Meditations' [PG] |
| 11626 | 1650: death of Descartes, in Stockholm [PG] |
| 11627 | 1651: Hobbes publishes 'Leviathan' [PG] |
| 11633 | 1662: the Port Royal Logic is published [PG] |
| 11634 | 1665: Spinoza writes his 'Ethics' [PG] |
| 11643 | 1676: Leibniz settled as librarian to the Duke of Brunswick [PG] |
| 11649 | 1687: Newton publishes his 'Principia Mathematica' [PG] |
| 11652 | 1690: Locke publishes his 'Essay' [PG] |
| 11654 | 1697: Bayle publishes his 'Dictionary' [PG] |
| 11659 | 1713: Berkeley publishes his 'Three Dialogues' [PG] |
| 11666 | 1734: Voltaire publishes his 'Philosophical Letters' [PG] |
| 11667 | 1739: Hume publishes his 'Treatise' [PG] |
| 11675 | 1762: Rousseau publishes his 'Social Contract' [PG] |
| 11682 | 1781: Kant publishes his 'Critique of Pure Reason' [PG] |
| 11683 | 1785: Reid publishes his essays defending common sense [PG] |
| 11687 | 1798: the French Revolution [PG] |
| 11694 | 1807: Hegel publishes his 'Phenomenology of Spirit' [PG] |
| 11701 | 1818: Schopenhauer publishes his 'World as Will and Idea' [PG] |
| 11710 | 1840: Kierkegaard is writing extensively in Copenhagen [PG] |
| 11713 | 1843: Mill publishes his 'System of Logic' [PG] |
| 11715 | 1848: Marx and Engels publis the Communist Manifesto [PG] |
| 11717 | 1859: Darwin publishes his 'Origin of the Species' [PG] |
| 11721 | 1861: Mill publishes 'Utilitarianism' [PG] |
| 11724 | 1867: Marx begins publishing 'Das Kapital' [PG] |
| 11733 | 1879: Peirce taught for five years at Johns Hopkins University [PG] |
| 17907 | 1879: Frege invents predicate logic [PG] |
| 17909 | 1892: Frege's essay 'Sense and Reference' [PG] |
| 17908 | 1884: Frege publishes his 'Foundations of Arithmetic' [PG] |
| 11735 | 1885: Nietzsche completed 'Thus Spake Zarathustra' [PG] |
| 17911 | 1888: Dedekind publishes axioms for arithmetic [PG] |
| 11740 | 1890: James published 'Principles of Psychology' [PG] |
| 11742 | 1895 (roughly): Freud developed theories of the unconscious [PG] |
| 11745 | 1900: Husserl began developing Phenomenology [PG] |
| 17917 | 1953: Wittgenstein's 'Philosophical Investigations' [PG] |
| 17919 | 1956: Place proposed mind-brain identity [PG] |
| 11804 | 1962: Kuhn's 'Structure of Scientific Revolutions' [PG] |
| 17921 | 1967: Putnam proposed functionalism of the mind [PG] |
| 11808 | 1971: Rawls's 'A Theory of Justice' [PG] |
| 11810 | 1972: Kripke publishes 'Naming and Necessity' [PG] |
| 11813 | 1975: Singer publishes 'Animal Rights' [PG] |
| 17920 | 1975: Putnam published his Twin Earth example [PG] |
| 11820 | 1986: David Lewis publishes 'On the Plurality of Worlds' [PG] |
| 11746 | 1903: Moore published 'Principia Ethica' [PG] |
| 11747 | 1904: Dewey became professor at Columbia University [PG] |
| 17910 | 1908: Zermelo publishes axioms for set theory [PG] |
| 11752 | 1910: Russell and Whitehead begin publishing 'Principia Mathematica' [PG] |
| 11756 | 1912: Russell meets Wittgenstein in Cambridge [PG] |
| 11762 | 1921: Wittgenstein's 'Tractatus' published [PG] |
| 11765 | 1927: Heidegger's 'Being and Time' published [PG] |
| 11768 | 1930: Frank Ramsey dies at 27 [PG] |
| 11770 | 1931: Gödel's Incompleteness Theorems [PG] |
| 11773 | 1933: Tarski's theory of truth [PG] |
| 11783 | 1942: Camus published 'The Myth of Sisyphus' [PG] |
| 11784 | 1943: Sartre's 'Being and Nothingness' [PG] |
| 11787 | 1945: Merleau-Ponty's 'Phenomenology of Perception' [PG] |
| 17918 | 1947: Carnap published 'Meaning and Necessity' [PG] |
| 11794 | 1950: Quine's essay 'Two Dogmas of Empiricism' [PG] |
| 2666 | Carneades' pinnacles of philosophy are the basis of knowledge (the criterion of truth) and the end of appetite (good) [Carneades, by Cicero] |
| 4465 | Note that "is" can assert existence, or predication, or identity, or classification [PG] |
| 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] |
| 4686 | Fallacies are errors in reasoning, 'formal' if a clear rule is breached, and 'informal' if more general [PG] |
| 7415 | Question-begging assumes the proposition which is being challenged [PG] |
| 7414 | What is true of a set is also true of its members [PG] |
| 6696 | The Ad Hominem Fallacy criticises the speaker rather than the argument [PG] |
| 21390 | Future events are true if one day we will say 'this event is happening now' [Carneades] |
| 21672 | We say future things are true that will possess actuality at some following time [Carneades, by Cicero] |
| 13634 | Satisfaction is 'truth in a model', which is a model of 'truth' [Shapiro] |
| 4687 | Minimal theories of truth avoid ontological commitment to such things as 'facts' or 'reality' [PG] |
| 13643 | Aristotelian logic is complete [Shapiro] |
| 10206 | Modal operators are usually treated as quantifiers [Shapiro] |
| 13651 | A set is 'transitive' if contains every member of each of its members [Shapiro] |
| 13647 | Choice is essential for proving downward Löwenheim-Skolem [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] |
| 10301 | The axiom of choice is controversial, but it could be replaced [Shapiro] |
| 13631 | Are sets part of logic, or part of mathematics? [Shapiro] |
| 13640 | Russell's paradox shows that there are classes which are not iterative sets [Shapiro] |
| 13654 | It is central to the iterative conception that membership is well-founded, with no infinite descending chains [Shapiro] |
| 13666 | Iterative sets are not Boolean; the complement of an iterative set is not an iterative sets [Shapiro] |
| 13653 | 'Well-ordering' of a set is an irreflexive, transitive, and binary relation with a least element [Shapiro] |
| 10207 | Anti-realists reject set theory [Shapiro] |
| 13642 | Logic is the ideal for learning new propositions on the basis of others [Shapiro] |
| 13627 | There is no 'correct' logic for natural languages [Shapiro] |
| 13667 | Skolem and Gödel championed first-order, and Zermelo, Hilbert, and Bernays championed higher-order [Shapiro] |
| 13668 | Bernays (1918) formulated and proved the completeness of propositional logic [Shapiro] |
| 13669 | Can one develop set theory first, then derive numbers, or are numbers more basic? [Shapiro] |
| 13662 | First-order logic was an afterthought in the development of modern logic [Shapiro] |
| 13624 | The 'triumph' of first-order logic may be related to logicism and the Hilbert programme, which failed [Shapiro] |
| 13660 | Maybe compactness, semantic effectiveness, and the Löwenheim-Skolem properties are desirable [Shapiro] |
| 13673 | The notion of finitude is actually built into first-order languages [Shapiro] |
| 10588 | First-order logic is Complete, and Compact, with the Löwenheim-Skolem Theorems [Shapiro] |
| 15944 | Second-order logic is better than set theory, since it only adds relations and operations, and nothing else [Shapiro, by Lavine] |
| 13629 | Broad standard semantics, or Henkin semantics with a subclass, or many-sorted first-order semantics? [Shapiro] |
| 13650 | Henkin semantics has separate variables ranging over the relations and over the functions [Shapiro] |
| 10298 | Some say that second-order logic is mathematics, not logic [Shapiro] |
| 13645 | In standard semantics for second-order logic, a single domain fixes the ranges for the variables [Shapiro] |
| 13649 | Completeness, Compactness and Löwenheim-Skolem fail in second-order standard semantics [Shapiro] |
| 10299 | If the aim of logic is to codify inferences, second-order logic is useless [Shapiro] |
| 10300 | Logical consequence can be defined in terms of the logical terminology [Shapiro] |
| 10259 | The two standard explanations of consequence are semantic (in models) and deductive [Shapiro] |
| 13637 | If a logic is incomplete, its semantic consequence relation is not effective [Shapiro] |
| 13626 | Semantic consequence is ineffective in second-order logic [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] |
| 13632 | Finding the logical form of a sentence is difficult, and there are no criteria of correctness [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] |
| 13674 | We might reduce ontology by using truth of sentences and terms, instead of using objects satisfying models [Shapiro] |
| 10290 | Second-order variables also range over properties, sets, relations or functions [Shapiro] |
| 10268 | Maybe plural quantifiers should be understood in terms of classes or sets [Shapiro] |
| 13633 | 'Satisfaction' is a function from models, assignments, and formulas to {true,false} [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] |
| 13644 | Semantics for models uses set-theory [Shapiro] |
| 10214 | Theory ontology is never complete, but is only determined 'up to isomorphism' [Shapiro] |
| 13636 | An axiomatization is 'categorical' if its models are isomorphic, so there is really only one interpretation [Shapiro] |
| 13670 | Categoricity can't be reached in a first-order language [Shapiro] |
| 10238 | The set-theoretical hierarchy contains as many isomorphism types as possible [Shapiro] |
| 10296 | The Löwenheim-Skolem Theorems fail for second-order languages with standard semantics [Shapiro] |
| 13648 | The Löwenheim-Skolem theorems show an explosion of infinite models, so 1st-order is useless for infinity [Shapiro] |
| 13675 | Substitutional semantics only has countably many terms, so Upward Löwenheim-Skolem trivially fails [Shapiro] |
| 10234 | Any theory with an infinite model has a model of every infinite cardinality [Shapiro] |
| 10590 | Up Löwenheim-Skolem: if natural numbers satisfy wffs, then an infinite domain satisfies them [Shapiro] |
| 10292 | Downward Löwenheim-Skolem: if there's an infinite model, there is a countable model [Shapiro] |
| 13658 | Downward Löwenheim-Skolem: each satisfiable countable set always has countable models [Shapiro] |
| 13659 | Upward Löwenheim-Skolem: each infinite model has infinite models of all sizes [Shapiro] |
| 10297 | The Löwenheim-Skolem theorem seems to be a defect of first-order logic [Shapiro] |
| 13635 | 'Weakly sound' if every theorem is a logical truth; 'sound' if every deduction is a semantic consequence [Shapiro] |
| 13628 | We can live well without completeness in logic [Shapiro] |
| 13630 | Non-compactness is a strength of second-order logic, enabling characterisation of infinite structures [Shapiro] |
| 13646 | Compactness is derived from soundness and completeness [Shapiro] |
| 13661 | A language is 'semantically effective' if its logical truths are recursively enumerable [Shapiro] |
| 6516 | Monty Hall Dilemma: do you abandon your preference after Monty eliminates one of the rivals? [PG] |
| 10201 | Virtually all of mathematics can be modeled in set theory [Shapiro] |
| 13641 | Complex numbers can be defined as reals, which are defined as rationals, then integers, then naturals [Shapiro] |
| 8763 | The number 3 is presumably identical as a natural, an integer, a rational, a real, and complex [Shapiro] |
| 13676 | Only higher-order languages can specify that 0,1,2,... are all the natural numbers that there are [Shapiro] |
| 13677 | Natural numbers are the finite ordinals, and integers are equivalence classes of pairs of finite ordinals [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] |
| 13652 | The 'continuum' is the cardinality of the powerset of a denumerably infinite set [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] |
| 13657 | First-order arithmetic can't even represent basic number theory [Shapiro] |
| 10202 | Natural numbers just need an initial object, successors, and an induction principle [Shapiro] |
| 10294 | Second-order logic has the expressive power for mathematics, but an unworkable model theory [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] |
| 13656 | Some sets of natural numbers are definable in set-theory but not in 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] |
| 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] |
| 13664 | Logicism is distinctive in seeking a universal language, and denying that logic is a series of abstractions [Shapiro] |
| 13625 | Mathematics and logic have no border, and logic must involve mathematics and its ontology [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] |
| 13663 | Some reject formal properties if they are not defined, or defined impredicatively [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] |
| 13638 | Properties are often seen as intensional; equiangular and equilateral are different, despite identity of objects [Shapiro] |
| 10591 | Logicians use 'property' and 'set' interchangeably, with little hanging on it [Shapiro] |
| 10272 | The notion of 'object' is at least partially structural and mathematical [Shapiro] |
| 10275 | A blurry border is still a border [Shapiro] |
| 15825 | Carneades denied the transitivity of identity [Carneades, by Chisholm] |
| 21389 | Carneades distinguished logical from causal necessity, when talking of future events [Long on Carneades] |
| 10258 | Logical modalities may be acceptable, because they are reducible to satisfaction in models [Shapiro] |
| 24054 | Everything has a probability, something will happen, and probabilities add up [PG] |
| 10266 | Why does the 'myth' of possible worlds produce correct modal logic? [Shapiro] |
| 3875 | If reality is just what we perceive, we would have no need for a sixth sense [PG] |
| 3876 | If my team is losing 3-1, I have synthetic a priori knowledge that they need two goals for a draw [PG] |
| 8725 | Rationalism tries to apply mathematical methodology to all of knowledge [Shapiro] |
| 10203 | We apprehend small, finite mathematical structures by abstraction from patterns [Shapiro] |
| 21671 | Voluntary motion is intrinsically within our power, and this power is its cause [Carneades, by Cicero] |
| 21391 | Some actions are within our power; determinism needs prior causes for everything - so it is false [Carneades, by Cicero] |
| 21674 | Even Apollo can only foretell the future when it is naturally necessary [Carneades, by Cicero] |
| 7734 | Maybe a mollusc's brain events for pain ARE of the same type (broadly) as a human's [PG] |
| 7735 | Maybe a frog's brain events for fear are functionally like ours, but not phenomenally [PG] |
| 10229 | Simple types can be apprehended through their tokens, via abstraction [Shapiro] |
| 9626 | A structure is an abstraction, focussing on relationships, and ignoring other features [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] |
| 7398 | Carneades said that after a shipwreck a wise man would seize the only plank by force [Carneades, by Tuck] |
| 3877 | Utilitarianism seems to justify the discreet murder of unhappy people [PG] |
| 21392 | People change laws for advantage; either there is no justice, or it is a form of self-injury [Carneades, by Lactantius] |
| 6126 | Life is Movement, Respiration, Sensation, Nutrition, Excretion, Reproduction, Growth (MRS NERG) [PG] |
| 3874 | How could God know there wasn't an unknown force controlling his 'free' will? [PG] |
| 3873 | An omniscient being couldn't know it was omniscient, as that requires information from beyond its scope of knowledge [PG] |