119 ideas
| 10237 | Coherence is a primitive, intuitive notion, not reduced to something formal [Shapiro] |
| 20910 | Everything happens necessarily, and for a reason [Democritus] |
| 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] |
| 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] |
| 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] |
| 16146 | Two can't be a self-contained unit, because it would need to be one to do that [Democritus, by Aristotle] |
| 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] |
| 20901 | True Being only occurs when it is completely full, with atoms and no void [Democritus, by Aristotle] |
| 20902 | Being does not exist more than non-being [Democritus, by Aristotle] |
| 20904 | The non-existent exists as much as the existent, because it has causal powers [Democritus] |
| 20903 | The only distinctions are Configuration (shape), Disposition (order) and Turning (position) [Democritus, by Aristotle] |
| 20893 | Nothing comes from non-existence, or passes into it [Democritus, by Diog. Laertius] |
| 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] |
| 20896 | It is not possible to know what sort each thing is [Democritus] |
| 10272 | The notion of 'object' is at least partially structural and mathematical [Shapiro] |
| 3357 | Democritus denies reality to large objects, because atomic entities can't combine to produce new ones [Benardete,JA on Democritus] |
| 598 | Democritus said that substances could never be mixed, so atoms are the substances [Democritus, by Aristotle] |
| 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] |
| 1532 | Sensible qualities can't be real if they appear different to different creatures [Democritus, by Theophrastus] |
| 20894 | Man is separated from reality [Democritus] |
| 517 | All evidence comes from senses, so they are indispensable to the mind [Democritus] |
| 20897 | Obscure knowledge belongs to the five senses, and genuine knowledge is the other type [Democritus] |
| 577 | Democritus says there is either no truth, or it is concealed from us [Democritus, by Aristotle] |
| 20895 | We actually know nothing, and opinions are mere flux [Democritus] |
| 1528 | We in fact know nothing, but we each restructure our reality with beliefs [Democritus] |
| 492 | It is obviously impossible to understand the reality of each thing [Democritus] |
| 515 | We know nothing in reality; for truth lies in an abyss [Democritus] |
| 20892 | Democritus was devoted to discovering causal explanations [Democritus, by Eusebius] |
| 5882 | Democritus says soul consists of smooth round bodies brought together in accidental collision [Democritus, by Cicero] |
| 6034 | Atomists say soul has a rational part in the chest, and a diffused non-rational part [Democritus, by Aetius] |
| 20912 | The soul is the same as the mind [Democritus, by Aristotle] |
| 20916 | Animals have a share of reason [Democritus, by Porphyry] |
| 20914 | The directive centre is located in the whole head [Democritus, by Ps-Plutarch] |
| 10203 | We apprehend small, finite mathematical structures by abstraction from patterns [Shapiro] |
| 6033 | Democritus said everything happens of necessity, by natural motion of atoms [Democritus, by Cicero] |
| 5088 | Some say there is a determinate cause for every apparently spontaneous event [Democritus, by Aristotle] |
| 21670 | Democritus said atoms only move by their natural motions, which are therefore necessary [Democritus, by Cicero] |
| 24041 | Democritus says spherical atoms are fire, and constitute the soul [psuche] [Democritus, by Aristotle] |
| 20913 | Democritus says the soul is the body, and thinking is thus the mixture of the body [Democritus, by Theophrastus] |
| 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] |
| 1540 | Pleasure and pain guide our choices of good and bad [Democritus] |
| 468 | Musical performance can reveal a range of virtues [Damon of Ath.] |
| 495 | Wisdom creates a healthy passion-free soul [Democritus] |
| 1537 | Happiness is identifying and separating the pleasures [Democritus, by Stobaeus] |
| 20917 | Contentment comes from moderation and proportion in life [Democritus, by Stobaeus] |
| 13551 | Democritus says wealth is a burden to the virtuous mind [Democritus, by Seneca] |
| 20899 | Atoms cling together, until a stronger necessity disperses them [Democritus, by Aristotle] |
| 20898 | Atoms are irregular, hooked, concave, convex, and many other shapes [Democritus, by Aristotle] |
| 17542 | 'Full' and 'Void' secularised Parmenides's Being and Not-being [Democritus, by Heisenberg] |
| 1525 | Atomists say there are only three differences - in shape, arrangement and position [Democritus, by Aristotle] |
| 493 | Experiences are merely convention; only atoms and the void are real [Democritus] |
| 5947 | If only atoms are real and the rest is convention, we wouldn't bother to avoid pain [Democritus, by Diogenes of Oen.] |
| 13219 | When atoms touch, why don't they coalesce, like water drops? [Aristotle on Democritus] |
| 1533 | Because appearance is infinitely varied, atomists assume infinitely many shapes of atom [Democritus, by Aristotle] |
| 20908 | There could be an atom the size of the world [Democritus, by Ps-Plutarch] |
| 1527 | There must be atoms, to avoid the absurdity of infinite division down to nothing [Democritus, by Aristotle] |
| 20909 | The basic atoms are without qualities - which only arise from encounters between atoms [Democritus, by Galen] |
| 1536 | If a cone is horizontally sliced the surfaces can't be equal, so it goes up in steps [Democritus] |
| 23314 | Greeks explained regularity by intellectual design, not by laws [Democritus, by Frede,M] |
| 20905 | Growth and movement would not exist if there were no void to receive them [Democritus] |
| 24059 | Democritus is wrong: in a void we wouldn't see a distant ant in exact detail [Aristotle on Democritus] |
| 5101 | Movement is impossible in a void, because nothing can decide the direction of movement [Aristotle on Democritus] |
| 20911 | There are unlimited worlds of varying sizes, some without life or water [Democritus, by Hippolytus] |
| 1535 | Democritus said people imagined gods as the source of what awed or frightened them [Democritus, by Sext.Empiricus] |
| 20915 | The soul is destroyed with the body [Democritus, by Ps-Plutarch] |