81 ideas
23890 | For Plato true wisdom is supernatural [Plato, by Weil] |
3060 | Plato never mentions Democritus, and wished to burn his books [Plato, by Diog. Laertius] |
23891 | Two contradictories force us to find a relation which will correlate them [Plato, by Weil] |
13520 | A 'tautology' must include connectives [Wolf,RS] |
13524 | Deduction Theorem: T∪{P}|-Q, then T|-(P→Q), which justifies Conditional Proof [Wolf,RS] |
13522 | Universal Generalization: If we prove P(x) with no special assumptions, we can conclude ∀xP(x) [Wolf,RS] |
13521 | Universal Specification: ∀xP(x) implies P(t). True for all? Then true for an instance [Wolf,RS] |
13523 | Existential Generalization (or 'proof by example'): if we can say P(t), then we can say something is P [Wolf,RS] |
13529 | Empty Set: ∃x∀y ¬(y∈x). The unique empty set exists [Wolf,RS] |
13526 | Comprehension Axiom: if a collection is clearly specified, it is a set [Wolf,RS] |
13534 | In first-order logic syntactic and semantic consequence (|- and |=) nicely coincide [Wolf,RS] |
13535 | First-order logic is weakly complete (valid sentences are provable); we can't prove every sentence or its negation [Wolf,RS] |
13531 | Model theory reveals the structures of mathematics [Wolf,RS] |
13532 | Model theory 'structures' have a 'universe', some 'relations', some 'functions', and some 'constants' [Wolf,RS] |
13519 | Model theory uses sets to show that mathematical deduction fits mathematical truth [Wolf,RS] |
13533 | First-order model theory rests on completeness, compactness, and the Löwenheim-Skolem-Tarski theorem [Wolf,RS] |
13537 | An 'isomorphism' is a bijection that preserves all structural components [Wolf,RS] |
13539 | The LST Theorem is a serious limitation of first-order logic [Wolf,RS] |
13538 | If a theory is complete, only a more powerful language can strengthen it [Wolf,RS] |
13525 | Most deductive logic (unlike ordinary reasoning) is 'monotonic' - we don't retract after new givens [Wolf,RS] |
13530 | An ordinal is an equivalence class of well-orderings, or a transitive set whose members are transitive [Wolf,RS] |
13518 | Modern mathematics has unified all of its objects within set theory [Wolf,RS] |
15682 | Even fairly simple animals make judgements based on categories [Gelman] |
15691 | Children accept real stable categories, with nonobvious potential that gives causal explanations [Gelman] |
14502 | Plato's idea of 'structure' tends to be mathematically expressed [Plato, by Koslicki] |
20906 | Platonists argue for the indivisible triangle-in-itself [Plato, by Aristotle] |
3039 | When Diogenes said he could only see objects but not their forms, Plato said it was because he had eyes but no intellect [Plato, by Diog. Laertius] |
17948 | Plato's Forms meant that the sophists only taught the appearance of wisdom and virtue [Plato, by Nehamas] |
556 | If there is one Form for both the Form and its participants, they must have something in common [Aristotle on Plato] |
563 | If gods are like men, they are just eternal men; similarly, Forms must differ from particulars [Aristotle on Plato] |
565 | The Forms cannot be changeless if they are in changing things [Aristotle on Plato] |
557 | A Form is a cause of things only in the way that white mixed with white is a cause [Aristotle on Plato] |
9607 | The greatest discovery in human thought is Plato's discovery of abstract objects [Brown,JR on Plato] |
13263 | We can grasp whole things in science, because they have a mathematics and a teleology [Plato, by Koslicki] |
13261 | Plato sees an object's structure as expressible in mathematics [Plato, by Koslicki] |
13265 | Plato was less concerned than Aristotle with the source of unity in a complex object [Plato, by Koslicki] |
593 | Plato's holds that there are three substances: Forms, mathematical entities, and perceptible bodies [Plato, by Aristotle] |
13260 | Plato says wholes are either containers, or they're atomic, or they don't exist [Plato, by Koslicki] |
15700 | In India, upper-castes essentialize caste more than lower-castes do [Gelman] |
15685 | Essentialism is either natural to us, or an accident of our culture, or a necessary result of language [Gelman] |
15684 | Children's concepts include nonobvious features, like internal parts, functions and causes [Gelman] |
15681 | Essentialism: real or representational? sortal, causal or ideal? real particulars, or placeholders? [Gelman] |
11237 | Only universals have essence [Plato, by Politis] |
15678 | Essentialism says categories have a true hidden nature which gives an object its identity [Gelman] |
15683 | Sortals are needed for determining essence - the thing must be categorised first [Gelman] |
15697 | Kind (unlike individual) essentialism assumes preexisting natural categories [Gelman] |
11238 | Plato and Aristotle take essence to make a thing what it is [Plato, by Politis] |
15687 | Kinship is essence that comes in degrees, and age groups are essences that change over time [Gelman] |
15679 | Essentialism comes from the cognitive need to categorise [Gelman] |
15698 | We found no evidence that mothers teach essentialism to their children [Gelman] |
15709 | Essentialism is useful for predictions, but it is not the actual structure of reality [Gelman] |
15696 | Peope favor historical paths over outward properties when determining what something is [Gelman] |
15707 | There is intentional, mechanical, teleological, essentialist, vitalist and deontological understanding [Gelman] |
15703 | Memories often conform to a theory, rather than being neutral [Gelman] |
15708 | Inductive success is rewarded with more induction [Gelman] |
15694 | Children overestimate the power of a single example [Gelman] |
15695 | Children make errors in induction by focusing too much on categories [Gelman] |
15692 | People tend to be satisfied with shallow explanations [Gelman] |
17085 | A good explanation totally rules out the opposite explanation (so Forms are required) [Plato, by Ruben] |
1651 | Plato wanted to somehow control and purify the passions [Vlastos on Plato] |
15680 | Folk essentialism rests on belief in natural kinds, in hidden properties, and on words indicating structures [Gelman] |
15686 | Labels may indicate categories which embody an essence [Gelman] |
15690 | Causal properties are seen as more central to category concepts [Gelman] |
15688 | Categories are characterized by distance from a prototype [Gelman] |
15689 | Theory-based concepts use rich models to show which similarities really matter [Gelman] |
15699 | Prelinguistic infants acquire and use many categories [Gelman] |
3324 | Plato's whole philosophy may be based on being duped by reification - a figure of speech [Benardete,JA on Plato] |
7503 | Plato never refers to examining the conscience [Plato, by Foucault] |
2173 | As religion and convention collapsed, Plato sought morals not just in knowledge, but in the soul [Williams,B on Plato] |
9274 | Plato's legacy to European thought was the Good, the Beautiful and the True [Plato, by Gray] |
94 | Pleasure is better with the addition of intelligence, so pleasure is not the good [Plato, by Aristotle] |
17947 | Plato decided that the virtuous and happy life was the philosophical life [Plato, by Nehamas] |
6015 | Plato, unusually, said that theoretical and practical wisdom are inseparable [Plato, by Kraut] |
2912 | Plato is boring [Nietzsche on Plato] |
15693 | One sample of gold is enough, but one tree doesn't give the height of trees [Gelman] |
15701 | Nouns seem to invoke stable kinds more than predicates do [Gelman] |
15705 | Essentialism encourages us to think about the world scientifically [Gelman] |
15702 | Essentialism doesn't mean we know the essences [Gelman] |
15704 | Essentialism starts from richly structured categories, leading to a search for underlying properties [Gelman] |
15706 | A major objection to real essences is the essentialising of social categories like race, caste and occupation [Gelman] |
1526 | Almost everyone except Plato thinks that time could not have been generated [Plato, by Aristotle] |