47 ideas
354 | Wisdom makes virtue and true goodness possible [Plato] |
370 | Philosophy is a purification of the soul ready for the afterlife [Plato] |
350 | In investigation the body leads us astray, but the soul gets a clear view of the facts [Plato] |
362 | The greatest misfortune for a person is to develop a dislike for argument [Plato] |
17749 | Post proved the consistency of propositional logic in 1921 [Walicki] |
17765 | Propositional language can only relate statements as the same or as different [Walicki] |
17764 | Boolean connectives are interpreted as functions on the set {1,0} [Walicki] |
17752 | The empty set is useful for defining sets by properties, when the members are not yet known [Walicki] |
17753 | The empty set avoids having to take special precautions in case members vanish [Walicki] |
17610 | The Axiom of Choice paradoxically allows decomposing a sphere into two identical spheres [Maddy] |
17759 | Ordinals play the central role in set theory, providing the model of well-ordering [Walicki] |
17741 | To determine the patterns in logic, one must identify its 'building blocks' [Walicki] |
17620 | Critics of if-thenism say that not all starting points, even consistent ones, are worth studying [Maddy] |
17747 | A 'model' of a theory specifies interpreting a language in a domain to make all theorems true [Walicki] |
17748 | The L-S Theorem says no theory (even of reals) says more than a natural number theory [Walicki] |
17605 | Hilbert's geometry and Dedekind's real numbers were role models for axiomatization [Maddy] |
17625 | If two mathematical themes coincide, that suggest a single deep truth [Maddy] |
17761 | A compact axiomatisation makes it possible to understand a field as a whole [Walicki] |
17763 | Axiomatic systems are purely syntactic, and do not presuppose any interpretation [Walicki] |
17758 | Ordinals are transitive sets of transitive sets; or transitive sets totally ordered by inclusion [Walicki] |
17755 | Ordinals are the empty set, union with the singleton, and any arbitrary union of ordinals [Walicki] |
17756 | The union of finite ordinals is the first 'limit ordinal'; 2ω is the second... [Walicki] |
17760 | Two infinite ordinals can represent a single infinite cardinal [Walicki] |
17757 | Members of ordinals are ordinals, and also subsets of ordinals [Walicki] |
13155 | If you add one to one, which one becomes two, or do they both become two? [Plato] |
17615 | Every infinite set of reals is either countable or of the same size as the full set of reals [Maddy] |
17762 | In non-Euclidean geometry, all Euclidean theorems are valid that avoid the fifth postulate [Walicki] |
17754 | Inductive proof depends on the choice of the ordering [Walicki] |
17618 | Set-theory tracks the contours of mathematical depth and fruitfulness [Maddy] |
17614 | The connection of arithmetic to perception has been idealised away in modern infinitary mathematics [Maddy] |
21347 | If Simmias is taller than Socrates, that isn't a feature that is just in Simmias [Plato] |
360 | We must have a prior knowledge of equality, if we see 'equal' things and realise they fall short of it [Plato] |
1 | There is only one source for all beauty [Plato] |
368 | Other things are named after the Forms because they participate in them [Plato] |
16516 | The ship which Theseus took to Crete is now sent to Delos crowned with flowers [Plato] |
17742 | Scotus based modality on semantic consistency, instead of on what the future could allow [Walicki] |
357 | People are obviously recollecting when they react to a geometrical diagram [Plato] |
359 | If we feel the inadequacy of a resemblance, we must recollect the original [Plato] |
9343 | To achieve pure knowledge, we must get rid of the body and contemplate things with the soul [Plato] |
15859 | To investigate the causes of things, study what is best for them [Plato] |
13154 | Do we think and experience with blood, air or fire, or could it be our brain? [Plato] |
364 | One soul can't be more or less of a soul than another [Plato] |
361 | It is a mistake to think that the most violent pleasure or pain is therefore the truest reality [Plato] |
351 | War aims at the acquisition of wealth, because we are enslaved to the body [Plato] |
13156 | Fancy being unable to distinguish a cause from its necessary background conditions! [Plato] |
369 | If the Earth is spherical and in the centre, it is kept in place by universal symmetry, not by force [Plato] |
363 | Whether the soul pre-exists our body depends on whether it contains the ultimate standard of reality [Plato] |