88 ideas
| 19073 | True philosophy aims at absolute unity, while our understanding sees only separation [Hegel] |
| 15624 | Free thinking has no presuppositions [Hegel] |
| 15631 | The ideal of reason is the unification of abstract identity (or 'concept') and being [Hegel] |
| 15612 | Older metaphysics naively assumed that thought grasped things in themselves [Hegel] |
| 21768 | Logic is metaphysics, the science of things grasped in thoughts [Hegel] |
| 21984 | We must break up the rigidity that our understanding has imposed [Hegel] |
| 22081 | Let thought follow its own course, and don't interfere [Hegel] |
| 15626 | Categories create objective experience, but are too conditioned by things to actually grasp them [Hegel] |
| 15616 | If truth is just non-contradiction, we must take care that our basic concepts aren't contradictory [Hegel] |
| 15615 | Older metaphysics became dogmatic, by assuming opposed assertions must be true and false [Hegel] |
| 21767 | Dialectic is seen in popular proverbs like 'pride comes before a fall' [Hegel] |
| 15638 | Dialectic is the moving soul of scientific progression, the principle which binds science together [Hegel] |
| 15639 | Socratic dialectic is subjective, but Plato made it freely scientific and objective [Hegel] |
| 8721 | An 'impredicative' definition seems circular, because it uses the term being defined [Friend] |
| 8680 | Classical definitions attempt to refer, but intuitionist/constructivist definitions actually create objects [Friend] |
| 3678 | Reductio ad absurdum proves an idea by showing that its denial produces contradiction [Friend] |
| 19070 | Superficial truth is knowing how something is, which is consciousness of bare correctness [Hegel] |
| 5644 | In Hegel's logic it is concepts (rather than judgements or propositions) which are true or false [Hegel, by Scruton] |
| 19072 | In the deeper sense of truth, to be untrue resembles being bad; badness is untrue to a thing's nature [Hegel] |
| 8705 | Anti-realists see truth as our servant, and epistemically contrained [Friend] |
| 19071 | The deeper sense of truth is a thing matching the idea of what it ought to be [Hegel] |
| 8713 | In classical/realist logic the connectives are defined by truth-tables [Friend] |
| 8708 | Double negation elimination is not valid in intuitionist logic [Friend] |
| 8694 | Free logic was developed for fictional or non-existent objects [Friend] |
| 8665 | A 'proper subset' of A contains only members of A, but not all of them [Friend] |
| 8672 | A 'powerset' is all the subsets of a set [Friend] |
| 8677 | Set theory makes a minimum ontological claim, that the empty set exists [Friend] |
| 8666 | Infinite sets correspond one-to-one with a subset [Friend] |
| 8682 | Major set theories differ in their axioms, and also over the additional axioms of choice and infinity [Friend] |
| 21595 | Excluded middle is the maxim of definite understanding, but just produces contradictions [Hegel] |
| 8709 | The law of excluded middle is syntactic; it just says A or not-A, not whether they are true or false [Friend] |
| 8711 | Intuitionists read the universal quantifier as "we have a procedure for checking every..." [Friend] |
| 15628 | The idea that contradiction is essential to rational understanding is a key modern idea [Hegel] |
| 15629 | Tenderness for the world solves the antinomies; contradiction is in our reason, not in the essence of the world [Hegel] |
| 15630 | Antinomies are not just in four objects, but in all objects, all representations, all objects and all ideas [Hegel] |
| 8675 | Paradoxes can be solved by talking more loosely of 'classes' instead of 'sets' [Friend] |
| 8674 | The Burali-Forti paradox asks whether the set of all ordinals is itself an ordinal [Friend] |
| 8667 | The 'integers' are the positive and negative natural numbers, plus zero [Friend] |
| 8668 | The 'rational' numbers are those representable as fractions [Friend] |
| 8670 | A number is 'irrational' if it cannot be represented as a fraction [Friend] |
| 8661 | The natural numbers are primitive, and the ordinals are up one level of abstraction [Friend] |
| 8664 | Cardinal numbers answer 'how many?', with the order being irrelevant [Friend] |
| 8671 | The 'real' numbers (rationals and irrationals combined) is the Continuum, which has no gaps [Friend] |
| 8663 | Raising omega to successive powers of omega reveal an infinity of infinities [Friend] |
| 8662 | The first limit ordinal is omega (greater, but without predecessor), and the second is twice-omega [Friend] |
| 8669 | Between any two rational numbers there is an infinite number of rational numbers [Friend] |
| 8676 | Is mathematics based on sets, types, categories, models or topology? [Friend] |
| 8678 | Most mathematical theories can be translated into the language of set theory [Friend] |
| 8701 | The number 8 in isolation from the other numbers is of no interest [Friend] |
| 8702 | In structuralism the number 8 is not quite the same in different structures, only equivalent [Friend] |
| 8699 | Are structures 'ante rem' (before reality), or are they 'in re' (grounded in physics)? [Friend] |
| 8696 | Structuralist says maths concerns concepts about base objects, not base objects themselves [Friend] |
| 8695 | Structuralism focuses on relations, predicates and functions, with objects being inessential [Friend] |
| 8700 | 'In re' structuralism says that the process of abstraction is pattern-spotting [Friend] |
| 8681 | The big problem for platonists is epistemic: how do we perceive, intuit, know or detect mathematical facts? [Friend] |
| 8712 | Mathematics should be treated as true whenever it is indispensable to our best physical theory [Friend] |
| 8716 | Formalism is unconstrained, so cannot indicate importance, or directions for research [Friend] |
| 8706 | Constructivism rejects too much mathematics [Friend] |
| 8707 | Intuitionists typically retain bivalence but reject the law of excluded middle [Friend] |
| 22078 | Even simple propositions about sensations are filled with categories [Hegel] |
| 15634 | Thought about particulars is done entirely through categories [Hegel] |
| 8704 | Structuralists call a mathematical 'object' simply a 'place in a structure' [Friend] |
| 21981 | The one substance is formless without the mediation of dialectical concepts [Hegel] |
| 15637 | Essence is the essential self-positing unity of immediacy and mediation [Hegel] |
| 15613 | Real cognition grasps a thing from within itself, and is not satisfied with mere predicates [Hegel] |
| 15636 | The Cogito is at the very centre of the entire concern of modern philosophy [Hegel] |
| 22300 | Existence is just a set of relationships [Hegel] |
| 15609 | The sensible is distinguished from thought by being about singular things [Hegel] |
| 15625 | Sense perception is secondary and dependent, while thought is independent and primitive [Hegel] |
| 15619 | Empiricism made particular knowledge possible, and blocked wild claims [Hegel] |
| 15620 | Empiricism contains the important idea that we should see knowledge for ourselves, and be part of it [Hegel] |
| 15622 | Empiricism unknowingly contains and uses a metaphysic, which underlies its categories [Hegel] |
| 15621 | Empiricism of the finite denies the supersensible, and can only think with formal abstraction [Hegel] |
| 15632 | The Humean view stops us thinking about perception, and finding universals and necessities in it [Hegel] |
| 15623 | Humean scepticism, unlike ancient Greek scepticism, accepts the truth of experience as basic [Hegel] |
| 15617 | In abstraction, beyond finitude, freedom and necessity must exist together [Hegel] |
| 8685 | Studying biology presumes the laws of chemistry, and it could never contradict them [Friend] |
| 15608 | The act of thinking is the bringing forth of universals [Hegel] |
| 21986 | Hegel's system has a vast number of basic concepts [Hegel, by Moore,AW] |
| 15607 | We don't think with concepts - we think the concepts [Hegel] |
| 15610 | Active thought about objects produces the universal, which is what is true and essential of it [Hegel] |
| 8688 | Concepts can be presented extensionally (as objects) or intensionally (as a characterization) [Friend] |
| 15614 | Old metaphysics tried to grasp eternal truths through causal events, which is impossible [Hegel] |
| 13440 | Causation is the power of one property to produce another, and this gives time its direction [Esfeld] |
| 15635 | The older conception of God was emptied of human features, to make it worthy of the Infinite [Hegel] |
| 21980 | God is the absolute thing, and also the absolute person [Hegel] |
| 15618 | If God is the abstract of Supremely Real Essence, then God is a mere Beyond, and unknowable [Hegel] |
| 15633 | We establish unification of the Ideal by the ontological proof, deriving being from abstraction of thinking [Hegel] |