69 ideas
| 2930 | The main aim of philosophy must be to determine the order of rank among values [Nietzsche] |
| 20143 | Scientific knowledge is nothing without a prior philosophical 'faith' [Nietzsche] |
| 23722 | Objectivity is not disinterestedness (impossible), but the ability to switch perspectives [Nietzsche] |
| 4417 | Only that which has no history is definable [Nietzsche] |
| 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] |
| 23715 | Psychologists should be brave and proud, and prefer truth to desires, even when it is ugly [Nietzsche] |
| 8705 | Anti-realists see truth as our servant, and epistemically contrained [Friend] |
| 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] |
| 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] |
| 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] |
| 8704 | Structuralists call a mathematical 'object' simply a 'place in a structure' [Friend] |
| 4421 | Philosophers have never asked why there is a will to truth in the first place [Nietzsche] |
| 23719 | Forgetfulness is a strong positive ability, not mental laziness [Nietzsche] |
| 4420 | There is only 'perspective' seeing and knowing, and so the best objectivity is multiple points of view [Nietzsche] |
| 4414 | Philosophers invented "free will" so that our virtues would be permanently interesting to the gods [Nietzsche] |
| 8685 | Studying biology presumes the laws of chemistry, and it could never contradict them [Friend] |
| 4419 | People who think in words are orators rather than thinkers, and think about facts instead of thinking facts [Nietzsche] |
| 8688 | Concepts can be presented extensionally (as objects) or intensionally (as a characterization) [Friend] |
| 4411 | It is a delusion to separate the man from the deed, like the flash from the lightning [Nietzsche] |
| 8113 | Art is like understanding a natural language, and needs a grasp of a symbol system [Goodman, by Gardner] |
| 3793 | We must question the very value of moral values [Nietzsche] |
| 4408 | The concept of 'good' was created by aristocrats to describe their own actions [Nietzsche] |
| 23716 | A strong rounded person soon forgets enemies, misfortunes, and even misdeeds [Nietzsche] |
| 20129 | All animals strive for the ideal conditions to express their power, and hate any hindrances [Nietzsche] |
| 4409 | Only the decline of aristocratic morality led to concerns about "egoism" [Nietzsche] |
| 3259 | Nietzsche rejects impersonal morality, and asserts the idea of living well [Nietzsche, by Nagel] |
| 4416 | Basic justice is the negotiation of agreement among equals, and the imposition of agreement [Nietzsche] |
| 4418 | A masterful and violent person need have nothing to do with contracts [Nietzsche] |
| 4407 | Plato, Spinoza and Kant are very different, but united in their low estimation of pity [Nietzsche] |
| 4415 | Guilt and obligation originated in the relationship of buying and selling, credit and debt [Nietzsche] |
| 23718 | If we say birds of prey could become lambs, that makes them responsible for being birds of prey [Nietzsche] |
| 23717 | Modern nihilism is now feeling tired of mankind [Nietzsche] |
| 23721 | Old tribes always felt an obligation to the earlier generations, and the founders [Nietzsche] |
| 20142 | The state begins with brutal conquest of a disorganised people, not with a 'contract' [Nietzsche] |
| 23720 | Punishment makes people harder, more alienated, and hostile [Nietzsche] |
| 4410 | The truly great haters in world history have always been priests [Nietzsche] |