61 ideas
| 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] |
| 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] |
| 4913 | Brain lesions can erase whole categories of perception, suggesting they are hard-wired [Carter,R] |
| 8704 | Structuralists call a mathematical 'object' simply a 'place in a structure' [Friend] |
| 4910 | Sense organs don't discriminate; they reduce various inputs to the same electrical pulses [Carter,R] |
| 4911 | The recognition sequence is: classify, name, locate, associate, feel [Carter,R, by PG] |
| 4919 | There seems to be no dividing line between a memory and a thought [Carter,R] |
| 4908 | No one knows if animals are conscious [Carter,R] |
| 4902 | Pain doesn't have one brain location, but is linked to attention and emotion [Carter,R] |
| 4904 | Proper brains appear at seven weeks, and neonates have as many neurons as adults do [Carter,R] |
| 4915 | In primates, brain size correlates closely with size of social group [Carter,R] |
| 4917 | Consciousness involves awareness, perception, self-awareness, attention and reflection [Carter,R] |
| 4916 | There is enormous evidence that consciousness arises in the frontal lobes of the brain [Carter,R] |
| 4905 | Normal babies seem to have overlapping sense experiences [Carter,R] |
| 4918 | In blindsight V1 (normal vision) is inactive, but V5 (movement) lights up [Carter,R] |
| 4912 | Out-of-body experiences may be due to temporary loss of proprioception [Carter,R] |
| 4903 | Scans of brains doing similar tasks produce very similar patterns of activation [Carter,R] |
| 8685 | Studying biology presumes the laws of chemistry, and it could never contradict them [Friend] |
| 4920 | Thinking takes place on the upper side of the prefrontal cortex [Carter,R] |
| 4906 | Babies show highly emotional brain events, but may well be unaware of them [Carter,R] |
| 4909 | The only way we can control our emotions is by manipulating the outside world that influences them [Carter,R] |
| 4914 | A frog will starve to death surrounded by dead flies [Carter,R] |
| 8688 | Concepts can be presented extensionally (as objects) or intensionally (as a characterization) [Friend] |
| 4907 | The 'locus coeruleus' is one of several candidates for the brain's 'pleasure centre' [Carter,R] |
| 3031 | The greatest good is not the achievement of desire, but to desire what is proper [Menedemus, by Diog. Laertius] |