102 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] |
3137 | Varieties of singular terms are used to designate token particulars [Rey] |
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] |
3143 | Physics requires the existence of properties, and also the abstract objects of arithmetic [Rey] |
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] |
3145 | The Indiscernibility of Identicals is a truism; but the Identity of Indiscernibles depends on possible identical worlds [Rey] |
3172 | Empiricism says experience is both origin and justification of all knowledge [Rey] |
3166 | Animal learning is separate from their behaviour [Rey] |
3232 | Abduction could have true data and a false conclusion, and may include data not originally mentioned [Rey] |
3128 | It's not at all clear that explanation needs to stop anywhere [Rey] |
3136 | The three theories are reduction, dualism, eliminativism [Rey] |
5960 | When the soul is intelligent and harmonious, it is part of god and derives from god [Plutarch] |
3141 | Is consciousness 40Hz oscillations in layers 5 and 6 of the visual cortex? [Rey] |
3148 | Dualist privacy is seen as too deep for even telepathy to reach [Rey] |
3164 | Intentional explanations are always circular [Rey] |
3138 | Arithmetic and unconscious attitudes have no qualia [Rey] |
3142 | Why qualia, and why this particular quale? [Rey] |
3224 | If qualia have no function, their attachment to thoughts is accidental [Rey] |
3227 | Are qualia a type of propositional attitude? [Rey] |
3226 | Are qualia irrelevant to explaining the mind? [Rey] |
3229 | If colour fits a cone mapping hue, brightness and saturation, rotating the cone could give spectrum inversion [Rey] |
3223 | Self-consciousness may just be nested intentionality [Rey] |
3162 | Experiments prove that people are often unaware of their motives [Rey] |
3163 | Brain damage makes the unreliability of introspection obvious [Rey] |
3195 | If reason could be explained in computational terms, there would be no need for the concept of 'free will' [Rey] |
3196 | Free will isn't evidence against a theory of thought if there is no evidence for free will [Rey] |
3180 | Maybe behaviourists should define mental states as a group [Rey] |
3165 | Behaviourism is eliminative, or reductionist, or methodological [Rey] |
3167 | Animals don't just respond to stimuli, they experiment [Rey] |
3173 | How are stimuli and responses 'similar'? [Rey] |
3179 | Behaviour is too contingent and irrelevant to be the mind [Rey] |
3186 | If a normal person lacked a brain, would you say they had no mind? [Rey] |
3127 | Dualism and physicalism explain nothing, and don't suggest any research [Rey] |
3188 | Homuncular functionalism (e.g. Freud) could be based on simpler mechanical processes [Rey] |
3216 | Is the room functionally the same as a Chinese speaker? [Rey] |
3220 | Searle is guilty of the fallacy of division - attributing a property of the whole to a part [Rey] |
3206 | One computer program could either play chess or fight a war [Rey] |
8685 | Studying biology presumes the laws of chemistry, and it could never contradict them [Friend] |
3140 | If you explain water as H2O, you have reduced water, but not eliminated it [Rey] |
3134 | Human behaviour can show law-like regularity, which eliminativism can't explain [Rey] |
3200 | Pattern recognition is puzzling for computation, but makes sense for connectionism [Rey] |
3201 | Connectionism explains well speed of perception and 'graceful degradation' [Rey] |
3202 | Connectionism explains irrationality (such as the Gamblers' Fallacy) quite well [Rey] |
3199 | Connectionism assigns numbers to nodes and branches, and plots the outcomes [Rey] |
3150 | Can identity explain reason, free will, non-extension, intentionality, subjectivity, experience? [Rey] |
3129 | Physicalism offers something called "complexity" instead of mental substance [Rey] |
3139 | Some attitudes are information (belief), others motivate (hatred) [Rey] |
3174 | Good grammar can't come simply from stimuli [Rey] |
3171 | Children speak 90% good grammar [Rey] |
3213 | Animals may also use a language of thought [Rey] |
3170 | We train children in truth, not in grammar [Rey] |
3215 | Images can't replace computation, as they need it [Rey] |
3194 | CRTT is good on deduction, but not so hot on induction, abduction and practical reason [Rey] |
3147 | Problem-solving clearly involves manipulating images [Rey] |
3175 | Animals map things over time as well as over space [Rey] |
3207 | Simple externalism is that the meaning just is the object [Rey] |
8688 | Concepts can be presented extensionally (as objects) or intensionally (as a characterization) [Friend] |
3176 | Anything bears a family resemblance to a game, but obviously not anything counts as one [Rey] |
3181 | A one hour gap in time might be indirectly verified, but then almost anything could be [Rey] |
3204 | The meaning of "and" may be its use, but not of "animal" [Rey] |
3205 | Semantic holism means new evidence for a belief changes the belief, and we can't agree on concepts [Rey] |
3209 | Causal theories of reference (by 'dubbing') don't eliminate meanings in the heads of dubbers [Rey] |
3210 | If meaning and reference are based on causation, then virtually everything has meaning [Rey] |
3149 | Referential Opacity says truth is lost when you substitute one referring term ('mother') for another ('Jocasta') [Rey] |
3169 | A simple chaining device can't build sentences containing 'either..or', or 'if..then' [Rey] |
3221 | Our desires become important when we have desires about desires [Rey] |