95 ideas
| 15945 | Second-order set theory just adds a version of Replacement that quantifies over functions [Lavine] |
| 15914 | An 'upper bound' is the greatest member of a subset; there may be several of these, so there is a 'least' one [Lavine] |
| 15921 | Collections of things can't be too big, but collections by a rule seem unlimited in size [Lavine] |
| 15937 | Those who reject infinite collections also want to reject the Axiom of Choice [Lavine] |
| 15936 | The Power Set is just the collection of functions from one collection to another [Lavine] |
| 15899 | Replacement was immediately accepted, despite having very few implications [Lavine] |
| 15930 | Foundation says descending chains are of finite length, blocking circularity, or ungrounded sets [Lavine] |
| 15920 | Pure collections of things obey Choice, but collections defined by a rule may not [Lavine] |
| 15898 | The controversy was not about the Axiom of Choice, but about functions as arbitrary, or given by rules [Lavine] |
| 15919 | The 'logical' notion of class has some kind of definition or rule to characterise the class [Lavine] |
| 15900 | The iterative conception of set wasn't suggested until 1947 [Lavine] |
| 15931 | The iterative conception needs the Axiom of Infinity, to show how far we can iterate [Lavine] |
| 15932 | The iterative conception doesn't unify the axioms, and has had little impact on mathematical proofs [Lavine] |
| 15933 | Limitation of Size: if it's the same size as a set, it's a set; it uses Replacement [Lavine] |
| 15913 | A collection is 'well-ordered' if there is a least element, and all of its successors can be identified [Lavine] |
| 15926 | Second-order logic presupposes a set of relations already fixed by the first-order domain [Lavine] |
| 15934 | Mathematical proof by contradiction needs the law of excluded middle [Lavine] |
| 3137 | Varieties of singular terms are used to designate token particulars [Rey] |
| 15907 | Mathematics is nowadays (thanks to set theory) regarded as the study of structure, not of quantity [Lavine] |
| 15942 | Every rational number, unlike every natural number, is divisible by some other number [Lavine] |
| 15922 | For the real numbers to form a set, we need the Continuum Hypothesis to be true [Lavine] |
| 18250 | Cauchy gave a necessary condition for the convergence of a sequence [Lavine] |
| 15904 | The two sides of the Cut are, roughly, the bounding commensurable ratios [Lavine] |
| 15912 | Counting results in well-ordering, and well-ordering makes counting possible [Lavine] |
| 15949 | The theory of infinity must rest on our inability to distinguish between very large sizes [Lavine] |
| 15947 | The infinite is extrapolation from the experience of indefinitely large size [Lavine] |
| 15940 | The intuitionist endorses only the potential infinite [Lavine] |
| 15909 | 'Aleph-0' is cardinality of the naturals, 'aleph-1' the next cardinal, 'aleph-ω' the ω-th cardinal [Lavine] |
| 15915 | Ordinals are basic to Cantor's transfinite, to count the sets [Lavine] |
| 15917 | Paradox: the class of all ordinals is well-ordered, so must have an ordinal as type - giving a bigger ordinal [Lavine] |
| 15918 | Paradox: there is no largest cardinal, but the class of everything seems to be the largest [Lavine] |
| 15929 | Set theory will found all of mathematics - except for the notion of proof [Lavine] |
| 15935 | Modern mathematics works up to isomorphism, and doesn't care what things 'really are' [Lavine] |
| 3143 | Physics requires the existence of properties, and also the abstract objects of arithmetic [Rey] |
| 15928 | Intuitionism rejects set-theory to found mathematics [Lavine] |
| 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] |
| 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] |
| 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] |
| 3171 | Children speak 90% good grammar [Rey] |
| 3174 | Good grammar can't come simply from stimuli [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] |
| 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] |
| 13304 | Learned men gain more in one day than others do in a lifetime [Posidonius] |
| 20820 | Time is an interval of motion, or the measure of speed [Posidonius, by Stobaeus] |