35 ideas
17926 | Rejecting double negation elimination undermines reductio proofs [Colyvan] |
17925 | Showing a disproof is impossible is not a proof, so don't eliminate double negation [Colyvan] |
17924 | Excluded middle says P or not-P; bivalence says P is either true or false [Colyvan] |
17929 | Löwenheim proved his result for a first-order sentence, and Skolem generalised it [Colyvan] |
17930 | Axioms are 'categorical' if all of their models are isomorphic [Colyvan] |
17928 | Ordinal numbers represent order relations [Colyvan] |
17923 | Intuitionists only accept a few safe infinities [Colyvan] |
17941 | Infinitesimals were sometimes zero, and sometimes close to zero [Colyvan] |
17922 | Reducing real numbers to rationals suggested arithmetic as the foundation of maths [Colyvan] |
17936 | Transfinite induction moves from all cases, up to the limit ordinal [Colyvan] |
17940 | Most mathematical proofs are using set theory, but without saying so [Colyvan] |
17931 | Structuralism say only 'up to isomorphism' matters because that is all there is to it [Colyvan] |
17932 | If 'in re' structures relies on the world, does the world contain rich enough structures? [Colyvan] |
17943 | Probability supports Bayesianism better as degrees of belief than as ratios of frequencies [Colyvan] |
17939 | Mathematics can reveal structural similarities in diverse systems [Colyvan] |
17938 | Mathematics can show why some surprising events have to occur [Colyvan] |
17934 | Proof by cases (by 'exhaustion') is said to be unexplanatory [Colyvan] |
17933 | Reductio proofs do not seem to be very explanatory [Colyvan] |
17935 | If inductive proofs hold because of the structure of natural numbers, they may explain theorems [Colyvan] |
17942 | Can a proof that no one understands (of the four-colour theorem) really be a proof? [Colyvan] |
4608 | Minds are hard-wired, or trial-and-error, or experimental, or full self-aware [Dennett, by Heil] |
4880 | Sentience comes in grades from robotic to super-human; we only draw a line for moral reasons [Dennett] |
4873 | What is it like to notice an uncomfortable position when you are asleep? [Dennett] |
17937 | Mathematical generalisation is by extending a system, or by abstracting away from it [Colyvan] |
4881 | Being a person must involve having second-order beliefs and desires (about beliefs and desires) [Dennett] |
4875 | We descend from robots, and our intentionality is composed of billions of crude intentional systems [Dennett] |
4879 | There is no more anger in adrenaline than silliness in a bottle of whiskey [Dennett] |
4876 | Maybe there is a minimum brain speed for supporting a mind [Dennett] |
4878 | The materials for a mind only matter because of speed, and a need for transducers and effectors [Dennett] |
4874 | The predecessor and rival of the language of thought hypothesis is the picture theory of ideas [Dennett] |
4882 | Concepts are things we (unlike dogs) can think about, because we have language [Dennett] |
13304 | Learned men gain more in one day than others do in a lifetime [Posidonius] |
4872 | Most people see an abortion differently if the foetus lacks a brain [Dennett] |
20820 | Time is an interval of motion, or the measure of speed [Posidonius, by Stobaeus] |
4877 | Maybe plants are very slow (and sentient) animals, overlooked because we are faster? [Dennett] |