45 ideas
2653 | If the parts of the universe are subject to the law of nature, the whole universe must also be subject to it [Cicero] |
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] |
11211 | If a sound conclusion comes from two errors that cancel out, the path of the argument must matter [Rumfitt] |
17924 | Excluded middle says P or not-P; bivalence says P is either true or false [Colyvan] |
11212 | The sense of a connective comes from primitively obvious rules of inference [Rumfitt] |
11210 | Standardly 'and' and 'but' are held to have the same sense by having the same truth table [Rumfitt] |
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] |
17937 | Mathematical generalisation is by extending a system, or by abstracting away from it [Colyvan] |
2628 | Why would mind mix with matter if it didn't need it? [Cicero] |
20814 | Eloquence educates, exhorts, comforts, distracts and unites us, and raises us from savagery [Cicero] |
11214 | We learn 'not' along with affirmation, by learning to either affirm or deny a sentence [Rumfitt] |
2640 | We have the death penalty, but still have thousands of robbers [Cicero] |
2652 | Some regard nature simply as an irrational force that imparts movement [Cicero] |
2645 | Why shouldn't the gods fear their own destruction? [Cicero] |
2627 | I wonder whether loss of reverence for the gods would mean the end of all virtue [Cicero] |
2651 | God doesn't obey the laws of nature; they are subject to the law of God [Cicero] |
2634 | It seems clear to me that we have an innate idea of the divine [Cicero] |
2636 | Many primitive people know nothing of the gods [Cicero] |
2647 | It is obvious from order that someone is in charge, as when we visit a gymnasium [Cicero] |
2650 | If a person cannot feel the power of God when looking at the stars, they are probably incapable of feeling [Cicero] |
2655 | If the barbarians of Britain saw a complex machine, they would be baffled, but would know it was designed [Cicero] |
2656 | Chance is no more likely to create the world than spilling lots of letters is likely to create a famous poem [Cicero] |
2657 | If everything with regular movement and order is divine, then recurrent illnesses must be divine [Cicero] |
2638 | Either the gods are identical, or one is more beautiful than another [Cicero] |
2635 | The gods are happy, so virtuous, so rational, so must have human shape [Cicero] |
2641 | Why believe in gods if you have never seen them? [Cicero] |
2659 | The lists of good men who have suffered and bad men who have prospered are endless [Cicero] |
2658 | The gods blame men for having vices, but they could have given us enough reason to avoid them [Cicero] |