49 ideas
| 326 | For relaxation one can consider the world of change, instead of eternal things [Plato] |
| 315 | Philosophy is the supreme gift of the gods to mortals [Plato] |
| 306 | Nothing can come to be without a cause [Plato] |
| 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] |
| 324 | Before the existence of the world there must have been being, space and becoming [Plato] |
| 20364 | The apprehensions of reason remain unchanging, but reasonless sensation shows mere becoming [Plato] |
| 12042 | Plato's Forms were seen as part of physics, rather than of metaphysics [Plato, by Annas] |
| 307 | Something will always be well-made if the maker keeps in mind the eternal underlying pattern [Plato] |
| 318 | In addition to the underlying unchanging model and a changing copy of it, there must also be a foundation of all change [Plato] |
| 321 | For knowledge and true opinion to be different there must be Forms; otherwise we are just stuck with sensations [Plato] |
| 317 | The universe is basically an intelligible and unchanging model, and a visible and changing copy of it [Plato] |
| 9103 | A universal is not a real feature of objects, but only a thought-object in the mind [William of Ockham] |
| 334 | Only bird-brained people think astronomy is entirely a matter of evidence [Plato] |
| 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] |
| 5962 | Plato says the soul is ordered by number [Plato, by Plutarch] |
| 17937 | Mathematical generalisation is by extending a system, or by abstracting away from it [Colyvan] |
| 330 | No one wants to be bad, but bad men result from physical and educational failures, which they do not want or choose [Plato] |
| 9104 | A universal is the result of abstraction, which is only a kind of mental picturing [William of Ockham] |
| 316 | Music has harmony like the soul, and serves to reorder disharmony within us [Plato] |
| 332 | One should exercise both the mind and the body, to avoid imbalance [Plato] |
| 328 | Everything that takes place naturally is pleasant [Plato] |
| 322 | Intelligence is the result of rational teaching; true opinion can result from irrational persuasion [Plato] |
| 331 | Bad governments prevent discussion, and discourage the study of virtue [Plato] |
| 311 | The cosmos must be unique, because it resembles the creator, who is unique [Plato] |
| 310 | The creator of the cosmos had no envy, and so wanted things to be as like himself as possible [Plato] |
| 325 | We must consider the four basic shapes as too small to see, only becoming visible in large numbers [Plato] |
| 327 | There are two types of cause, the necessary and the divine [Plato] |
| 314 | Heavenly movements gave us the idea of time, and caused us to inquire about the heavens [Plato] |
| 312 | Time came into existence with the heavens, so that there will be a time when they can be dissolved [Plato] |
| 309 | Clearly the world is good, so its maker must have been concerned with the eternal, not with change [Plato] |
| 308 | If the cosmos is an object of perception then it must be continually changing [Plato] |