49 ideas
| 15845 | It seems absurd that seeing a person's limbs, the one is many, and yet the many are one [Plato] |
| 8729 | Intuitionists deny excluded middle, because it is committed to transcendent truth or objects [Shapiro] |
| 9867 | It is absurd to define a circle, but not be able to recognise a real one [Plato] |
| 8763 | The number 3 is presumably identical as a natural, an integer, a rational, a real, and complex [Shapiro] |
| 18249 | Cauchy gave a formal definition of a converging sequence. [Shapiro] |
| 9865 | Daily arithmetic counts unequal things, but pure arithmetic equalises them [Plato] |
| 8764 | Categories are the best foundation for mathematics [Shapiro] |
| 8762 | Two definitions of 3 in terms of sets disagree over whether 1 is a member of 3 [Shapiro] |
| 8760 | Numbers do not exist independently; the essence of a number is its relations to other numbers [Shapiro] |
| 8761 | A 'system' is related objects; a 'pattern' or 'structure' abstracts the pure relations from them [Shapiro] |
| 8744 | Logicism seems to be a non-starter if (as is widely held) logic has no ontology of its own [Shapiro] |
| 8749 | Term Formalism says mathematics is just about symbols - but real numbers have no names [Shapiro] |
| 8750 | Game Formalism is just a matter of rules, like chess - but then why is it useful in science? [Shapiro] |
| 8752 | Deductivism says mathematics is logical consequences of uninterpreted axioms [Shapiro] |
| 8753 | Critics resent the way intuitionism cripples mathematics, but it allows new important distinctions [Shapiro] |
| 8731 | Conceptualist are just realists or idealist or nominalists, depending on their view of concepts [Shapiro] |
| 8730 | 'Impredicative' definitions refer to the thing being described [Shapiro] |
| 14503 | If a mixture does not contain measure and proportion, it is corrupted and destroyed [Plato] |
| 15857 | Any mixture which lacks measure and proportion doesn't even count as a mixture at all [Plato] |
| 4447 | If the good is one, is it unchanged when it is in particulars, and is it then separated from itself? [Plato] |
| 15856 | A thing can become one or many, depending on how we talk about it [Plato] |
| 374 | If one object is divided into its parts, someone can then say that one are many and many is one [Plato] |
| 389 | How can you be certain about aspects of the world if they aren't constant? [Plato] |
| 8725 | Rationalism tries to apply mathematical methodology to all of knowledge [Shapiro] |
| 23029 | Knowledge is secured by the relations between its parts, through differences and identities [Green,TH, by Muirhead] |
| 390 | If goodness involves moderation and proportion, then it seems to be found in beauty [Plato] |
| 392 | Neither intellect nor pleasure are the good, because they are not perfect and self-sufficient [Plato] |
| 391 | The good involves beauty, proportion and truth [Plato] |
| 393 | Good first, then beauty, then reason, then knowledge, then pleasure [Plato, by PG] |
| 385 | Some of the pleasures and pains we feel are false [Plato] |
| 387 | A small pure pleasure is much finer than a large one contaminated with pain [Plato] |
| 373 | Pleasure is certainly very pleasant, but it doesn't follow that all pleasures are good [Plato] |
| 379 | The good must be sufficient and perfect, and neither intellect nor pleasure are that [Plato] |
| 371 | Reason, memory, truth and wisdom are far better than pleasure, for those who can attain them [Plato] |
| 376 | Would you prefer a life of pleasure without reason, or one of reason without pleasure? [Plato] |
| 382 | It is unlikely that the gods feel either pleasure or pain [Plato] |
| 381 | We feel pleasure when we approach our natural state of harmony [Plato] |
| 386 | Intense pleasure and pain are not felt in a good body, but in a worthless one [Plato] |
| 388 | Hedonists must say that someone in pain is bad, even if they are virtuous [Plato] |
| 377 | If you lived a life of maximum pleasure, would you still be lacking anything? [Plato] |
| 378 | A life of pure pleasure with no intellect is the life of a jellyfish [Plato] |
| 23046 | States only have full authority if they heed the claims of human fellowship [Green,TH] |
| 23051 | Equality also implies liberty, because equality must be of opportunity as well as possessions [Green,TH] |
| 23028 | The highest political efforts express our deeper social spirit [Green,TH, by Muirhead] |
| 23054 | Communism is wrong because it restricts the freedom of individuals to contribute to the community [Green,TH, by Muirhead] |
| 23047 | Original common ownership is securing private property, not denying it [Green,TH, by Muirhead] |
| 23042 | National spirit only exists in the individuals who embody it [Green,TH, by Muirhead] |
| 23048 | The ground of property ownership is not force but the power to use it for social ends [Green,TH, by Muirhead] |
| 23049 | Property is needed by all citizens, to empower them to achieve social goods [Green,TH] |