29 ideas
| 1922 | Spiritual qualities only become advantageous with the growth of wisdom [Plato] |
| 23889 | Among the Greeks Aristotle is the only philosopher in the modern style [Weil] |
| 8729 | Intuitionists deny excluded middle, because it is committed to transcendent truth or objects [Shapiro] |
| 11259 | How can you seek knowledge of something if you don't know it? [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] |
| 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] |
| 20219 | True opinions only become really valuable when they are tied down by reasons [Plato] |
| 5985 | Seeking and learning are just recollection [Plato] |
| 5986 | The slave boy learns geometry from questioning, not teaching, so it is recollection [Plato] |
| 8725 | Rationalism tries to apply mathematical methodology to all of knowledge [Shapiro] |
| 1923 | As a guide to action, true opinion is as good as knowledge [Plato] |
| 1919 | You don't need to learn what you know, and how do you seek for what you don't know? [Plato] |
| 1927 | It seems that virtue is neither natural nor taught, but is a divine gift [Plato] |
| 1913 | Is virtue taught, or achieved by practice, or a natural aptitude, or what? [Plato] |
| 1921 | If virtue is a type of knowledge then it ought to be taught [Plato] |
| 1916 | Even if virtues are many and various, they must have something in common to make them virtues [Plato] |
| 1918 | How can you know part of virtue without knowing the whole? [Plato] |
| 23892 | The only legitimate proof of God by order derives from beauty [Weil] |