50 ideas
| 17641 | Discoveries in mathematics can challenge philosophy, and offer it a new foundation [Russell] |
| 20981 | What justifies reliance on reason? Is it just a tool? Why is it better than blind belief? [Sen] |
| 20982 | In politics and ethics, scrutiny from different perspectives is essential for objectivity [Sen] |
| 17638 | If one proposition is deduced from another, they are more certain together than alone [Russell] |
| 17632 | Non-contradiction was learned from instances, and then found to be indubitable [Russell] |
| 8729 | Intuitionists deny excluded middle, because it is committed to transcendent truth or objects [Shapiro] |
| 17629 | Which premises are ultimate varies with context [Russell] |
| 17630 | The sources of a proof are the reasons why we believe its conclusion [Russell] |
| 17640 | Finding the axioms may be the only route to some new results [Russell] |
| 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] |
| 17627 | It seems absurd to prove 2+2=4, where the conclusion is more certain than premises [Russell] |
| 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] |
| 17628 | Arithmetic was probably inferred from relationships between physical objects [Russell] |
| 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] |
| 17637 | The most obvious beliefs are not infallible, as other obvious beliefs may conflict [Russell] |
| 8725 | Rationalism tries to apply mathematical methodology to all of knowledge [Shapiro] |
| 17639 | Believing a whole science is more than believing each of its propositions [Russell] |
| 17631 | Induction is inferring premises from consequences [Russell] |
| 20990 | Rationality is conformity to reasons that can be sustained even after scrutiny [Sen] |
| 21005 | A human right is not plausible if public scrutiny might reject it [Sen] |
| 20983 | The original position insures that the agreements reached are fair [Sen] |
| 20987 | The veil of ignorance encourages neutral interests, but not a wider view of values [Sen] |
| 20984 | A social contract limits the pursuit of justice to members of a single society [Sen] |
| 20986 | A person's voice may count because of their interests, or because of their good sense [Sen] |
| 21001 | Famines tend to be caused by authoritarian rule [Sen] |
| 21002 | Effective democracy needs tolerant values [Sen] |
| 20999 | Democracy as 'government by discussion' now has wide support [Sen] |
| 20979 | Democracy needs more than some institutions; diverse sections of the people must be heard [Sen] |
| 20993 | Eradicating smallpox does not impoverish nature [Sen] |
| 20995 | Capabilities are part of freedom, involving real opportunities [Sen] |
| 20998 | Freedom can involve capabilities, independence and non-interference [Sen] |
| 20997 | The need for equality among people arises from impartiality and objectivity [Sen] |
| 20996 | All modern theories of justice demand equality of something [Sen] |
| 20988 | Freedom from torture or terrorist attacks is independent of citizenship [Sen] |
| 20980 | You don't need a complete theory of justice to see that slavery is wrong [Sen] |
| 20978 | Practical justice concerns not only ideals, but ways to achieve them [Sen] |
| 20985 | Our institutions should promote justice, rather than embodying it [Sen] |
| 20994 | We must focus on removing manifest injustice, not just try to design a perfect society [Sen] |
| 21000 | If justice needs public reasoning, which needs democracy, then justice and democracy are linked [Sen] |
| 17633 | The law of gravity has many consequences beyond its grounding observations [Russell] |