7 ideas
| 10882 | Predicative definitions only refer to entities outside the defined collection [Horsten] |
| Full Idea: Definitions are called 'predicative', and are considered sound, if they only refer to entities which exist independently from the defined collection. | |
| From: Leon Horsten (Philosophy of Mathematics [2007], §2.4) |
| 10884 | A theory is 'categorical' if it has just one model up to isomorphism [Horsten] |
| Full Idea: If a theory has, up to isomorphism, exactly one model, then it is said to be 'categorical'. | |
| From: Leon Horsten (Philosophy of Mathematics [2007], §5.2) |
| 10885 | Computer proofs don't provide explanations [Horsten] |
| Full Idea: Mathematicians are uncomfortable with computerised proofs because a 'good' proof should do more than convince us that a certain statement is true. It should also explain why the statement in question holds. | |
| From: Leon Horsten (Philosophy of Mathematics [2007], §5.3) |
| 10881 | The concept of 'ordinal number' is set-theoretic, not arithmetical [Horsten] |
| Full Idea: The notion of an ordinal number is a set-theoretic, and hence non-arithmetical, concept. | |
| From: Leon Horsten (Philosophy of Mathematics [2007], §2.3) |
| 8840 | There are five possible responses to the problem of infinite regress in justification [Cleve] |
| Full Idea: Sceptics respond to the regress problem by denying knowledge; Foundationalists accept justifications without reasons; Positists say reasons terminate is mere posits; Coherentists say mutual support is justification; Infinitists accept the regress. | |
| From: James Van Cleve (Why coherence is not enough [2005], I) | |
| A reaction: A nice map of the territory. The doubts of Scepticism are not strong enough for anyone to embrace the view; Foundationalist destroy knowledge (?), as do Positists; Infinitism is a version of Coherentism - which is the winner. |
| 8841 | Modern foundationalists say basic beliefs are fallible, and coherence is relevant [Cleve] |
| Full Idea: Contemporary foundationalists are seldom of the strong Cartesian variety: they do not insist that basic beliefs be absolutely certain. They also tend to allow that coherence can enhance justification. | |
| From: James Van Cleve (Why coherence is not enough [2005], III) | |
| A reaction: It strikes me that they have got onto a slippery slope. How certain are the basic beliefs? How do you evaluate their certainty? Could incoherence in their implications undermine them? Skyscrapers need perfect foundations. |
| 22307 | Propositions don't name facts, because two opposed propositions can match one fact [Russell] |
| Full Idea: It is perfectly evident that a proposition is not the name for a fact, from the mere circumstance that there are two propositions corresponding to each fact. 'Socrates is dead' and 'Socrates is not dead' correspond to the same fact. | |
| From: Bertrand Russell (Papers of 1918 [1918], VIII.136), quoted by Michael Potter - The Rise of Analytic Philosophy 1879-1930 42 'Prop' | |
| A reaction: He finally reaches in 1918 what now looks fairly obvious. The idea that a proposition is part of the world is absurd. We should call the parts of the world 'facts' (despite vagueness and linguistic dependence in such things). Propositions are thoughts. |