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) |
| 14797 | Vagueness is a neglected but important part of mathematical thought [Peirce] |
| Full Idea: Logicians have too much neglected the study of vagueness, not suspecting the important part it plays in mathematical thought. It is the antithetical analogue of generality. | |
| From: Charles Sanders Peirce (Critical Common-Sensism [1905], I) |
| 14798 | All communication is vague, and is outside the principle of non-contradiction [Peirce] |
| Full Idea: The 'vague' might be defined as that to which the principle of contradiction does not apply. For it is false neither that an animal (in a vague sense) is male, nor that an animal is female. No communication between persons can be entirely non-vague. | |
| From: Charles Sanders Peirce (Critical Common-Sensism [1905], I) | |
| A reaction: Note that he makes vagueness largely a matter of the way we talk, which is David Lewis's approach, and looks right to me. |
| 6587 | It is always wrong to believe things on insufficient evidence [Clifford] |
| Full Idea: It is wrong always, everywhere, and for anyone, to believe anything upon insufficient evidence. | |
| From: William K. Clifford (works [1870]), quoted by Robert Fogelin - Walking the Tightrope of Reason Ch.4 | |
| A reaction: This is a famous remark, but is in danger of being tautological unless one gives some account of what 'insufficient' means. If Clifford means the evidence must be conclusive, this is nonsense. 'Never believe if there is no evidence' is better. |