9 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) |
| 7334 | Anti-realism needs an intuitionist logic with no law of excluded middle [Dummett, by Miller,A] |
| Full Idea: Dummett argues that antirealism implies that classical logic must be given up in favour of some form of intuitionistic logic that does not have the law of excluded middle as a theorem. | |
| From: report of Michael Dummett (works [1970]) by Alexander Miller - Philosophy of Language 9.4 | |
| A reaction: Only realists can think every proposition is either true or false, even if it is beyond the bounds of our possible knowledge (e.g. tiny details from remote history). Personally I think "Plato had brown eyes" is either true or false. |
| 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) |
| 3303 | For anti-realists there are no natural distinctions between objects [Dummett, by Benardete,JA] |
| Full Idea: Dummett says that anti-realism offers us a picture of reality as an amorphous lump not yet articulated into discrete objects. | |
| From: report of Michael Dummett (works [1970]) by José A. Benardete - Metaphysics: the logical approach Ch.2 | |
| A reaction: This might be called 'weak' anti-realism, where 'strong' anti-realism is the view that reality is quite unknowable, and possibly non-existent. |
| 14802 | Physical and psychical laws of mind are either independent, or derived in one or other direction [Peirce] |
| Full Idea: The question about minds is whether 1) physical and psychical laws are independent (monism, my neutralism), 2) the psychical laws derived and physical laws primordial (materialism), 3) physical law is derived, psychical law primordial (idealism). | |
| From: Charles Sanders Peirce (The Architecture of Theories [1891], p.321) | |
| A reaction: I think you are already in trouble when you start proposing that there are two quite distinct sets of laws, and then worry about how they are related. Assume unity, and only separate them when the science forces you to (which it won't). |
| 14800 | The world is full of variety, but laws seem to produce uniformity [Peirce] |
| Full Idea: Exact law obviously never can produce heterogeneity out of homogeneity; and arbitrary heterogeneity is the feature of the universe the most manifest and characteristic. | |
| From: Charles Sanders Peirce (The Architecture of Theories [1891], p.319) | |
| A reaction: This is the view of laws of nature now associated with Nancy Cartwright, but presumably you can explain the apparent chaos in terms of the intersection of vast numbers of 'laws'. Or, better, there aren't any laws. |
| 14801 | Darwinian evolution is chance, with the destruction of bad results [Peirce] |
| Full Idea: Darwinian evolution is evolution by the operation of chance, and the destruction of bad results. | |
| From: Charles Sanders Peirce (The Architecture of Theories [1891], p.320) | |
| A reaction: The 'destruction of bad results' is a much better slogan for Darwin that Spencer's 'survival of the fittest'. It is, of course, a rather unattractive God who makes progress by endlessly destroying huge quantities of failed (but living) experiments. |