16 ideas
| 6299 | Axioms are often affirmed simply because they produce results which have been accepted [Resnik] |
| Full Idea: Many axioms have been proposed, not on the grounds that they can be directly known, but rather because they produce a desired body of previously recognised results. | |
| From: Michael D. Resnik (Maths as a Science of Patterns [1997], One.5.1) | |
| A reaction: This is the perennial problem with axioms - whether we start from them, or whether we deduce them after the event. There is nothing wrong with that, just as we might infer the existence of quarks because of their results. |
| 6304 | Mathematical realism says that maths exists, is largely true, and is independent of proofs [Resnik] |
| Full Idea: Mathematical realism is the doctrine that mathematical objects exist, that much contemporary mathematics is true, and that the existence and truth in question is independent of our constructions, beliefs and proofs. | |
| From: Michael D. Resnik (Maths as a Science of Patterns [1997], Three.12.9) | |
| A reaction: As thus defined, I would call myself a mathematical realist, but everyone must hesitate a little at the word 'exist' and ask, how does it exist? What is it 'made of'? To say that it exists in the way that patterns exist strikes me as very helpful. |
| 6300 | Mathematical constants and quantifiers only exist as locations within structures or patterns [Resnik] |
| Full Idea: In maths the primary subject-matter is not mathematical objects but structures in which they are arranged; our constants and quantifiers denote atoms, structureless points, or positions in structures; they have no identity outside a structure or pattern. | |
| From: Michael D. Resnik (Maths as a Science of Patterns [1997], Three.10.1) | |
| A reaction: This seems to me a very promising idea for the understanding of mathematics. All mathematicians acknowledge that the recognition of patterns is basic to the subject. Even animals recognise patterns. It is natural to invent a language of patterns. |
| 6303 | Sets are positions in patterns [Resnik] |
| Full Idea: On my view, sets are positions in certain patterns. | |
| From: Michael D. Resnik (Maths as a Science of Patterns [1997], Three.10.5) | |
| A reaction: I have always found the ontology of a 'set' puzzling, because they seem to depend on prior reasons why something is a member of a given set, which cannot always be random. It is hard to explain sets without mentioning properties. |
| 6302 | Structuralism must explain why a triangle is a whole, and not a random set of points [Resnik] |
| Full Idea: An objection is that structuralism fails to explain why certain mathematical patterns are unified wholes while others are not; for instance, some think that an ontological account of mathematics must explain why a triangle is not a 'random' set of points. | |
| From: Michael D. Resnik (Maths as a Science of Patterns [1997], Three.10.4) | |
| A reaction: This is an indication that we are not just saying that we recognise patterns in nature, but that we also 'see' various underlying characteristics of the patterns. The obvious suggestion is that we see meta-patterns. |
| 6295 | There are too many mathematical objects for them all to be mental or physical [Resnik] |
| Full Idea: If we take mathematics at its word, there are too many mathematical objects for it to be plausible that they are all mental or physical objects. | |
| From: Michael D. Resnik (Maths as a Science of Patterns [1997], One.1) | |
| A reaction: No one, of course, has ever claimed that they are, but this is a good starting point for assessing the ontology of mathematics. We are going to need 'rules', which can deduce the multitudinous mathematical objects from a small ontology. |
| 6296 | Maths is pattern recognition and representation, and its truth and proofs are based on these [Resnik] |
| Full Idea: I argue that mathematical knowledge has its roots in pattern recognition and representation, and that manipulating representations of patterns provides the connection between the mathematical proof and mathematical truth. | |
| From: Michael D. Resnik (Maths as a Science of Patterns [1997], One.1) | |
| A reaction: The suggestion that patterns are at the basis of the ontology of mathematics is the most illuminating thought I have encountered in the area. It immediately opens up the possibility of maths being an entirely empirical subject. |
| 6301 | Congruence is the strongest relationship of patterns, equivalence comes next, and mutual occurrence is the weakest [Resnik] |
| Full Idea: Of the equivalence relationships which occur between patterns, congruence is the strongest, equivalence the next, and mutual occurrence the weakest. None of these is identity, which would require the same position. | |
| From: Michael D. Resnik (Maths as a Science of Patterns [1997], Three.10.3) | |
| A reaction: This gives some indication of how an account of mathematics as a science of patterns might be built up. Presumably the recognition of these 'degrees of strength' cannot be straightforward observation, but will need an a priori component? |
| 16756 | Substantial forms must exist, to explain the stability of metals like silver and tin [Albertus Magnus] |
| Full Idea: There is no reason why the matter in any natural thing should be stable in its nature, if it is not completed by a substantial form. But we see that silver is stable, and tin and other metals. Therefore they will seem to be perfected by substantial forms. | |
| From: Albertus Magnus (On Minerals [1260], III.1.7), quoted by Robert Pasnau - Metaphysical Themes 1274-1671 24.2 | |
| A reaction: Illuminating. This may be the best reason for proposing substantial forms. Once materialism arrives, the so-called 'laws' of nature have to be imposed on the material to do the job - but what the hell is a law supposed to be? |
| 16718 | Primary qualities are the cause of all the other sensible qualities [Albertus Magnus] |
| Full Idea: The primary qualities of tangible things are the cause of all the other sensible qualities. | |
| From: Albertus Magnus (On 'Generation and Corruption' [1261], II.1.1), quoted by Robert Pasnau - Metaphysical Themes 1274-1671 21.2 | |
| A reaction: This makes the primary qualities sound suspiciously like the essence. |
| 2850 | How can emotivists explain someone who recognises morality but is indifferent to it? [Brink] |
| Full Idea: It is not clear how the emotivist can accommodate the amoralist - one who recognises moral requirements but is indifferent to them. | |
| From: David O. Brink (Emotivism [1995], p.224) | |
| A reaction: Nietzsche recognised current morals, but was indifferent to them. It is hard to imagine, though, an amoralist who lacked all the feelings which imply morality. |
| 2848 | Two people might agree in their emotional moral attitude while disagreeing in their judgement [Brink] |
| Full Idea: Critics of emotivism claim that moral agreement need not track agreement in attitude; moralists with the same attitude can disagree in their views, and they can hold the same view while disagreeing in attitude. | |
| From: David O. Brink (Emotivism [1995], p.224) | |
| A reaction: Thus two racists can disagree about how racists should behave. Sounds like a good criticism. |
| 2851 | Emotivists find it hard to analyse assertions of moral principles, rather than actual judgements [Brink] |
| Full Idea: It is hard for the emotivist to give an analysis of the occurrence of moral ideas in unasserted contexts, such as "IF he did wrong, then he should be punished". | |
| From: David O. Brink (Emotivism [1995], p.224) | |
| A reaction: This is the 'Frege-Geach Problem'. |
| 2853 | Emotivists claim to explain moral motivation by basing morality on non-cognitive attitudes [Brink] |
| Full Idea: By stressing the intimate connection between moral judgements and the agent's non-cognitive attitudes, emotivists claim to capture the motivational properties of moral judgement. | |
| From: David O. Brink (Emotivism [1995], p.223) | |
| A reaction: The same claim is made by contractarians, who start from our universal self-interest. Emotivists also nicely capture the motivation properties of immoral judgements. |
| 2852 | Emotivists tend to favour a redundancy theory of truth, making moral judgement meaningless [Brink] |
| Full Idea: If you want to recognise the truth of some moral judgements, perhaps to make room for the possibility of moral mistakes, then one may not be satisfied with the emotivists' tendency to appeal to the redundancy theory of truth. | |
| From: David O. Brink (Emotivism [1995], p.224) | |
| A reaction: Probably thinking of Simon Blackburn. People who adopt a redundancy view of truth for semantics are left floundering when discussing what is true in the rest of philosophy. |
| 2849 | Emotivism implies relativism about moral meanings, but critics say disagreements are about moral reference [Brink] |
| Full Idea: Emotivism suggests that different feelings lead to different individual meanings for moral terms, but critics say that meanings are the same, and disagreement is about the extension (range of reference) of the terms. | |
| From: David O. Brink (Emotivism [1995], p.224) | |
| A reaction: It's hard to see how 'ought to p' could have quite different meanings for an emotivist and (say) a theistic moralist. 'Ought' is an obvious and simple word. Good criticism. |