display all the ideas for this combination of texts
6 ideas
| 21644 | Numbers are used as singular terms, as adjectives, and as symbols [Hofweber] |
| Full Idea: Number words have a singular term use, and adjectival (or determiner) use, and the symbolic use. The main question is how they relate to each other. | |
| From: Thomas Hofweber (Ontology and the Ambitions of Metaphysics [2016], 05.1) | |
| A reaction: Thus 'the number four is even', 'there are four moons', and '4 comes after 3'. |
| 21646 | The Amazonian Piraha language is said to have no number words [Hofweber] |
| Full Idea: The now famous Piraha language, of the Amazon region in Brazil, allegedly has no number words. | |
| From: Thomas Hofweber (Ontology and the Ambitions of Metaphysics [2016], 05.6) | |
| A reaction: Two groups can be shown to be of equal cardinality, by one-to-one matching rather than by counting. They could get by using 'equals' (and maybe unequally bigger and unequally smaller), and intuitive feelings for sizes of groups. |
| 21665 | The fundamental theorem of arithmetic is that all numbers are composed uniquely of primes [Hofweber] |
| Full Idea: The prime numbers are more fundamental than the even numbers, and than the composite non-prime numbers. The result that all numbers uniquely decompose into a product of prime numbers is called the 'Fundamental Theorem of Arithmetic'. | |
| From: Thomas Hofweber (Ontology and the Ambitions of Metaphysics [2016], 13.4.2) | |
| A reaction: I could have used this example in my thesis, which defended the view that essences are the fundamentals of explanation, even in abstract theoretical realms. |
| 21649 | How can words be used for counting if they are objects? [Hofweber] |
| Full Idea: Number words as singular terms seem to refer to objects; numbers words in determiner or adjectival position are tied to counting. How these objects are related to counting is what the application problem is about. | |
| From: Thomas Hofweber (Ontology and the Ambitions of Metaphysics [2016], 06.1.3) | |
| A reaction: You can't use stones for counting, so there must be more to numbers than the announcement that they are 'objects'. They seem to have internal relations, which makes them unusual objects. |
| 21647 | Logicism makes sense of our ability to know arithmetic just by thought [Hofweber] |
| Full Idea: Frege's tying the objectivity of arithmetic to the objectivity of logic makes sense of the fact that can find out about arithmetic by thinking alone. | |
| From: Thomas Hofweber (Ontology and the Ambitions of Metaphysics [2016], 06.1.1) | |
| A reaction: This assumes that logic is entirely a priori. We might compare the geometry of land surfaces with 'pure' geometry. If numbers are independent objects, it is unclear how we could have any a priori knowledge of them. |
| 21648 | Neo-Fregeans are dazzled by a technical result, and ignore practicalities [Hofweber] |
| Full Idea: A major flaw of the neo-Fregean program is that it is more impressed by the technical result that Peano Arithmetic can be interpreted by second-order logic plus Hume's Principle, than empirical considerations about how numbers come about. | |
| From: Thomas Hofweber (Ontology and the Ambitions of Metaphysics [2016], 06.1.3) | |
| A reaction: This doesn't sound like a problem that would bother Fregeans or neo-Fregeans much. Deriving the Peano Axioms from various beginnings has become a parlour game for modern philosophers of mathematics. |