### Ideas for Michael Burke, Rayo,A/Uzquiasno,G and Thomas Hofweber

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13 ideas

###### 6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / a. Numbers
 9998 What is the relation of number words as singular-terms, adjectives/determiners, and symbols? [Hofweber] Full Idea: There are three different uses of the number words: the singular-term use (as in 'the number of moons of Jupiter is four'), the adjectival (or determiner) use (as in 'Jupiter has four moons'), and the symbolic use (as in '4'). How are they related? From: Thomas Hofweber (Number Determiners, Numbers, Arithmetic , §1) A reaction: A classic philosophy of language approach to the problem - try to give the truth-conditions for all three types. The main problem is that the first one implies that numbers are objects, whereas the others do not. Why did Frege give priority to the first?
 10002 '2 + 2 = 4' can be read as either singular or plural [Hofweber] Full Idea: There are two ways to read to read '2 + 2 = 4', as singular ('two and two is four'), and as plural ('two and two are four'). From: Thomas Hofweber (Number Determiners, Numbers, Arithmetic , §4.1) A reaction: Hofweber doesn't notice that this phenomenon occurs elsewhere in English. 'The team is playing well', or 'the team are splitting up'; it simply depends whether you are holding the group in though as an entity, or as individuals. Important for numbers.
 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 , 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 , 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.
###### 6. Mathematics / A. Nature of Mathematics / 4. Using Numbers / f. Arithmetic
 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 , 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.
###### 6. Mathematics / A. Nature of Mathematics / 4. Using Numbers / g. Applying mathematics
 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 , 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.
###### 6. Mathematics / C. Sources of Mathematics / 1. Mathematical Platonism / a. For mathematical platonism
 10003 Why is arithmetic hard to learn, but then becomes easy? [Hofweber] Full Idea: Why is arithmetic so hard to learn, and why does it seem so easy to us now? For example, subtracting 789 from 26,789. From: Thomas Hofweber (Number Determiners, Numbers, Arithmetic , §4.2) A reaction: His answer that we find thinking about objects very easy, but as children we have to learn with difficulty the conversion of the determiner/adjectival number words, so that we come to think of them as objects.
###### 6. Mathematics / C. Sources of Mathematics / 1. Mathematical Platonism / b. Against mathematical platonism
 10008 Arithmetic is not about a domain of entities, as the quantifiers are purely inferential [Hofweber] Full Idea: I argue for an internalist conception of arithmetic. Arithmetic is not about a domain of entities, not even quantified entities. Quantifiers over natural numbers occur in their inferential-role reading in which they merely generalize over the instances. From: Thomas Hofweber (Number Determiners, Numbers, Arithmetic , §6.3) A reaction: Hofweber offers the hope that modern semantics can disentangle the confusions in platonist arithmetic. Very interesting. The fear is that after digging into the semantics for twenty years, you find the same old problems re-emerging at a lower level.
###### 6. Mathematics / C. Sources of Mathematics / 4. Mathematical Empiricism / c. Against mathematical empiricism
 10005 Arithmetic doesn’t simply depend on objects, since it is true of fictional objects [Hofweber] Full Idea: That 'two dogs are more than one' is clearly true, but its truth doesn't depend on the existence of dogs, as is seen if we consider 'two unicorns are more than one', which is true even though there are no unicorns. From: Thomas Hofweber (Number Determiners, Numbers, Arithmetic , §6.2) A reaction: This is an objection to crude empirical accounts of arithmetic, but the idea would be that there is a generalisation drawn from objects (dogs will do nicely), which then apply to any entities. If unicorns are entities, it will be true of them.
###### 6. Mathematics / C. Sources of Mathematics / 5. Numbers as Adjectival
 10000 We might eliminate adjectival numbers by analysing them into blocks of quantifiers [Hofweber] Full Idea: Determiner uses of number words may disappear on analysis. This is inspired by Russell's elimination of the word 'the'. The number becomes blocks of first-order quantifiers at the level of semantic representation. From: Thomas Hofweber (Number Determiners, Numbers, Arithmetic , §2) A reaction: [compressed] The proposal comes from platonists, who argue that numbers cannot be analysed away if they are objects. Hofweber says the analogy with Russell is wrong, as 'the' can't occur in different syntactic positions, the way number words can.
###### 6. Mathematics / C. Sources of Mathematics / 6. Logicism / a. Early logicism
 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 , 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.
###### 6. Mathematics / C. Sources of Mathematics / 6. Logicism / c. Neo-logicism
 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 , 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.
###### 6. Mathematics / C. Sources of Mathematics / 6. Logicism / d. Logicism critique
 10006 First-order logic captures the inferential relations of numbers, but not the semantics [Hofweber] Full Idea: Representing arithmetic formally we do not primarily care about semantic features of number words. We are interested in capturing the inferential relations of arithmetical statements to one another, which can be done elegantly in first-order logic. From: Thomas Hofweber (Number Determiners, Numbers, Arithmetic , §6.3) A reaction: This begins to pinpoint the difference between the approach of logicists like Frege, and those who are interested in the psychology of numbers, and the empirical roots of numbers in the process of counting.