Combining Philosophers

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

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6 ideas

6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / a. Numbers
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 [2005], 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?
'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 [2005], 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.
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'.
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.
6. Mathematics / A. Nature of Mathematics / 4. Using Numbers / f. Arithmetic
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.
6. Mathematics / A. Nature of Mathematics / 4. Using Numbers / g. Applying mathematics
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.