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Single Idea 10008

[filed under theme 6. Mathematics / C. Sources of Mathematics / 1. Mathematical Platonism / b. Against mathematical platonism ]

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.

Clarification

See Idea 10007 for the two roles of quantifiers

Gist of Idea

Arithmetic is not about a domain of entities, as the quantifiers are purely inferential

Source

Thomas Hofweber (Number Determiners, Numbers, Arithmetic [2005], §6.3)

Book Ref

-: 'Philosophical Review 114' [Phil Review 2005], p.219

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.

Related Idea

Idea 10007 Quantifiers for domains and for inference come apart if there are no entities [Hofweber]

The 47 ideas from Thomas Hofweber

16413 Science has discovered properties of things, so there are properties - so who needs metaphysics? [Hofweber]
16415 Esoteric metaphysics aims to be top science, investigating ultimate reality [Hofweber]
16416 The quantifier in logic is not like the ordinary English one (which has empty names, non-denoting terms etc) [Hofweber]
17988 Quantification can't all be substitutional; some reference is obviously to objects [Hofweber]
17989 Since properties have properties, there can be a typed or a type-free theory of them [Hofweber]
17990 Instances of minimal truth miss out propositions inexpressible in current English [Hofweber]
17991 Holism says language can't be translated; the expressibility hypothesis says everything can [Hofweber]
9998 What is the relation of number words as singular-terms, adjectives/determiners, and symbols? [Hofweber]
10000 We might eliminate adjectival numbers by analysing them into blocks of quantifiers [Hofweber]
10001 An adjective contributes semantically to a noun phrase [Hofweber]
10002 '2 + 2 = 4' can be read as either singular or plural [Hofweber]
10003 Why is arithmetic hard to learn, but then becomes easy? [Hofweber]
10004 Our minds are at their best when reasoning about objects [Hofweber]
10005 Arithmetic doesn’t simply depend on objects, since it is true of fictional objects [Hofweber]
10006 First-order logic captures the inferential relations of numbers, but not the semantics [Hofweber]
10008 Arithmetic is not about a domain of entities, as the quantifiers are purely inferential [Hofweber]
10007 Quantifiers for domains and for inference come apart if there are no entities [Hofweber]
21634 Metaphysics is (supposedly) first the ontology, then in general what things are like [Hofweber]
21635 Without propositions there can be no beliefs or desires [Hofweber]
21636 'Singular terms' are not found in modern linguistics, and are not the same as noun phrases [Hofweber]
21637 If two processes are said to be identical, that doesn't make their terms refer to entities [Hofweber]
21638 Syntactic form concerns the focus of the sentence, as well as the truth-conditions [Hofweber]
21639 'Background deletion' is appropriately omitting background from an answer [Hofweber]
21640 'It's true that Fido is a dog' conjures up a contrast class, of 'it's false' or 'it's unlikely' [Hofweber]
21641 Inferential role semantics is an alternative to semantics that connects to the world [Hofweber]
21643 The inferential quantifier focuses on truth; the domain quantifier focuses on reality [Hofweber]
21644 Numbers are used as singular terms, as adjectives, and as symbols [Hofweber]
21645 'Semantic type coercion' is selecting the reading of a word to make the best sense [Hofweber]
21646 The Amazonian Piraha language is said to have no number words [Hofweber]
21647 Logicism makes sense of our ability to know arithmetic just by thought [Hofweber]
21648 Neo-Fregeans are dazzled by a technical result, and ignore practicalities [Hofweber]
21649 How can words be used for counting if they are objects? [Hofweber]
21652 Our perceptual beliefs are about ordinary objects, not about simples arranged chair-wise [Hofweber]
21653 Maybe not even names are referential, but are just by used by speakers to refer [Hofweber]
21654 The "Fido"-Fido theory of meaning says every expression in a language has a referent [Hofweber]
21655 Compositonality is a way to build up the truth-conditions of a sentence [Hofweber]
21656 Proposition have no content, because they are content [Hofweber]
21657 Since properties can have properties, some theorists rank them in 'types' [Hofweber]
21658 Properties can be expressed in a language despite the absence of a single word for them [Hofweber]
21659 'Being taller than this' is a predicate which can express many different properties [Hofweber]
21660 Reality can be seen as the totality of facts, or as the totality of things [Hofweber]
21661 There are probably ineffable facts, systematically hidden from us [Hofweber]
21662 Do there exist thoughts which we are incapable of thinking? [Hofweber]
21664 Supervenience offers little explanation for things which necessarily go together [Hofweber]
21663 Counterfactuals are essential for planning, and learning from mistakes [Hofweber]
21665 The fundamental theorem of arithmetic is that all numbers are composed uniquely of primes [Hofweber]
21666 'Fundamentality' is either a superficial idea, or much too obscure [Hofweber]