Abstraction with domain expansion

Jonathan Payne

Paris-Nancy PhilMath workshop28th September 2011

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Outline

Neo-Logicism and abstraction

Two interpretations: static/orthodox vs. dynamic/creative

Implementing ‘creative’ abstraction

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Outline

Neo-Logicism and abstraction

Two interpretations: static/orthodox vs. dynamic/creative

Implementing ‘creative’ abstraction

1.

2.

3.

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Neo-Logicism – abstraction principles

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!o*!o+!§(o*) = §(o+) " >(o*, o+)

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∀,∃

Abstraction operatorAbstract term

Abstraction relation

Abstraction domain

Neo-logicism – abstraction principles

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!?!@(G? = G@" ? # @)

!?!@!{q : ?q} = {q : @q} " !q(?q " @q)

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Direction principle (DP)

Basic Law V (BLV)

Hume’s principle (HP)

!e*!e+!=(e*) = =(e+) " e*||e+

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Neo-logicism – the aim

Start from:

No understanding of mathematical concepts

No knowledge of mathematics/existence of infinitely many objects

Get to:

An understanding of mathematical concepts

Knowledge of mathematics

Abstraction principles are implicit definitions of numerical concepts

The context principle/syntactic priority thesis

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Outline

Neo-Logicism and abstraction

Two interpretations: static/orthodox vs. dynamic/creative

Implementing ‘creative’ abstraction

1.

2.

3.

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Fine (2002) on definitions

‘Definitions of a standard sort are made from a standpoint in whichthe existence of the objects or items that are to assigned to thedefined terms is presupposed. The purpose of the definition is notto introduce new objects into the domain but to make anappropriate assignment of the objects already in the domain to theterms that are to be defined.’

‘Creative definitions […] are made from a standpoint in which theexistence of the objects that are to be assigned to the terms is notpresupposed. The purpose of the definition may indeed be toassign objects to the terms. But these objects are not selectedfrom a previously given domain, Rather the objects are introducedinto the discourse simultaneously with their assignment to theterms.’ (p.56)

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Hero

‘[I]ntroduce a faultlessly rational subject … and call him Hero.

[…]

Hero needs to be able to determine the truth-conditions of each of

the infinite series of statements,

t=Nx:x≠x,

(whereby he understands Frege’s term for 0 and predicate for 1);

t=Ny:[y=Nx:x≠x],

(whereby he understands Frege’s term for 1);

t = Nx:x≠x ∨ t=Ny:[y=Nx:x≠x].

(whereby he understands Frege’s predicate for 2);

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and so on’ (Wright 1998)

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A middle way – epistemic vs. semantic domains

Understanding a domain

so that it is determinate that it is the domain being quantified over

‘semantic domain’

Knowing about a domain

through explicit bits of knowledge/axioms etc.

‘epistemic domain’

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Against expansionism as an interpretation

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!?!@(G? = G@" ? # @)

Contains ∀,∃

(m)�B �q

:(m) :�q(q = m)

Outline

Neo-Logicism and abstraction

Two interpretations: static/orthodox vs. dynamic/creative

Implementing ‘creative’ abstraction

An external/metalanguage characterisation

Internal/object language characterisation

Metaphysics (metametaphysics?)

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An ‘external’ characterisation/picture

[see board]•

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The need for an internal characterisation

This presupposes sets/other objects.

Epistemological

Foundational/metaphysical

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Internally characterising

Go modal

◊φ = ‘it is possible to expand the domain to one for which φ comes outtrue‘ ‘it is possible to reinterpret the quantifiers so that φ is true‘ ‘it is legitimate to quantify in such a way so that φ is true under that’

Replacement for E!-I:

Non-rigid abstract terms and trans-world equivalence with actuality/scopeexemption:

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:(m) :� ��q(q = m)

!!?!@({q : ?q} = @{q : @q} " !q(?q " @@q))

Metaphysics

Objections:

Internal characterisation is either creationist or incoherent

How is ◊E!-I justified?

Reply:

A(t) ⇒ ‘t’ refers ⇒ can expand domain to include referent

Syntactic priority/context principle for quantifiers

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