abstraction reconceptualised

Neologicism Static Abstraction Bad Company Dynamic abstraction

Abstraction ReconceptualisedA dynamic approach to the bad company problem

James Studd

The Philosophy of Mathematics SeminarUniversity of Oxford

6 May 2013


Outline

1 Neologicism

2 How abstraction works I (static account).

3 Bad Company: Scylla/Charybdis generalised.

4 How abstraction works II (dynamic account)


Scottish neologicismNeologicists ground mathematical knowledge in abstraction.

(1) Abstraction principles (APs) enjoy a special epistemic status

e.g. Hume’s Principle (HP): #X = #Y ↔ X ≈ YThe number of Xs = the number of Ys

iff the Xs are just as many as the Ys.

(2) APs suffice to naturally recover standard mathematics

e.g. Frege’s TheoremHP + SOL interprets second-order PA (i.e. Z2)

(1) + (2) provide access to knowledge about abstracta. . .. . . but both are open to doubt.

James Studd Abstraction Reconceptualised 1/17


(Q1) Do APs enjoy a special epistemic status?Two sub-questions:

(Q1a) Why can we stipulate HP but not PA? (Or ∃x(God(x))!)

(A1a) There’s something special about APs.

(A1a): not the whole story: not all APs can be stipulated true.

e.g. (BLV): ext(X) = ext(Y)↔ X ≡ Y Inconsistent!The extension of X = the extension of Y

iff the Xs are the same things as the Ys.

(Q1b) What distinguishes ‘bad’ APs from ‘good’ ones?

(A1b) (i) Specify a criterion C that precisely good APs meet.e.g. C = Stable, Strongly Stable, Irenic, etc.

(ii) Defend a philosophical rationale for C.



(Q2) Do APs suffice to recover mathematics?What about theories other than arithmetic?(Q2a) Do APs interpret analysis? e.g. Hale (2000), Cook (2002)

(Q2b) Do APs interpret ZFU? Early attempt: Boolos (1989)

New V: ext(X) = ext(Y)↔ ((Un(X) ∧ Un(Y)) ∨ X ≡ Y)where Un(X) =df ∃U(X ≈ U ∧ ∀xUx) (X is universe-sized.)

New V permits us to naturally recover a fair tranche of ZFU.

Boolos (1989)New V + SOL interprets: Extensionality + Empty Set + Pairing+ Choice + Separation + Pure Union + Pure Regularity

(But not: Powerset or Infinity.)



DilemmaC fails to rule out bad company or rules out good company.

Scylla/Charybdis (very first approximation)Suppose C = Stable, Strongly Stable or Irenic.

Scylla (too strong): Without C, Neologicism ` ⊥BLV is singly inconsistent; other APs jointly inconsistent.

Charybdis (too weak): With C, Neologicism loses route to ZFU.New V is neither Stable, nor Strongly Stable, nor Irenic.

Possible responses:(i) seek other Cs

(ii) seek other APs.



Outline

1 Neologicism





(Q1a) Why can we stipulate APs? What is special about APs?(A1a.i): The process of ‘abstraction’ makes them true.

Frege on abstractionThe judgement . . . a//b, can be taken as an identity. If we do this,we obtain the concept of direction, and say: ‘the direction of linea is identical with the direction of line b’. Thus we replace thesymbol // with the more generic symbol =, through removingwhat is specific in the content of the former and dividing itbetween a and b. We carve up the content in a way differentfrom the original way, and this yields us a new concept.(Grundlagen, §66)

‘Recarving contents’Contents of equivalence statements attached to identity statements.

e.g. The content of a//b is attached to Da = Db



Two neologicist desiderataTwo desiderata are apparent if abstraction is to help neologicists.

Abstraction must result in an interpretation I1 meeting ND1.

ND1. APs are true under I1 (at least in good cases)

We restrict ourselves to purely logical second-order APs:

Static abstraction principlesThese are the universal closures of the following:

(AP) §X = §Y ↔ X ∼ Y

§ a function symbol; §X a singular term; X ∼ Y purely logical.

Second desideratum: for Frege’s thm. to ensure truth of PA

ND2. Logic is (materially) sound under I1.James Studd Abstraction Reconceptualised 6/17


How static abstraction worksSuppose I0 standardly interprets logical expressions over M0.How does abstraction result in I1?

Wright (1997), Hale (1997) (and some rational reconstruction):

S1. ‖§X = §Y‖σI1

= ‖X ∼ Y‖σI0

.Notation: ‖φ‖σ

Iis the content of φ under I and σ.

Comments:Syntax is to be taken at face value.Sentential content is ‘coarse grained’ or ‘unstructured’.Non-standard interpretation: S1 has a top-down character.

Worry: do we meet ND2? Is logic sound?Yes: for fragment interpreted by S1 provided C0 is met.C0: X ∼ Y expresses an equivalence relation under I0

No: for whole language.James Studd Abstraction Reconceptualised 7/17


But might we not supplement S1 with further stipulations?

Staticness stipulatedNeologicists seem to assume S2 in addition to S1

S1. ‖§X = §Y‖σI1

= ‖X ∼ Y‖σI0

.S2. Logical expressions retain antecedent interpretations over M0

a. ‖∀v‖I1

= ‖∀v‖I0

b. ‖=‖I1

= ‖=‖I0

c. ‖∧‖I1

= ‖∧‖I0

d. ‖¬‖I1

= ‖¬‖I0

NB: bottom up character: S2 operates on subsentential content.

Can we make both stipulations?Each of S1 and S2 seem individually innocuous.But can both be stipulated simultaneously?



The source of inflation.S1 and S2 make conflicting demands on the size of the universe.

Suppose: M0 universe of I0 has cardinality κ0

M1 universe of I1 has cardinality κ1

e.g. BLV

S1. ‖ext(X) = ext(Y)‖σI1

= ‖X ≡ Y‖σI0

κ1 ≥ 2κ0 > κ0

S2. ‖∀v‖I1

= ‖∀v‖I0

κ1 = κ0

Similar trouble whenever X ∼ Y is ‘inflationary’ on M0.(i.e. |℘(M0)/ ∼ | > |M0|)e.g. HP is inflationary on finite M0



Outline

1 Neologicism





Bad company: inconsistent APsThe neologicists needs C to rule out inconsistent APs

Notn: Σ-C =df = SOL + APs that meet C.

First try: C0: ∼ expresses an equivalence relationMotivated by neologicist account of abstraction.

Scylla: Σ-C0 3 BLV ` ⊥.

Second try: C1: APs must be satisfiableRules out BLV (and other inconsistent APs)

Scylla: Σ-C1 ⊃ {HP, PP} � ⊥Parity Principle (PP) only satisfiable on finite universes.

Further tries: C: APs must be ‘conservative’ (+ X)Roughly: APs must yield no new consequences for old ontology

This idea has been regimented and refined in numerous ways.James Studd Abstraction Reconceptualised 10/17


Putative conditions CsSay that a sentences φ is κ-satisfiable if it has a model whose universehas cardinality κ. Sentence φ is said to be:

(i) Unbounded if for any γ, there is κ ≥ γ s.t. φ is κ-satisfiable

(ii) Stable if there is some γ s.t. for all κ ≥ γ φ is κ-satisfiable ifκ ≥ γ

(iii) Strongly Stable if there is some γ s.t. for all κ ≥ γ φ isκ-satisfiable iff κ ≥ γ

(iv) Field Conservative if for any theory Γ and sentence χ that eachlack §: Γ¬A, φ � χ¬A only if Γ � χ.

(v) Irenic if Field Conservative and jointly satisfiable with everyField Conservative sentence.



Good company: New V and friendsThe good company are APs used in the recovery of ZFU.

(RV): ext(X) = ext(Y)↔ ((Un(X) ∧ Un(Y)) ∨ X ≡ Y)

Various proposals for ‘The Xs are Collectable (¬Un(X))’:

New V (Boolos): The Xs are fewer than everything¬Un(X) =df ∃U(|X| < |U| ∧ ∀xUx)

Small2 V (Hale): The Xs are fewer than fewer than everything¬Un(X) =df ∃U∃Y(|X| < |Y | < |U| ∧ ∀xUx)

Newer V (Cook): The Xs are all available at some ordinal stage.¬Un(X) =df ∃α∀x(Xx→ x ∈S Stg(α))

Such RV-based attempts recover portions of ZFU(New + Newer V interpret all of ZF)



Scylla and Charybdis (refined version)Scylla: Σ-C is inconsistent.

SatisfiableScylla↑ Unbounded Field Conservative ↑Scylla

Charybdis↓ Stable Irenic ↓CharybdisStrongly Stable

Charybdis: C rules out the recovery of unrestricted ZFUΣ-C 0 ZFU (given: df (ß) + df (∈))Σ-C 0 Countable Plenitude ∧ Extensionality (given: df (ß) + df (∈))

Countable Plenitude: Any (up to) countably many things form a set

ZFU ` ∀X(|X| ≤ ℵ0 → ∃x(ßx ∧ ∀z(z ∈ x↔ Xz)))

Responses: (ii) Restricted universe? (ii) new C?James Studd Abstraction Reconceptualised 13/17


Outline

1 Neologicism





Dynamic abstractionTwo key differences to the static account:

(1) We don’t assume the universe remains static.We reject S2, which ensured M0 = M1

Releases cardinality pressure: Inflationary APs need not be bad

Instead they cause the universe to increase: M0 ⊂ M1

(2) This allows abstraction to be iterated.Inflationary abstraction: new abstracts engender newintermediaries

e.g. Suppose M0 has cardinality κ.Abstracting with BLV:M1 contains 2κ setsM2 contains 22κ sets . . .



S1F. For β > α: ‖§X = §Y‖σIβ

= ‖X ∼ Y‖σIα

Adequacy condition if logic is to be sound.CF0 . (i) X ∼ Y is an equivalence relation over ℘(Mα) under Iα

(ii) ‘Stability’: ‖X ∼ Y‖σIα1

= ‖X ∼ Y‖σIα2

for σ over both Mα1 and Mα2 .

S2F. Logic is rendered sound by a suitable distribution ofunstructured sentential contents, e.g.:

‖φ ∧ ψ‖σIα

= {w | w ∈ ‖φ‖σIα

and w ∈ ‖φ‖σIα};

‖∀vφ‖σIα

= {w | w ∈ ‖φ‖σ[a/v]Iα

for every a in M1,w}

S3F. Subsentential semantic values over Mα are selected asthose that compositionally recover the distribution ofsentential contents specified in S1F and S2F.



Dynamic APsWe give up ND1: dynamic abstraction does not render APs true.

Static: ‖§X = §Y‖σI1

=S1 ‖X ∼ Y‖σI0

=S2 ‖X ∼ Y‖σI1

Dynamic: ‖§X = §Y‖σIβ

=S1F ‖X ∼ Y‖σIα

, for β > α

Mathematical modalityDescribe the result of dynamic abstraction in a modal language:

�>ψ is true under Iα iff ψ is true under Iβ for all β > α

�<ψ is true under Iα iff ψ is true under Iβ for all β < α

Dynamic APsThese are the necessitated universal closures of the following:

�>§X = §Y ↔ X ∼ Y(APF1 )

�∀x(^∃X(x = §X)→ ^<∃Y^(x = §Y))(APF2 )James Studd Abstraction Reconceptualised 16/17


Scylla/Charybdis RevisitedHow does dynamic abstraction fare with Scylla/Charybdis?Notn: ΣF-CF = SOL + modal axioms + dynamic APs meeting CF

Natural first try: CF0 : ∼ stably expresses an equivalence relation

Scylla (being too strong )Static: Σ-C0 is inconsistent.Dynamic: ΣF-CF0 is consistent

Charybdis (being too weak)Static: Cs avoiding Scylla rule out recovering unrestricted ZFU.Dynamic: ΣF-CF0 interprets unrestricted ZFU(provided we iterate far enough.)


abstraction reconceptualised

Spiritual