Theoretical Equivalence and Invariant Structure

Hans Halvorson

May 18, 2013

• If T1 and T2 are equivalent, then what must they share incommon? What are the “invariants” of theoreticalequivalence?

• What is preserved in theory change? What are the“invariants” of theory change?

• What parts of our current theories ought we to believe?

Purported theoretical equivalences

Schrodinger ∼= Heisenberg (von Neumann)

Lagrangian ∼= Hamiltonian (Legendre)

Newtonian Gravity ∼= Newton-Cartan Gravity (Weatherall)

Spacetimes ∼= Leibniz Algebras

Geometry of Points ∼= Geometry of Lines

Mereological Nihilism ∼= Compositionalism (Hirsch)

Relativism ∼= Expressivism (Dreier)

Naive accounts of equivalence

• T1 and T2 are equivalent just in case they say the same thing,i.e. they have the same content.

• Correct but uninformative. How to specify content?

• T1 and T2 are equivalent just in case they make the samepredictions.

• T1 and T2 are equivalent just in case they are true in exactlythe same possible worlds.

• T1 and T2 are equivalent just in case they have the same setof models.

“Theories are extralinguistic entities which may bedescribed or characterized by a number of differentlinguistic formulations.” Suppe 1977

“While a theory may have many different formulations,its set of models is what is important.” van Fraassen 2008

T1 T2

M

Spectrum of equivalence

same stringof symbols

same language

same axioms

theoreticalequivalence?

empiricalequivalence

finer coarser

consistent?(Putnam)

everybodydisagrees

nobodydisagrees

Definitional equivalence

• Empirical equivalence was a suitable notion for verificationists,circa 1920–1950.

• The idea of definitionally equivalent or synonymous theoriesdue to Tarski or Montague, circa 1960.

• Co-opted for philosophy of science by Glymour: “Theoreticalrealism and theoretical equivalence” 1970.

Definitional equivalence

T1 T2

Language {c} {P}

Axioms ∅ ∃!xPx

Definitional equivalence

T1 T2

Language {◦,−1, e} {◦, e}

Axioms associativity associativitye is an identity e is an identityx ◦ x−1 = e = x−1 ◦ x ∃!y(x ◦ y = e = y ◦ x)

Interpretations and model maps

DefinitionAn interpretation or translation of T1 into T2 consists of a map Fthat takes n-place relation symbols of L1 to n-place open formulasof L2, and such that:

T1 ` φ =⇒ T2 ` φF

T1

T2

F

M1

M2

F ∗

Definitional extension

DefinitionT ′ is a definitional extension of T if:

• T ′ explicitly defines each predicate symbol R of L′ in terms ofthe language L:

T ′ |= ∀x(R(x)↔ φR(x))

• T ′ is a conservative extension of T .

Definitional extension

T ′

T

PI

M′

M

I∗ P∗

• T and T ′ are mutually interpretable.• For each sentence φ of L′

T ′ |= (IPφ)↔ φ

• For each sentence ψ of L

T |= (PIψ)↔ ψ

• The category of models of T is isomorphic to the category ofmodels of T ′.

Common definitional extension

DefinitionTwo theories T1 and T2 are definitionally equivalent just in casethey have a common definitional extension.

T ′

T1 T2

P1 P2

I1 I2

M′

M1 M2F

G

Invariants of definitional equivalence

M1 M2

M0

These models describe the same worldA

A|L1 A|L2

[φ(x)]

[φ(x)]

[ψ(x)]

• F :M1 →M2 is an equivalence of categories.• F(A) and A are isomorphic

• have the same symmetry groups;• domains have same cardinality;• same definable sets.

Cutting nature at its joints

Model Isomorphism Criterion: If two theories are equivalentthen their models are pairwise isomorphic.

• Hamiltonian mechanics and Lagrangian mechanics are notequivalent.

• Spacetime theories and Leibniz algebra theories are notequivalent.

Structural Realism: A theory is true if the WORLD is isomorphicto one of its models.

Model IsomorphismCriterion

Against the model isomorphism criterion

Against the model isomorphism criterion

<{z}

={z}

−4

−4i

−3

−3i

−2

−2i

−1

−1i

11i

2

2i

3

3i

c 7−→ 〈Re(c), Im(c)〉r + is ←− [ 〈r , s〉

Against the model isomorphism criterion

• A theory already implicitly quantifies over n-tuples:

∃x1∃x2 · · · ∃xnφ(x1, . . . , xn)

• Tarski, Szczerba: we must recognize equivalence of geometrieswith points as primitives and geometries with lines asprimitives

Multi-sorted logic

• Begin with a collection A,B,C , . . . of sorts.• Each symbol (relations, functions, variables) of L is required

to be coded by sorts. example R : A,A,B• Quantifiers only range over particular sorts.• Equality is sort-relative: =A

• An L structure assigns a set/domain/universe to each sort.

Defining new universes

Product sorts (products): The formula (x1 = x1) ∧ · · · ∧ (xn = xn)defines a product sort S1 × · · · × Sn

Reduced sorts (subobjects): A formula φ(x) of L defines areduction of the sort S1 × · · · × Sn.

Proposal: T+ should be considered a definitional extension of T .

Defining new universes

Suppose that the formula φ(x , y) defines an equivalence relationin T . Let T eq be the theory that adds a new sort E , functionsymbol f : A;E , and axioms

∀xA∀yA(φ(x , y)↔ f (x) = f (y))∀zE∃xA(f (x) = z)

Proposal: T eq should be considered a definitional extension of T .

Morita equivalence

The Hard Way: Each theory T has a classifying topos E [T ]. T1is Morita equivalent to T2 just in case E [T1] and E [T2] areequivalent toposes.

The Easy Way: Theories T1 and T2 are Morita equivalent just incase they have a common definitional extension — in the extendedsense where new sorts are definable from old.

Logical accounts of equivalence

samemodels

definitionallyequivalent

mutuallyinterpretable

finer coarser

Moritaequivalent

Invariants

Morita equivalence: equivalent categories of modelsDefinitional equivalence: cardinalities of individual models,

definable subsets

Note: “Equivalent categories of models” is necessary but notsufficient for theoretical equivalence.

Example: X versus X ∪ {∗}.

The case for Morita equivalence

• Argument by squeezing: Mathematical examples suggest thatother criteria are too fine.

• Argument from mathematical convergence: It’s a naturalnotion, both in logic and in category theory.

• Realizes Putnam’s dream of sophisticated realism. (He himselfabandoned hope!)

Concluding unscientific proposal

Proposal: If T1 ∼= T2, then one should commit only to theinvariants, i.e. features on which T1 and T2 agree.

Problem: The models of T1 and T2 might not agree on anythinginteresting.

Prospect: There is a more nuanced way in which the modelsdescribe the same world.