Theoretical Equivalence and Invariant Structure Hans Halvorson May 18, 2013
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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.
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