Is Structural Underdetermination Possible?

Is StructuralIs Structural

UnderdeterminationUnderdetermination

Possible?Possible?

Holger LyrePhilosophy DepartmentUniversity of Augsburg/Bonn

Theoretical Frameworks and Empirical Underdetermination

International Workshop, University of Düsseldorf,

April 10-12, 2008

Theoretical Frameworks and Empirical Underdetermination, Düsseldorf, 10-12 April 2008

OverviewOverview

1. The Thesis of Theory Underdetermination by Empirical Evidence (TUD)

2. Intermediate Structural Realism

3. Structural Underdetermination(and Mathematical Overdetermination)


Part 1:Part 1:

The Thesis of Theory UnderdeterminationThe Thesis of Theory Underdeterminationby Empirical Evidence (TUD)by Empirical Evidence (TUD)


The TUD thesis and its „neighbors“The TUD thesis and its „neighbors“

Thesis of Theory Underdetermination (TUD)For any theory T and any body of observation O there exists another theory T‘, such that T and T‘ are empirically equivalent (but ontologically different).

main intuition behind TUD: „T>O“, theory exceeds observation

TUD-„neighbors“:Duhem-Quine holism (confirmational holism)Hume‘s problem of inductionConventionalism


TUD, Duhemian holism, inductionTUD, Duhemian holism, induction

Duhemian confirmational holism: no scientific hypothesis can be tested in isolation it is possible to adhere to a thesis (in the face of adverse

observations) by revising other theses theories can only be confirmed as a whole small gap to TUD: generate rivaling theories by readjusting

the total system of hypotheses BUT: even the total system is underdetermined by all

possible observationsInduction problem as „Humean underdetermination“: underdetermination of theory by past evidence BUT: underdetermination even in the case of all possible

(past and future) observations


TUD and the problem of missing examplesTUD and the problem of missing examples

Earman (1993): „Are there interesting cases of empirically indistinguishable theories?... Here I find the philosophical literature disappointing... [but] I claim ... That what we have is a shortcoming of the philosophical literature and not a failure of the underdetermination thesis.“

Is Earman right? Given the generality of the TUD thesis, the fact that there doesn‘t exist (many) „interesting“ examples is indeed a pressing problem for TUD!


Classes of TUD examplesClasses of TUD examples

Pathological cases

Cases of „epistemic TUD“

Conventionalism-borderline cases

History of science cases

Examples from factual, mature science...!?


Pathological cases of TUDPathological cases of TUD

Consider a law G(X;t) = 0 versus G(X;t) + g(t) = 0,where g(t) = go(t) (t - t1) (t - t2) ...with arbitrary go(t) and measuring times ti.

„The world has been created ... just 5 minutes ago / ... in 7 days“ (creationism vs. evolution)

always possible, but not really `thrilling‘...

take notice of simplicity, economy, coherence...


„„Epistemic TUD“Epistemic TUD“

1. General boundaries of technical feasability in experiments (e.g. string energies?)

2. Glymour (1977) and Malament (1977) consider cosmologies where even idealized observers (who live forever) cannot decide about certain global topological features

BUT: As an anti-realistic argument we should be interested in „ontological TUD“


Conventionalism-borderline casesConventionalism-borderline cases

Poincaré: space as infinite versus space as finite with suitably shrinking measuring rods

„Hollow earth“: inversion at

the sphere:r' = R2/r

with earth radius R

nice example for Duhemian holism: avert possibility to attack the theory by a `journey to the center of the earth‘ by introducing an additional „wrap around“-hypothesis


Conventionalism-borderline cases (II)Conventionalism-borderline cases (II) rather a reconstrual of predicates:

convex picture concave picturestraight line circlepoint at infinite distance centercenter of earth point at infinite distance

Quine (1975): „[The] case ... due to Poincaré ... is less simple than the mere switch of ‚electron‘ and ‚molecule‘, but it presents no serious challenge. The two formulations are formulations, again, of a single theory.“

BUT: Poincaré and „hollow earth“ assume distinguished points

(center of earth and point of infinity)violation of the „Principle of Homogeneity of Space“Quine (1990): „...drastically unlike theory formulations“sufficient for „ontological difference“?


History of science examples of TUDHistory of science examples of TUD

1. TN: Newtonian mechanics and gravityH(v): hypothesis of absolute space

with center of mass velocity vTN+ H(v): infinitely many theories for v R

2. Lorentz‘ ether theory vs. Einstein‘s Special Relativity

BUT: in retrospect historic cases appear mainly as artefacts of incomplete scientific knowledge!


Examples from actual, mature science...?!Examples from actual, mature science...?!

Gauge Theories

Quantum Mechanics (part 3, briefly)

General Relativity (part 3)


Ontology of gauge theories (I)Ontology of gauge theories (I)

Explanation of gauge theoretic holonomy effects (e.g. Aharonov-

Bohm) by different entities and different locality assumptions:


Locality assumptionsLocality assumptions

Point-like interactionInteracting entities can be defined within arbitrarily small spacetime regions (usually idealized as spacetime points), couple to each other in that regions and are non-zero in overlapping regions.

Local actionAll causes of an event propagate via some continuous physical processes.

SeparabilityGiven a physical system S and its exhaustive, disjoint decomposition into spatiotemporally divided subsystems, it is possible to retrieve the properties of S from the properties of these subsystems.


Ontology of gauge theories (II)Ontology of gauge theories (II)

Interference shift of AB effect by Stokes formula:

Potentials Field strengths Holonomies

Point-like int. yes no no

Local action yes no yes

Separability no yes no

Measurability no yes yes

Good example of TUD in mature science? Perhaps… …BUT: Holonomy view is presumably the most coherent one with

the biggest potential for further developments...


Conclusion part 1: lots of problems with TUDConclusion part 1: lots of problems with TUD

it’s quite doubtful that the problem of missing TUD examples is a mere „... shortcoming of the philosophical literature“ (Earman) – it rather strongly undermines the plausibility of TUD itself!

The rare TUD cases in the actual practice of science may rather be considered artefacts of incomplete knowledge! as such they could be used as an indication of open

scientific problems! The TUD thesis is at best a highly speculative thesis with a

pressing problem of missing examples!


Part 2: Intermediate Structural RealismPart 2: Intermediate Structural Realism


What is a “structure”?What is a “structure”?“Define a system to be a collection of objects with certain

relations among them. [...] Define a pattern or structure to be the abstract form of the system, highlighting the interrelationships among the objects, and ignoring any features of them that do not affect how they relate to other objects in the system.”

Shapiro (2000)

Working definition: structures as domains (= sets of objects) with sets of relations imposed on them:

set a = {a1, a2, ...an} of n objects ai and arbitrary k-ary

relations R(a), then structure ∑ = a,R(a) Structural realism (SR): roughly the view that we should be

committed in the structural rather than object-like content of our mature scientific theories


SR-views about structureSR-views about structure

Epistemic SR:There are relations and (maybe) relata, but we have epistemic access to relations only

Ontic SR:non-eliminative (moderate) OSR:

There are relations and relata, but there is nothing more to the relata than the relations in which they stand

eliminative OSR:There are only relations and no relata

Note: The widely debated question whether the slogan „structures is all there is“ leads to the problem of “relations without relata” doesn’t depend on the ESR/OSR distinction


An example: Michael Esfeld’s Moderate SR (2004)An example: Michael Esfeld’s Moderate SR (2004)

“Quantum entanglement shows that there are [cor-] relations among physical systems … [which] … amount to the whole having intrinsic properties that do not supervene on intrinsic properties of the parts.”

this in turn suggests “replacing a metaphysics of intrinsic properties with a metaphysics of relations”

From the metaphysics of relations to Moderate SR:things exist, but “the relations in which they stand are all there is to the things at the basic level”


Problems of Moderate SRProblems of Moderate SR

taken literally, the idea to individuate theories by means of their pure structural content is far too weak

problem of “unintended domains”! physical examples of “structural equivalents”:

Classical electrodynamics vs. hydrodynamics(continuity equation, currents, theorems of Gauss and Stokes…)

SU(2) of strong or weak isospinU(1) as temporal or gauge groupand many more…

The examples already show: the structural content of modern physics theories is mostly given by their symmetry structure


Symmetries in physicsSymmetries in physics

Symmetry of a domain D: set of one-to-one mappings of D onto itself (symmetry transformations), such that the structure of D is preserved

Symmetry transformations form a group and exemplify equivalence relations (partitioning of D into equivalence classes)

Note the distinction between symmetries with real instantiations (e.g. space-time transformations) as opposed to symmetries without real instantiations:scale transformationscoordinate transformationsgauge transformations!

only invariants allow for a realistic interpretation!


Gauge transformations: no real instantiations!Gauge transformations: no real instantiations!

Well-known for global gauge transformations:

AA

iA e

expectation value

of some observable does not change under

y y

y ycy y

=

®

Less well-known for local gauge transformations: x x

{ }

i (x) ˆx e x U x

wavefunction in position representation

with spanning an abstract Hilbert space

local gauge transformations are changes in the

position representation

Hilbert space op

ˆ ˆˆ ˆO ' UOU

ˆ ˆ ˆp p ' p (x)

erators transform as ,

in particular


SR and gauge symmetriesSR and gauge symmetries

Gauge theories are the most important case of symmetry structure in modern physics

BUT: Gauge symmetries do not posses real instantiations

only invariants allow for a realistic interpretation! eigenvalues of Casimir operators m, s, q ...

General feature of a “symmetry-based” physics Mass, spin, charge as paradigmatic cases of …

… intrinsic properties(“A thing has its intrinsic properties in virtue of the way that thing itself, and nothing else, is.“ - D. Lewis)

… but “structurally derived”: do not suppose objects independently of structure!


„„Structurally derived intrinsic properties…“Structurally derived intrinsic properties…“

Note: group invariants lead to object classes only! Such „structurally derived intrinsic properties“ do not

individuate objects, but may nevertheless apply to lone objects

Hence, variants of SR including structurally derived intrinsic properties don‘t collapse into entity realism

Possible solution to the problem of unintended domains:Theory models are individuated via structurally derived intrinsic properties connected with those particular phenomena which make up our data models


Conclusion Part 2Conclusion Part 2: “Intermediate SR”: “Intermediate SR”

ESR: There are relations and (maybe) relata, but we have epistemic access to relations only

OSRnon-eliminative variants:

„Moderate“ SR:Only relational, but no intrinsic properties exist

Intermediate SR (ISR):Relational and structurally derived intrinsic properties exist (as invariants of structures)

eliminative variant („relations without relata“)


Part 3:Part 3:Structural UnderdeterminationStructural Underdetermination(and Mathematical Overdetermination)(and Mathematical Overdetermination)


SR as an antidote against TUD?SR as an antidote against TUD?

SR is sometimes considered an antidote against the anti-realistic threat of TUD

TUD undermines entity content (only) SR seems to avoid the threat of TUD by not

committing us to entities BUT:

Can we make sure that the structural content of theories is not underdetermined either?

Are there cases of structural TUD?


Is structure unique? – The case of QMIs structure unique? – The case of QM

QM not only provides a case for TUD (with its different interpretations), but perhaps for “structural TUD”

Different interpretations, but perhaps not different mathematical structure

BUT YES: QM in Hilbert space, phase space, operator algebras, lattices…

BUT NO: only a certain structural core is connected with the empirical evidence of QM e.g. non-commutative algebra as structural core

(Stone-von Neumann-theorem: uniqueness of the

Heisenberg CCR’s) SR should focus on the relevant structural core…


Is structure unique? – The case of GR…Is structure unique? – The case of GR…

“... there is no unique “gauge theory of gravitation.” ... this is due to the fact that gravitation is a “rich” theory from the geometrical point of view: it contains several invariants which may be used to build the kinetic part of the gravitational Lagrangian.”

Andrzej Trautman (1980)


Four versions of General RelativityFour versions of General Relativity

Consider the following four formulations of General Relativity:

• GR in Riemannian spacetime entities: g, R

• GR in flat Minkowskian spacetime entities: g = + h

• GR as Lorentz gauge theory / gauge group SO(1,3) entities: tetrads

, curvature tensor R

• GR as translational gauge theory / gauge group G = R1,3

entities: tetrads , torsion tensor F


Gronwald and Hehl (1996): On the Gauge Aspects of Gravity, gr-qc9602013


Remarks on the translational gauge theoryRemarks on the translational gauge theory


Is structure unique? – The case of GRIs structure unique? – The case of GR

Reconsider the ontology of the AB effectRivaling interpretations, but the U(1)-gauge group structure

is invariantSR should focus on this core structure only (or rather on the

gauge group invariants)SR seems to provide a solution to this TUD scenario

In GR, however: there is even dispute about the gauge group itself!No structural core between the four rivaling variants of GR!Apparently not only a case for entity, but for structural TUD!


Interim: mathematical overdeterminationInterim: mathematical overdetermination

Erhard Scheibe: “mathematical overdetermination” (MOD) of modern physical theories T:

“M>P”, the mathematical exceeds the physical (parts of T)

Michael Redhead: “surplus structure” General problem: Is there a non-circular way to distinguish

between relevant physical and surplus structure?

SR needs an account of distinguishing M and P, otherwise the position transforms into Platonism(SR proponents commit themselves to physical in re-structure in the world only)

to focus on “relevant” structure is a special task for SR!


SR options in the face of structural TUD in GRSR options in the face of structural TUD in GR

There are basically two options for SR in order to deal with the case to structural TUD in GR:

1. Dismiss structural differences as mere cases of mathematical overdetermination and identify the relevant structural core

2. Declare the case of GR as a less interesting case since GR is not expected to be fundamental (TUD scenarios of GR can be dismissed as artifacts of incomplete knowledge)

Both options can of course never guarantee the impossibility of any structural TUD, but would help to undermine the power of an anti-SR argument based on it


General ConclusionGeneral Conclusion Gauge or, more generally, surplus structure from mathematical

overdetermination is a problem for SR SR needs an account of “relevant” as opposed to surplus structure SR should take relational and structurally derived intrinsic

properties (as invariants of structures) into account TUD is a speculative thesis with a pressing problem of missing

examples (from factual, mature science) “Practical TUD” rather an indicator for incomplete knowledge Two of three supposedly still existing interesting cases (QM and

U1-gauge theory) can be circumvented by adopting SR (by focusing on the structural core)

The third case, GR, seems to provide a case of structural TUD and, hence, a serious threat to SR!

Two options have been indicated to circumvent this threat

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Is Structural Underdetermination Possible?

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