Is Structural Underdetermination Possible?

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Is Structural Underdetermination Possible?
Theoretical Frameworks and Empirical
Underdetermination International Workshop,
University of Düsseldorf, April 10-12, 2008

• Holger Lyre
• Philosophy Department
• University of Augsburg/Bonn

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Overview

• The Thesis of Theory Underdetermination by
  Empirical Evidence (TUD)
• Intermediate Structural Realism
• Structural Underdetermination(and Mathematical
  Overdetermination)

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Part 1 The Thesis of Theory Underdetermination
by Empirical Evidence (TUD)
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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 TgtO, theory exceeds
  observation
• TUD-neighbors
• Duhem-Quine holism (confirmational holism)
• Humes problem of induction
• Conventionalism

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TUD, 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 observations
• Induction problem as Humean underdetermination
• underdetermination of theory by past evidence
• BUT underdetermination even in the case of all
  possible (past and future) observations

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TUD 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 doesnt exist
  (many) interesting examples is indeed a
  pressing problem for TUD!

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Classes of TUD examples

• Pathological cases
• Cases of epistemic TUD
• Conventionalism-borderline cases
• History of science cases
• Examples from factual, mature science...!?

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Pathological cases of TUD

• Consider a law G(Xt) 0 versus G(Xt) 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...

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Epistemic TUD

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

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Conventionalism-borderline cases

• Poincaré space as infinite versus space as
  finite with suitably shrinking measuring rods
• Hollow earth
• inversion atthe sphere r' R2/rwith 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

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Conventionalism-borderline cases (II)

• rather a reconstrual of predicates convex
  picture concave picture straight
  line circle point at infinite
  distance center center 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?

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History of science examples of TUD

• TN Newtonian mechanics and gravityH(v)
  hypothesis of absolute space with center of
  mass velocity vTN H(v) infinitely many
  theories for v ? R
• Lorentz ether theory vs. Einsteins Special
  Relativity
• BUT in retrospect historic cases appear mainly
  as artefacts of incomplete scientific knowledge!

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Examples from actual, mature science...?!

• Gauge Theories
• Quantum Mechanics (part 3, briefly)
• General Relativity (part 3)

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Ontology of gauge theories (I)

• Explanation of gauge theoretic holonomy effects
  (e.g. Aharonov-
• Bohm) by different entities and different
  locality assumptions

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Locality 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.

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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...

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Conclusion part 1 lots of problems with TUD

• its 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!

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Part 2 Intermediate Structural Realism
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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

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SR-views about structure

• Epistemic SRThere are relations and (maybe)
  relata, but we have epistemic access to relations
  only
• Ontic SR
• non-eliminative (moderate) OSRThere are
  relations and relata, but there is nothing more
  to the relata than the relations in which they
  stand
• eliminative OSRThere 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 doesnt
  depend on the ESR/OSR distinction

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An example Michael Esfelds 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
  SRthings exist, but the relations in which
  they stand are all there is to the things at the
  basic level

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Problems 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(conti
  nuity equation, currents, theorems of Gauss and
  Stokes)?
• SU(2) of strong or weak isospin
• U(1) as temporal or gauge group
• and many more
• The examples already show the structural content
  of modern physics theories is mostly given by
  their symmetry structure

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Symmetries 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 transformations
• coordinate transformations
• gauge transformations!
• ? only invariants allow for a realistic
  interpretation!

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Gauge transformations no real instantiations!

• Well-known for global gauge transformations

• Less well-known for local gauge transformations

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SR 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!

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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 dont collapse into
  entity realism?
• Possible solution to the problem of unintended
  domainsTheory models are individuated via
  structurally derived intrinsic properties
  connected with those particular phenomena which
  make up our data models

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Conclusion Part 2 Intermediate SR

• ESR There are relations and (maybe) relata, but
  we have epistemic access to relations only
• OSR
• non-eliminative variants
• Moderate SROnly 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)?

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Part 3 Structural Underdetermination (and
Mathematical Overdetermination)
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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?

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Is 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 CCRs)
• SR should focus on the relevant structural core

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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)

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Four 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???

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

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Remarks on the translational gauge theory
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Is structure unique? The case of GR

• Reconsider the ontology of the AB effect
• Rivaling interpretations, but the U(1)-gauge
  group structure is invariant
• SR 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!

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Interim mathematical overdetermination

• Erhard Scheibe mathematical overdetermination
  (MOD) of modern physical theories T MgtP, 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!

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SR 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
• Dismiss structural differences as mere cases of
  mathematical overdetermination and identify the
  relevant structural core
• 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

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General 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