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Is Structural Underdetermination Possible?

Holger Lyre Philosophy Department University of Augsburg/Bonn. Theoretical Frameworks and Empirical Underdetermination International Workshop, University of Düsseldorf, April 10-12, 2008. Is Structural Underdetermination Possible?. Overview.

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Is Structural Underdetermination Possible?

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  1. Holger Lyre Philosophy Department University of Augsburg/Bonn Theoretical Frameworks and Empirical Underdetermination International Workshop, University of Düsseldorf, April 10-12, 2008 Is Structural Underdetermination Possible? Theoretical Frameworks and Empirical Underdetermination, Düsseldorf, 10-12 April 2008

  2. Theoretical Frameworks and Empirical Underdetermination, Düsseldorf, 10-12 April 2008 Overview The Thesis of Theory Underdetermination by Empirical Evidence (TUD) Intermediate Structural Realism Structural Underdetermination(and Mathematical Overdetermination)

  3. Theoretical Frameworks and Empirical Underdetermination, Düsseldorf, 10-12 April 2008 Part 1: The Thesis of Theory Underdetermination by Empirical Evidence (TUD)

  4. Theoretical Frameworks and Empirical Underdetermination, Düsseldorf, 10-12 April 2008 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 induction • Conventionalism

  5. Theoretical Frameworks and Empirical Underdetermination, Düsseldorf, 10-12 April 2008 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 allpossible (past and future) observations

  6. Theoretical Frameworks and Empirical Underdetermination, Düsseldorf, 10-12 April 2008 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 doesn‘t exist (many) „interesting“ examples is indeed a pressing problem for TUD!

  7. Theoretical Frameworks and Empirical Underdetermination, Düsseldorf, 10-12 April 2008 Classes of TUD examples • Pathological cases • Cases of „epistemic TUD“ • Conventionalism-borderline cases • History of science cases • Examples from factual, mature science...!?

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

  9. Theoretical Frameworks and Empirical Underdetermination, Düsseldorf, 10-12 April 2008 „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“

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

  11. Theoretical Frameworks and Empirical Underdetermination, Düsseldorf, 10-12 April 2008 Conventionalism-borderline cases (II) • rather a reconstrual of predicates:convex pictureconcave picturestraight 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“?

  12. Theoretical Frameworks and Empirical Underdetermination, Düsseldorf, 10-12 April 2008 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. Einstein‘s Special Relativity BUT: in retrospect historic cases appear mainly as artefacts of incomplete scientific knowledge!

  13. Theoretical Frameworks and Empirical Underdetermination, Düsseldorf, 10-12 April 2008 Examples from actual, mature science...?! • Gauge Theories • Quantum Mechanics (part 3, briefly) • General Relativity (part 3)

  14. Theoretical Frameworks and Empirical Underdetermination, Düsseldorf, 10-12 April 2008 Ontology of gauge theories (I) Explanation of gauge theoretic holonomy effects (e.g. Aharonov- Bohm) by different entities and different locality assumptions:

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

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

  17. Theoretical Frameworks and Empirical Underdetermination, Düsseldorf, 10-12 April 2008 Conclusion 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!

  18. Theoretical Frameworks and Empirical Underdetermination, Düsseldorf, 10-12 April 2008 Part 2: Intermediate Structural Realism

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

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

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

  22. Theoretical Frameworks and Empirical Underdetermination, Düsseldorf, 10-12 April 2008 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(continuity 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

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

  24. Theoretical Frameworks and Empirical Underdetermination, Düsseldorf, 10-12 April 2008 Gauge transformations: no real instantiations! Well-known for global gauge transformations: • Less well-known for local gauge transformations:

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

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

  27. Theoretical Frameworks and Empirical Underdetermination, Düsseldorf, 10-12 April 2008 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“ 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“)‏

  28. Theoretical Frameworks and Empirical Underdetermination, Düsseldorf, 10-12 April 2008 Part 3: Structural Underdetermination (and Mathematical Overdetermination)

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

  30. Theoretical Frameworks and Empirical Underdetermination, Düsseldorf, 10-12 April 2008 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 CCR’s) • SR should focus on the relevant structural core…

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

  32. Theoretical Frameworks and Empirical Underdetermination, Düsseldorf, 10-12 April 2008 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

  33. Theoretical Frameworks and Empirical Underdetermination, Düsseldorf, 10-12 April 2008 Gronwald and Hehl (1996): On the Gauge Aspects of Gravity, gr-qc9602013

  34. Theoretical Frameworks and Empirical Underdetermination, Düsseldorf, 10-12 April 2008 Remarks on the translational gauge theory

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

  36. Theoretical Frameworks and Empirical Underdetermination, Düsseldorf, 10-12 April 2008 Interim: 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!

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

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

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