��Structural Realism �and the Foundations of Physics

A Summary of Research and Work in Progress

Dr. William M. Kallfelz

Department of Philosophy and Religious Studies

Department of Mathematics and Statistics

Mississippi State University

January 26, 2012

Overview

• I. Basic Definitions, Motivation
• II. Some Concrete Examples (Kallfelz 2009a, 2009b, 2009c, 2010b ): Applying Clifford (geometric) algebra to some physical theories: Unifying geometric and clarifying ontological content
• III. Some General Considerations (Ladyman 1998, 2007, Parsons 2004a, 2004b)
• IV. Concluding Points (Kallfelz 2010a, 2011): : Can there be a Constructive Structural Realism?

I. Basic Definitions, Motivation �

• Structural Realism-A research program in the foundations of mathematics, foundations of physics, and the philosophy of science.

Foundations of physics

Foundations of mathematics

Philosophy of science

Structural Realism

James Ladyman (1997, 2009)

Johnathan Bain (2006)

Charles Parsons (1993, 2004)

I. Basic Definitions, Motivation (cont.) �

• Structural Realism-Makes an essentially metaphysical claim about abstract objects:
• I.e., (in the foundations of mathematics): sets, numbers, classes, etc.—e.g. in the foundations of mathematics,
• I.e. (in the foundations of physics): theoretical entities—e.g., wavefunctions, fields, metric tensors, etc.
• Note 1: For a brief overview on the ontology (i.e., what we mean by their “existence”) of abstract objects, see Rosen (2001).
• Basic Claim: The objective status of abstract objects (whether purely mathematical or otherwise) is fundamentally determined by the ‘background structure’  that characterizes them.

I. What Makes Structural Realism “Structural”�

• Note 2: By “objective status” we are referring to the general truth-conditions that accompany any abstract object’s propositional structure.
• For example, most present-day mathematics (whether pure or applied) is characterized by the axiomatic system of Zermelo-Fraenkel-Cantor Set Theory (ZFC for short).
• So, you can think of ZFC as an instance of a structure  constituting the domain of abstract objects of most (but not all) of the field of present-day mathematics.
• “That there is a single comprehensive structure within which all of mathematics can be constructed can still be questioned on the ground that there is no absolute universe of sets. This claim…raises large and difficult issues.
• But nearly all actual mathematical work…can be framed in a rather limited set theory. ZFC is more than sufficient.” (Charles Parsons, 2004b, n. 47, p. 110)

I. What Makes Structural Realism “Realist” �

• Note 3: In terms of mathematical truth and mathematical existence, the structural realist attempts to aim for a “middle path” between (what she considers) are two undesirable extremes.
• I. Mathematical Platonism- The argument that abstract objects have an “intrinsic nature,” determining therefore their truth and existence conditions.
• The structural realist disagrees: There can be no abstract object without a ‘background’ framework !
• Whether it’s ZFC, non-standard set theory, Category Theory, etc., it makes little difference, since a’ la Bourbaki “[b]y a structure is usually meant [none other than] a domain of objects with certain functions and relations on the domain, satisfying certain conditions.” –Parsons, 2004b, p. 43

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I. What Makes Structural Realism “Realist” �

• Note 3: In terms of mathematical truth and mathematical existence, the structural realist attempts to aim for a “middle path” between (what she considers) are two undesirable extremes.
• II. Mathematical Constructivism- The argument that abstract objects’ truth and existence conditions reduce to their constructibility (which in the case of truth-conditions, means directly constructive provability). (L.E.J. Brouwer, Errett Bishop, 1972, etc.)
• The structural realist (again) disagrees: The truth conditions of an abstract object’s propositional structure remains an objective fact (within ), independent of provability!
• “On its face, the necessity of mathematics is not epistemic…[even though] mathematical knowledge is characteristically obtained by proof.
• The fact that Goldbach’s conjecture is not known to be true or false does not alter the fact that if true, it is necessary; if false, it is necessarily false.” –Parsons, 2004b, p. 82.

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II. Some Concrete Examples (Kallfelz 2009a, 2009b, 2009c, 2010b ):

An historical analogy:

• The algebraic characterization of geometric notions by Descartes fundamentally methodologically transformed the emerging science of modern mechanics.
• David Hestenes (1985, 1986) suggests the honorific ‘geometric’ to Clifford algebra, reflecting his self-professed ‘Cartesian’ intuition to ‘geometrize’ a physical concept whenever possible.
• “[T]he vector algebra of Gibbs…was effectively the end of the search for a unifying mathematical language and the beginning of a proliferation of novel algebraic systems, created as and when they were needed; for example, spinor algebra, matrix and tensor algebra, differential forms, etc.” (Lansenby, et. al. (2000), 21)
• See also Sir Michaeal Atiyah (2001) “Geometry and Physics: A Marriage Made in Heaven”

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II.a) Clifford Algebra-Overview

• Using results of Hamilton and Grassmann, W. K Clifford developed his algebra (1878-1882) to characterize rotations and spin, (generalized to any n –dimensional vector space V). �
• “Geometric:” A systematic collection of directed line segments (vectors), areas (bivectors), volumes (trivectors),…, n-dimensional hypervolumes (n-vectors or n-blades) n = dimV.

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II.a) Clifford Algebra-Overview

• The notion of directed line segment survives today in the notion of a vector:

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• The notion of directed area survives today in the notion of a vector cross product:

II.a) Clifford Algebra-Overview (cont.)

• Cross-product

II.a) Clifford Algebra-Overview (cont.)

• An Example CL(R³),
• In 3-dimensional Euclidean space (R³), three vectors generate its vector algebra V(R³).
• However, eight (2³) Clifford elements generate its Clifford algebra CL(R³), the Clifford algebra is 8-dimensional.

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II.a) Clifford Algebra-Overview (cont.)

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• These 8 (linearly) independent generators consist of:
• 1 “grade 0” element (the real scalars)

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R

II.a) Clifford Algebra-Overview (cont.)

• 3 “grade 1” elements (three orthogonal directed segments)

II.a) Clifford Algebra-Overview (cont.)

• 3 “grade 2” elements (three orthogonally directed planes or 2-blades)

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ij

jk

ki

II.a) Clifford Algebra-Overview (cont.)

• 1 “grade 3” element (the “unit volume” or pseudoscalar)

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II.a) Clifford Algebra-Overview CL(R^N)

• In general, for any n-dimensional vector space X, its vector algebra V(X) is likewise n-dimensional, but its Clifford algebra CL(X) is 2ⁿ-dimensional.
• Clifford algebras possess a richer geometric structure than their vector algebraic counterparts
• In addition, algebraically division is defined via the Clifford unit pseudoscalar.
• So the linear equation AX = B has the formal solution X = A^-1B in CL(V).
• “Much of the power of geometric (Clifford) algebra lies in this property of invertibility.” (Lasenby, et. al. (2000), 25)

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II.a) Clifford Algebra-Overview (cont.) The Clifford Product

• For any A, B  CL(V) their (associative) Clifford product is defined by: AB = AB + AB

AB is their (commutative and associative) inner product.

AB is their anti-commutative (AB = -BA) and associative exterior (or Grassmann) product.

• CL is equipped with an adjoint † and grade operator
• < >_r

where < >_r is defined as isolating the rth grade of a Clifford element A . For any Clifford elements A, B:

<AB > ^† _r = (-1)^C(r,2) <B ^† A ^† >_r

(where: C(r ,2) = ^r!/_{(2!(r – 2)!)} = ^{r(r – 1)} /₂ )

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II.b) Clifford Algebra and Spinor Algebra-Unifying Geometric and Ontological Content

• Clifford algebras offer a reformulation of quantum theory characterizing its terms with “a natural semantic isomorphism” between its algebraic and geometric meaning.
• On the other hand in the case of standard Hilbert space in QM mathematical formalism) “[o]ne can distinguish three fundamentally different geometric roles tacitly assigned to the unit imaginary i , namely
• The generator of rotations in a plane :

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• The generator of duality transformations :

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• The indicator of an indefinite metric:

II.b) Clifford Algebra and Spinor Algebra-(cont.)

• “ Confusion is difficult to avoid when i is required to perform more than one of these roles in a single system. Worse yet, in physics all three possibilities are sometimes realized in a single theory confounded with problems of physical interpretation…
• In the interest of fidelity to geometric interpretation, the convention that complex numbers are scalars should be abandoned in favor of…a system in which each basic geometric distinction has a unique algebraic representation. Geometric [Clifford] Algebra has this property.” (David Hestenes,1984, xii – xiii).

II.b) Clifford Algebra and Spinor Algebra-(cont.)

• Physicists generally regard the ^k [Pauli spin matrices] as three components of a single vector, instead of an orthonormal frame of three vectors…Consequently, they write:

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• …and to facilitate manipulation they employ the identity:

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… a good example of the redundancy in the language of physics which complicates the manipulations and obscures the meanings unnecessarily.” (Hestenes, 1986, 323)

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II.b) Clifford Algebra and Spinor Algebra-(cont.)

• The redundancy in the above identity is due to its ‘overlapping geometric content’: The (vector) dot and cross products of course comprise the binary operations of standard (Gibbs’) vector algebra in R³, while the Pauli spin matrices acting as the ‘vector coefficients’ belong to the spinor algebra C².

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• The geometric contents of R³ and C² can be unified, however, when one instead considers ^k as generators of a Clifford algebra, thereby “eliminat [ing] all redundancy incorporating both languages into a single coherent language.”

II.b) Clifford Algebra and Spinor Algebra-(cont.)

• To see this, simply write for any 3-vector:

• Then the above identity with its (otherwise geometrically overlapping content) now simply is represented as: .

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• This is just precisely the definition of the Clifford product of two 3-vectors!

II.c) Other Examples (Discussed in Kallfelz, 2007, 2009a,c, 2010b)

• Finkelstein (1996, 2001) presents a unification of field theories (quantum and classical) and space-time theory based fundamentally on finite dimensional Clifford algebraic structures, since:
• i.) A Clifford algebra can express a fully quantum space-time. (Finkelstein et. al. 2001, 5; Kallfelz 2009a, p. 68)
• ii.) Clifford statistics (the simplest statistics supporting a 2-valued representation of S_N, the symmetry group on N objects) adequately expresses the distinguishability of events as well as the existence of half-integer spin. (ibid.)
• iii.) Drawing on I.E. Segal (1951)—Quantum topology characterized via CL is regularizable (singularity free), which implies: a simple relativity group, and a stable Lie algebra over its tangent manifold.

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II.c) Other Examples (Discussed in Kallfelz, 2007, 2009a,c, 2010b)

• Outline of the procedure:
• 1) The “atomic” quantum dynamical unit (represented by a generator of a Clifford algebra) is the chronon , analogous to the tangent or cotangent vector (forming an 8-dimensional manifold) and not the space-time point (forming a 4-dimensional manifold). (Kallfelz, 2009a, p. 68)
• 2.) A Clifford statistical ensemble of chronons can factor as a Maxwell-Boltzmann ensemble of Clifford subalgebras, becoming a Bose-Einstein aggregate in the N   limit (where N is the number of factors).
• 3.) This Bose-Einstein aggregate condenses into an 8-dimensional manifold M, which is isomorphic to the tangent bundle of space-time.
• 4.) M is a Clifford manifold, i.e. a manifold provided with a Clifford ring: (where: C₀(M), C₁(M),…,C_N(M) represent the scalars, vectors,…, N-vectors on the manifold). For any tangent vectors ^(x), ^(x) on (Lie algebra dM) then: ^(x) _ ^(x) = g^(x)
• where: _ is the scalar product. Hence the space-time manifold is a singular limit of the Clifford algebra representing the global dynamics of chronons in an experimental region.

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III. Some General Considerations (Ladyman 1998, 2007, Parsons 2004a, 2004b)�

• The previous examples surveyed reveal that characterizing physical theories by way of particular background structures (in this case, Clifford Algebra) poses some distinct advantages, which include:
• 1.) Theoretical (Example II.b-spinors): Resolving, to a certain extent, certain interpretative difficulties.
• 2.) Practical (Example II.c-quantum topology): Being able to extend particle physics into space-time physics (general relativity) in a way that is regularizable (singularity free) and admitting a natural interpretation of large scale space-time structure based on (Clifford) statistics of elementary quantum spacetime processes (chronons).

III. Some General Considerations (Ladyman 1998, 2007, Parsons 2004a, 2004b)�

• These particular instances (and many more—discussed in Kallfelz 2007, 2009a, c 2010b)
• Offer (I claim in Kallfelz 2010a, 2011) substantive evidence for structural realism—applied in the domain of the foundations of physics.
• Recall: (slide 4 above)
• Basic Claim: The objective status of abstract objects (whether purely mathematical or otherwise) is fundamentally determined by the ‘background structure’  that characterizes them.
• Corollary: The abstract objects in a physical theory are precisely its theoretical entities!

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III. Some General Considerations (Ladyman 1998, 2007, Parsons 2004a, 2004b)�

• As “an alternative to both traditional [scientific] realism and [scientific] instrumentalism, structural realism must incorporate epistemic commitment to more than the empirical content of a scientific theory, namely to the structure [ ] of the theory
• …while stopping short of the realists’ commitment to the full ontology postulated by the theory.” –James Ladyman, 1998, pp. 415-416.
• In other words, just like in the case of structural realism in pure mathematics, a middle course is steered between two undesirable extremes:
• I. (Analogous to Platonism)-The realists’ “full commitment to the full ontology postulated by the theory.”
• II. (Analogous to Constructivism)-The instrumentalists’ “agnostic” approach to any content in the theory which is not empirical (directly observable).

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III. Some General Considerations (Ladyman 1998, 2007, Parsons 2004a, 2004b)�

• Recall in the previous examples (II.b, II.c):
• The theoretical entity of spinor is fundamentally structurally characterized by the Clifford Product in CL(R³) (in the case of non-relativistic quantum theory) and (respectively) in CL(R⁴) in the case of QFT (relativistic quantum theory)
• The theoretical entity of chronon , is fundamentally structurally characterized by a generator for Clifford Algebra in CL(R^N) which (by way of the factorization procedure illustrated in the “Octad Lemma” in Finkelstein (2001)) one may recover the Lie algebra characteristic of the tangent manifold for the diffeomorphism group (in General Relativity)!
• The instrumentalist would shrug and say that these formal maneuvers just give the scientist better predictions –as that’s all that science can do!
• The structural realist disagrees! Progress in physics is partially determined by more reliable theories with greater explanatory power—some structures are better equipped at doing that than others!

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Concluding Points (Kallfelz 2010a, 2011):

• “A structuralist understanding of arithmetic [for example] belongs to sophisticated mathematics; it is not part of the layman’s understanding…or even the mathematician’s before a certain amount of foundational reflection has been undertaken.” (Parsons, 2004b, p. 77)
• One can (of course) say the same in the case of physics!
• Open question: Can (in the case of mathematics and/or physics)a structure (suitably sophisticated) provide all of the important formal content (including the truth-conditions)—i.e., is a chronon nothing but a generator of CL(R^N), are integers nothing but roles or instances of a total ordering?

Concluding Points (Kallfelz 2010a, 2011):

• To this open question—the eliminative structural realist says “yes” while the non-eliminative structural realist says “no”:
• “[E]liminative structuralism, to the extent that it can do with just…second-order logic, can do without structures as entities: talk of entities is just a heuristic device…
• [for example] a statement about the complex numbers [with nontrivial automorphism i - i] is viewed as nothing but a general statement about realizations of the structure [of the complex plane].” (Parsons, 2004a, p. 66)

Concluding Points (Kallfelz 2010a, 2011): Can there Be a Constructive Structural Realism?

• My present work (which I summarize in Kallfelz 2010a, 2011) is to adopt modal logic and modal considerations in an effort to resolve this issue.
• “The question whether even classical [i.e. characterized by ZFC] mathematics is sufficient for applications remains in a peculiar but genuine sense open…. Is [it] inherently inapplicable, or rather because we have not yet been clever enough to apply it?” (John Burgess 439, italics added)
• In the case of my previous examples (II.b, II.c) they can be characterized as models in some (second-order) modal logic—”the appropriate interpretation of modality [is] that mathematical truths are necessary…and mathematical constructions [express]… a kind of possibility.” (Parsons, 2004b, 89)

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References

• Atiyah, Sir Michael (2001) “Geometry and Physics: A Marriage Made in Heaven.” (lecture) Michigan Center for Theoretical Physics, April 3, 2001.
• Bain, Johnathan (2006). “Towards Structural Realism,” (manuscript)

ls.poly.edu/~jbain/papers/SR.pdf

• Bishop, Errett (1967). Foundations of Constructive Analysis. New York: McGraw-Hill.
• Burgess, John (1992). “How Foundational Work in Mathematics Can Be Relevant to Philosophy of Science.” Proceedings of the Biennial Meeting of the PSA, vol II., 433-441.
• Curd, M. & Cover, J. A., eds. (1998). Philosophy of Science: The Central Issues. London: W. W. Norton & Co.

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References (cont.)

• Falkenstein, Lorne (1998). “Was Kant a Nativist?” in Kitcher, Patricia, ed. Kant’s Critique of Pure Reason: Critical Essays. Oxford: Rowan & Littlefield, 21-47
• Finkelstein, David (1996). Quantum Relativity: A Synthesis of the Ideas of Einstein and Heisenberg. Berlin: Springer-Verlag.
• Finkelstein D., Baugh, J., Saller, H (2001) “Clifford Algebra as Quantum Language,” J. Math. Phys 42, 1489.
• Finkelstein, D. & Kallfelz, W. (1997). “Organism And Physics,” Process Studies vol. 27 n. 3, pp. 279-292
• Hestenes, David (2003). “Reforming the mathematical language of physics.” (Oersted Medal Lecture 2002). Am. J. Phys. 71 (2) Feb 2003
• Kallfelz, William M., “Embedding Fundamental Aspects of the Relational Blockworld Interpretation in Geometric (or Clifford) Algebra” (posted: April 5^th, 2007)  http://philsci-archive.pitt.edu/archive/00003278/

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References (cont.)

• Kallfelz, William (2009a). Clifford Algebra: A Case for Geometric and Ontological Unification. Saarbruecken, Germany: VDM Verlagsservicegesellschaft MbH.
• Kallfelz, William (2009b ), "A Response to G.B. Bagci's 'Ghirardi-Rimini-Collapse Theory and Whiteheadean Process Philosophy',” Process Studies, Natural Sciences Focus Section, Pete Gunter, ed., vol. 38 n.2, 2009, pp. 394-411.
• Kallfelz, William M. (2009c), “Physical Emergence and Process Ontology,” World Futures: The Journal of General Evolution, special issue on process thought and natural science, special editors: Franz Riffert and Timothy Eastman, vol. 65 issue 1, 2009, pp. 42-60.
•  Kallfelz, William M. (2010a), "Modal Rationalism and Constructive Realism: Models and Their Modality,” (posted: July 27, 2010) philsci-archive.pitt.edu/archive/00005489/

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References (cont.)

• Kallfelz, William M. (2010b), "Clifford Algebraic Computational Fluid Mechanics: A New Class of Experiments,” Philosophy of Scientific Experimentation: A Challenge to Philosophy of Science, Center for Philosophy of Science, University of Pittsburgh (vol. 640, Oct. 16, 2010)

philsci-archive.pitt.edu/archive/8348/

• Kallfelz, William M. (2011) American Philosophical Association-Eastern Division Meeting, "Whitehead's Natural Philosophy: A Meta-Physical Framework for Productive Physics," –Dec. 28, 2011. (Session GIII-8, Proceedings and Addresses of the American Philosophical Association, Sept., 2011, vol. 85, Issue I, p. 53)
• Kuhn, Thomas (1977). “Objectivity, Value Judgment, and Theory Choice,” in The Essential Tension: Selected Studies in Scientific Tradition and Theory Change. (Chicago: University of Chicago Press), 320-339.

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References (cont.)

•  Ladyman, James (1998). “What is Structural Realism?” Studies in History and Philosophy of Science, vol. 29, n. 3, pp. 409-424.
• Ladyman, James (2009). “Structural Realism,” Stanford Encyclopedia of Philosophy (on-line).
• Lasenby, J., Lasenby, A., Doran, C. (2000) “A Unified Mathematical Language for Physics and Engineering in the 21^st Century,” Phil. Trans. R. Soc. Lond. A, 358, 21-39.
• Mugur-Schaechter, Mioara & Van der Merwe, eds. (2002). Quantum Mechanics, Mathematics, Cognition and Action: Proposals for a Formalized Epistemology. Fundamental Theories of Physics, vol. 129. Dordrecht, NL: Kluwer Academic Publishers
• Parsons, Charles (1993). “On Some Difficulties Concerning Intuitive Knowledge and Intuition,” Mind, vol. 102, n. 406, pp. 233-246.

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References (cont.)

• Parsons, Charles (2004a). “Structuralism and Metaphysics,” The Philosophical Quarterly, vol. 54, n. 214, pp. 56-77.
• Parsons, Charles (2004b). Mathematical Thought and Its Objects, Cambridge: Cambridge University Press, 2004.
• Rohrlich, Fritz (1988). “Pluralistic Ontology and Theory Reduction in the Physical Sciences.” British Journal for the Philosophy of Science, vol. 39, no. 3 (September), 295-312.
• Rosen, Gideon (2001). “Abstract Objects.” The Stanford Encyclopedia of Philosophy (on-line) http://plato.stanford.edu/entries/abstract-objects/

References (cont.)

• Segal, I. E. (1951) A Class of Operator Algebras which are Determined by Groups. Duke Mathematical Journal 18, 221.
• Sneed, Joseph (1971) The Logical Structure of Mathematical Physics, (Synthese Library) D Reidel.
• Teller, Paul (2004). “How We Dapple the World,” Philosophy of Science, 71 (October), 425-447.
• Whitehead, A. N. (1929). Process and Reality (PR). Corrected Edition, edited by David Ray Griffin & Donald Sherburne. New York: The Free Press, 1978.

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