BM
Copyright © 2007 by Kevin Sharpe. All rights reserved.
Submitted for publication.
by
Kevin Sharpe
Graduate College, Union Institute &
University,
Harris
10 Shirelake Close,
www.ksharpe.com
ABSTRACT.
A need exists for a mathematical language in which to express hypotheses involving qualities, without reducing them to quantities. Achieving this could allow the humanities to access mathematical functions with which to express the relationships between elements of their subject matter, and hence a flow of ideas between the humanities – including spiritual thought – and the sciences may eventuate. This paper suggests a family of topological algebras as such a medium, based on suggestions from David Bohm and his colleagues, and then explores it theologically.
KEYWORDS.
David Bohm, entanglement, mathematical metaphysics, qualities, quantities, the science-religion relationship, topological algebras.
CONTENTS.
Topologizing
the Algebra: A Mathematical Metaphysics
A
Family of Topological Algebras
A
Mathematical System of Spiritual Thought
Western culture usually assumes that matters objective
(scientific, material, technological) form a different and distinct class from
matters subjective (feeling, spiritual, humanistic, moral). The physical sciences
and the humanities (to place labels on the pair) have deeply absorbed this
world view and over the centuries have striven to create and uphold the
distinction. As a positive outcome, the distinction enables the growth of an
objective, scientific method, opposing closed and dogmatic systems of religious
beliefs. As a negative outcome, the distinction can lead to social and
environmental chaos, unchecked technological expansion, and a lack of moral
conviction (Sharpe
We should, rather, seek a relationship in which science and the humanities share a basic set of knowledge and assumptions (including investigatory methods and a foundational metaphysics) while specializing in their particular emphases (respectively, knowledge of the physical world and knowledge of the human world). The sharing would include knowledge on subjects of mutual import. Further, the interchange would not primarily travel one-way – with, perhaps, the humanities seeking to adopt scientific method and adapting themselves to scientific knowledge – while science stands objectively firm; science needs the humanities. On the other hand, for science to learn from the humanities need not mean that it relinquish its method. Rather, it means recognizing science as a human endeavor that needs to learn from other human endeavors and to explore as hypotheses what the others have found true or pertinent.
To help this project of reconciliation and reconstruction,
this paper seeks a mathematical description of both the qualitative and
quantitative properties of the world, a common language in which they each can
form their hypotheses and exchange their knowledge. It seeks, in other words,
to create a mathematical metaphysics, an underlying plank, on which both the
sciences and the humanities can stand and which allows the transport of ideas
between them. The mathematics may also offer a richer vocabulary and a greater
choice of ideas for the humanities, and these may help scholars in the
humanities to express more adequately their ideas and the relationships among
humanistic realities (by opening the possibility of using mathematical
functions) than with current linguistic ideas.
The humanities might, in their project of reconciliation and
reconstruction with the sciences, adopt a scientific or empirical approach to
the accruing of their knowledge. This is not an obvious path to take. A chief
stumbling-block preventing it from happening comes from the objective and quantitative
language of mathematics, which science often uses and for which it aims.
Quantitative mathematics appears foreign to the qualitative humanities. Science
assumes it can adequately describe the world using quantifiable observables and
mathematical theory; the humanities believe they cannot. The type of
mathematics this paper seeks, therefore, ought to avoid the reduction of the
humanities to quantitative studies, trying to explain their subject matter only
in terms of quantitative variables. A key to the challenge is this: How might
mathematics express qualitative language? How might mathematics rigorously
express subjective and humanistic ideas?
Early in my career, I wrote a paper that outlines how to
mathematicize the world topologically, so bringing me closer to my goal (Sharpe
I realized later that Bohm’s work provided a mathematical metaphysics as well as a mathematical system of spiritual thought. This paper states this realization and starts to expound it. My early topological explorations also become part of this current work.
Classical physics studies each part of the universe as
separate, the parts coming together to explain the whole. Bohm and Hiley (who,
with their colleagues from Birkbeck College, University of London, form the ‘Birkbeck
School’) disagree with this approach and take the relationships between the
parts and the qualities of a part to depend on the whole (Bohm and Hiley
With the idea of the ‘implicate
order’ (or the ‘holomovement’), the
1.
The implicate order is what is basic to reality. It is
what is.
2.
The implicate order is undivided or unbroken wholeness. Each
region of space and time contains in it the total order of the universe,
including the past, the present, and the future. The word ‘implicate’ comes
from the verb ‘to implicate,’ ‘to fold inward.’ Reality as implicate means that
each portion contains information on every other portion within it.
3. In the implicate order, movement continuously takes place. The Birkbeck School makes activity – rather than something essentially static and rigid – the basis for their proposed order.
The implicate order unfolds. Certain aspects of it lift into attention, come into relief to produce parts that appear independent. Bohm calls the reality made of these items the ‘explicate order.’ They create the stable, independent and lasting world of parts that humans experience. Then, having unfolded to become the explicate order, the implicate folds back into itself. The structured activity of reality is not the movement of objects; it is the ongoing process of implicate to explicate to implicate.
In most contexts, the implicate order does not fully become an explicate order. Not all the properties of an object appear at once. What does unfold from the implicate order to become the explicate appears as incomplete and independent parts. That does not mean, however, that the explicate order loses all of the wholeness of the implicate. Holonomy (the law of the whole) limits the breaking of a situation into independent parts; the total environment of the object restricts the process of unfolding. The parts come from a more basic whole and in the end are not separate.
One of the significant and novel features of quantum theory
for Bohm appears in the Einstein-Podolsky-Rosen (or EPR) thought experiment
(Einstein, Podolsky, and Rosen
If they move beyond the formalism, most physicists now avoid Einstein’s conclusion that a connecting signal links the two simultaneous events in the EPR experiment, but merely say or imply that a correlation exists between them. ‘Connection’ suggests that something connects them; ‘correlation’ suggests a mutual relationship between them and it happens that, on changing one, the other simultaneously changes. Quantum mechanics does not explain with causes or connections why the EPR phenomenon occurs. It just predicts, through its mathematics, that it happens.
The Birkbeck physicists interpret the EPR result as representing an essential new feature, ‘nonlocality’ (now usually called ‘entanglement,’ referring to the mathematics behind the theory), in the quantum world. Entanglement suggests the opposite of locality. Establishing the entanglement of all parts of a quantum system has superseded the older approach to physics with its image of individual particles and the absoluteness of locality. The EPR experiment, generally stated, shows that when something changes, instantaneously something else happens some distance away. The change in the first object correlates through the entanglement of quantum states with the change in the second.
Bohm uses the idea of the ‘implicate order’ for the EPR situation and to explain entanglement: entanglement arises, he believes, from the implicate order whose unfolding underlies distant events in a way that unifies them. The implicate order relates them by entangling them. The system forms a changing whole that the unfolding organizes by containing and instantaneously carrying information about the whole environment. The unfolding thus influences the outcomes of all events, correlating those separated in space and time (Maroney and Hiley 1999). No direct causal connection links them – one event does not itself influence another – they just relate to each other because they both unfold from the implicate order.
Bohm, Hiley, and the other members of
the
The above ideas about the implicate order are philosophical and metaphysical. Bohm also needs to express them mathematically and in a 1973 paper begins to do this.
The explicate order often arises in ways that scientific
instruments can observe and report in terms of Euclidean geometry, its
measures, and order. Following Felix Klein (1849-1925), Bohm takes particular
general transformations to determine the features of this geometry. Three
displacement operators, Di, apply to three-dimensional Euclidean
space, each of which defines a set of parallel lines that it transforms into
themselves. Three rotational operators, Ri, also apply, and each of
these defines a set of concentric circles that it transforms into themselves.
Taken together, the operators define concentric spheres that the set of Ri
transform into themselves. Bohm adds to these a dilatation operator R
A displacement, Dj,
acting on one set of operators, Ri, R
(Ri', R
Rotations on the displacement Di provide displacements, Di', in new directions:
Di'
= RjDiRj-
A displacement, Di,
rotated n similar times produces the displacement, (Di)n.
This allows the similar ordering and numerically description of displacements
as the ordering of the integers. The process of successive displacements of the
same size also provides a measure, along with an order. Just as each rotation,
Ri, produces a collection of rotations with an order and a measure,
so each dilatation, R
Bohm builds from his mathematical description of an explicate
order to one for an implicate order. Because of the difference between
operations in an implicate order and those in an explicate order, however, he
prefers the word ‘transformation’ (with the symbol T) for a simple geometric
change within an explicate order, and ‘metamorphosis’ (with the symbol M) for
change within an implicate order. He uses E to denote a set of transformations
in a particular explicate order (
E' = MEM-
For an implicate order, applying a metamorphosis M repeatedly produces (M)n, the n-times enfolding of the structure. Defining Qn = (M)n, then
Qn : Qn-
This produces an order because of the similarity (in fact, equality) in the differences between the Qn; it constitutes an implicate order because the differences mark the degree of implication. Further, the equivalence of the operations M provides a measure with n as an ‘indicator of implication.’
This leaves:
·
a set of operations (metamorphoses and
transformations);
·
the possibility of adding the operations,
multiplying them, and multiplying them by a number;
·
a unit operation (which leaves any operation
unaltered when multiplied with it); and
· a zero operation (which leaves any operation unaltered when added to it).
These four things together define an
algebra. Writes Hiley (
The laws of physics have
up until now referred, Bohm writes, to the explicate order. The
Cartesian coordinate frames in fact function chiefly to provide a clear and
precise description of the explicate order (Bohm
In
Bohm’s work on the algebraic form of the implicate-explicate
orders first appeared in
1. An
algebra need not be based on a continuum-like space with its division of space
into points with no volume.
2. Algebras
lie at the mathematical root of quantum physics.
3. An algebra allows for two different operations, addition and multiplication, which together can describe sequences of events as well as interactions between them.
Bohm, Hiley, and their partners in
the
In particular, the
Birkbeck School wants an algebra to represent the quantum world. After their
attempts with combinatorial topology, they turned their attention to the
structures supporting the Clifford and simplictic algebras. They ask whether
they can package quantum theory completely in these algebraic forms, and
whether this can be done so that overpowering distinctions do not show between
the observer and the observed. If so, they claim, they could understand quantum
phenomena using the implicate order. They conclude that they can do this
(Frescura and Hiley
Despite the failure or stalling of the Birkbeck School’s attempts to found the quantum world on the above algebras, the basic model of an algebraic structure still remains a possibility for metaphysically modeling the implicate and explicate orders. This more general pursuit stands independent of the physics so that, if the physics falters, the metaphysics might remain. The object is to develop the general algebraic model into a metaphysics and then to ask about its possible base for a theology.
An algebra comprises a set, two operations, a zero, and a
one. To discuss Bohm’s algebra, though, we need to specify to what the set, the
operations, the zero, and the one refer. The set is assumedly Euclidean space, R
How is the set R
How to represent qualities mathematically thus becomes the challenge. Once a mathematical representation for qualities and for quantities has emerged, one that can assimilate an algebraic structure, then the basic vocabulary exists with which mathematically to ask and answer questions concerning the metaphysics’ vision of reality.
The metaphysics proposed to answer this challenge involves a topological mathematical description. A topology for a space X is a family τ of subsets of X such that the union of any number of members of τ, and the intersection of a finite number of members of τ, are members of τ. The mathematical understanding of the world this metaphysics proposes says that each of the various properties (qualitative and quantitative) that the world or parts of the world appears to have uniquely defines (as elucidated below) a topology for its space.
The task initially constructs the topologies and spells out
the relationship that can exist between a property and its equivalent topology.
What follows introduces the steps in the development; the original paper on
this subject contains the details, plus points and questions that these moves
might raise (Sharpe
1. Isolate
the properties (qualitative and quantitative) of the world (assumed to be
Euclidean R
2. Divide
the (spatial) world up into volumes Pn with the same value (or equal
realizations) of a particular property; for instance, the outline of everything
yellow in front of the observer.
3. Order
this set of volumes, P, each with a different value of the property, to form a
‘partially ordered set’ (or ‘poset’): for two volumes P
A unique relationship exists between the partial ordering of
subsets of a space (like R
In summary, this mathematical model states that a topology
on Euclidean R
The above discussions outlined the construction of a Bohmian
algebra and the topology for a property. These together form a topological
algebra if the topology renders the operations of the algebra continuous. An
algebra and a topology for a property form a topological algebra if the
operations + and . are continuous on R
Continuity can be seen from the way a metamorphosis of implicate orders transforms local relationships. Hiley writes about holograms:
consider the relation between the object that is being
hologrammed and the hologram itself. We find that the local features of the
object no longer appear as local features on the hologram. The locality
relation of each region of the object has been transferred to every portion of
the hologram. Locality is thus ‘stored’ in a non-local way, but notice the
original local relations have not been lost. They can be recovered, using the appropriate
laser light (Hiley
Bohm’s image of enfolding-unfolding (for instance, his
image of droplets of dye dropped onto glycerol and folded into it by stirring,
and then unfolded from it by stirring in the opposite direction (Bohm
The above develops a mathematical framework with which to describe the world in both its quantitative and qualitative aspects, and to which the algebraic model for the essential elements of the implicate order applies. The above also develops a family of topological algebras: a mathematics of the implicate order in which each property refers to a topology under which metamorphoses are continuous. It offers a mathematical language with which to express Bohm’s metaphysics in a general way.
The implicate order approach proposes a form of holism, because the whole lies in any of its parts and what happens to any part affects all others. The popularity of Bohm’s metaphysics derives in part because of its holistic vision: implicate wholes reach a level of explanatory importance that explanations at the explicate level of parts cannot take away.
In that this holism carries over with the mathematics of the algebra, an algebra (and, in particular, the algebraic model of the implicate order) is holistic; Bohm develops a mathematics of wholes. Wholes tie to the implicate order and algebras offer a way to relate wholes to parts. The implicate order models the whole-parts relationships and the algebra models an implicate order.
How adequate is this mathematics of holism? Does Bohm’s implicate-explicate theory and its algebraic mathematicization picture the whole-parts relationship well?
The model of the implicate order may offer more than holism needs. The implicate order has properties – for instance, that every part contains the whole – that wholes in general need not have.
Second, one aspect of holism (a whole exceeds the sum of its parts) may not apply in the algebra model because, in any algebra, the product of two terms equates to the sum of scalar multiples of other terms:
AiAj = ∑K λKij
AK
for numbers λKij. So a whole equals a sum of
parts. This suggests the reductionist nature of algebraic holism, and raises
the question of what constitutes holism. Can it say only that any of the parts
of a whole can interrelate, while allowing that the whole may equal the sum of
parts (see Sharpe and Walgate
Besides as a vehicle for Bohm’s metaphysics, this framework may have relevance for spiritual thought and hence complement Bohm’s forays (e.g., 1980) from his physics and mathematics into the spiritual. It may, for instance, enable the humanities (including spiritual thought) and the theories of some sciences to dialogue via a mathematical language. It also offers to the humanities a conceptual array of mathematical concepts with which to express their ideas, an array that is wealthy, is as yet largely unexplored by them, and that supplements their usual linguistic concepts. They could use this metaphysics by posing theories in it and so express them mathematically while remaining fully humanistic and not mechanical or arithmetic.
For instance, suppose Ultimate Reality is or is like the
implicate order, then the above may provide a mathematics with which to
conceptualize Ultimate Reality (see Sharpe 1993b; 1997;
·
Does the implicate order algebra display
behaviors or attributes one might associate with Ultimate Reality? In
particular, does it make sense to speak – as does the implicate order model –
of anything as containing everything else (including Ultimate Reality), though
with less detail than the original?
·
One way Ultimate Reality relates to everything
is through entanglement. What does this mean and to what does it lead?
· Do the two algebraic operations, + (succession) and . (interference), correspond to something spiritually significant?
##Is this mathematics really a system of spiritual ideas, at least rudimentarily? To help answer this question, one might ask what essentially makes a system of ideas spiritual. When does a metaphysics become spiritual? Perhaps an adequate system of spiritual ideas ought at least to say:
1. Ultimate
Reality exists;
2. people
live in relationship with Ultimate Reality; and
3. the nature of Ultimate Reality extrapolates from human attributes.
With regard to the proposed mathematical metaphysics and each of the above three points:
1. If Ultimate Reality constitutes the implicate order, then this point occurs automatically.
This idea of Ultimate Reality as the implicate order compares with Paul Tillich’s idea of the ‘Ground of Being.’ In the metaphysics suggested here, this is Ultimate Reality ‘sustaining’ the universe.
2. This also automatically holds with Ultimate Reality as the implicate order because everything exists in continuous folding into and out of Ultimate Reality.
Ultimate Reality
acts in ways that mathematical functions acting on the family of topological
algebras may model. Anything possible in the world, when it occurs, is Ultimate
Reality at work.
Hints
of Ultimate Reality’s existence surface when wholeness appears, in entanglement
phenomena for example. This may affect humans through their brains, according
to the theory of Roger Penrose and Stuart Hameroff (Penrose
3. This would also obtain because of the whole-part relationship of humans to reality as implicate. What constitutes the human is the human part of the whole (Ultimate Reality) that contains everything. What these properties comprise in their extrapolated Ultimate Reality and implicate form asks a relevant question, but unanswerable at present. What one can know of them will only emerge as they become explicated in relation to specific situations.
How does this topological algebraic way of describing the world, a way that allows discussion of both the quantitative and the qualitative in one language, offer itself for use and testing?
Consider first the formulation of testable hypotheses and
theories in this language. Such hypotheses would center on the relationships
between properties of the world. For instance, suppose that, if an object has
property P
Put in topological terms, two topologies, τ
Specific sorts of topological
algebras (plus specific restrictions on the algebras) might help model specific
aspects or realms of reality, as the
The testing of this proposed mathematical metaphysical tool and model differs from but could relate to the testing of hypotheses framed in its language. How adequate is the topological algebra model in general?
1. The assumptions behind the metaphysics could seek support. If they show themselves inadequate, wrong, or inconsistent, then the language derived from them becomes questionable. Are all the assumptions necessary? Bohm suggests something even broader than an algebra:
Of course, algebra is in itself
a limited form of mathematization. There is no reason in principle why we
should not ultimately go on to other sorts of mathematization (involving, for
example, rings and lattices or still more general structures which have yet to
be created) (Bohm
Thus, one may ask about the adequacy of an algebra as a mathematical base for the metaphysics. Does emphasizing wholeness versus the more specific idea of the implicate order (because of which, Bohm requires an algebraic form) still require the two operations? Listing the properties of a whole and of the implicate order, and then rendering them into a mathematical form might help answer this. It would show what mathematical structure fulfills them.
2. Consider its adequacy in (at least) two operational ways: to see if the proposal allows the adequate expression of known quantitative laws, and to ask if the proposal can produce further rational theories that prove successful in prediction and verification. The metaphysical framework will find support or justification if it allows the adequate statement of qualitative theories that prove themselves under testing.
3. Similarly, consider its inadequacy: does it produce theories or predictions that prove unsuccessful against observation?
4. Questions need asking that push the proposed system of thought to see if it proposes reasonable answers. For instance, its implications for boundaries – including the very large and the very small – may also help in its evaluation. In addition:
A need exists for a mathematical language in which to express hypotheses involving qualities, without reducing them to quantities. To achieve this might allow the humanities access to mathematical functions with which to express the relationships between elements of their subject matter. The functions might also aid a flow of ideas between the humanities and the sciences, including theology and the sciences. A family of topological algebras is offered as such a medium.
It remains to see whether an algebra (with or without topologies) offers an adequate or appropriate model for reality, where it works and where it does not, and whether the Bohmian algebras allow further explication.
Aspect, Alain, Philippe Grangier, and Gérard Roger.
Aspelmeyer, Markus, Hannes R. Böhm, Tsewang Gyatso, Thomas Jennewein, Rainer Kaltenbaek, Michael Lindenthal, Gabriel Molina-Terriza, Andreas Poppe, Kevin Resch, Michael Taraba, Rupert Ursin, Philip Walther, and Anton Zeilinger. 2003. Long-Distance Free-Space Distribution of Quantum Entanglement. Science 301:5633 (1 August), pp. 621-623.
Bohm, David.
_________.
_________.
_________.
_________.
_________.
_________.
Bohm, David, and B. J. Hiley.
_________.
_________. 1981. On a Quantum Algebraic Approach to a Generalized Phase Space. Foundations of Physics 11:3-4 (April), pp. 179-203.
Bohm, David, B. J. Hiley, and Allan E. G. Stuart.
Einstein, A., B. Podolsky, and N. Rosen.
Frescura, F. A. M., and B. J. Hiley.
_________.
_________.
Hiley, B. J.
_________.
_________.
_________. 2001. Non-Commutative Geometry, the Bohm Interpretation, and the Mind-Matter Relationship. AIP Conference Proceedings 573, pp. 77-88.
Hiley, B. J., and N. Monk. 1993. Quantum Phase Space and the Discrete Weyl Algebra. Modern Physics Letters A 8:38 (14 December), pp. 3625-3633.
Hiley, B. J., and A. E. G. Stuart. 1971. Phase Space, Fiber Bundles, and Current Algebras. International Journal of Theoretical Physics 4:4 (August), pp. 247-265.
Mann, Charles, and Robert Crease.
Maroney, O., and B. J. Hiley. 1999. Quantum State Teleportation Understood through the Bohm Interpretatation. Foundations of Physics 29:9 (September), pp. 1403-1415.
Monk, N. A. M., and B. J. Hiley. 1998. A Unified Algebraic Approach to Quantum Theory. Foundations of Physics Letters 11:4 (August), pp. 371-377.
Peacocke, Arthur R. 1993. Theology for a Scientific Age: Being and Becoming – Natural, Divine,
and Human.
Penrose, Roger.
Polkinghorne, John C. 1991. The Nature of Physical Reality. Zygon: Journal of Religion and Science 26:2 (June), pp. 221-236.
Sharpe, Kevin.
_________.
_________.
_________. 1987b. David Bohm’s Physics and Religion. In Science and Theology in Action,
ed. Chris Bloore and Peter Donovan (Palmerston North,
_________.
_________.
_________. 1993b. A Holomovement Metaphysics and Theology. Zygon: Journal of Religion and Science 28:1 (March), pp. 47-60.
_________. 1997. A Holomovement Metaphysics and Theology. Bridges: An Interdisciplinary Journal
of Theology, Philosophy, History, and Science
_________.
_________, and Jonathan Walgate.