1. INTRODUCTION
EINSTEIN established a relation of
equivalence between matter and energy [1].
An equivalence relation is composed of three
distinct and separate terms. To the knowledge of
this writer, so far there is no identified third term
to which both matter and energy are equivalent.
Hence the relationship is not formally valid yet.
This paper proposes that the third term of the
equivalence be posited to be spirit.
After observing some of the canonical
requirements of the equivalence relation and the
fundamental advantages of casting our thought
processes into this format for the force it brings to
our reasoning, we shall first note the
shortcomings of some potential solutions to the
lack of formal validity of the equivalence of matter
to energy and then we shall try to obtain an
operational definition for the word spirit. Only
then shall we observe some of the consequences
of accepting the proposal of making spirit the
third term of the equivalence.
If the proposal stands to all the tests of validity,
this solution will eventually yield two considerable
benefits. It will transform physics from a linear
into a relational discipline. It will also tend toward
the reunification of the physical with the social
sciences.
2. PROBLEM STATEMENT
Matter and energy are two terms. E = mc2 is
not an equivalence relation; c2 is not a third term:
c2 is a unit of measure (of speed). As logicians
know, to be valid, an equivalence relation must
be composed of three terms. The three terms
have to be reflexive (namely identical to
themselves throughout the discourse), symmetric
(one observes the same entity from two points of
view in order to obtain a deeper understanding of
both entities), and transitive (a third term must
exist to which both terms are equivalent in order
to eschew the confines of circular reasoning and
to complete the analysis). With the assistance of
the equivalence relation the analysis does not
start from an arbitrary point nor does it end at an
arbitrary point, but is rigorously interlocked.
These observations can be made more evident
by specifying the progress of our thought
processes and by casting them into a set of
figures. Science eschews all singularities. There
is a good reason for this practice. A single point,
a single observation does not lead to an
objective, replicable analysis or experiment.
Analysis begins with the observation of two
events. Yet, the observation of two events
necessarily leads to circularity of reasoning.
Once we are faced with only two observations,
we are obliged to observe all possibilities. Hence
the mind is led back to the exploration of all
potential outcomes of the position of Point B on
the circumference of the circle in relation to Point
A at the center of the circle. This is a process that
eventually leads to a reversal of one’s position
and then to a return to the original position—and
no certainty is necessarily acquired in the
process. Therefore, science asks for a third term.
The third term points the research in the right
direction. However, if the third term is placed in a
linear position, the end result might be a
dispersal of the thought process into the empty
infinity of an enlarged circle. Linearity leads to
progressio ad infinitum.
It is the equivalence relation that restrains the
analysis from collapsing into infinity by
constraining the terms into an interlocked
relationship as in its standard configuration:
A ↔ B ↔ C. The equivalence relation starts in logic
and has the widest possible range of
applications. All forms of syllogism are based on
the equivalence relation. Hence the relation of
equivalence is well known to the literati. The
equivalence relation is also part and parcel of all
mathematics textbooks. It stands at the very
foundation of mathematics, in which three fingers
of my hand (3 of base 10 number system) are
equivalent to the word/number/symbol (three, 3,
or III) and to the three apples in front of my eyes.
A triangle is based on the equivalence relation.
The whole of trigonometry is based on the
equivalence relation. Indeed, as R. G. D. Allen
pointed out, the rules of equivalence “hold” also
for the relation of “equality (=)” [2].
In brief, there are many reasons why it is
essential to cast any scientific analysis in the
format proposed by the rules of logic in general,
and the principle of equivalence in particular. A
few of them, not necessarily in their order of
importance, are as follows. Logic, as a whole,
provides objective criteria for the evaluation of
any proposition; most disagreement, as is well
known, disappears as soon as the magic words
are pronounced: “But that is not logically
tenable.” Logic provides guidance to our
analysis; without it, we are rudderless. Guided by
rules of logic, we know whether or not we have
completed our analysis. Logic makes it possible
to replicate the reasoning or the experiment.
From the above it inexorably follows that the
fundamental relationship that Einstein
established between matter and energy is yet
incomplete. Two terms do not make an
equivalence relation. The relationship between
matter and energy is completed only when a third
element is found to which both matter and energy
are equivalent.
3. INADEQUACY OF SOME POSSIBLE SOLUTIONS
There are no explicit formulations of a third
term to which both matter and energy are
equivalent. As pointed out above, c2 is not a third
term, but a unit of measurement of speed that
has nearly nothing to do with light. It happens to
be the speed of light; hence, at best, it is an
attribute of light. By extension, it might be
assumed that mc2 contains in it, not just the
meaning of matter, but also—implicitly—the
meaning of light. Even if c2 stood for light, it
cannot be the third term because light is a form of
energy (clearly in the wave conception of light; or
a form of matter in the particle conception of
light). Thus, whether light is an intrinsic
component of E or m, it cannot at the same time
be an extrinsic term to which either E or mc2
might be equivalent. It cannot appear as an
addition to either side of the equation, without
creating double counting and without violating the
first requirement that each term of the
equivalence must be reflexive, namely identical
to itself throughout the observation. The addition
of the term light does not make the construction
symmetric; one cannot change the term light with
the term energy (or matter) and obtain positive
results: one does not gain a better understanding
of either matter or energy. Neither does that
addition make the terms of the construction
transitive: from light one necessarily goes back to
either matter or energy—not to both. These
considerations can also be put in common
language: a part cannot be confused with the
whole. If light is part of energy or part of matter,
light cannot be equivalent either to energy or to
matter, because this definition would run into the
impossibility of equating a part with the whole.
Since matter and energy, to be equivalent to
each other, must be whole units, namely units or
entities all complete in themselves, the third term
must also be a whole unit, a whole entity. It
cannot be a part of a whole.
The same considerations apply if the term third
is assumed to be derived from the equation E =
hv, where E is energy, h is Planck's constant
(which is equal 1 and thus disappears from the
equations of physics), and v is the measure of
the frequency of energy radiation emitted as
photons, rather than the speed of light.
A more abstract set of considerations are
necessary to dispel the notion that space (like the
old ether and the futuristic “higher order”) might
be the third element of the equivalence. The third
element has to have an existence of its own.
Take away matter and/or energy and space
disappears from our field of observation. Hence it
cannot be the third element that would make the
equivalence of matter to energy a valid
relationship.
We must search for a third term to which both
matter and energy are equivalent.
4. FINDINGS
This paper proposes that the search for the
third term to complete the equivalence of matter
to energy is exhausted with the introduction of
spirit into the relationship. This is the answer that
Fritjof Capra [3] inspired. One then obtains the
following equivalence: matter ↔ spirit ↔ energy.
This is a relationship that reads: matter is
equivalent to spirit and spirit is equivalent to
energy. This is a complete relationship of
equivalence, which can be defined as the
Relational Reality, and it can be diagrammed
using these established protocols:
Figure 1: The Relational Reality
Figure 1 can be interpreted not only to mean
that matter transforms itself into energy and
energy into matter, but—at the very least—
especially along these lines: The physical world
in which we live has to be observed first from the
point of view of matter and then from the point of
view of energy. The essential prerequisite is to
see these two aspects of reality not in linear
fashion, but in a relational mode, namely as two
separate and distinct viewpoints of the same
reality. When that is done, one can also see that
the total reality in which our daily existence is
immersed can be grasped only if it is observed,
not only from the viewpoint of matter and energy,
but also from the viewpoint of spirit. One enters
into the stone with a hammer; into the energy of
the stone with a cyclotron; and into its spirit with
prayer.
Thus we come back to the very roots of our
civilization. Our ancestral ancestors—not unlike
many brothers and sisters in many civilizations of
today—started their analysis of the world neither
from matter nor, certainly, from energy. It is fairly
certain that they started their analysis of the
world form the point of view of spirit.
But what is spirit? Can we obtain a precise
definition of this term? Can we obtain an
operational definition of this term?
5. ON THE DEFINITION OF SPIRIT
Spirit is incommensurable. Therefore, it is
difficult to define. Once it is realized, as we shall
more clearly see below, that mathematics, the
most precise of all sciences proceeds on the
basis of two incommensurable entities, namely
zero and infinity, this inherent difficulty that is
presented by the word spirit ought not to be of
much concern to a physicist. That said, we shall
try to identify some of the characteristics of spirit.
As used in this paper, spirit is a relation, the
relation that binds matter to energy. It keeps
them both factually together and intellectually
separate from each other. With the word spirit,
we can stop thinking of the universe as a linear
relationship in which matter somehow passes
into energy, and we can start conceiving of the
universe of matter being in organic relationship
with the world of energy. We can study the
objective reality first as a world of matter and
then as a world of energy. These are all enclosed
worlds of their own. If we conceive of both matter
and energy as two entities, indeed as two worlds,
in their own, without their individual link to spirit,
they would both be in fatal conflict with each
other. Instead, we notice near perfect and
continuous harmony between the two entities.
This we might say is an attempted definition of
spirit in the small, as in “the spirit of this stone”:
spirit is the link, the glue that holds matter and
energy together.
By trying to define spirit in the large, as an
infinite entity into which both matter and energy
are encompassed, and indeed as an infinite
entity in which we—observers—are all
encompassed, we might gain a greater control
over the forces of this world by regaining the
sense of what used to be called the “sacred”.
Only if the earth is seen as sacred again will we
feel obliged to respect its inner existence. It is
through the word spirit that we reach a better
understanding of both matter and energy.
Through that word, we enter deeply into their
essence and we get in close contact with each of
them. In an age in which we are discovering the
essential importance of a sound ecological
management of the planets, the word spirit will
incite us to gain a greater respect for the world of
matter as well as the world of energy than we
have at present.
Thus the word spirit has a theoretical as well
as an operational validity. And then it can
inexorably be observed that the infinity of spirit
manifests itself to us most clearly as both matter
and energy. Hence, the preeminence of the study
of physics in today’s culture is no longer
surprising.
6. AN EXTENSION OF THE WORD SPIRIT
But spirit does not manifest itself only as matter
and energy. It also manifests itself, indeed, as
spirit. Man’s mind has forever been engaged in
the attempt to define “spirit”. We must admit that
the task has eluded us. And there is a very good
reason why the task is destined to elude us
forever. Spirit is not an intellectual affair, hence it
can never be caught by the intellect. Since it is an
intensely personal relation, indeed an intensely
personal affair, the essence of the word spirit can
only be caught by our feelings. This is the
fundamental reason why approximation to the
understanding and explanation of spirit have
been in the past the prerogative of mystics,
theologians, philosophers, literati, and musicians.
As the practitioners of these disciplines have
forever made an attempt to convey their
understanding of spirit to all other people who
may be interested in the topic, so physicists in
the future—as they have indeed done in the past
(see, e.g., Aristotle and Thomas Aquinas)—have
to try to convey to the practitioners of the spirit
the goodness, the truth, and the beauty that they
discover in both matter and energy.
7. SOME LIMITATIONS OF THE MEANING OF SPIRIT
One could define spirit as Spirit, namely as
God. However, this definition might be
misleading; it might lead into the old pitfalls of
pantheism. To avoid such dangers, it is
necessary to distinguish God from God’s spirit; it
might be necessary to say that God is also spirit;
and then one must be careful to limit God’s
presence in matter and energy by saying that the
spirit of God is also in matter and energy, also in
the stone and its energy. If God is by definition
everywhere, then—given the above
qualifications—it is possible to say that God is
also in the stone and in its energy. And then one
surprising result ensues: a very practical
consequence indeed. It appears that all three
entities of thought, namely matter, energy, and
spirit might share the same unit of measure:
degrees of heat. One of God’s characteristics is
to be in essence love, Love par excellence. And
is not warmth and heat one of the most
endearing physical manifestations of love?
8. SOME IMPLICATIONS FOR PHYSICS
Not being a physicist, this writer can suggest
some of the implications for physics of
establishing a true equivalence between matter,
energy, and spirit only at a very broad level of
generality. When one multiplies the mass by the
square of the speed of light, when one spins
matter at the squared speed of light, one no
longer observes matter but energy. One is no
longer in the world of matter, but in the world of
energy. One has made such a definite break
between the two worlds that, in order to achieve
clarity of mind and expression, one must
accordingly design a new nomenclature. Using
words from one world and applying them to the
other leads to analogical thought, but not to
innovative and incisive thought.
The second consequence that this writer can
envisage is the need to jettison the old
attachment to absolute quantification.
Quantification in physics has always taken place
within sharply defined limits. One has simply to
resign to the nature of things that this is the only
type of quantification that might forever be viable
in physics. In order to reduce the level of
apprehension about this condition, physicists will
want to notice that mathematics too has always
been subjected to this condition. If one does not
see the number system as a linear but a
relational organization of numbers, it becomes
clear that mathematics is based on the following
foundational equivalence:
0 ↔ 1 ↔ ∞. The first
impression is that mathematics has been able
always to proceed with the quantification of only
one of its terms: namely, the number 1.
Mathematics does not, and cannot quantify either
zero or infinity. And it does not matter. Indeed, on
second thought, mathematics does not quantify
the third of its foundational terms either;
mathematics does not present us with an
absolute quantification of one, but a relative
quantification of one. Numbers proceed from
(plus or minus) one to infinity, but they never
touch infinity; the conception of the limit is there
to recognize this deficiency and to allow us to
work within the limits offered by reality. Thus,
taking a leaf from the transition from Galileo and
Newton to Einstein through Hume in relation to
space and time [4], we shall not be concerned
with absolute but with relative quantification.
Hence, we can safely maintain that
If the universe is infinite, we shall never weigh
its mass;
If the universe is infinite, we shall never
measure its length;
What we measure is its mass and its length in
relation to man.
Then, man—indeed, every man and woman—
is again positioned at the center of the universe.
9. SOME CONCEPTUAL CONSEQUENCES
In 1946 Einstein remarked: "The unleashed
power of the atom has changed everything save
our modes of thinking” [5]. With the establishment
of the equivalence of matter to spirit and to
energy, everything changes. Technically, Figure
1 establishes that while any element of reality
occupies its own distinctive position, everything is
in full relationship with everything else. Hence, as
proved by the Internet, everything is indeed
directly related to everything else. This
complexity is better observed by rotating at ever
increasing speed, not only the entire Figure 1,
but also each rectangle inside Figure 1 about its
geometric center. One then obtains the image of
four circles: one, the circle of matter; two, the
circle of spirit; three, the circle of energy; four, the
circle of the relational reality as a whole. This is a
Venn diagram delimited by a circle. And what is a
circle, if not a two-dimensional image of a
sphere? Ultimately, one is thus presented with a
construction composed of four interpenetrating
concentric spheres, one for each point of view
from which reality can be observed: the point of
view of matter, spirit, energy, and the system as
a whole. An analysis of this type of construction
can be followed in detail in the humbler reality of
the world of economic justice [6] and the world of
economics [7]. The mathematics of this
construction is well-known [8] and it might be
useful to reproduce it here in a very abstract form
as follows:
a. = fa(a,b,c)
b. = fb(a,b,c)
c. = fc(a,b,c),
where a. = rate of change in the first element of
the relationship, b. = rate of change in the second
element of the relationship, and c. = rate of
change in the third element of the relationship.
From the linear world of rationalism, thus
everything is transformed into the organic world
of relationalism. Above all, beyond changes of
perspective in physics, if this construction of
reality is accepted, the warlike relation between
the “two cultures” is expected to change and
eventually to come to a screeching halt; with
time, this war—with its multifarious
manifestations of reductionism, materialism, and
atheism, and, above all, mutual
misunderstandings—will unavoidably come to a
screeching halt.
While waiting for a response to these
observations from the people of science, we
already know the response from the people of
spirit. Poetry and philosophy have spoken
forcefully about the evident relationship between
matters of the earth and matters of the spirit [9].
Since this writer is more familiar with the Catholic
tradition, he will limit himself to one quotation
from within this belief system. But many other
expressions come easily to mind. "Every culture,"
Christopher Dawson wrote, "is like a plant. It
must have its roots in the earth, and for sunlight it
needs to be open to the spiritual. At the present
moment we are busy cutting its roots and
shutting out all light from above” [10].
If mathematicians and physicists, following
strict rules of logic that they already obey in all
steps of their reasoning, can be convinced that
their own fields—as moral theologians insist—are
all immersed into the world of spirit, all other
scientists, especially social scientists, will not
take long to follow suit. After all, it was Einstein
who said: “Science without religion is lame,
religion without science is blind” [11].
10. CONCLUSION
There are many indications that the world of
linear, rational, Cartesian logic has come to an
end—see, e.g., John Lukacs, At the End of an
Age [12]. This is a world in which reality is
reduced to isolated atoms. The principle of
equivalence is a ready-made tool that allows us
to escape the strictures of Cartesian logic and
leads us into the world of relational logic, a world
in which everything is naturally related to
everything else. This paper has used this
principle and reached some novel conclusions in
relation to physics and mathematics. In the
process, it has laid the groundwork for healing
the ongoing schism between the “two cultures”.
A POSTSCRIPT
The reader might be interested to know that
this paper was not written with the Shroud of
Turin in mind. Yet, at one point it became
apparent to this writer that the paper makes the
Shroud a logical and “natural” necessity. Even
the Transfiguration and the appearance of Jesus
in the Cenacle become understandable,
because—if this reasoning is right—Jesus is,
was, and will forever be the perfect union of
matter and energy and spirit. And, of course,
accepting this reasoning one can see that the
consecrated communion host is real.
If this reasoning is accepted to be theologically
and logically valid, it leads to a further
observation. The study of singularities is not
concluded by the study of matter alone, or energy
alone, or spirit alone. It is the integration of the
three worlds that might yield a better
understanding of singularities as well as a better
understanding of the world as a whole.
The study of singularities cannot be eschewed
by science. Science cannot thus limit itself.
Indeed, as various technical studies of the
Shroud of Turin prove, science has an essential
role to play in the analysis and the distinction of
true from false singularities.
ACKNOWLEDGMENT
The author wishes to acknowledge the
technical assistance received from his longstanding
collaborator, Louis J. Ronsivalli, an MIT
food science technologist, and a most positive
feedback from Dr. F. Hadi Madjid, a Harvard
physicist. The paper has greatly benefited from
comments and recommendations from six
referees on an earlier draft. Thanks also go to
Jonathan F. Gorga for invaluable editorial
assistance.
REFERENCES
[1] Einstein, A. (1905). “Ist die Trägheit eines Körpers von
seinem Energieinhalt abhängig?” (Does the Inertia of a
Body Depend upon its Energy-Content?). Annalen der
Physik. Ser. 4, Vol. 18, pp. 639-641. See also, [1935
(1971)]. “Elementary Derivation of the Equivalence of
Mass and Energy”. American Mathematical Society,
Bulletin. Vol. 41, pp. 223-30. In R. W. Clark. Einstein—
The Life and Times. New York and Cleveland: World
Publishing, p. 538; see also pp. 98-103, 115-120, 126,
141, 144, 268, 537, and 587.
[2] Allen, R.G. D. (1970), Mathematical Economics, 2nd
edn. London and New York: Macmillan, St. Martin’s, p.
748.
[3] Capra, F. (1980), The Tao of Physics, New York:
Bantam Books.
[4] Cf. Feinstein, J. S. (2006), The Nature of Creative
Development, Stanford: Stanford Business Books, esp.
pp. 303-315 and 322-328.
[5] Einstein, A. (1946). In Nathan O., and Norden, H. eds.,
Einstein on Peace. New York: Avnet Books, 1981 ed, p.
376, from a pamphlet published by Beyond War in 1985
entitled A New Way of Thinking.
[6] Gorga, C. (1999). “Toward the Definition of Economic
Rights”, Journal of Markets and Morality Vol. 2, No. 1,
pp. 88-101.
[7] Gorga, C. (2002). The Economic Process: An
Instantaneous Non-Newtonian Picture. University Press
of America, Lanham, Md., and Oxford.
[8] Thompson, J. M. T. (1986), Nonlinear Dynamics and
Chaos, Geometric Methods for Engineers and
Scientists, New York: Wiley.
[9] For poetry, Walt Whitman’s work should suffice. For
philosophy, see. e.g., Hegel, G. W. (1807). The
Phenomenology of Spirit and Emerson, R. W. The
Natural History of the Intellect (or The Natural History of
the Spirit) unpublished.
[10] Dawson, C. In Catholic Educator's Resource Center
(CERC), Bi-Weekly Update, November 5, 2004. At
www.catholiceducation.org.
[11] Einstein, A. (1941). "Science, Philosophy and Religion: a
Symposium". From The Quotation Page at
http://www.quotationspage.com/quote/24949.html.
[12] Lukacs, J. (2002), At the End of an Age, New Haven
and London: Yale University Press.
ABOUT THE AUTHOR
Carmine Gorga is a former Fulbright scholar and the recipient of a Council of Europe
Scholarship for his dissertation on "The Political Thought of Louis D.
Brandeis." With a book titled The Economic Process and a series of papers, Dr. Gorga
has transformed the linear world of economics into a relational discipline in
which everything is related to everything else, This characteristic of
Concordian economics has been recognized by JEL in December 2017 (p. 1642). Gorga was assisted for 27 years by Professor Franco Modigliani, a Nobel laureate in
economics at MIT. For a full understanding of Concordian economics, Gorga has
gradually realized that we need to go beyond Individualism and Collectivism,
toward Somism (men and women in the social context)—see www.somistinstitute.org—and then we need to pass from Rationalism to Relationalism: see www.relationalism.org. See also Wikipedia and Google Scholar.
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