The original publication is available at

*International Journal of Mathematics, Game Theory and Algebra,*

*2010, vol. 19, issue 4, pp. 235-244.*

ON
THE
TRANSFORMATION OF MATHEMATICS

FROM A

LINEAR TO A RELATIONAL DISCIPLINE

**Toward the Reunification of the Physical and the Social Sciences**

Carmine Gorga

**Abstract**

While the structure of mathematics is built on an
uninterrupted series of equivalence relations, the number system forms a linear
progression. To eliminate this dichotomy, while leaving the functionality of the number system intact, the
present paper proposes to transform the relation between zero, one, and
infinity into a relation of equivalence. This relation breaks the empty linear

*progressio ad infinitum*of the number system; selects zero, one, and infinity as the most important elements in the series; and locks them into a position of mutual relationships. The terms turn out to be reflexive, symmetric, and transitive. It is thus possible to see that each element represents a concrete world of its own, a condition that sheds lights of understanding on each one of the other two and yields the derivation and the definition of each term in relation to the others. If this application of the principle of equivalence is accepted as valid, mathematics is transformed into a relational discipline. Then everything changes in the spirit of both the physical and the social sciences. Mutual adjustments will eventually facilitate the re-unification of the physical with the social sciences and abate the warlike relation between the “two cultures.”**Keywords:**number system; linearity; equivalence; rationalism; relationalism

**Brief Bio**

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.” Using age-old principles of logic and
epistemology, in a book and a series of papers Dr. Gorga has transformed the
linear world of economic theory into a relational discipline in which
everything is related to everything else—internally as well as externally. He
was assisted in this endeavor by many people, notably for twenty-seven years by
Professor Franco Modigliani, a Nobel laureate in economics at MIT. The
resulting work,

*The Economic Process: An Instantaneous Non-Newtonian Picture,*was published in 2002 and it is currently being reissued in an expanded version. For reviews, see http://www.carmine-gorga.us/id18.htm. During the last few years, Mr. Gorga has*concentrated his attention on matters of methodology for the reunification of the sciences.*
Introduction

While the
structure of mathematics is built on an uninterrupted series of equivalence
relations, the number system is constructed as a linear progression. To eliminate
this dichotomy, the present paper, without affecting the inner workings of the number system, attempts to
consider zero, one, and infinity as three distinct entities, unlike the numbers
2, 3, or 4, and ties them together into an equivalence relation. This pivotal operation transforms
mathematics from a linear to a relational discipline: Each foundational element
is related to the rest of mathematics and to the outside world.

After observing
some of the canonical requirements of the equivalence relation and the
fundamental advantages of casting thought processes into this format, the paper
calls attention to some of the major links in the series of equivalence
relations on which mathematics is built. Equivalence is not composed of a
mechanical addition of one element to another, as in the linear relation of 0,
1, 2, 3… ∞: 0 + 1 + ∞ is meaningless. Rather, it searches
for the most important elements in a set and interlocks them. This approach transforms
a linear mode of thinking into a relational one. Hence the paper proposes that
the most fundamental relation in mathematics is the equivalence of Zero to One
to Infinity; in traditional notation, 0 ≡ 1 ≡ ∞; or, 0 = 1 = ∞; and,
in different notation, 0 ↔ 1 ↔
∞. As can be seen, nothing changes but our mode of thinking about the
number system: The system is all there, but changed from a linear to a
relational apparatus.

If the present
proposal stands all the tests of validity, this solution will eventually yield
two considerable benefits. This internal transformation (1) reveals some inner
characteristics of mathematics that are shielded from plain view and (2) tends
to facilitate the eventual re-unification of the physical with the social
sciences.

Problem Statement

The number system
is conceived as proceeding from zero to one to infinity. This is a progression
that leaves the three fundamental entities of mathematics—namely zero, one, and
infinity—unrecognized and unrelated to each other or related to each other in a
linear mode of thought. This linear linkage leads the mind to an empty

*progressio ad infinitum*. This condition leaves mathematics serving, yet conceptually isolated from, all other mental disciplines. The proof is that each one of the three terms, namely zero, one, and infinity are conceived as being identical in form to the number two, three, and all other numbers (the ever incomplete listing of which yields one of the approaches to the meaning of infinity). As such, zero, one, and infinity are left unidentified as in the Hegelian night in which all cows are black. In reality, all numbers are not equal. As will be seen, those entities are three fundamentally distinct building blocks of mathematics.
While the relationship
tying zero to one and to infinity is conceived as being linear, the equivalence
relation stands as part and parcel of all of mathematics. It stands at the very
foundation of the number system, in which three fingers of the hand (3 of base
10 number system) are equivalent to a word/number/symbol—namely, three, 3, or
III—and to the three apples in front of our eyes. All algebraic relations are
equivalence relations. A system of equations is based on the equivalence
relation. A triangle is based on the equivalence relation. The whole of
trigonometry is based on the equivalence relation. Indeed, as R. G. D. Allen
points out, the rules of equivalence “hold” also for the relation of “equality
(=)”.

^{1 }Hence, 1 = 1 is an equivalence relation, because its validity stands on the proof that 1 = (6 – 5) or any other such relation. Hence, in*extenso*, 1 = 1 ought to be written 1 = (6 – 5) = 1 or 1 = (6 – 5) = (7 – 6).
The equivalence
relation starts in logic and has the widest possible range of application
outside of mathematics as well. All forms of syllogism are based on an
equivalence relation; most religions are based on an equivalence relation.
Hence the relation of equivalence is well known to theologians, philosophers,
and the literati. As logicians—and mathematicians—know, to be valid an
equivalence relation must be composed of three terms. The three terms have to
be reflexive (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, to observe the same entity from three points of view or have a
triple check on our reasoning, and to complete the analysis). With the
assistance of the equivalence relation the terms of the analysis do not start
from an arbitrary point and end at an arbitrary point, but are strictly
interlocked.

These observations
can be made more certain by specifying why science eschews all singularities.
There is a good reason for this practice. Punctilionism, the defense to the
death of a single point unrelated to the rest of the universe, is not analysis.
A single event 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. Faced
with two observations, one is obliged to observe all possible relationships
between them. Hence, the mind is led back to the exploration of all potential
outcomes of the position of Point B on the circumference in relation to Point A
at the center of the circle. This is a process that eventually leads to the
reversal of one’s position (an 180

^{0}turn) and then to a return to one’s original position—and no certainty is acquired in the meantime. Therefore, science asks for a third term. The third term points the research in the right direction. If the third term is placed in a linear relation-position-alignment, however, the end result might be a dispersal of the thought process into the empty infinity of an enlarged circle. Again, 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. 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 the analyst; without it,
the analyst is rudderless. Thanks to the rules of logic, it becomes apparent whether
or not the analysis is complete. Logic makes it possible to replicate the
reasoning or the experiment.

From the above
it inexorably follows that, in the number system is commonly conceived, the relationship
that exists between zero, one, and infinity is linear and unspecified; namely, it
is 0 → 1 → ∞. The terms do not make an equivalence relation. The terms are
thoroughly specified when it is recognized that the relationship linking them
is a relation of equivalence, namely when they are linked in this form: 0 ↔ 1 ↔ ∞.

Inadequacy
of Present Conception

There are various
reasons why the present linear conception of the number system is inadequate.
The most important one perhaps is that placing zero, one, and infinity in a
linear relationship to each other condemns the number system, and by extension
the whole of mathematics, to remain a closed entity separate from all other
forms of thought. Where is the relationship between mathematics and poetry? Or
philosophy? Or religion? Also, by
placing the three terms in linear succession with each other, they become an
indistinct part of the number system: 0 is a number just as 1 is a number; and
0 occupies a position just as 1 occupies a position on the number list. Thus
they lose their distinct identity, and cannot be defined: More specifically, they cannot be defined in
a manner that, ideally, might satisfy everyone once and for all. In
addition, it will be seen that the closed world of mathematics provides a
faulty and misguided sense of certainty to the rest of the intellectual community;
instead, by ascertaining and affirming the truth about its own modus operandi,
mathematics could indeed offer much useful guidance.

Findings

This paper
proposes that the search for the relationship among Zero, One, and Infinity is
completed when it is realized that what links the three terms to each other is
a relationship of equivalence. One then obtains this equivalence: Zero ↔ One ↔ Infinity. This is a relationship that
allows the number system and by extension the whole of mathematics to be
classified as Relational Mathematics. The relationship can be diagrammed interchangeably
using these established protocols:

Figure 1. Relational Mathematics

Figure 1 can be interpreted not only to mean
that Zero is a different aspect of One and One is a different aspect of
Infinity, but also along these lines: The mathematical world which controls so
much of our lives has to be observed first from the point of view of Zero, then
from the point of view of One, and then from the point of view of Infinity. The
essential prerequisite is to see these three elements of the number system not
in a linear fashion, but in a relational mode, namely as three separate and
distinct viewpoints of the same system. The easiest method to realize
that the three entities are inextricably related to each other is to
superimpose the three rectangles forming Figure 1 upon each other at once,
alternatively by separately placing each one of the three rectangles on top of
the other two. Two rectangles then are obstructed from view, but they remain
stubbornly there. Indeed, it is then that we come to the full realization that
only by distinguishing the three entities from each other can we hope to
understand all three of them. Otherwise, we reduce the construction to a
singularity; or lock it into circular reasoning, if we were to deny either the
reality of Infinity or the reality of Zero.

Technically, Figure 1 establishes that while
any element of the mathematical reality occupies its own distinctive position,
everything is in full relationship with everything else. This complexity is
better observed by rotating about its geometric center at ever increasing
speed, not only the entire Figure 1, but also each rectangle inside Figure 1.
One then obtains the image of four circles: one, the circle of Zero (A); two,
the circle of One (B); three, the circle of Infinity (C); four, the circle of the
mathematical relational reality as a whole (M). This is a Venn diagram of sets
A, B, C, and M delimiting four circles. 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 the mathematical world can be observed: the point of
view of Zero, One, Infinity, 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

^{2}, economic theory^{3}, and economic policy^{4}—as well as, in outline, in the parallel world of physics.^{5}The mathematics of this construction is well-known^{6}and it might be useful to reproduce it here in a more 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. (M is there to suggest that the equivalence relation is and will always be open to form the next equivalence and to understand the rest of the world.)
When the analysis is completed, it is
possible to see that the total reality in which the number system is immersed
can be grasped only if it is observed, not only from the viewpoint of One, but
also from the viewpoint of Zero and Infinity. Through a set of equivalence
relations, in the night of time we built the number system. We posited 1 + 1 =
2. This last entity, 2, is a convention. The magic—or creative synthesis, as
Benedetto Croce pointed out—is in the sum of 1 plus 1; and then another magic
is in the universal acceptance of the convention that 2 equals the sum of 1
plus 1.

Following the same reasoning, we built the
entire number system. On the positive side. The negative side is not a new
chain, but a symmetric reversed chain: -1, -2, etc. The proof can be
constructed by following this evidence: 1 – 1 = 0, just as (-1) – (-1) = 0.

The number system forms a well recognized
unit of thought. The seed of this system is all contained in the entity that we
call One. And from One, as briefly seen above, we gradually pass to the initial
understanding of Infinity through both the positive and the negative chain of
numbers. A fuller understanding of Infinity is contained in the realization
that, just like One, the entity that we call Infinity is a whole unit in
itself; it is a complete system: it is a “full” set—a set whose understanding
is approached by the inclusion in it of the entire number system. Similarly,
the entity that we call Zero is the “empty” set—a set whose understanding is
approached by subtracting one number at a time from the number system until we
are left with no more numbers. This empty set, this null set, is a whole entity
in itself. The briefest possible proof that Zero is not “another” number is
contained in the realization that Zero is not the half point between all
positive and all negative integers: Since both chains are infinite, every point
on them is the half point. Furthermore, Zero is not another number because we
are not able to distinguish whether it is derived from 1 – 1 or from (-1) –
(-1). Were it necessary, the practical function of Zero as indicating position
could be replaced by any other symbol, such as a forward slash, without
altering the structure or the integrity of the number system.

In other words, we
reach a fuller understanding of the set we call One by putting it in relation
with the set we call Infinity as well as the set we call Zero. We enter into
the world of One precisely as we have done ever since the dawn of civilization,
by observing things and counting them with the help of our fingers. Once we
begin to recognize that there is no end to the number system we enter the world
of Infinity. 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
with the understanding of the number One. Then they jumped off to the
understanding of such an esoteric concept as the world of Infinity. Only
relatively recently in the history of civilization have we discovered the need
to understand the world of Zero.

How do we define these entities?

On the Definition of
Zero, One, and Infinity

One is
incommensurable. Therefore, it is difficult to define. Indeed, by itself we
might never be able to define it. One represents a whole world of its own. And
we can be sure of its existence, not only because we have created it, but
especially because it functions as the seed of the entire number system. The
number/word “two” would be a complete abstraction if it were not possible to
say that it results from the addition of One plus One or, in standard notation,
2 = 1 + 1. (And we can be sure that this equality is valid, because the two
sides of the equation are equivalent to each other: they have the same meaning,
the same value, the same weight.)

Indeed, it might
be forever impossible to define One if it were not for the concept of Infinity.
Thus, calling for assistance on the principle of non-contradiction, it can be
said that One is not-Infinity. Yet, both One and Infinity would forever remain
locked into a circular relationship if both concepts were not anchored in the
reality of Zero. With the inclusion of this third element, we can now
distinguish One from Infinity, not simply by saying that One is not-Infinity,
but by specifying that One is the beginning of a set that separates itself from
Zero and tends to Infinity.

Similarly, it
might be forever impossible to define Infinity if it were not for the concept
of Zero. Thus, calling again for assistance on the principle of
non-contradiction, it can be said that Infinity is not-Zero. By extension, we
can now say that Infinity is a full set, while Zero is an empty set. Yet, both
Infinity and Zero would forever remain locked into a circular relationship if
both concepts were not anchored in the reality of One. With the inclusion of
this third element, we can now distinguish Infinity from Zero, not simply by
saying that Infinity is not-Zero, but by specifying that Infinity is the
end/beginning of a set that starts with One.

In turn, since
Zero is not-One, Zero is also not-Infinity. It is by subtracting, one by one,
all the content of Infinity—one by one all the numbers from the number
system—that we reach the concept of Zero. Zero then is the symmetric, but
negative, image of Infinity. It is only by understanding Zero that we can
conceive of Infinity; and only by understanding Infinity that we can conceive
of Zero—and indeed One.

Zero, One, and
Infinity can be understood only in relation to each other. Zero is the other
face of Infinity; One is the other face of Zero; Zero is the emptiness of
Infinity; One is the fullness of Infinity; Infinity is the fullness of One and
the emptiness of Zero; One is the limit between Zero and Infinity or,
conversely, One is the separation of Zero from Infinity. The terms are
reflexive, symmetric, and transitive.

As used in this
paper, One is a relation, the relation that binds Zero to Infinity. One is the
link, the glue that holds the world of Zero and the world of Infinity together.
It keeps both Zero and Infinity factually together and intellectually separate
from each other. With the word One, we stop thinking of the universe of
mathematics as a linear relationship in which Zero loses its distinction and
its existence when it inexplicably passes into One and eventually into
Infinity, and we start conceiving the three terms of the universe of
mathematics as being three whole entities—complete in themselves and in organic
relationships with each other. We can then study the objective reality of the
number system first as the world of One, then as the world of Zero, and then as
the world of Infinity. Each term is an enclosed world of its own.

Some Implications
for Mathematics

Observed at a very
broad level of generality, some of the implications for mathematics of locking
Zero, One, and Infinity into a relation of equivalence are as follows. The
number system is no longer composed of an infinite series of elements; it is
composed of a comprehensible series of three elements, the world of One, the
world of Zero, and the world of Infinity. These three elements are no longer
independent of each other but are strictly interconnected. It is through the
word One that we reach a better understanding of both Zero and Infinity.
Through the concept of One, we enter deeply into the essence of both Zero and
Infinity and we extend our intellectual grasp to encompass both of these
totally abstract entities.

More importantly
perhaps, by trying to define One as an infinite entity into which both Zero and
Infinity are encompassed, and indeed are One, we realize that this is an
infinite world in which we—observers—are all encompassed. Conversely, we can
look at the world of Zero or the world of Infinity by realizing that they also
encompass the world of One. Thus the utmost hope is that, having reached this
level of understanding of the number system, we might gain greater control over
the forces of this world by regaining the sense of interconnectedness that
links everything to everything else.

The most evident
and practical consequence of transforming mathematics into a relational
discipline is the need to jettison the old attachment to absolute
quantification. Quantification in mathematics has always taken place within
sharply defined limits. Thus, taking a leaf from the transition from Galileo
and Newton to Einstein through Hume in relation to space and time,

^{7}we shall not be concerned with absolute but with relative quantification. We shall conclude that quantification is never absolute; it is always relative. And it might forever remain relative. In order to reduce a possible level of apprehension about this condition, we might notice that mathematics has always been subject to this condition. Mathematics, the most precise of all sciences, proceeds on the basis of two incommensurable entities, namely Zero and Infinity. Indeed, on second thought, mathematics presents us, not with an absolute but with a relative quantification of its third foundational term as well: One, One as symbolic representation of the number system as a whole. The number system is infinite; hence, it is incommensurable. One is incommensurable.
Of course, we must
not confuse the instrument of measurement with the object being measured. Yet,
reference to the object of measurement confirms the validity of the observation
that quantification is never absolute. We can number the apples in front of our
eyes as being three because they are in a relative relationship with us. But
can we dream of measuring all the apples that ever existed and all the apples
that will ever exist? Did not Mandelbrot conclude that even the coastline of Britain
is infinite?

^{8}
Needless to say,
this is not an advocacy of not measuring things that can be measured.

From these
considerations there ensues a surprising and interesting consequence. Blind
believers in the powers of mathematics have imparted a wrong sense of security
to the world. From every corner it is heard: Measure things, quantify them,
number them—and you shall acquire certainty. Through the equivalence relation
of Zero to One and Infinity, it becomes utterly clear that mathematics does not
impart any sense of certainty to the world. None of its three foundational
terms can ultimately be numbered or transformed into a measurable or
quantifiable entity. What is invaluable is that mathematicians do know it and
have learned how to cope with their world of uncertainty. They have invented
the instrument of the

*limit*—whereby numbers, proceeding from (plus or minus) One, approach Zero and Infinity, but never quite reach them. Thus mathematicians, from being the most abstract of all people, turn out to be the most concrete and practical ones indeed.
Some Conceptual
Consequences for Other Disciplines

In 1946 Einstein remarked:

*"*The unleashed power of the atom has changed everything save our modes of thinking*”*.^{9}With the establishment of the equivalence of Zero to One and to Infinity, while the number system remains as intact as before, our modes of thinking about mathematics change and transform this discipline from a linear to a relational entity. And then (potentially) everything changes.
From
the linear world of Cartesian logic and rationalism, everything is transformed
into the organic world of relationalism in which—as proved by the
Internet—everything is indeed related to everything else. Above all, beyond
changes of perspective in mathematics, if the proposed construction of the
number system is accepted as valid, with time, the warlike relation between the
“two cultures”—with its multifarious manifestations of reductionism,
materialism, and atheism, and, above all, mutual misunderstandings—will,
through mutual adjustments, unavoidably come to a screeching halt. Mathematics
is no longer the fount of all certainty; if neither Zero, nor One, nor Infinity
is a measurable entity, mathematics is as imponderable as all other mental
disciplines. This is a verity that is well-known to insiders (Kline 1980);

^{10}yet, it does not seem to have yet percolated among non-mathematicians. If there is such a thing as certainty, it exists outside the realm of mathematics.
While
waiting for a response to these observations from the people of science, we
already know the response from the people of the spirit. Poetry

^{11}and philosophy^{12}have spoken forcefully about the evident relationship between matters of this earth, including man’s mind, and matters of the spirit; between ponderable and imponderable entities. 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”.^{13}
If mathematicians, 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
unmeasurable world of the 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.”

^{14}
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*^{15}or Morris Kline,*Mathematics: The Loss of Certainty.*^{16}That was a world in which reality was reduced to isolated atoms. The principle of equivalence is a ready-made tool that allows us to escape the strictures of linear, rational logic and leads us into the world of relationalism, a world in which everything is naturally related to everything else. This paper has used this principle to link Zero to One and to Infinity. In the process, it has reached conclusions in relation to the world of mathematics that lay the groundwork for eventually healing the ongoing schism between the “two cultures.”
ACKNOWLEDGMENTS

The author wishes to
acknowledge the technical assistance received from his long-standing
collaborator, Louis J. Ronsivalli, an MIT food science technologist, and a most
positive feedback from Dr. F. Hadi Madjid, a Harvard physicist. This
presentation has greatly benefited from comments and recommendations from six
referees on an earlier draft of this paper. Thanks also go to Jonathan F. Gorga
for his invaluable editorial assistance.

NOTES

1.
R.G.
D. Allen,

*Mathematical Economics*, 2nd edn, p. 748 (London and New York: Macmillan, St. Martin’s, 1970).
2.
C. Gorga, ‘Toward the
Definition of Economic Rights’,

*Journal of Markets and Morality***2**, 88-101 (1999).
3.
C.
Gorga, Oxford : University Press
of America , 2002). An
expanded edition is in press.

*The Economic Process: An Instantaneous Non-Newtonian Picture*(Lanham, Md., and
4.
C.
Gorga, “Concordian economics: tools to return relevance to economics”,

*Forum for Social Economics*(forthcoming).*5.*C. Gorga, “On the Equivalence of Matter to Energy and to Spirit”,

*Transactions on Advanced Research*3, 2, 40-47 (2007).

6.
J.
M. T. Thompson,

*Nonlinear**Dynamics and Chaos, Geometric Methods for Engineers and Scientists (*New York: Wiley, 1986).
7.
Cf.
J. S. Feinstein,

*The Nature of Creative Development,*esp. pp. 303-315 and 322-328 (Stanford: Stanford Business Books, 2006).
8.
B.
B. Mandelbrot,

*The Fractal Geometry of Nature,*p. 25 (New York: W. H. Freeman and Company, 1982).
9.
A. Einstein. In Nathan
O., and Norden, H. (eds),

*Einstein on Peace,*p. 376 (New York: Avnet Books, 1981)*and a pamphlet published by Beyond War in 1985 entitled**A New Way of Thinking*.
10. M. Kline,

*Mathematics: The Loss of Certainty*(New York: Oxford University Press, 1980).
11. For poetry, W. Whitman’s work should
suffice.

12. For philosophy, see. e.g., G. W. Hegel,

*The Phenomenology of Spirit*(1807) and R. W. Emerson,*The Natural History of the Intellect*(or*The Natural History of the Spirit)*unpublished*.*
13. C. Dawson. In Catholic Educator's Resource Center (CERC), November 5, 2004 . At www.catholiceducation.org.

*Bi-Weekly Update*,
14. A. Einstein,

*‘Science, Philosophy and Religion: a Symposium’*(1941). In*The Quotation Page*at*http://www.quotationspage.com/quote/24949.html**.*
15.
J. Lukacs, New Haven and London : Yale University Press, 2002).

*At the End of an Age*(
16.
Kline,

*op. cit.*(note 8) p.7.**Short title:**Mathematics as a Relational Discipline

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