Relationalism

The greatest WordPress.com site in all the land!

In mathematics

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 1800 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 justice2, economic theory3, and economic policy4—as well as, in outline, in the parallel world of physics.5 The mathematics of this construction is well-known6 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. Poetry11 and philosophy12 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 Age15 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, The Economic Process: An Instantaneous Non-Newtonian Picture (Lanham, Md., and Oxford: University Press of America, 2002). An expanded edition is in press.

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 titled 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 ResourceCenter (CERC), Bi-Weekly Update, November 5, 2004. At http://www.catholiceducation.org.

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, At the End of an Age (New Haven and London: Yale University Press, 2002).

16. Kline, op. cit. (note 8) p.7.

Short title: Mathematics as a Relational Discipline

Advertisements

Single Post Navigation

Leave a Reply

Fill in your details below or click an icon to log in:

WordPress.com Logo

You are commenting using your WordPress.com account. Log Out / Change )

Twitter picture

You are commenting using your Twitter account. Log Out / Change )

Facebook photo

You are commenting using your Facebook account. Log Out / Change )

Google+ photo

You are commenting using your Google+ account. Log Out / Change )

Connecting to %s

%d bloggers like this: