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A TOPOLOGICAL APPROACH TO AUDIT DYNAMICS

AND AUDITING RESEARCH

Borislav Dimitrov1

1Borislav Dimitrov, Lawyer /Civil Law LL.B/, Internal Audit Consultant, with corporative and government /as

state expert at Internal Audit Directorate of MoD of The Republic of Bulgaria/ expertise in IP law, internal audit,

legal auditing and compliance. Research Associate at Institute for Philosophical Research, Bulgarian Academy

of Science, 1989; Founding Director of Ariadne Topology and Cultural Dynamics Institute for Cultural

Phenomenology of Qualitative quantity, 2011. Researcher with Bulgarian Auditing Research Center, Registered

member of CARP, Center for Advanced Research in Phenomenology, Inc.,Department of Philosophy, University

of Memphis, USA - Center for Advanced Research in Phenomenology, Inc., Member of European Auditing

Research Network (EARNet)


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A Topological Approach To Audit Dynamics and Auditing Research

Abstract

The purpose of the paper is to open up discussions on the opportunity to implement

topological approach to audit dynamicsboth to the issues of auditing practice and themes in

auditing research.

In the context of auditing research structure and dynamics, the proposition of Qualitativequantity mode of Inquiry as third paradigm in the Classic Debate Qualitative versus

Quantitative researchis introduced.

On the base of the Topological mode of Qualitative quantity as homeomorphism of change

and transformation, the transition within the system of auditing research themes is proposed

and demonstrated, as transition from typology to topology.

Audit and auditing research are duscuussed as topoplogical dynamical sysytem. Topological

data analysis methods are proposed in data analysis in auditing.

Topology based on the Qualitative quantity method of research is proposed as innovative

means of verifying and enhancing the value of the audit dynamics and system of audit

experience.

In conclusion our claim is that topological approach to audit dynamics will contribute to the

empirical resolution of various issues within the typology of audit and auditing research, and

will enhance the quality and synergie of audit experience.


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A TOPOLOGICAL APPROACH TO AUDIT DYNAMICS

AND AUDITING RESEARCH

Topology provides the synergetic means of ascertaining the values of any system of experiences.

Operating Manual for Spaceship Earth, R. Buckminster Fuller

Topology! The stratosphere of human thought!

In the twenty-fourth century it might possibly be of use to someone...

The First Circle, A. Solzhenitsyn1. Background of the topological approach, based on Qualitative quantity research

method, to audit dynamics

1.1. The New Direction in Auditing Research

There are opportunities for auditing research to benefits form variety of disciplines employing

the qualitative research methododlogy. These opportunities are subject of discussion in

numbers of research papers in area of auditing research. Acknowledging the methodological

problems associated with the debate qualitative vs quantitative in the analytical research,

the so called arbitrarily tinkering with the traditional approach in analytical auditing

research, we should recognize the fact that there is a search for the new direction in auditing

research dynamics.2According to Penno, the traditional modeling approach in the analytical

auditing research has made important contributions but it does not adequately represent the

auditor's role.3 Acknowledging the fact that the economic paradigm has been the dominant

paradigm in analytical auditing research, Mark Penno examines how economic theory has

benefited from recent developments in sociology, psychology, philosophy, artificial

intelligence, and jurisprudence, and how research from these fields is closely related to better

describing the auditor's role. These scientific disciplines mentioned by Penno are all related

with the qualitative research method.

In the issue of qualitative and quantitative methods in auditing and auditing research the

impact of the disciplinary divide on the auditing research is emphasizing on thequalitative dimension discussed as and described in the context. 4 The increasing advocacy

for qualitative research methods as a rich alternative to the conventional scientific model

within the social sciences generally, is recognized widly in research papers.

2 Penno, Mark C., Analytical Auditing Research: New Directions from Several Disciplines , 2005

3 Ibid

4 Humphrey, Christopher., Auditing research: a review across the disciplinary divide, Accounting, Auditing &

Accountability Journal, Vol. 21 Iss: 2, pp.170203, 2008


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Research in accounting particularly during the period following World War II emphasized an

almost universal devotion to quantitative approaches. Qualitative research during this period

was generally demoted to low level research work Thus, qualitative research techniques,

which should normally address social phenomena from the subjectivist or anti-positivist

perspective, were noticeably infrequent in auditing and accounting research literature.

Research in auditing is concerned with the measure of the quality of audit performance, the

procedures and the professional ethics that must be followed in the conduct of an audit. The

disciplines traditionally foreign to auditing research that employs the qualitative research

methods, such as the Ethnography and Cultural Anthropology are now considered as

benefitial for the dynamics of auditing research. In the spirits of advocating this qualitative

methodology, it is not surprising that these two disciplines are directed to the auditing

research. A paper published in American Journal of Scientific Research entitled

Ethnography: A Basis for Conducting Research in Auditing and Marketing, American

Journal of Scientific Research proposed the ethnography as a basis for research in auditing:

With this focus in the mind of the auditing researcher we hereby propose that ethnography, a

fairly new qualitative research approach in the management sciences, and promises to be a

significant research methodology in investigating and understanding behaviour in the auditing

field. 5 The opportunities for combining psychological and economic research in auditing

are introduced and discussed by Dr. Christopher Koch and Prof. Dr. Jens Wstemann, in their

paper A Review of Bias Research in Auditing: Opportunities for Combining Psychological

and Economic Research.6 Following this new horizon in auditing research, the aim of my

paper is to introduce in the dynamics of auditing research an interdisciplinary approachthe

topological approach - related with my personal research interest and experience in

philosophy of science, topological philosophies, cultural phenomenology, law and semioticsof law, audit and auditing research qualitative quantity /research method/ and cultural

phenomenology of qualitative quantity implemented as cultural phenomenology ofauditing

research dynamics.

The intended result of this article is to establish the base for implementation of topological

approach to the dynamics of auditing research, focusing on the gradual and continuous

/topological notion/ of the category of qualitative quantity, proposed here as research method

in analytical /auditing/ research. The aim of this article is to contribute to the empirical

resolution of various issues within the typology of auditing research, and to enhance the

quality and synergie of audit dynamics.

5Bariyima D. Kiabel, Elizabeth I Ugoji, N. Gladson Nwokah, Ethnography: A Basis for Conducting Research in

Auditing and Marketing, American Journal of Scientific Research ISSN 1450-223X Issue 4 (2009), pp 28-35,

EuroJournals Publishing, Inc. 2009 (Hopper and Powell, 1985; Chua, 1986; Berelson, 1952; Morgan and

Smircich, 1980).

6 Dr. Christopher Koch, Prof. Dr. Jens Wstemann, in their paper A Review of Bias Research in Auditing:

Opportunities for Combining Psychological and Economic Research


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It is known that generally, there are two major and largely opposing views about the nature of

social inquiry and analytical researchsociological positivism also called the objectivist

or positivist approach and the German Hegelian Idealism also called the subjectivist or

relativist approach. Due to the nature of the category and method of qualitative quantity

employed in this paper and my research, the topological approach to auditing research could

be defined as grounded on the known relativistic approach, within the Hegelian thesis. In

accordance with the proponents of the German school of thought or the relativist approach, I

am viewing the research in social and management sciences as research that involves the

investigation and interpretation of the social world using qualitative techniques in the analysis

of data. The thesis and dialectics of qualitative quantity includes topological ontology and

epistemology, cobsidering complex social and cultural dynamics issues with regard to the

human nature. The innovative element introduced with the topological approach in auditing

research is the opportunity to enhance both the qualitative and quantitative research methods

with the qualitative quantity research method /with the application and implementation in

topology/. In this relativistics direction of qualitative research method to the study of human

affairs, the cultural phenomenology of auditing research and audit dynamics is possible as a

cultural phenomenology of qualitative quantity. The qualitative quantity research method is

an enhanced implementation of the qualitative research methods including as well the

implementation of the quantitative research methods. The amalgama of qualitative and

quantitative research methods is recognazied in the filds of studies such as the visual

mathematics, visual statistics, cognitive science and AI.

The qualitative research approach widely used by researchers in anthropology and sociology

involves a situation where the researcher takes active part in activities involving the

phenomenon under investigation. Ethnography approach is widely used by researchers in

cultural anthropology. The arguments and issues discussed in the paper Ethnography: A

Basis forConducting Research in Auditing and Marketing are basis and reason to consider

application of the topological approach in the auditing research. Topological approach in

cultural anthropology and ethnography is supported by the British social anthropologist

Edmund Leach, Claude Levi-Strauss, Gregory Bateson. In the paper Ethnography: A Basis

for Conducting Research in Auditing, the auhors offers wide discussion on the research

approaches in the Social and Management Sciences: The Quantitative/Qualitative

Distinction. This discussion is rich with philosophical assumptions related with ontology and

epistemology, touching some issues which we may regard as cultural phenomenology. As it isasserts in the paperontological assumptions enable us to understand the very essence of the

phenomenon being studied. Is the phenomenon to be studied, imposed on the individual from

without and as such having an existence of its own; or is it a product of the individuals mind?

Epistemological assumptions are assumptions about the grounds of knowledge - how

knowledge is developed and communicated to fellow human beings, how one can sort out

what is true and what is to be regarded as false. 7Assumptions relating to human nature

7Bariyima D. Kiabel, Elizabeth I Ugoji, N. Gladson Nwokah, Ethnography: A Basis for Conducting Research in

Auditing and Marketing, American Journal of Scientific Research ISSN 1450-223X Issue 4 (2009), pp 28-35,

EuroJournals Publishing, Inc. 2009 (Hopper and Powell, 1985; Chua, 1986; Berelson, 1952; Morgan and

Smircich, 1980).


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/cultural phenomenology/ identify the relationship between human beings and the

environment in which they operate. The kind of ontologies, epistemologies and nature of

human beings determines the type of methodology one adopts in social inquiry. Both

quantitative and qualitative research approaches are rooted in phenomenal experience, and in

this sense both are empirical. They share a common ground. As Dunn 8points out: although

qualitative research is popularly perceived as antithetical to quantitative work, it is actually

complimentary to it. Most quantitative projects possess qualitative elements, and vice versa.

The authors of the Ethnography: A Basis for Conducting Research in Auditing established

important thesis as evidence in our research project /the relevance of topological approach and

qualitative quantitity research methodology for audit dynamics and auditing research in the

field of internal and external auditing/ the innovative thesis of the Relevance of Ethnography

in Auditing Research.

The general definition of an audit is an evaluation of a person, organization, system, process,

enterprise, project or product and the mode of dialectics of qualitative quantity due to the

gradualness as notion of the exhibit form of this category is the mode of the evolution. The

topological notion of qualitative quantity is the notion of gradualness representing continuous

transformation. In the context of cultural phenomenology I am approaching the the audit and

auditing research as an dynamic system, examinating this system from the standpoint of

dialectics and self-organisation.

1.2. Qualitative quantity and the proposition of Qualitative quantity mode of Inquiry, as

third paradigm in The Classic Debate Qualitative vs Quantitative research The

paradigm shift from "typological" meaning-making to topological meaning-making

The purpose in this article is not to provide a wide discussion on the debate between the two

paradigms in research methods - qualitative vs quantitative. The relevance of these two

paradigm is widely recognized in the research culture as the fact that the two dominant

research paradigms have resulted in two research cultures, one professing the superiority of

deep, rich observational data and the other the virtues of hard, generalizable . . . data.9

According to Johnson and Onwuegbuzie, both sets of purists qualitative and quantitative -

view their paradigms as the ideal for research, and, implicitly if not explicitly, they advocate

the incompatibility thesis. 10 Johnson and Onwuegbuzie provides in their study an extensive

account of the research papers for the both side of the debate and proposed the so calledmixed methods research as the third research paradigm. Johnson and Onwuegbuzie

viewed the pragmatism as an attractive philosophical partner for mixed methods research, and

framework for designing and conducting mixed methods research. Providing an excellent

8Dunn, D.S.(2001), Statistics And Data Analysis For The Behavioral Science. New York: MCGraw Hill.

9 R. Burke Johnson and Anthony J. Onwuegbuzie, Mixed Method s Research: A Research Paradigm Whose

Time Has Come, Educational Researcher, Vol. 33, No. 7, pp. 1426.

10Ibid


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review on theparadigm wars - qualitative vs quantitative - and incompatibility thesis /which

posits that qualitative and quantitative research paradigms, including their associated

methods, cannot and should not be mixed/, Johnson and Onwuegbuzie claimed that there are

some commonalities between quantitative and qualitative research. The authorss thesis is

based on the tenets of pragmatism in their explanation of the fundamental principle of mixed

research and how to apply it. Johnson and Onwuegbuzie offered their own understanding and

definition of the term research paradigm coined by Thomas Kuhn who popularized the idea

of a paradigm. Khun pointed out that paradigm is a general concept including a group of

researchers having a common education and an agreement on exemplars of high quality

research or thinking. 11According to Johnson and Onwuegbuzie, the research paradigm is

set of beliefs, values, and assumptions that a community of researchers has in common

regarding the nature and conduct of research. 12 There are ontological, epistemological,

axiological and methododlogical levels in the research paradigm: the beliefs include, but are

not limited to, ontological beliefs, epistemological beliefs, axiological beliefs, aesthetic

beliefs, and methodological beliefs. 13 According to the the authors their use of the term

research paradigm refers to a research culture. The argument of Johnson and Onwuegbuzie

is that there is now a trilogy of major research paradigms: qualitative research, quantitative

research, and mixed methods research.14

For the purpose of this article I will focus on the existence of the third approach, establishing

that the qualitative quantity method research is the possible development of this third

paradigm of integrative research. The proposition of qualitative quantity method of research is

the foundation of the topological approach to dynamics of research culture. Although both

qualitative and quantitative philosophies continue to be highly useful the advantage of the

philosophy of qualitative quantity is laid by the topological notion of this category and

method.

In Mixed Methods Research: A Research Paradigm Whose Time Has Come, Johnson and

Onwuegbuzie offered the Development of a Mixed Methods Research Typology.

I will approach the issue of the third paradigm the mixed methods research with the

topological paradigm shift illustrated by the mixed-mode Semiosis and the transition

11Kuhn, T. S. (1962). The structure of scientific revolutions. Chicago, IL: University of Chicago Press

12R. Burke Johnson and Anthony J. Onwuegbuzie, Mixed Methods Research: A Research Paradigm Whose







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from Typological Mode to the Topological Mode proposed by Jay L. Lemke.15

Jay L. Lemke

introduced the mixed-mode Semiosis - Typological vs. Topological Semiosis. 16 According to

Lemke there are two fundamentally different kinds of meaning-making: "typological"

meaning-making - meaning-by-kind /natural language/ and topological meaning-making -

meaning-by-degree /visual language/, which is more easily presented by means of motor

gestures or visual figures -- the meaning of continuous variation or "topological" meaning.

Meaning-by-kind is qualitative and meaning-by-degree is quantitative. 17

Figure by Jay L. Lemke, Typological vs. Topological Semiosis18

.http://www-personal.umich.edu/~jaylemke/webs/wess/tsld002.htm

For Lemke Typological semiosis is qualitative semiosis and Topological semiosis is

quantitative semiosis. Qualitative character of typological semiosis determines the discretevariant and Quantitative character of topological semiosis and the continuous variation. I

support the idea that qualitative quantity is possible to appear as categorical gradiant /mixture

from categories in typological semiosis and gradients in topological semiosis/. The

topological and continuous notion of qualitative quantity as gradualness and continuous

variation or "topological" meaning can be seen in Figure above given in Lemkes work

Mathematics in the middle: measure, picture, gesture, sign, and word. In Opening Up

Closure: Semiotics Across Scales, Lemke proposes the Mixed-mode Semiosis.

15Jay L. Lemke, Topological Semiosis and the Evolution of Meaning http://www-

personal.umich.edu/~jaylemke/webs/wess/index.htm

16Jay L. Lemke, Typological vs. Topological Semiosis.http://www-

personal.umich.edu/~jaylemke/webs/wess/tsld002.htm

17Jay L. Lemke, Mathematics in the middle: measure, picture, gesture, sign, and word, and Opening Up

Closure: Semiotics Across Scales

18Jay L. Lemke, Typological vs. Topological Semiosis.http://www-

personal.umich.edu/~jaylemke/webs/wess/tsld002.htm


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In this mixed-mode of typological and topological semiosis, the domain of qualitative

quantity could be seen, as categorical gradiant /Lemke/ of the unified system for

meaningmaking /Lemke/.

The Qualitative quantity Method Research due to the gradual and nondiscursive notion of the

category /qualitative quantity/ and the ability to exhibit topological homeomorphism is themethod /within the system of meaning-making/ with the ability to transform the form or the

structure from typology to topology. The aim of the The Qualitative quantity Method

Research is to create Topology of meaning.

The term topology of meaning emerged at the proceedings of the Einstein meets Magritte

Conference, Brussels, Belgium /1995/. The topology of meaning was introduced by R. Ian

Flett and Donald H. McNeil in their paper Whats Wrong with this Picture? Towards a

Systemological Philosophy of Science with Practice. 19

Typology of research topics in a specific area of research in research culture are necessary

very useful to understand the relationships between the research topics. Since Aristotle, theorganization of knowledge is based on the conceptualization, classification, typology or

taxonomy. Typology is instrument of hierarchical system.

Topology of research topics in a specific area of research in research culture is instrument of

heterarchical relation of types in the the dynamics of research culture.

Topology of the heterarchical relationship in the structure is associated with the work of

Warren McCullochs A Heterarchy of Values Determined by the Topology of Nervous

Nets"20 and also the works of Gotthard Gnther. In 1960 Gnther met Warren McCulloch and

worked with him and Heinz von Foerster and Humberto Maturana. Gnther's work was based

upon Hegel, Heidegger and Spengler. From 1976 to 1980 Gnther completed three studies

Contributions to the Foundation of an Operational Dialectic /"Beitrge zur Grundlegung

einer operationsfhigen Dialektik/. Gnther aim was to make dialectics operationable and

contributed with his work and influence to the fields of cybernetics. The qualitative notion of

numbers or the qualitative quantity in Gotthard Gnthers thinking is unfolded in his Number

and Logos, devoted to his frend and one of the fathers of cybernetics Warren McCulloch,

bearing Gnthers note: Unforgettable Hours with Warren St. McCulloch. Warren

McCulloch was the one who influenced Gregory Batesons topological thinking. Bateson took

McCullochs idea ofheterarchy and adapted this to Bertrand Russsells hierarchy of logical

types in a rather peculiar way: logical types could systematically unravel circularity of

information and how it becomes entangled in a multi-level universe, but logical typing cannotitself demonstrate appropriate communication.

The semiotic bridge in qualitative vs quantitative debate is Qualitative quantity the

categorical gradiant which unite in one these two fundamentally different kinds of

19Donald H. McNeil, Whats going on with the topology of recursion?, Science and Art: The Red Book of

`Einstein Meets Magritte': The Red Book Vol 2 (Einstein Meets Magritte: An Interdisciplinary Reflection on

Science, Nature, Art, Human Action and Society), Kluwer, 1999 - Flett, R. Ian, and Donald H. McNeil. 1995).

20

Warren McCullochs A Heterarchy of Values Determined by the Topology of Nervous Nets" /In: Bulletin ofMathematical Biophysics, 7, 1945, 8993


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meaningmaking - meaning-by-kind and meaning-by-degree in the continuous variation of

"topological" meaning.

The proposed qualitative quantity method of research is not opposing the pragmatism of

Charles Sanders Peirce, William James, and John Dewey or postpragmatism school of though.

As to Johnson and Onwuegbuzie established in their work the Pragmatism is thePhilosophical Partner for Mixed Methods Research. Something more, the category of

Qualitative quantity and the exhibit form of this category - topological notion - could be seen

and unfold in the works of Charles Sanders Peirce.

2. Qualitative quantityTopology and Topological Dynamics

2.1. Qualitative quantity

From his reading of Hegel, Engels elucidated the three laws of dialectics in his Dialectic of

Nature. The second law of dialectics, the law of transformation, established by Engels is the

law of the passage of quantitative changes into qualitative changes. This law states that

continuous quantitative development results in qualitative "leaps" in nature whereby a

completely new form or entity is produced. This is how "quantitative development becomes

qualitative change". The new quality develops quantitatively through a step-by-step process of

quantitative change, qualitative changes begin with the quantitative introduction of the new

quality into the quantitative development of the old measure. Qualitative changes occur as

leaps. This Engelss law of transformation and the passage of quantitative changes into

qualitative changes, as all of the three laws of Engelss dialectics become clich in the mode

of thinking of quality and quantity. The three laws of dialectics are not only oversimplified,

but also misleading at best, establishing something quite self-evident, trivial andcommonplace. Gradualness and gradual changes which are not leading to turning points,

where one force overcomes the other and quantitative change leads to qualitative change,

remained inapparent just like Hegels dialectics of Qualitative quantity.

Approaching the domain of topology from the standpoint of the dialectics of qualitative

quantity, we should conclude that the interplay of quality and quality is associated with the

development and growth. Both the classical and non-classical approaches to the dialectics of

quality and quantity are addressing the dialectical nature of change.

The known quality, from the second law of dialectics, defined by Hegel as determined quality,implies discontinuous change, a leap, and the transformation is discursive. The exhibit form

of determined quality is abrupt displacements in the equilibrium - revolution.

The quality of the quantity implies gradual and continuous changes, and transformation is

non-discursive. The exhibit form of qualitative quantity is transformation without leap or

abrupt displacements in the equilibrium - evolution.

The recent developments of topology and its applications as topological dynamics in social

science and social research are not only illustrating the relevance of the so called topolog ical

philosophies and topological thinking in social and cultural dynamics, but also directing to


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the resurgence of one dialectical category and concept established by G.F.W. Hegel the

Qualitative quantity.

Hegel proclaims the Qualitative quantity in the both of his Logics The Science of Logic

/Wissenschaft der Logik, referred to as the Greater Logic /1812, 1813, 1816/ and The Lesser

Logic /Part One of the Encyclopedia of Philosophical Sciences / Enzyclopdie der

philosophischen Wissenschaften im Grundrisse /1817/.

Hegel established the Qualitative quantity in 105 and 106 of his Lesser Logic 21, where

Hegel claimed the Qualitativequantity, or Measure:

The two sides of the ratio are still immediate quanta: and the qualitative and quantitat ive

characteristics still external to one another. But in their truth, seeing that the quantitative itself

in its externality is relation to self, or seeing that the independence and the indifference of the

character are combined, it is Measure. The result of the dialectic however is not a mere

return to quality, as if that were the true and quantity the false notion, but an advance to theunity and truth of both, to qualitative quantity, or Measure. / 106, The Lesser Logic/.

According to David Gray Carlson 22 Measure is the third and last province in the kingdom

of Quality, which itself comprises the first third kingdom in the empire of the Science of

Logic. When Measure concludes, we will have arrived at the portal of the negative,

correlative underworld of shadowy Essence. David Gray Carlson states that Hegel

proclaims the development of Measure to be extremely difficult, and many commentators

have concurred. We can nevertheless describe the theme of Measure easily enough--change;

more precisely, an exploration of the difference between qualitative and quantitative

change.23

The Qualitative quantity appears in Hegels early work The Science of Logic /The Greater

Logic/. 24 In the first chapter, the Specific Quantity, 711Qualitative quantity in thefirst

place an immediate, specific quantum., also in 731, 774 /in Nodal Line of Measure

Relations /B/ in Chapter2 /Real Measure/. The specific notion of the qualitative quantity

Hegel explores in the Second Chapter Real Measure - The Relation of Self-Substistent

Measures /A/ in Combinationof Two Measures /a/, in Measure of a series of Measure

Relations /b/, and in ElectiveAffinity /c/. The notion of qualitative quantity is the reason

for Hegel to quote Carl LinnaeussNature Does Not Make Leaps/ 774/.

Hegels emphasis on the leap and nodal line in the relationship of qualitative and quantitative

and the reason to set in brakets the Carl Linnaeus aphorism Nature Does NotMake Leaps

21Hegels Logic, translated by William Wallace, with Foreword by J N Findlay, Clarendon Press 1975. First

published 1873

22David Gray Carlson, Hegels Theory of Measure, Benjamin N. Cardozo School of Law, 2003, Working

Paper No. 66

23Ibid

24Hegels Science of Logic, tr. A. V. Miller, George Allen & Unwin, 1969


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/natura non facit saltus/, comes from the point of observeability and perceptability. This

consideration is clear in Hegel:

In thinking about the gradualness of the coming-to-be of something, it is ordinarily assumed

that what comes to be is already sensibly oractually in existence; it is not yet perceptible only

because of its smallness. Similarly with the gradual disappearance of something, the non-

being or other which takes its place is likewise assumed to be really there, only not

observable, and there, too, not in the sense of being implicitly or ideally contained in the first

something, but really there, only not observable. / 777/.

It looks like Hegels emphasis on the leap and quality /that breaks in/per saltum is considerd

with our human ability to percept and observe easy qualitative difference of something

apparent, that lack smallness large enough to be noticed, something visible and

observable. It seems that qualitative quantity is ignored because of its notion of gradualness

and lack of ability to leap as qualityper saltum. Emphasizing on the transformation of quality

to quality by leaps per saltum in the nodal Line of Measure Relations, in 777 Hegel

defines the attempt to explain coming-to-be or ceasing-to-be on the basis of gradualness of

the alteration as tedious like any tautology. Hiding the qualitative quantity behind the

curtains of tautology, in his liturgy of quality per saltum, Hegel is misleading the

philosophers following him, specially the dialectical materialism of Engels and Marx, who

created the clishe of first law of dialectic of transition of quantity into quality and visa versa.

Hegels notion of the qualitative quantity became subject of discussion in the very

beginning of the Twenty century by John Grier Hibben in his Hegels Logic: An Essay in

Interpretation /1902/.

25

Discussing quantitative relation, in Chapter VIII:Quantity, JohnGrier Hibben asserted:

the concept of quantity will not explain it /the nature of quantitative relation - my note/

satisfactorily, and we fall back again upon the idea of quality in order to account for it. Thus

the idea of quality was found to be partial, and when developed to its utmost limit, carried our

thought over into the sphere of quantity. Then the idea of quantity when fully developed

brought us back again to that of quality.Is the movement of thought only a circle that merely

brings us back to the starting-point? According to Hegels method, the incompleteness of

thought at this stage is overcome by the dialectic process which combines these two ideas of

quality and of quantity into one complete relation representing an advanced and higher pointof view. This relation Hegel calls that of qualitative quantity, or of measure (das Maass).

This is the third and last stage in the development of the idea of quantity, and represents, as

Hegel insists, both the unity and the truth of quality and of quantity combined. 26

To my knowledge, in contemporary dialectics and philosophical research the significant

notion of Hegels category qualitative quantity remained inapparent /In Heideggers sense

25John Grier Hibben in his Hegels Logic: An Essay in Interpretation, 1902

26 Ibid


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of his Phenomenology of Inapparent.27

Reason for this is probably Hegels on warning

about the intellectual difficulty / 777 of the The Science of Logic, The Greater Logic/,

accusing the attempt to explain coming-to-be or ceasing-to-be on the basis of gradualness of

the alteration as tedious like any tautology. 28 In contemporary philosophical research the

creative power of tautology was definitely recognized. 29

For Gregory Bateson evolution and tautology are dialectically linked. If there are two realms,

that of tautology (whose essence is predictable repetition and replication) and that of

evolution (whose essence is creativity, exploration and change), then life entails an

alternation, a dialectics, between the two. As mental processes and phenomenal happenings

the two may be adversarial, but a zigzag between them whereby each determines the other

would appear to be necessary for the continuation of life. Homeostasis and adaptation,

structure and process, form and function, status and learning, conservatism and radicalism,

quantity and pattern, homology and analogy, calibration and feedbackthese are 'dialectical

plural necessities of the living world30

For Bateson, explanation is the mapping ofdescription onto tautology. The new form of knowledge is generated by a principled mapping

of a description of a phenomenon onto a tautology. As he states in Mind and Nature: An

explanation has to provide something more than a description provides, and in the end, an

explanation appeals to a tautology, which as I have defined it, is a body of propositions so

linked together that the links between the propositions are necessarily valid. A tautology in its

simplest form is If P is true, then P is true.31

Bateson asserted that knowledge is both evolutionary and tautological. As Nigel Rapport and

Joanna Overing established: The organism deals with entropy in two contrasting ways.

Bateson calls them evolutionary versus tautological: an embracing of the implications andramifications of possible change versus a homeostatic eschewing of them. The watchwords of

the latter tautological process (also to be known as 'epigenetic' and 'embryological') are:

coherence, steady state, rigour and compatibility. The process acts as a critical filter,

demanding certain standards of conformity in the perceiving and thinking individual. Left to

itself it proceeds towards tautology: towards nothing being added once the initial arbitrary

axioms and definitions of order have been laid down. Hence, the first test of a new idea is: is

it consistent with the status quo ante? is it entirely latent in the original axioms which supply

27 Martin Heidegger, Phenomenology of the inapparent /Phnomenologie des Unscheinbaren/ Seminar inZhringen 1973

28For Hegelian and Heidegerian Tautologies, see: Tze-Wan Kwan, Hegelian and Heidegerian Tautologies,

Analecta Husserlliana, The yearbook of Phenomenological research, Logos of Phenomenology and

Phenomenology of Logos, Volume LXXXVIII, 2005, The World Institute for Advanced Phenomenological

Research and Learning

29In the works of Gregory Bateson, also see: Allen Thiher, The power of tautology: The roots of leterary theory,

Associated University Press, 1997

30Gregory Bateson, Mind and Nature: A Necessary Unity. London: Fontana, (1980:237).

31Ibid


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the 'proof of its correctness? In the tautological procedure, in short, every 'becoming' is tied

back to existing conditions.32

The Hegels tautology about tautology regarding the relevance of the qualitative quantity

should be accepted as the "true," new tautology.

The resurgence of G.F.W. Hegels category qualitative quantity, historically left

inapparent as the real breackthrough from the clich of the known laws of dialectics the

law of transformation of quantity to quality, and the establishment of the thesis of topological

notion of qualitative quantity is subject of few research papers between the years of 1989-

2011. 33

In 1988 with the first publication ofL'tre et l'vnement /Beingand Event/, translated in

English only in 2005, in Mediation Fifteen on Hegel, Alain Badiou recognizes the

qualitative quantity as the core of the domain of quantitative infinity, claiming that

Quantitative infinity is quantity qu a quantity, the prliferator of proliferation, which is to say,quite simply, the quality of quantity, the quantitative such as discerned qualitatively from any

other determination. 34The true resurgence of Hegels Qualitative quantity can be unfolded

in the works of Charles Sanders Peirce. The notion of Qualitative quantity and the presence of

the Topological Peirce could be discovered in Peirces Abduction, Peirces Theory of the

Continuum and his concept of Hypostatic abstraction. The Topological approach of Charles

Sanders Peirces qualitative-ness is recognized by recent research works 35 and giving us the

ground to speak about The Topological Peirce. The proposed qualitative quantity method of

research is relativistic approach based on the The Topological Hegel /after Arkady Plotinsky/,

but not opposing the pragmatism of Charles Sanders Peirce, positivism and postpositivism.

32Nigel Rapport and Joanna Overing, Social and Cultural Antropology, Routledge, 2000, p.110

33 Borislav Dimitrov, Quality of quantity, Philosophic Thought Magazine, Institute of Philosophical

Sciences, Bulgarian Academy of Science. March, 1989; Borislav Dimitrov, Quality and Time, presented at the

conference The Fundamental Knowledge between Ontology Dilemma and Cognitive Problems, published in

1990, by The Institute for Philosophical Research at the Bulgarian Academy of Science; Borislav Dimitrov, The

TopologicalNotion of Qualitative quantity, 2011); Borislav Dimitrov, The Relevance of Topological Approach

based on Qualitative quantity research method to audit dynamics and auditing research, 2011.

34Alain Badiou, Being and Event, Oliver Feltham (tr.), Continuum, 2006, see p. 168-169, The Arcana of

Quantity

35See:

- Robert W. Burch, "A Peircean Reduction Thesis: The Foundations of Topological Logic". Texas Tech

University Press,Lubbock, TX, 1991;

- Helmut Pape, "Abduction and the Topology of Human Cognition"http://user.unifrankfurt.

de/~wirth/texte/pape.html;

- Arnold Johanson, "Modern Topology and Peirce's Theory of the Continuum", Transactions of the Charles S.Peirce Society, Vol.37, No..1, Indiana University Press, 2001,pp.1-12 http://www.jstor.org/pss/40320822.
http://user.unifrankfurt/http://user.unifrankfurt/http://user.unifrankfurt/http://user.unifrankfurt/


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Charles Peirce is the Semiotical Partner for Qualitative quantity methods research due to the

topological notion of the category - qualitative quantity.

After the establishment of the thesis of the relevance of Hegels category qualitative

quantity and the argument that topology is the field of qualitative quantity and topological

homeomorphism is exhibit form of this category (1989), for the past fifteen years, the

significant developments of the science and philosophy of science related to the concept of

qualitative quantity emerged, evidently enriching the grounds of the thesis qualitative

quantitytopology and topological dynamics. The topology and topological thinking is now

undergoing a resurgence, with some highly practical research and applications. The new area

of interdependence between the quality and quantity is open now for rethinking and re-

conceptualization. In Qualitative Spatial Representation in the field of Qualitative Spatial

Reasoning, the term qualitative quantitative space emerged and the concept of qualitative

quantity is vitally accepted and utilized.36 Qualitative quantity approach is aplicable in

General Design Theory, proposed by Yoshikawa in which design and design knowledge aremathematically represented by using topology. Qualitative quantity seen in Poincares

topology and his development of the qualitative theory of differential equations is enhanced

by the new science of Mereotopology, whichbegan with theories A.N. Whitehead articulated

in several books and articles he published between 1916 and 1929. Mereotopology is a branch

of metaphysics, and ontological computer science, a first-order theory, embodying

mereological and topological concepts, of the relations among wholes, parts, parts of parts,

and the boundaries between parts. The Qualitative quantity is the fundamental category of

Qualitative research /QR/, due to the aim of this discipline to utilise methods that seek to

discern the quality as opposed to the quantity of its subject. Qualitative quantity is

critical in Spatial-temporal reasoning applicable in computer science as Visual thinking and

Visual music, Visual statistic /the series of works by David James Krus/. Topological

approaches penetrated cybernetics and AI, after Warren McCullochs work A Heterarchy of

Values Determined by the Topology of Nervous Nets".The interdisciplinary amalgam of

dialectics and cybernetics is recognized in the works of Gotthard Gnther (Gotthard Gnther,

Cybernetics and the Dialectic Materialism of Marx and Lenin, published at

http://www.thinkartlab.com) as an enlarged representation of a lecture Gotthard Gnther did

deliver at the University of Cologne (Kln, Germany) July 17, 1964.)

The topological approaches are used widely in psychology after Kurt Lewin work Principlesof topological psychology 37 The topology of qualitative quantity could be found in Jean

Piagets Genetic Epistemology, especially in the known Piagets topological primacy thesis.

Piaget and Inhelder /1959/ claimed that the young childs intristic geometry was first of all

topological and then, later, projective Euclidean. The term topology of meaning emerged at

the proceedings of the Einstein meets Magritte Conference, Brussels, Belgium /1995/,

36See: A. G. Cohn, Qualitative Spatial Representation and Reasoning Techniques, Division of Artificial

Intelligence, School of Computer Studies, University of Leeds.-Note 1.footnote about qualitative quantity

37Kurt Lewin work (1936) Principles of topological psychology. New York: McGraw-Hill.
http://www.thinkartlab.com/http://www.thinkartlab.com/


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introduced by R. Ian Flett and Donald H. McNeil in their paper Whats Wrong with this

Picture? Towards a Systemological Philosophy of Science with Practice.

2.2. Topology

Topology is a major area of mathematics concerned with properties that are preserved undercontinuous deformations of objects, such as deformations that involve stretching, but no

tearing or gluing. In topology, any continuous change which can be continuously undone is

allowed. So a circle is the same as a triangle or a square, because you just `pull on' parts of the

circle to make corners and then straighten the sides, to change a circle into a square. Then you

just `smooth it out' to turn it back into a circle. These two processes are continuous in the

sense that during each of them, nearby points at the start are still nearby at the end. In

topology we can transform a spatial body such as a sheet of rubber in various ways which do

not involve cutting or tearing. We can invert it, stretch or compress it, move it, bend it, twist

it, or otherwise knead it out of shape. Certain properties of the body, the properties of the

qualitative quantity, will in general be invariant under such transformations - which is to say

under transformations which are neutral as to shape, size, motion and orientation. The

qualitative quantity transformations can be defined as being those which do not affect the

possibility of our connecting two points on the surface or in the interior of the body by means

of a continuous line.

Our world is undergoing profound change due to the advanced research in topology and

significant importance of topological thinking and topological meaning making. Topology

itself is the science of the change. Topology is extremely applicable to the complex dynamics

systems. The unique and modern treatment of topology today is employing a cross-disciplinary approach.

Topology has been transformed from a theoretical field that highlights mathematical theory to

a subject that plays a growing role in nearly all fields of scientific investigation.

The new qualitative geometry of thinking - topology - penetrates not only the areas of

scientific research but the real-world phenomena and our whole personal and social life. From

the so called philosophycal topologies the topological thinking and topological methods has

great impact on the wide areas such as psychology /topological psychology after Kurt Lewins

1936 Principles of Topological Psychology/, sociology /social topology of Gregory Bateson/ -identity and behaviour, culture and cultural dynamics. The ATACD project /A Topological

Approach to Cultural Dunamics/ is a project funded by EU and now a research network based

on the mathematical theories of topology/, management, financial management, internal

control and audits, law and politics /topological law, legal research and topology of legal

systems, AI and Law/, security and defence, situation design. The father of the term and the

contemporary content of Cultural Dynamics is Pitirim Aleksandrovich Sorokin /18891968/.

His magnum opus, the monumental Social and Cultural Dynamics /Social and Cultural

Dynamics: A Study of Change in Major Systems of Art, Truth, Ethics, Law and Social

Relationships (1957 (reprinted 1970) ed.)/ spanned 2,500 years and attempted to isolate the

principles of social change as they were manifested in his studies of art, philosophy, science,

law ethics, religion and psychology.


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The topology and topological thinking is now undergoing a resurgence, with some highly

practical research and applications. Today topology became the new philosophy and an active

practise in our contemporary world.

Topology is linked genuinely with the quality and quantity. A first introduction to the basic

concepts of topology takes as its starting point the notion of transformation. The qualitative

and quantitative approaches and methods become cliche as well as the well known first law of

dialecticsthe law of transformation of quantity to quality and the apparent appearance of the

new quality in qualitative leap.

This topological qualitative quantity is the real breakthrough in dialectical thinking and

systems for creating meaning. The notion of the qualitative quantity is topological and can be

illustrated with the well known example ofa continuous deformation (homeomorphism) of a

coffee cup into a doughnut (torus) and back. The notion of the qualitative quantity is

continuous and the appearance of this new quality is not the qualitative leap but qualitative

gradualness.

2.3. Topological Dynamics and Qualitative quantity

In mathematics, topological dynamics is a branch of the theory of dynamical systems in

which qualitative, asymptotic properties of dynamical systems are studied from the viewpoint

of general topology. The central object of study in topological dynamics is a topological

dynamical system, i.e. a topological space, together with a continuous transformation, a

continuous flow, or more generally, a semigroup of continuous transformationsof that

space.

Qualitative quantity is category and research method of Topological Dynamics due to the

exhibit form of this categorythe gradual changes and the continuous transformations.

The American mathematician George David Birkhoff is considered the founder of

Topological Dynamics. In 1913, Birkhoffproved Poincares "Last Geometric Theorem," a

special case of the three-body problem, a result that made him world famous. 38 In 1927, he

published his Dynamical Systems.39

The link between Topology and Qualitative quantity, between Henri Poincares AnalysisSitus /topology/ and Hegels qualitative quantity, was established in the research back in

1989th with the argument that the exhibit form and notion of the Qualitative quantity is

topological homeomorphism due to the continuous transformation typical for the notion of

qualitative quantity. 40 In Quality of quantity (1989), Borislav Dimitrov, grounded his

38Birkhoff, George David. 1913. "Proof of Poincar's geometric theorem" Trans. Amer. Math. Soc. 14: 1422..

39Birkhoff, George David. 1917. "Dynamical Systems with Two Degrees of Freedom" Trans. Amer. Math. Soc.

18: 199300

40

Borislav Dimitrov, Quality of quantity, Philosophic Thought Magazine, Institute of PhilosophicalSciences, Bulgarian Academy of Science. March, 1989
http://upload.wikimedia.org/wikipedia/commons/2/26/Mug_and_Torus_morph.gifhttp://upload.wikimedia.org/wikipedia/commons/2/26/Mug_and_Torus_morph.gifhttp://upload.wikimedia.org/wikipedia/commons/2/26/Mug_and_Torus_morph.gifhttp://upload.wikimedia.org/wikipedia/commons/2/26/Mug_and_Torus_morph.gif


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thesis on the exploration of D'Arcy W. Thompsons Growth and Form /1917/, and Hermann

Hakens findings and examples that illustrates the qualitative quantity notion in structural

stability. (Hermann Hakens Synergetics: Introduction and Advanced Topics, Springer,

1983, Chapter 1.13. Qualitative Changes: General approach, p. 434-435).The concept of

structural stability is related with the topological homeomorphism, thus topological

homeomorphism was proposed as exhibit form of the category qualitative quantity.

In Synergetics: Introduction and Advanced Topics, Qualitative Changes: General

approach, Haken explores and illustrate the structural stability with an example of D'Arcy

W. Thompsons On Growth and Form, /1917/ - the porcupine fish and the sun fish can be

transformed into each other by a simple grid transformation. Here I am illustrating Hermann

Hakens example with the original Thompsons illustration of the transformation of the fish

Argyropelecus olfersi into the fish Sternoptyx diaphana by applying a 70 shear mapping. The

reverse transformation is possible simply with manipulating the grid and shear mapping.

In his book Synergetics: Introduction and Advanced Topics, 41 in the Chapter 1.13.

Qualitative Changes: General approach, Hermann Haken explores and illustrate the

structural stability with an example /figure 1.13, p.434 in Haken/ given by the Scottish

biologist, mathematician and classics scholar D'Arcy W. Thompson, the author of the book,

On Growth and Form, /1917/. The quality of the quantity could be seen in the Herman

Hakens citation on the D'Arcy W. Thompson. Exploring the invariance in deformation and

transformation of the forms against spatial or temporal deformation, Haken wrote:

Figure 1.13, p.434 /Synergetics: Introduction and Advanced Topics/ shows two different

kind of fish, namely, porcupine fish and sun fish. According to the studies by D'Arcy W.Thompson of the beginning of the twentieth century, the two kinds of fish can be transformed

into each other by a simple grid transformation. While from the biological point of view such

a grid transformation is a highly interesting phenomenon, from the mathematical point of

view, we are dealing here with an example of structural stability. In a mathematicians

interpretation the two kinds of fish are the same. They are just deformed copies of each other.

A fin is transformed into a fin, an eye into an eye and etc. In other words, no new qualitative

features such a new fin, occur. In the following we shall have structural changes /in the widest

sense of word/ in the mind.42

Under the illustration set in Figure 1.13, p.434 /Synergetics: Introduction and Advanced Topics/, Haken wrote the porcupine fish and the sun fish can be transformed into each

other by a simple grid transformation. /After D'Arcy W. Thompson: On Growth and the

Form, ed. By J.T. Bonner, University Press, Cambridge, 1981/.

41Hermann Hakens Synergetics: Introduction and Advanced Topics, Springer, 1983, Chapter 1.13.

Qualitative Changes: General approach, p. 434-435

42

Hermann Hakens Synergetics: Introduction and Advanced Topics, Springer, 1983, Chapter 1.13.Qualitative Changes: General approach, p. 434-435


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Haken concluded that The concept of structural stability seems to play a fundamental role in

biology in a still deeper sense than in the formation of different species by way of

deformation. Namely, it seems that, say within the species, organisms exhibit a pronounced

invariance of their functions against spatial or temporal deformation. This invariance property

seems to hold for the most complicated organs, the human brain. For example, this property

enables us to recognize the letter a even if it is strongly deformed. From th is ability an artout of writing letters developed in China /and in old Europe/.43

Two objects are homeomorphic if they can be transformed /or deformed/ into each other by a

continuous inverible mapping, continuous one-to-one and having continuous inverse. The two

fish are two objects with the same topological properties. They are said to be homeomorphic.

There are properties that are not destroyed by stretching and desorting an object.

2.4. Audit and Auditing Research as Topological Dynamical Systems

Topological Approach to Audit Dynamics is based on the understanding that Audit and

Auditing Research are Dynamical Systems thus Topological Dynamical Systems.

Establishing the relevance of Qualitative quantity research methods in auditing research, we

are approaching and the typology of auditing research issues or themes from the viewpoint of

general topology proposing and focusing on the study of continuous transformation, a

continuous flow, and a semigroup of continuous transformationsin audit practice and

auditing research.

The topological approach to audit dynamics and auditors role focuses on the betweeness of

these twothe organization and it objectives.

The aim of internal audit, by it definition is to help organization to accomplish its

objectives. One of the purposes of the International Standards for the Professional Practice of

Internal Auditing is to foster improved organizational processes and operations.

According to Standard 2100 / Nature of Work/, The internal audit activity must evaluate and

43

Hermann Hakens Synergetics: Introduction and Advanced Topics, Springer, 1983, Chapter 1.13.Qualitative Changes: General approach, p. 434-435
http://www.theiia.org/guidance/standards-and-guidance/ippf/standards/standards-items/?C=3094&i=8266http://www.theiia.org/guidance/standards-and-guidance/ippf/standards/standards-items/?C=3094&i=8266http://www.theiia.org/guidance/standards-and-guidance/ippf/standards/standards-items/?C=3094&i=8266


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The main goal of auditing and role of auditor, the improvement of management and helping

the organization to accomplish its objectives, is related with ascertaining the data and quality

of data analysis. Audit analytics enable organization to analyse transactional data to obtain

fact based insights into their operations. Data analysis helps auditors to identify indicators of

risk, internal controls failures, and non-compliance to internal or external requirements.

The use of purpose build data analysis for audit as audit analytics has been identified by

audit industry serveys as a top-five enabling strategic priority. Over the past 20 years, data

analysis has become an essential part of the audit process for the vast majority of audit

organizations. Using data analysis in an audit (generally referred to as audit analytics) has

already provided significant benefits for audit organizations of all sizes across a broad range

of industries, but there is still much progress that can be made by optimizing the audit

analytics process. Data analysis in auditing is generally referred to as audit analytics. It has

already provided significant benefits for audit organizations of all sizes across a broad rangeof industries and there is a still much progress that can be made by optimizing audit analytics

process.

Topological approach to audit analytics or data analysis in auditing would contribute to this

process and enhance the quality and synergy of audit and auditing research. By leveraging

topological data analysis, internal auditor will be able to detect not only abrupt changes

related with emerging vulnerabilities in business process and weakness that could potentially

expose the business to risk, but also to detect discrete and continuous changes associated with

gradualness and not easily observable.

Topological data analysis is a new area of study, aimed at having applications in areas such

as data mining and computer vision. In other words this is a recent mathematical method for

analysing data that has had new, dramatic, and unexpected applications to robot motion,

sensor networks, statistics, and medicine, among other areas. Topological data analysis is

applicable to the audit dynamics and auditing research due to the fact that topological data

analysis uses a branch of mathematics called algebraic topology to capture the shape of a

point-cloud data set that "persists" in a dynamical setting. For this purpose the data - themes

or issues in audit practice and auditing research are represented as shape of a point-cloud data

set that "persists" in a dynamical setting.

The main problems subject of Topological Data analysis could be recognized in Audit

Analytics, and these are the following:

- How one infers high-dimensional structure from low-dimensional representations; and

- How one assembles discrete points into global structure.

Both of the problems are very related with the role of the auditor. The job of the auditor is to

assemble discrete point into global structure and to enhance or improve the low-dimensional

representations into the high dimensional structure.


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The focus in Topological Data Analysis is on continuous flow, recognizing that the human

brain can easily extract global structure from representations in a strictly lower dimension, i.e.

we infer a 3D environment from a 2D image from each eye. The inference of global structure

also occurs when converting discrete data into continuous images, e.g. dot-matrix printers and

televisions communicate images via arrays of discrete points.

The main method used by topological data analysis is:

1. Replace a set of data points with a family of simplicial complexes, indexed by a proximity

parameter. This converts the data set into global topological objects.

2. Analyse these topological complexes via algebraic topology specifically, via the new

theory ofpersistent homology.

3. Encode the persistent homology of a data set in the form of a parameterized version of a

Betti number which will be called a barcode.

In Topological Data Analysis, the primary mathematical tool considered is a homology theory

for point-cloud data setspersistent homologyand a novel representation of this algebraic

characterizationbarcodes. Topological Data Analysis considered the shape of data.

Robert Ghrist in his article Barcodes: The Persistent Topology of Data 46 concluded:

When a topologist is asked, How do you visualize a four-dimensional object? the

appropriate response is a Socratic rejoinder: How do you visualize a threedimensional

object? We do not see in three spatial dimensions directly, but rather via sequences of planarprojections integrated in a manner that is sensed if not comprehended.

We spend a significant portion of our first year of life learning how to infer three-dimensional

spatial data from paired planar projections. Years of practice have tuned a remarkable ability

to extract global structure from representations in a strictly lower dimension.

The inference of global structure occurs on much finer scales as well, with regard to

converting discrete data into continuous images. Dot-matrix printers, scrolling LED tickers,

televisions, and computer displays all communicate images via arrays of discrete points which

are integrated into coherent, global objects. This also is a skill we have practiced from

childhood. No adult does a dot-to-dot puzzle with anything approaching anticipation.47

Similar discussion on the discrete presentation of data /information/ but our ability to perceive

gradualness of information and the continuous nature of the shape of data, offers Magnus

46 Robert Ghrist in his article Barcodes: The Persistent Topology of Data, 2007, Bulletin of American

Mathematical Society, Volume 45, Number 1, January 2008, Pages 6175

47

Robert Ghrist in his article Barcodes: The Persistent Topology of Data, 2007, Bulletin of AmericanMathematical Society, Volume 45, Number 1, January 2008, Pages 6175


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Bakke Botnan in his work Three Approaches in Computational Geometry and Topology -

Persistent Homology, Discrete Differential Geometry and Discrete Morse Theory48 :

One of the most remarkable properties of the human brain is the ability to infer the world as

a three-dimensional space. We do not see three spatial dimensions directly, but from

experience we know how to visualise three dimensions via sequences of paired planar

projections. In other words, we know how to extract global structures by studying

representations from a strictly lower dimension. Another skill developed is how to infer a

continuum from discrete data. As an example, consider the painting The Seine at La Grande

Jatte by the French artist Georges Seurat. This painting consists of discrete data points and is

obviously noisy. Nonetheless, we have no problems perceiving the tree by the waterline, the

person in the kayak or the sailboat. Rather than altering out noise qualitatively it is favourable

to have a quantitative measure.49

The paintings of Georges Seurat, as his Seine at La Grande Jatte, are probably one of the

best representation of the interplay of quality and quantity, in particular the gradual and

continuous notion of qualitative quantity. Pixel-based raster graphics, as in television and flat-

panel screens, mimics the pointillist method are associated with the artist Georges Seurat, who

covered the canvas with tiny spots of paint.

Shmuel Weinbergerin his work Persistent Homology also emphasized on the human ability

to take sense data of individual points and assemble them into a coherent image of a

continuum. Weinberger illustrating his explanation of persistent homology again with

Seurats painting: Consider the art of Seurat or a piece of old newsprint. The eye, or the

brain, performs the marvelous task of taking the sense data of individual points andassembling them into a coherent image of a continuumit infers the continuous from the

discrete. Difficult issues of a similar sort occur in many problems of data

analysis.Onemighthave samples that are chosen nonuniformly (e.g., not filling a grid), and,

moreover, one is constantly plagued by problems of noisethe data can be corrupted in

various ways. Puremathematicians have problems of this sort as well. One is often interested

in inferring properties of an enveloping space from a discrete object within it or, in reverse,

seeking commonalities of all the discrete subobjects of a given continuous one. To give one

example, this theme is a central one in geometric group theory, in which a typical problem,

going back to Furstenberg and Mostow, asks to reconstruct a connected Lie group from a

lattice in it. And Weinberger concluded again with the approach of topology: Because

topology is essentially a qualitative field, it is perhaps not surprising that there has been a

development of some common topological technology for these problems.50

48Magnus Bakke Botnan in his work Three Approaches in Computational Geometry and Topology - Persistent

Homology, Discrete Differential Geometry and Discrete Morse Theory, 2011

49Magnus Bakke Botnan in his work Three Approaches in Computational Geometry and Topology - Persistent

Homology, Discrete Differential Geometry and Discrete Morse Theory, 2011

50Shmuel Weinberger, Persistent Homology


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Speaking of clouds of data, Ghrist claimed that: Very often, data is represented as an

unordered sequence of points in a Euclidean n-dimensional space En. Data coming from an

array of sensor readings in an engineering testbed, from questionnaire responses in a

psychology experiment, or from population sizes in a complex ecosystem all reside in a space

of potentially high dimension. The global shape of the data may often provide important

information about the underlying phenomena that the data represent. One type of data set for

which global features are present and significant is the so-called point cloud data coming

from physical objects in 3-d. Touch probes, point lasers, or line lasers sweep a suspended

body and sample the surface, recording coordinates of anchor points on the surface of the

body. The cloud of such points can be quickly obtained and used in a computer representation

of the object. A temporal version of this situation is to be found in motion-capture data, where

geometric points are recorded as time series. In both of these settings, it is important to

identify and recognize global features: where is the index finger, the keyhole, the fracture?

Following common usage, we denote by point cloud data any collection of points in E n,

though the connotation is that of a (perhaps noisy) sample of points on a lower-dimensional

subset.51

Vin de Silva and Robert Ghrist in their work Homological Sensor Networks, directed to

topology as the rapidly evolving area of applied computational topology:

The need to move from local to global is one which a large spectrum of engineers and

scientists are finding to be prevalent. Very few of the calculus-based tools with which they are

most familiar prove sufficient. Recently, it has been demonstrated that homology theory is

useful for problems in data analysis and shape reconstruction, computer vision, robotics,

rigorous dynamics from experimental data, and control theoryTopology is especially keenat giving criteria for when one can or cannot find a particular global object (a

homeomorphism, a nonzero section, an isotopy, etc.): this falls under the rubric ofobstruction

theory. This perspective is one which has not yet permeated the applied sciences, in which the

question, What is possible? is usually approached from the top-down, Heres something

we can build, as opposed to the bottom-up approach that topological methods yield. A

brilliant example of this obstruction-theoretic viewpoint in an applied context is Farbers

topological complexity for robot motion planning. In this article, we use homology theory to

give coverage criteria for networked sensors which are nearly senseless. It seems

counterintuitive that one can provide rigorous answers for a network with neither localizationcapabilities nor distance measurements. A topologist is not surprised that such coarse data can

be integrated into a global picture. Some engineers are. Homological methods have the

pleasant consequence that they may allow engineers to focus on designing simpler sensors

which are nevertheless useful in a security network. Why bother miniaturizing GPS for smart

dust if you can solve the problem without it? If topological methods can determine the

51Robert Ghrist in his article Barcodes: The Persistent Topology of Data, 2007, Bulletin of American

Mathematical Society, Volume 45, Number 1, January 2008, Pages 6175


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minimal sensing needed to solve a global problem, then such methods may have significant

impact on the way systems and sensors are developed and deployed.52

3.2. The Crumpled Handkerchief of the Auditing Research

If you take a handkerchief and spread it out in order to iron it, you can see in it certain fixeddistances and proximities. If you sketch a circle in one area, you can mark out nearby points

and measure far-off distances. Then take the same handkerchief and crumple it, by putting it

in your pocket. Two distant points suddenly are close, even superimposed. If, further, you tear

it in certain places, two points that were close can become very distant. This science of

nearness and rifts is called topology, while the science of stable and well-defined distances is

called metrical geometry. Classical time is related to geometry, having nothing to do with

space, as Bergson pointed out all too briefly, but with metrics. On the contrary, take your

inspiration from topology, and perhaps you will discover the rigidity of those proximities and

distances you consider arbitrary. And their simplicity, in the literal sense of the wordpli[fold]:

it's simply the difference between topology (the handkerchief is folded, crumpled, shredded)

and geometry (the same fabric is ironed out flat). [] Sketch on the handkerchief some

perpendicular networks, like Cartesian coordinates, and you will define the distances.

But, if you fold it, the distance from Madrid to Paris could suddenly be wiped out, while, on

the other hand, the distance from Vincennes to Colombes could become infinite.53

- Serres, Michel (1995).

The methaphor of the handkerchief /of the Auditing Research/ stands here for the space of

audit as topological space. The auditing research handkerchief stand in for the canvas forpainting of auditing research themes or issues of auditing. The audit dynamics could be

approached as geometric landscape, where audit research themes are fixed distances and

welldefined proximities.

The well folded fabric of auditing research we may find in the study of Cdric Lesage and

Heidi Wechlter, Typology of research topics in audit: a content analysis. 54 In this study the

crumpled canvas of the auditing research is ironed with the extensive computerized content

analysis representing typology of research topics /in auditing research/. The data from the

abstracts (2099) selected from 23 main academic journals in auditing research are collected,

ranging the year of creation to year 2005. The typology offered identifies 17 majors themes in

audit research. The authors investigate the importance and trend of these 17 themes, as well as

the contribution of the major journals to their development.

52Vin de Silva and Robert Ghrist in their work Homological Sensor Networks,

http://www.math.upenn.edu/~ghrist/preprints/noticesdraft.pdf

53Conversations on science, culture, and time, Michel Serres with Bruno Latour; translated from French by

Roxanne Lapidus, The University of Michigan Press, p.60, 61

54

Cdric Lesage and Heidi Wechlter, Typology of research topics in audit: a content analysis,http://www.iae.univ-poitiers.fr/afc07/Programme/PDF/p88.pdf
http://www.math.upenn.edu/~ghrist/preprints/noticesdraft.pdfhttp://www.iae.univ-poitiers.fr/afc07/Programme/PDF/p88.pdfhttp://www.iae.univ-poitiers.fr/afc07/Programme/PDF/p88.pdfhttp://www.math.upenn.edu/~ghrist/preprints/noticesdraft.pdf


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In figure 3, page 16 of their study55

, Lesage and Wechlter offered the map of Typology of

Research Themes in Auditing, where 17 clusters identified by the software Spad have been

named by the authors, based on the characteristic vocabulary of each cluster and on the most

representative papers. We could easily recognize in the authorss research method how

qualitative quantity is illustrated. The approach of the used content analysis is based on the

qualitative mode keeping in track the quanitative elements as well. The descriptive analysis of

the "speech" or the words in the collected abstracts is analysed in their /words/ frequency as

associated with each item. According to Lesage and Wechlter, from this new contingencies

table (items*words), a simple correspondence analysis can be made. This table is by its

construction very sparse, but it makes it possible to outline some interesting associations and

to support some models already present in the literature. The analysis was carried out on table

items*words, with for illustrative variables the journal and the period. As expected, the

analysis is slightly explanatory6. However, it renders possible a rather interesting and

finetuned classification based on the factor analysis7 (17 clusters). This classification enables

us to measure the evolution of the topics in time, and the impact of each journal, which will

deepen the former analysis of the characteristic vocabulary.56

The 17 clusters57

representing the Typology of Research Themes in Auditing:

Audit Procedures: error, audit procedures account, approach statistics, test, audit risk

detection, balance;

Audit Sampling: statistics, sampling Bayesian, error, sample population, audit sample

method, test;

Corporate Governance: audit committee, corporate governance board, governance director,

financial report;

Internal Auditing: internal auditing, objectivity, reliance, function, department, organization,

employment;

Earmings Quality: earnings, accruals, discretionary, return, audit firm, big 4, disclosure,

resignation, fees;

Audit Engagement: client, risk, fees, lawbailing, cost, contract, bidding arrangements,

business risk;

Liability Fraud & Litigation: fraud, liability, court misstatement, detection, auditee strategy,

legal, audit failure;

Audit Report & Going-Concern Opinion: audit report, qualification, bankrupt, going

concern, qualified opinion, audit, opinion; Profession & Regulation: accountant, government certified public accountants profession,

regulation, public;

Internal Regulation: world, regulation, international, social, political, country, law,

harmonization;

Education: student, course, education, problem, accounting, cash, records, statements,

instruction;

55Ibid

56Ibid

57Ibid


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Audit Markets: discount, audit fee, big four, concentration, merger, audit market, market,

premium, price, fees;

Tax Audit: tax, price, transfer, compliance, specialist, penalties, income, game, purchase,

revenue;

Informal Processing: system, computer, evaluation, audit procedures, expert, decision

making, inernal control; Judgement: task, knowledge, judgement, experience, cue, audit judgement, consensus,

cognitive;

Audit Behaviour: behaviour, ethics, misreporting, moral, time budget, budget, intention,

commitment, perssonel;

Audit Review: persuasiveness, audit review, working paper, senior, style, audit managers,

memory.

According to Lesage and Wechlter, typologies of research topics in a specific area are

necessary because they enable the organization of knowledgethey are very useful to

understand the relationships between the research topics, leading to the analysis of the main

topics, their time evolution, etc.58 Lesage and Wechlter recognized that sociological sciences

assume that typologies are a kind a weak theoretical framework. As given the large

diversity of issues and theoretical approaches used in audit research, it would therefore seem

particularly important to have a classification based on a rigorous analysis of research

literature, organizing knowledge in audit research from the actual scientific production, and

not according to an a priori practical framework. The objective of Lesage and Wechlter s

contribution is thus to identify the research topics in audit, assess the evolution of their

importance during the 20th century, and determine the contribution of the main journals.

Offering an discussuion on the usefulness of typologies to describe and understand a specific

complex problemq and following the old scientific tradition, conceptualized by Aristotle:classification, or typology or taxonomy, Lesage and Wechlters aim is to present the

advantages of typologies for the comprehension of a field of research using taxonomy versus

typology. In contrast with Lesage and Wechlter, I am focused on the the interest of the

topology to understand a phenomenon.

According to Aristotle, taxonomy is a hierarchical system which describes the downward

relations between the species and the genres. The species derive from a common genre and

inside taxonomy, If taxonomy is hierarchical classification in typology, I am interested in

topology as the heterarchical relation of types in the sense of Gotthard Gnther and Warren

McCullochs A Heterarchy of Values Determined by the Topology of Nervous Nets". As I

menshioned above, the approach I will follow is based on the issue of mixed-mode Semiosis -

Typological vs. Topological Semiosis, introduced by Jay L. Lemke - The transition from

Typological Mode to the Topological Mode.

The semiotic bridge in mathematics is qualitative quantity the categorical gradiant which

unite in one these two fundamentally different kinds of meaning-making - meaning-by-kind

and meaning-by-degree in the continuous variation of "topological" meaning.

58Ibid


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The anthropologist Edmund Leach back in the 1960s, described topology as a geometry of

elastic rubber sheeting precisely because the shape and size of things or the distance between

them is less significant than what holds them together; that is, the ways in which they are

connected, the nature of their relatedness, so to speak. In this, one can perhaps see the initial

attraction of topology for Bruno Latour and other actor-network theorists, where object

integrity, what holds networks together, is of paramount importance.

In the topological sense we may approach the typology of research topics in audit using the

topological methaphor of Latour with the analogy of the handkerchief as surface. How it is

possible?

The Figure 1 is example introduced by Norman MacLeod 104 in his Paleo -Math 101, Shape

Models II: The Thin Plate Spline59.

Figure 1. Deformational modes of theAcaste-Calymeme geometric transformation.

(A) Uniform (affine) TPS surface. (B) Non-uniform (non-affine) TPS surface.60

The Figure above represents the deformational modes of the Acaste-Calymeme geometric

transformation. (A) Uniform (affine) TPS /Thin plate splines/ surface. (B) Non-uniform (non-

affine) TPS surface, where the 17 clusters of Typology of Research Themes in Auditing are

represented in the geometric transformation of the surface / handkerchief/. The flat and well-ironed surfaces of a handkerchief stand in for a geometric landscape of fixed distances and

welldefined proximities. In the figure above, this is the (A): Uniform (affine) TPS surface.

The crumplet handkerchief is represented in the figure with (B): Non-uniform (nonaffine)

TPS surface.

59(Norman MacLeod, Paleo-Math 101, Shape Models II: The Thin Plate Spline,

http://www.palass.org/modules.php?name=palaeo_math&page=26)

60(Norman MacLeod, Paleo-Math 101, Shape Models II: The Thin Plate Spline,

http://www.palass.org/modules.php?name=palaeo_math&page=26)
http://www.palass.org/modules.php?name=palaeo_math&page=26http://www.palass.org/modules.php?name=palaeo_math&page=26http://www.palass.org/modules.php?name=palaeo_math&page=26http://www.palass.org/modules.php?name=palaeo_math&page=26http://www.palass.org/modules.php?name=palaeo_math&page=26http://www.palass.org/modules.php?name=palaeo_math&page=26


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Figure 2 represents the 17 clusters of Typology of Research Themes in Auditing from

Cdric Lesage and Heidi Wechlter, Typology of research topics in audit: a content analysis,

placed on Deformational modes of the Acaste-Calymeme geometric transformation /after

After Norman MacLeods example about the Uniform (affine) TPS surface and Non-uniform

(non-affine) TPS surface.

Figure 2. The 17 clusters of Typology of Research Themes in Auditing on /Norman

MacLeods example/ The Thin Plate Spline Deformational modes of the Acaste-Calymeme

geometric transformation. (A) Uniform (affine) TPS surface. (B) Non-uniform (non-affine)

TPS surface.

Proposing the thin plate spline, MacLeod recalls DArcy Thompson, in concluding that:

Is the thin plate spline the long-sought realization of the Thompsonian transformation grid

concept? In some ways it is and in some ways it isnt. I suspect Thompson himself would

have absolutely loved thin plate splines. DArcy Thompson was a great believer in the

constraints materials and physical processes place on morphological arrangements. The idea

that the TPS algorithm involves a metaphorical concept of bending energy which is required

to be minimized by the resulting geometry would have spoken to one of his most deeply held

beliefs about the organic world. However, no data or morphological patterns have come to

light in the 93 years that have elapsed since On Growth and Forms publication to lend


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support the idea that evolutionary processes operate in such a way as to minimize physical

parameters such as bending energy. To be sure, organic design cannot exceed the performance

a topological approach to audit dynamics and auditing research 1 21[2]

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