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Title: A HISTORIC OVERVIEW OF THE INTERPLAY OF THEOLOGY AND PHILOSOPHY IN THE ARTS, 

MATHEMATICS AND SCIENCES

Article Type: Issue 2009 - 1

Keywords: 

Corresponding Author: Prof. Bharath Sriraman, Ph.D.

Corresponding Author's Institution: The University of Montana

First Author: Bharath Sriraman, Ph.D.

Order of Authors: Bharath Sriraman, Ph.D.

Abstract: The etymology of the word "mathematics" can be traced to Greek and Latin roots with meanings 

such as "to think or have one's mind aroused" or "the art of knowing". The natural philosophers of the 

Renaissance did not draw an explicit distinction between mathematics, the sciences and to an extent the 

arts. In this paper we first explore connections forged by the thinkers of the Renaissance between 

mathematics, the arts and the sciences, with attention to the nature of the underlying questions that call for a 

particular mode of inquiry. Second, we will examine both the relationship and individual differences between 

innovative behaviors across domains. Recently Robert Root-Bernstein (2003) introduced the construct of 

polymathy to suggest that innovative individuals are equally likely to contribute both to the arts and the 

sciences and either consciously or unconsciously forge links between the two. Several contemporary 

examples are presented of individuals who pursued multiple fields of research and were able to combine the 

aesthetic with the scientific. Finally, we will also discuss the possibilities for re-introducing university courses 

on natural philosophy as a means to integrate mathematics, the arts and the sciences.



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A HISTORIC OVERVIEW OF THE INTERPLAY OF THEOLOGY AND 

PHILOSOPHY IN THE ARTS, MATHEMATICS AND SCIENCES

Bharath Sriraman
The University of Montana (USA)

The etymology of the word “mathematics” can be traced to Greek and Latin roots with 
meanings such as “to think or have one's mind aroused” or “the art of knowing”. The 
natural philosophers of the Renaissance did not draw an explicit distinction between 
mathematics, the sciences and to an extent the arts. In this paper we first explore connections 
forged by the thinkers of the Renaissance between mathematics, the arts and the sciences, 
with attention to the nature of the underlying questions that call for a particular mode of 
inquiry. Second, we will examine both the relationship and individual differences between 
innovative behaviors across domains. Recently Robert Root-Bernstein (2003) introduced the 
construct of polymathy to suggest that innovative individuals are equally likely to contribute 
both to the arts and the sciences and either consciously or unconsciously forge links between 
the two. Several contemporary examples are presented of individuals who pursued multiple 
fields of research and were able to combine the aesthetic with the scientific. Finally, we will 
also discuss the possibilities for re-introducing university courses on natural philosophy as a 
means to integrate mathematics, the arts and the sciences.

INTRODUCTION
The history and development of mathematics is intricately connected to the rise and fall of 
ancient and modern civilizations. The progression of humanity from hunter-gather societies 
onto societies with sophisticated astronomical calendars, visually pleasing architectural forms 
(temples, mosques, cathedrals etc) reveals our quest to understand the cosmos, our attempts 
to represent and symbolize it via patterns, symmetries and structure. A common characteristic 
of many civilizations (both ancient and modern) is the quest to answer three basic 
philosophical questions:
 (1) What is reality? Or what is the nature of the world around us? 
This is linked to the general ontological question of distinguishing objects (real versus 
imagined, concrete versus abstract, existent versus non-existent, independent versus 
dependent and so forth).
(2) How do we go about knowing the world around us? [the methodological question, which 
presents possibilities to various disciplines to develop methodological paradigms] and, 
(3) How can we be certain in the “truth” of what we know? [the epistemological question]. 

The interplay of mathematics, arts and sciences is found in the attempts to answer these 
fundamental philosophical questions. In this sense Philosophy can be viewed as the 
foundational bridge unifying mathematics, arts and the sciences. In this paper we will first 
focus on the attempts of the thinkers of the Renaissance, who did not view themselves simply 
as theologians or mathematicians or inventors or painters or philosophers or political 
theorists, but who thought of themselves as philosophers in the pursuit of Knowledge, Truth 
and Beauty. Then we will try to pinpoint the thinking characteristics of these natural 
philosophers, particularly the trait of polymathy (Root-Bernstein, 2003) which explains 
innovative behavior across numerous subject specific domains. Finally we will examine the 
implications of these findings for present day high school and university education.

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REVISITING THE RENAISSANCE
Theology influencing Art and Science (resulting in powerful mathematics as a 

consequence) 

The great works of Art during the Renaissance, 
particularly those of Italian painters (Massacio, 
Brunelleschi , Leonardo da Vinci , Michelangelo , Titian, 
Giotto, Raphael etc)immediately reveal the interplay of 
church doctrine with art in that these painters inspired 
devotion in people by painting divine Christian icons.  In 
this process of trying to convey these divine images as 
realistically and beautifully as possible, these painters 
moved away from the medieval style of painting (very 2-
dimensional) and essentially created the rules of 
perspectivity that allowed 3-D images to be projected 
onto a flat surface. This suggests that art was instrumental 
in initiating the mathematical foundations of the true rules 
of perspective. Geometrical optics also played a major 
role in how artists experimentally arrived at the 
mathematical rules of perspectivity (e.g., Fig1. 
Massacio’s Holy Trinity1on previous page).

Calter (1998) traces a rich lineage of the interaction of 
optics with art: Euclid’s Optica (300 B.C) - Vitruvius' Ten 
Books on Architecture- Ptolemy's Optica, (c. 140 A.D) -
(the Islamic) Alhazen's Perspectiva, (c. 1000 A.D) -
Roger Bacon's Opus Majus, (c. 1260 A.D.), with sections 

on optics, “whose geometric laws, he maintained, reflected God's manner of spreading his 
grace throughout the universe – onto John Pecham's Perspectiva communis, (c. 1270 A.D). 
The amalgamation of mathematical ideas proposed in this lineage was formalized by 
Desargues (1593-1662) which is today studied in courses on projective geometry. Today the 
visual-artistic side of mathematics is completely lost under the rubble of formalization. 
However the visual side of mathematics has seen a revival in the 20th century in the area of 
fractal geometry due to the work of Benoit Mandelbrot. This is explored further at a later 
junction in the paper. 

The scientists-mathematicians-theologians of the Renaissance-Post Renaissance 

The relationship between science and theology can be traced back to the pre-Socratic Greeks. 
Pre-Socratic Greek society evolved from the typical “Sky-God” explanation of 
creation/reality onto a society that developed a rigorous and systematic philosophy to answer 
the three aforementioned questions. 

The Pythagorean School (c. 500 B.C) developed a mystical numeric system to designate and 
describe everything in the universe. They even went so far as to claim that “All things of the 
universe have a numerical attribute that uniquely describes them”. For instance numbers were 
designated abstract attributes. Such as one: the number of reason, two: the first even or 

                                        
1 Retrieved from http://www-groups.dcs.st-and.ac.uk/~history/HistTopics/Art.html

http://library.thinkquest.org/2838/artgal.htm#Brunelleschi#Brunelleschi
http://library.thinkquest.org/2838/artgal.htm#davinci#davinci
http://library.thinkquest.org/2838/artgal.htm#Michelangelo#Michelangelo
http://library.thinkquest.org/2838/artgal.htm#Titian#Titian
http://library.thinkquest.org/2838/artgal.htm#Giotto#Giotto


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Sriraman

female number (the number of opinion), three: the first true male number (the number of 
harmony), four: the number of justice or retribution, five: marriage, six: creation and so on…

Each number had its own personality - masculine or feminine, perfect or incomplete, 
beautiful or ugly. This feeling modern mathematics has deliberately eliminated, but 
we still find overtones of it in fiction and poetry. Ten was the very best number: it 
contained in itself the first four integers - one, two, three, and four [1 + 2 + 3 + 4 = 
10] - and these written in dot notation formed a perfect triangle. (Brumbaugh, 1981)

Further the motion of planets and musical notes were expressed as ratios of numbers. The 
Pythagorean School developed an elaborate numerical system consisting of even and odd 
numbers to describe the world around them. However Plato (429-327 B.C.E), and Aristotle 
(384-322 B.C.E) deviated from the mysticism of the Pythagoreans and instead attempted to 
understand the universe via reason. Plato suggested that the universe consisted of two realms, 
the visible realm which was deceptive because of its changing nature and an abstract realm 
which he believed was eternal and unchanging. Within this “dualistic” ontology of reality, 
Plato answered the epistemological question by suggesting that knowledge derived 
empirically from the changing world was fallible whereas knowledge derived from the 
abstract realm was infallible or absolute. Plato accorded a special place for mathematics in 
this pursuit of absolute knowledge by claiming that mathematics was derivable independent 
of the physical senses. Thus the purest form of “thought” was mathematical thought as it was 
deemed capable of deriving “eternal truths” or absolute knowledge. In spite of the alleged 
motto of the academy, Plato distinguished “numbers as ideas” from “numbers as 
mathematical objects”. Unfortunately over time this important distinction faded and 
Platonism approached Pythagoreanism, which in turn influenced Renaissance philosophers, 
then modern natural science and thereby again modern philosophy.

Aristotle on the other hand was an empiricist whose prodigious work left a lasting impression 
until the 13th century (namely the dawn of the Renaissance). The Aristotelian approach to 
science was empirical and placed a heavy emphasis on perception through the senses. 
Aristotle rejected the Platonic notion of the mind’s capacity to intuit /discern a priori reality 
and instead proposed an a posteriori or empirical methodology whereby knowledge is 
acquired by the mind. Aristotelian science was axiomatic and deductive in nature with the 
aim of explaining natural phenomenon. The underlying assumption of Aristotelian science 
was that all natural objects were fulfilling a potential determined by an actual prior natural 
object. For instance, a seed becomes a plant because it is merely fulfilling its potential of 
becoming a plant. Science historians today agree that Aristotle was an empiricist, who 
believed that knowledge is gained via observation, experimentation and experience.  The 
question of whether or not Aristotelian science was the origin of dualism is still a matter of 
present day debate2. Recent scholarship on post Renaissance science and natural philosophy 
traces a rich intellectual lineage centered on "scholasticized Aristotelianism" from 17th

century natural philosophy onto medieval thinkers like Aquinas onto Aristotle3 (Sriraman & 
Benesch, 2005, p. 42).

                                        
2 One could argue that Aristotle drew a distinction in the natural world between the animate and inanimate, 
whereas Descartes was more focused on the human. Descartes’ dualism dominated physics for a substantial 
period of time. Newtonian physics was one of the consequences of the Cartesian view point. Robinson, H. M. 
(1983) Aristotelian Dualism. In J. Annas (Ed.), Oxford studies in ancient philosophy (Vol. 1, pp. 123-144). 
Oxford: Clarendon Press.
3 The Cambridge History of Seventeenth-Century Philosophy. Edited by Daniel Garber and Michael Ayers. 
Vols. 1 and 2,Cambridge: Cambridge. University Press, 1998. p.1616



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Thomas Aquinas (1225-1274) synthesized “all that had been argued in Western thought up to his 
time and he showed it to be compatible with Christian beliefs” (Sharp, 2003, p.346). His 
argument is superbly summarized by Sharp (2003) as follows:
…Aquinas argued that all our rational knowledge of this world is acquired through sensory 
experience, on which our minds then reflect. When children are born, their minds are like a clean 
slate (tabula rasa). Aquinas developed a theory of knowledge which is uncompromisingly 
empirical. The world through which we gain our knowledge is God’s creation, and therefore it is 
impossible for this gained knowledge to conflict with religious revelation. (p.346)

The Greek philosophers also stumbled upon the idea of the “infinite”, a sophisticated 
mathematical abstraction, as evidenced in Zeno’s paradoxes4. Bertrand Russell observed that 
scholastic theology was one of the outcomes of mathematical abstraction (Russell, 1945, p. 37). 
His claim is supported by numerous historical examples, a handful of which are presented here. 
Sa’id ibn Yusuf (Saadia Gaon), a 10th century theologian and leader of the Babylonian Jews in 
his theological treatise Kitab al-Amanat wa-al Itiqadat (the Book of Beliefs and Opinions) wrote: 

How is it possible to establish this concept (of the Creator) in our minds when none of 
our senses have ever perceived Him?…It is done in the same way in which our minds 
recognize the impossibility of things being existent and nonexistent at the same time, 
although such a situation has never been observed by the senses. (Saadia Gaon,1948)

Saadia Gaon also cleverly reversed Zeno’s paradox of Achilles and the tortoise5 to prove that 
Creation occurred.

If the world were uncreated, then time would be infinite. But infinite time cannot be 
traversed. Hence, the present moment couldn’t have come about. But since the present 
moment exists…the world had a beginning.  (Saadia Gaon,1948)

The ideas of the Greeks also had a profound influence on post-Renaissance mathematicians like 
Descartes (1596-1650), Pascal(1623-1662 )and Leibniz (1646-1716 ) among others. One 
routinely comes across the use of mathematical analogies to prove the existence of God in the 
theological works of Descartes, Leibniz, and Pascal. For example, Descartes in the Fifth 
Meditation states:

Certainly the idea of God, or a supremely perfect being is one which I find within me, 
just as surely the idea of any shape or number. And my understanding that it belongs 
to his nature that he always exists is no less clear and distinct than is the case when I 
prove of any shape or number that some property belongs to its nature. Hence, even if 
it turned out that not everything on which I have meditated in these past days is true, I 
ought still to regard the existence of God as having at least the same level of certainty 
as I have hitherto attributed to the truths in mathematics.  (Descartes 1996, p.45)

                                        
4 Zeno (born around 495 B.C.E) was a Greek philosopher and logician, and a student of the philosopher 
Parmenides. Zeno is remembered for paradoxes that stumped mathematicians for centuries. Zeno’s paradoxes 
evolved from Parmenides’ ideas about the illusory nature of motion, change and time.
5 Achilles and the tortoise: The running Achilles can never catch a crawling tortoise ahead of him because he 
must first reach where the tortoise started. However, when he reaches there, the tortoise has moved ahead, and 
Achilles must now run to the new position, which by the time he reaches the tortoise has moved ahead, etc. 
Hence the tortoise will always be ahead.



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Leibniz in Theodicy argued that faith and reason were compatible.
Theologians of all parties, I believe (fanatics alone excepted), agree that no article of faith 
must imply contradiction or contravene proofs as exact as those of mathematics, where 
the opposite of the conclusion can be reached ad absurdum, that is, to contradiction. It 
follows thence that certain writers have been too ready to grant that the Holy Trinity is 
contrary to that great principle, which states that two things, which are the same as the 
third, are also the same as each other. For this principle is a direct consequence of that of 
contradiction, and forms the basis of all logic; and if it ceases, we can no longer reason 
with certainty.  (Leibniz 1985, 87)

Leibniz’s dissertation on the conformity of faith and reason can be interpreted to mean that it is 
logically contingent and intelligible for a human being to ask why an eternal being exists (Craig 
1995). 

Blaise Pascal, perhaps the most intriguing mathematical mystic argued for the use of 
“infinitesimal6 reasoning” (or reasoning in infinitely small quantities) by proclaiming that the 
infinitely large and the infinitely small were mysteries of nature that man stumbled on by divine 
inspiration. Pascal is also remembered for his famous wager, where he argued that if God’s 
existence has a probability of 0.5 (50/50 chance), then it is only rational for us to believe he does 
exist. Pascal in Pensees states: 

If there is a God, He is infinitely incomprehensible, since, having, neither parts nor 
limits, He has no affinity to us. We are then incapable of knowing either what He is or 
if He is ... you must wager. It is not optional. You are embarked. Which will you 
choose then? Let us weigh the gain and the loss in wagering that God is. Let us 
estimate these two chances. If you gain, you gain all; if you lose, you lose nothing.
Wager then without hesitation that he is (Pascal, in Popkin 1989)

The aforementioned historical examples, viz., the number mysticism of the Pythagoreans, the 
paradoxes of Zeno that brought forth the abstraction of the infinite, the attempts of medieval 
theologians like Saadia Gaon to systematize theology by constructing uniqueness proofs to 
theological theorems, and the use of mathematical arguments to prove the existence of a Creator 
by post-Renaissance mathematicians, illustrate that there has been a rich interplay between 
mathematics and theology (Sriraman, 2004a). 

While these aforementioned thinkers of the Renaissance and post Renaissance who strongly 
believed in the existence of a creator invoked mathematical arguments to prove their beliefs 
in their philosophical writings, others such Copernicus and Galileo, who were professed 
believers of the Catholic Church, found it increasingly difficult to believe in the prescribed 
view of the world (earth) as the center of the universe. Their model building during this time 
period reveals the interplay as well as the conflict between theology and science with 
mathematics replacing Aristotilean logic as the language of description. 

                                        
6 An infinitesimal is a number that is infinitely small but greater than zero. Infinitesimal arguments have 
historically been viewed as self-contradictory by mathematicians in the area of analysis. The infinitesimal 
Calculus of Newton and Leibniz was reformulated by Karl Weierstrass in the 19th century for the sole purpose 
of eliminating the use of infinitesimals. In the 20th century Abraham Robinson revived the notion of 
infinitesimals and founded the subject of Non-standard analysis to resolve the contradictions posed by 
infinitesimals within Calculus. Robinson attempted to use logical concepts and methods to provide a suitable 
framework for differential and integral Calculus.



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Modeling the Universe:  Copernicus-Galileo-Kepler

The Ptolemaic model (c. 87-150 A.D) of astronomy was 
based on the assumption that the earth was the center of 
the universe which was accepted by the Catholic Church 
as being compatible with its teachings. However this 
geocentric view of the world could not explain the 
curious planetary phenomenon observed by Nicholaus 
Copernicus (1473-1543). That is the retrograde motion 
(moving backwards and then forwards) of Mars, Jupiter 
and Saturn, in addition to nearly invariant times that 
Venus and Mercury appeared in the sky which is shortly
before sunrise and after sunset. 

Fig.2 Claudius Ptolemy7

However these queer motions were perfectly reasonable 
if one viewed the sun as the center of the “universe” as opposed to the earth. In such a model 
the peculiarities of the inner planets (Mercury and Venus) as well as the outer planets (Mars, 
Jupiter and Saturn) in relation to the earth make perfect sense. The retrograde motion of outer 
planets is due to the fact that they get overtaken by the earth in its orbitary motion. Similarly 
Venus and Mercury appear static and only before sunrise and after sunset because their 
orbitary motions do not allow them to get behind the earth and manifest in the night sky. It is 
amazing what a little change in perspective does to our perceptions! However the conflicts of 
Copernicus’ findings with Church dictum prevented a wider dissemination of his simpler 
planetary model until his death. Centuries later, the great German philosopher, writer, 
scientist Goethe (1749-1832) reflected8 on Copernicus’ new perspective of our reality:

"Of all discoveries and opinions, none may have exerted a greater effect on the human 
spirit than the doctrine of Copernicus. The world had scarcely become known as round 
and complete in itself when it was asked to waive the tremendous privilege of being the 
center of the universe. Never, perhaps, was a greater demand made on mankind - for by 
this admission so many things vanished in mist and smoke! What became of our Eden, our 
world of innocence, piety and poetry; the testimony of the senses; the conviction of a 
poetic - religious faith? No wonder his contemporaries did not wish to let all this go and 
offered every possible resistance to a doctrine which in its converts authorized and 
demanded a freedom of view and greatness of thought so far unknown, indeed not even 
dreamed of." 

Galileo Galilei (1564-1642) pushed things further by using mathematics to explain 
interplanetary motion. In fact many science historians claim that Galileo was the first person 
to systematically use mathematics as the language of science instead of Aristotilean logic. 
Aristotle’s conceptions of motion had several flaws which were rectified by Galileo by 
determining that velocity and acceleration were distinct. More importantly the question that 
vexed Copernicus of why the motion of the Earth was unfelt (if in fact it was moving) was 
answered by Galileo by suggesting that (1) only acceleration is felt whereas velocity is unfelt 

                                        
7
 Retrieved from http://obs.nineplanets.org/psc/img/ptolemybig.gif

8
 Retrieved from http://www.blupete.com/Literature/Biographies/Science/Copernicus.htm



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and invariant except when acted on by an external force (the notion of inertia). Thus, Galileo 
suggested that the Earth in addition to orbiting around the sun was also rotating on its own 
axis. Needless to say, his attempt to make his model public met with fierce resistance from 
the Church and led to his condemnation by the Inquisition. 

During this same time period the German astronomer Johannes Kepler (1571- 1630) 
confirmed and supported many of Galileo’s well formulated theories. Johannes Kepler was 
born in Weil der Stadt, Württemburg, (and not very far from here). While studying for 
Lutheran ministry at the University of Tübingen, he became familiar with the Copernican 
model, which he defended explicitly in the Mysterium Cosmographicum. The political forces 
of that time period with his unique personal circumstances, namely his strong adherence to 
the Augsburg Confession but rejection of several key Lutheran tenets, the use of the calendar 
introduced by Pope Gregory XIII, his rejection of the Formula of Concord and finally his 
snub to Catholicism led to him to exile in Prague where he worked for the Danish astronomer 
Tycho Brahe. With the help of Brahe’s data, Kepler made several seminal discoveries 
published in Astronomia Nova. The beauty of this work lies in the fact that Kepler arrived at 
the first two laws of planetary motion by working with incomplete/imperfect data (we must 
remember that this data was obtained before the invention of the telescope!). The first two 
laws were (1) Planets move in ellipses with the Sun at one focus, and (2) The radius vector 
describes equal areas in equal times. Finally the third law was published Harmonices Mundi
in 1619. The third law states that: the squares of the periodic times are to each other as the 
cubes of the mean distances. Incidentally Newton’s theory of gravitation grew out of Kepler’s 
third law (and not a fallen apple as suggested by myth). 

Szpiro (2003) recently suggested that among the forces driving Kepler’s work during his 
turbulent Tübingen years was to seek a theological explanation to his questions:

Since God had created a perfect world, he thought it should be possible to discover 
and understand the geometric principles that govern the universe. After much 
deliberation Kepler believed he had found God’s principles in the regular solids….His 
explanations of the universe were based on an imaginary system of cubes, spheres and 
other solids that he thought were fitted between the sun and other planets…published 
in Mysterium Cosmographicum. This tome did not unveil any mysteries of the 
planetary system…since no solids exist that are suspended in the universe. But the 
book came to the attention of Tycho Brahe. (Szpiro, 2003, p.13)

And the rest is history… 

Isaac Newton’s ( 1642-1727) prodigious work included a mathematical model of the planetary 
system, in a sense suggesting that the universe was governed by certain laws, expressible via 
mathematics and discernible by humans (Sharp, 2003). This led to the development of natural 
philosophy as an answer to the ontological, methodological and epistemological questions with 
mathematics becoming the medium of establishing truth. One consequence of Aristotle’s 
empiricist tradition was the acceptance of the notion that knowledge of the external world was 



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derived by an active soul9, which was in essence separate from that world. Salmon (1990) 
comments10:

“ It is illuminating to recognize that Cartesian dualism offered a way of resolving the 
conflict between science and religion-which had brought such great troubles to Galileo-
by providing each with its own separate domain. Physical science could deal with matter, 
while religion could handle whatever pertains to the soul. " (p.236). 

This led to a growing acceptance among 17th century natural philosophers in the notion of 
duality (or dualism). René Descartes (1596-1650) is considered the founder of this belief 
system since he initiated the mind-body problem. Cartesian dualism essentially proclaims that 
we are composed of two distinct and basic substances, namely the mind (soul) and matter. 
Matter was the material substance that extended into the world and took up space, whereas 
the mind (soul) was a thinking substance, which was not “localizable” in space. “If these two 
aspects (mind-matter) are to be held in equal balance, it seems that it will have to be in some 
way more subtle than mere juxtaposition11”. The problem of dualism can be reformulated as 
follows: “One can think of subject and object as two unique and separate natures, neither of 
which is reducible to each other. The question of course in such a dualistic assumption is 
‘how do these two natures relate to each other?’

THE MODERN DAY RENAISSANCE
Scientists in the 20th and 21st centuries have developed the technical tools and the analytic 
and theoretical maturity necessary for analyzing nature at unprecedented micro and 
macrocosmic levels. By doing so, they have reaffirmed the dynamic nature of the whole that 
was reflected in the paradoxes of the ancients.  The result is a view of nature in which 
processes have supplanted ‘things’ in descriptions and explanations.

By the end of the 19th Century, limitations of the classical Newtonian/Euclidean world-view 
had become increasingly problematic as physicists began exploring nature at the sub-atomic 
level. The paradoxes posed by uncertainty, incompleteness, non-locality, and wavicles, etc. 
let it seem apparent that in the sub-atomic world observations and observers are aspects of a 
whole.  The physicist John Wheeler comments:

“...In the quantum principle we’re instructed that the actual act of making an 
observation changes what it is that one looks at.  To me, this is a perfectly marvelous 
feature of nature…. So the old word observer simply has to be crossed off the books, 
and we must put in the new word participator.  In this way we’ve come to realize that 
the universe is a participatory universe.” 12  

Biologists have found that methodological reductionism, i.e. going to the parts to understand 
the whole, which was central to the classical physical sciences, is less applicable when 
dealing with living systems. Such an approach may lead to a study not of the ‘living’ but of 

                                        
9
 Polkinghorne comments that Aristotle’s view of the soul as the underlying “form” or pattern of the body was 

“taken up by Thomas Aquinas who rejected Platonic dualism that had dominated Western Christian thinking 
since Augustine.”  Sir John Polkinghorne, Science & Theology, Fortess Press, 1998, p.63.
10 Salmon, W.C.(1990).  Philosophy and the Rise of Modern Science" Teaching Philosophy, p.236.
11

 Sir Polkinghorne, Ibid, p. 54.
12

 Comment made by the physicist John Wheeler in Paul Buckley and F. David Peat: Conversations in physics 
and Biology, University of Toronto Press,  1979, p. 53-4



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the ‘dead’, because in the examination of highly complex living systems “only by ripping 
apart the network at some point can we analyze life. We are therefore limited to the study of 
‘dead’ things.”13

One of the most important shifts in the natural sciences in the modern period has been away 
from the view of a simple and complete separation between observer and observed to an 
awareness that an observer also represents a living aspect of that which is being observed--
both as a product of nature and as the mental possibility in nature of observing, as in the 
notion of the ‘participator universe14’. A synthesis of ‘product’ and ‘process’ are at the heart 
of the puzzles and paradoxes that we associate with ideas of  ‘indeterminacy’ in physics, and 
with genes in biology. The very concept of ‘objectivity’ maintains that the observed and 
observer are separate does not hold in the study of  “highly complex biological processes 
such as evolution or the functioning of the central nervous system ... we cannot distance 
ourselves from the object being considered; indeed, this is so at the very moment we start to 
think.”15

It is amazing how close in understanding, and that across six centuries, modern physics and 
biology are to the Neo-Confucian Philosopher, Wang Yang-Ming’s continuum view of 
‘innate knowledge’:

The innate knowledge of man is the same as that of plants and trees, tiles and 
stones…Heaven, Earth, the myriad things, and man form one body. The point at 
which this unity is manifested in its most refined and excellent form is the clear 
intelligence of the human mind.16

The very process of generalizing implies a belief in the unity of the world: “if the different 
parts of the universe were not like the members of one body, they would not act on one 
another…know nothing of one another, and we…would know only one of these parts.  We do 
not ask if nature is one, but how it is one.”17  The position on mind and nature of theoretical 
physicists seems consistent with that of the neo-Confucian philosopher. Another physicist 
suggest that the heliocentric universe is again becoming geo or human centered in that it is 
“... formless potentia  … and becomes manifest only when observed by conscious beings...Of 
course, we are not the geographical center, but that is not the issue. We are the center of the 
universe because we are its meaning.”18 This is yet another amazing shift in perspective.

Benoit Mandelbrot (1924- ), a very recognizable name in 20th century mathematics because 
of his seminal contributions to the development of fractal geometry has repeatedly 
emphasized the need to re-orient our perspectives to better understand the world around us.  
He has often very humbly characterized himself as an “accidental” mathematician. In spite of 
his early interest and precocity in the study of geometry he was “encouraged’ by the French 
university establishment to embrace formalization which led him to leave the École Normale 
Supériere. He writes

                                        
13

 Cramer, Ibid, 214
14

 John Wheeler’s participatory universe as quoted in footnote 12.
15

 Friedrich Cramer: Chaos and Order, VCH Publishers, New York,  1993, p. 212
16

 Wing-tsit Chan,  Wang Yang-Ming p.221
17 Henri Poincaré, The Foundations of Science, George Bruce Halsted trans., The Science Press, Lancaster, PA., 
1946, 
18 Amit Goswami, The Self-Aware Universe, G. P. Putnam Sons, N.Y.  1993, p. 141



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“I spent several years doing all kinds of things and became, in a certain sense, a 
specialist of odd and isolated phenomena…I did not know or care in which field I was 
playing. I wanted to find a place, a new field, where I could be the first person to 
introduce mathematics. Formalization had gone too far for my taste, in the 
mathematics favored by the establishment…” (Mandelbrot, 2001, p.192)

  
Mandelbrot made his astonishing mathematical discovery when working on an economics 
problem accidentally handed to him by a friend. Economists had long  attempted to make 
sense of (and predict) stock market fluctuations and had proposed theories based on existing 
data which did not hold up when tested with primitive computers. Mandelbrot viewed 
fluctuations from the perspective of changing scales. That is the time scale can be in days or 
months or years. He suggested that the interchangeable nature of the time scales was the key 
to understanding the fluctuations. 

I cooked up the simplest mathematical formula I thought could explain this 
phenomenon…[making] no assumptions about people, markets or anything in the real 
world. It was based on a ‘principle of invariance’, -the hypothesis that, somehow 
economics is a world in which things are the same in the small as they are in the large 
except, of course for a suitable change of scale (Mandelbrot, 2001). 

In 1960, Edward Lorenz, who was modeling the earth’s atmosphere with nonlinear equations 
at MIT, switched from rounding his equations to the sixth decimal point to doing so to the 
third.  What emerged was a totally different system! He attributed the difference to a 
combination of the iteration of his equations plus the sensitivity of the system to initial 
conditions---in this case, the changes in the terminal decimal points. Lorenz named this 
randomness within his non-random weather models the ‘butterfly effect’ in a paper he wrote 
entitled “Can the flap of a butterfly’s wing stir up a tornado in Texas?” The discovery of  
‘sensitive dependency on initial conditions’ coupled with the ‘iteration of patterns or data’ 
which produce random irregularities in deterministic systems is the beginning of the 
contemporary science of “deterministic chaos.”19

The term ‘fractal’ was coined from Latin fractua ‘irregular,’ to refer to the results of this 
combination of iteration and sensitivity. And it was Mandelbrot who provided the pictures of 
this deterministic chaos in his computer generated fractal images---what is described as “…a 
way of seeing infinity.”20  We discover these irregular nonlinear fractal structures and 
patterns throughout nature, in the iterations of buds in Romanesco broccoli, the arterial and 
venous systems of kidneys, lungs, brains, coast lines, mountain ranges, root systems, 
turbulences in fluids. For example, one might ask the length of a head of cauliflower or a 
coastline. At one level, the answer might be eight inches or 580 miles.  However, at the 
fractal level of iteration of growth patterns and/or ocean forces, both can be seen as infinite. 

In most, perhaps all of nature, we encounter a kind of deterministic chaos in a world 
described by ‘fractal geometries’ which have  “…become a way of measuring qualities that 
otherwise have no clear definition: the degree of roughness or brokenness or irregularity in an 
object.”21  This is the heterogeneous and nonlinear world of the branching of buds in the 

                                        
19 Heinz-Otto Peitgen, Harmut Juergens, Dietmar Saupe: Chaos and Fractals, NewFrontiers of Science, 
Springer Verlag, New York, 1992, p. 48
20 James Gleick, , Chaos, Making a New Science, Penguin Books, N.Y. 1987,  p. 98
21 Gleick, Ibid. 



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cauliflower head, the spongy tissue of the lungs, the indentation on the beach. “Chaos is more 
like the rule in nature, while order (=predictability) is more like the exception.”22

Fig3. Cubism23 2 variations24

An unpredictable 
consequence of fractal geometry 
coupled with advances in computer 
graphics was that it was now possible 
for machines to produce geometric 
“art” based on very simple formulae 
which “shows surprising kinship to 
Old Masters paintings, Cubist 
paintings or Beaux Arts architecture. 
An obvious reason is that classical 
visual arts, like fractals, involve very 
many scales of length and favor self-
similarity (Mandelbrot, 1981, 1989).

The discovery that self-similarity was an inherent property of nature as mathematically 
conceptualized by Mandelbrot was long written about and expressed  by poets, satirists, 
writers, philosophers and numerous religious traditions. For instance, in Southern India, 
Kolam is an art form used by women to decorate the entrance to homes and courtyards (see 
Fig.4). These art forms go back over 6000 years and consist of self-similar patterns repeated 
in different scales in very sophisticated fashion. 

                                        
22 Heinz-Otto Peitgen, Harmut Juergens, Dietmar Saupe:  p. 48
23 “Cubism is a more modern art movement in which forms are abstracted by using an analytical approach to the 
object and painting the basic geometric solid of the subject. Cubism is a backlash to the impressionist period in 
which there is more of an emphasis of light and color. Cubism itself follows Paul Cezanne’s statement that 
"Everything in nature takes its form from the sphere, the cone, and the cylinder." in which these 3 shapes are 
used to depict the object of the painting. Another way that the cubist expressed their painting was by showing 
different views of an object put together in a way that you can not actually see in real life. The Cubism period 
stated in Paris in 1908, reached its peak in 1914, and continued into the 20's. Major cubists were Pablo Picasso 
and Georges Braque” 
Retrieved from  http://abstractart.20m.com/cubism.htm
24 Retrieved from http://www.fractal-art.com/htmle/men-gallery01.html



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Fig.4 Anklets of Krishna Fig.5. Spirals in a Hindu temple25

Architecture in Hindu temples (see Fig.5) also reveal that the notion of self-similarity was 
used to create visually stunning forms.  The English writer, Jonathan Swift (1773), best 
known for Gulliver’s Travels wrote:

So, Nat’ralists observe, a Flea
Hath Smaller Fleas that on him prey,
And these have smaller Fleas to bit’em,
And so proceed as infinitum 

And it was Mandelbrot (2001) who remarked Swift was merely repeating a saying by the 
German philosopher Leibniz, who in turn was repeating Aristotle!

POLYMATHY
The numerous examples of thinkers given thus far represent a unique sample of individuals 
who made remarkable contributions to the arts, sciences and mathematics, and who also 
happened to be philosophers. These individuals are best characterized as polymaths.  The 
term polymath is in fact quite old and synonymous with the German term “Renaissance-
mensch26.” Although this term occurs abundantly in the literature in the humanities, very few 
(if any) attempts have been made to isolate the qualitative aspects of thinking that adequately 
describe this term.  Most cognitive theorists believe that skills are domain specific and 
typically non-transferable across domains. This implicitly assumes that “skills” are that which 

                                        
25 Retrieved from http://classes.yale.edu/fractals/IMA/FB/ArtFrac/ArtFractals.html
26 The author prefers the use of the German term to avoid the sexist overtones of the English equivalent 
“Renaissance man” 



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one learns as a student within a particular discipline. However such an assumption begs the 
question as to why polymathy occurs in the first place. Although the numerous historic and 
contemporary examples presented are of eminent individuals, it has been found that 
polymathy as a thinking trait occurs frequently in non-eminent samples (such as high school 
students) when presented with the opportunities to engage in trans-disciplinary behavior. In 
particular the use of unsolved classical problems and mathematics literature has been found 
to be particularly effective in fostering trans-disciplinary thinking (see Sriraman, 
2003a,2003b, 2004b, 2004c).  

Thinking traits of polymaths

Root-Bernstein (2003) has been instrumental in rekindling an interest in mainstream 
psychology in a systematic investigation of polymathy. That is the study of individuals, both 
historical and contemporary, and their trans-disciplinary thinking traits which enabled them 
to contribute to a variety of disciplines. His analysis of the works and biographies of 
numerous innovators both historical and contemporary reveals that arts advance the sciences 
and scientists are inspired by the arts.   

Fig.6 The impossible Penrose Tribar27

One recent example provided by Root- Bernstein (2003) is the effect of Escher’s 
drawings on a young Roger Penrose, the mathematical physicist, who visited one of Escher’s 
exhibitions in 1954. Stimulated by the seemingly impossible perspectives conveyed by 
Escher in 2-dimensions, Penrose began creating his own impossible objects such as the 
famous Penrose “impossible” tribar which shows a 3-dimensional triangle that twists both 
forwards and backwards in 2-dimensions. Root- Bernstein writes:

“Roger Penrose showed his tribar to his father L.S. Penrose, a biologist who dabbled 
in art… [who] invented the impossible staircase in which stairs appear to spiral both 
up and down simultaneously…[and] sent Escher a copy…[who] then developed 
artistic possibilities of the impossible staircase in ways that have since become 
famous” (p.274)

Another well known consequence of Escher’s artistic influence on mathematicians is the 
investigation of tiling problems (both periodic and aperiodic) popularized by both Roger 
Penrose and Martin Gardner, which helped cystallographers understand the structure of many 
metal alloys which are aperiodic (Peterson, 1985 as quoted by Root-Berstein, 2003, p.274).

                                        
27 Retrieved from http://mathworld.wolfram.com/PenroseTriangle.html



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Common thinking traits of the polymaths described in this paper in conjunction with the 
thousands of polymaths (historical and contemporary) as analyzed by Root-Bernstein, Dean 
Simonton and many others are: (1) Visual geometric thinking and/or thinking in terms of 
geometric principles, (2) Frequent shifts in perspective, (3) thinking in analogies, (4) 
Nepistemological awareness (that is, an awareness of domain limitations), (5) Interest in 
investigating paradoxes (which often reveal interplay between language, mathematics and 
science), (6) Belief in Occam’s Razor [Simple ideas are preferable to complicated ones], (7) 
Acknowledgment of Serendipity and the role of chance, and (8) the drive to influence the 
Agenda of the times. 

IMPLICATIONS
The tension between the disciplines that came out of the Renaissance, namely natural 
philosophy–art – alchemy (metallurgy/chemistry)- theology during the post Renaissance 
continues today in the modern day antipathy between the ever increasing sub-disciplines 
within arts, science, mathematics and philosophy. Many of the thinking processes of 
polymaths who unified disciplines are commonly invoked by artists, scientists, 
mathematicians and philosophers in their craft albeit the end products are invariably different. 
These disciplines explore our world for new knowledge. Literature is an excellent medium to 
create frequent shifts in perspective. Paradoxes can be easily investigated by exploring 
geometry motivated by Art. After all Art suggests new possibilities and pushes the limits of 
our imagination, whereas science verifies the actual limitations of these possibilities using 
mathematics. Both are driven by the need to understand reality with philosophy (and 
theology) often serving as the underlying framework linking the three.  Models and Theory 
building lie at the intersection of art-science-mathematics. The history of model building in 
science conveys nepistemological awareness of domain limitations. Arts imagine
possibilities, science attempts to generate models to test possibilities, mathematics serves as 
the tool.  The Implications for education today is to move away from the post Renaissance 
snobbery rampant within individual disciplines at the school and university levels. By 
building bridges today between disciplines, the greatest benefactors are the potential 
innovators of tomorrow. Our Symposium is a worthy start towards this endeavor.

ACKNOWLEDGEMENTS
The author would like to thank Astrid Beckmann and Claus Michelsen for inviting me to give 
this plenary talk. In the long list of others to thank for helping me formulate, understand and 
connect the myriad ideas are Harry Adrian, Walter Benesch, Benoit Mandelbrot, Robert 
Wheeler, the inspirational work of researchers on the psychology of creativity (Robert Root-
Bernstein, Mihaly Csikszentmihalyi, Dean Simonton & Robert Sternberg). The author would 
also like to thank Taylor & Francis for allowing me to use small sections of my published 
articles in the journal Theology and Science. Last but not least I express my gratitude to the 
great thinkers of the Renaissance who were instrumental in helping us understand the 
magnificence of reality.
  

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