(used with author's permission)
If an abiding message is to be gleaned from an examination of Leonardo’s scientific and artistic legacy, it is the insatiable curiosity, the persistent questioning that defined his life – from resolving every day problems to exploring grand-scale issues pertaining to the workings of nature. He observed and pondered: the commonplace became wondrous, the wondrous commonplace. “Nature is the best teacher,” he wrote. “Learn from nature, not from each other.” His curiosity encompassed diverse intellectual worlds: technical and nontechnical, scientific and artistic; and indeed it is by the conjoining of these intellectual worlds that he was able to produce works of dazzling quality and diversity.
The typical liberal arts curriculum offered in the American undergraduate education system gives students exposure to different intellectual cultures. But then the need to develop specialized skills in one field or another tends to discourage further forays across the cultural divide. The higher education systems in most other countries call for this specialization to begin even earlier. Beyond formal education, normal maturation or aging itself is sadly accompanied by the monotonic dimming of one’s curiosity. Once the specialization process begins, those individuals who are more technically oriented have greater facility communicating with others who are technical, and those who are artistic or less technical similarly feel comfortable in the company of individuals with skills and interests akin to their own. In short, birds of a feather most clearly flock together, communicating with each other in their own languages. …
Under the rubric “Leonardo’s model” this book presented the mathematical and scientific basis underlying both science and art, and the tenets of the laws of nature, albeit all of it in an essentially qualitative manner. The interdisciplinary approach, it was seen, invited a discussion of the human side of doing science and art. Among the most creative artists and scientists – especially those who launched revolutionary transformations in the prevailing order in their fields – there abound tortured souls, baffling personalities very different from those in everyday life. There are also the historical and social conditions that made certain times ripe for such unusually creative individuals to spring up and leave their marks.
Beethoven on his deathbed reportedly raised his fist in defiance at a thunderstorm raging outside and cried, “I know I am an artist!” In the last hours of his death, Leonardo with an air of forlorn resignation, remarked to an assistant, “Tell me, did anything get done?” For us, these are puzzling and poignant pleas for reassurance by two of the greatest creative individuals in history. In Leonardo’s case there also resonates the tone of frustration at unfulfilled ideas and unfinished work. One cannot deny that the reasoning out of solutions, especially in art, and the creating of mental inventions, especially theories in science and technology, were more important for Leonardo than bringing them to actuality. Thus his chronic problem of darting from one beckoning source of curiosity to the next was real. And as his reputation for failing to complete projects and commissions was well-deserved. Through it all, however, he was already breaking significant new ground. From the threshold of the twenty-first century we can view the dizzying array and quality of his accomplishments, and we can wonder: “Could there have existed a dozen different and supremely gifted individuals all operating with the same name – Leonardo da Vinci? …
Early in the book I had expressed, with some trepidation, the faith that presenting science through art and art through science would lead to an understanding of Leonardo’s mind. Who could be presumptuous enough to claim to fathom the mind of someone as complex as Leonardo? Complex he certainly was, but certain components of his psyche have indeed emerged that are surprisingly simple.
In the surviving paintings and notebooks are strewn a jumble of clues. Leonardo’s life can be viewed as a sort of kaleidoscope – creating art of sublime beauty and awesome power, abruptly darting to thoughts on turbulent flow in white water, the geometric patterns of polyhedra, the anatomical studies of man and beast metamorphosing into each other, human flight, spring-driven vehicles, then returning to create one more miraculous work of art. But amidst it all, there are the scattered unfinished projects, endlessly frustrating the most loyal patron.
Leonardo’s unusual style, dizzying in pace and diversity, defined his true modus operandi. The depiction of turbulent water in the daily notebooks is reflected in the golden curls of his subjects. The gently twisted helix, a product of mathematical musings, is suspiciously evocative of the poses struck by the subjects of his portraits. Regarding painting, he wrote, “It embraces all the forms that are and are not found in nature.” The Platonic solids informed his architectural drawings, were seen as rough sketches in the codices, and are formalized in De divina proportione. There is the golden pyramid, which he appears to have discovered independently of past civilizations, organizing the grouping of the characters in the Virgin of the Rocks (and most likely also in the unfinished lost mural, The Battle of Anghiari). In studying the flight of birds he is at once a naturalist, an aerodynamic engineer, and an artist capturing the motion of wings in serial animated drawings; but his thoughts ultimately speculated on human flight. The helical spiral used earlier to give dynamism to the subjects in his paintings returned as the aerial screw, giving lift to the Leonardo helicopter. Totally preoccupied with manned flight, he examined free-fall and gravitation, issues of fundamental significance for physics. The experiments he performed were magnificently framed, leading to conclusions of constant acceleration, independent of the weight of the falling body. In Leonardo’s entire process there was total symbiosis, total synergy. His personal growth can be seen in all areas of his creativity, but also seen is the evidence of the deliberate and systematic effort, the mental exercises he performed to attain this growth. An “unlettered man,” he left behind lists of daily words to be mastered in the manner of the autodidact striving for self-improvement….
Through the ages both the arts (including the visual arts and literature) and the sciences (the natural sciences and mathematics) have generally evolved gradually, incrementally. But there have also taken place, however infrequently, fundamental changes in the prevailing order – changes of kind rather than degree, akin to a phase transition between vapor and liquid, or liquid and solid. Initiating these “revolutions” have been a finite number of artistic and intellectual revolutionaries – better characterized as transformative geniuses or “magicians.” Their reasoning processes do not appear to be constrained by the familiar topography of logic, a topography that includes the slopes and valleys. Rather, reaching conclusions for them is reminiscent of leaping from mountaintop to mountaintop. But their works represent no less than the periodic syntheses of the accumulated knowledge in a field; they succeed by defining human limits and elevating the human spirit. Leonardo, Michelangelo, Shakespeare, Newton, Beethoven, and no more than three or four others – all magicians, all towering intellects, all consumed by a fire raging within – also appear to share one common quality: they rarely allow any profound penetration into their psyche; their creative processes are forever steeped in mystery. One thing is clear: the more time one devotes to understanding the mind of the magician, the deeper he recedes into mist. This is a curious phenomenon that distinguishes the magician from others, including the ordinary genius….
Leonardo Da Vinci, paragon artist-scientist-engineer, a cynosure of the Italian Renaissance, more so than any other magician manifests that quality. Does it really matter that Leonardo will always remain this mysterious and mythical creature, who as a part-time artist produced the epitome in art? His works number so few that even the expression “part-time artist” overstates the case. But it is certainly not the quantity that counts. Goethe’s throwaway line, “In art, the best is good enough,” encapsulates the circumstance. Leonardo also developed scientific methodology a century before Galileo, and anticipated future technologies centuries before they were realized. What really matters is that he demonstrated a model of supreme efficacy, a lesson for the ages: seeking connections, conjoining different intellectual worlds, is indeed the source of profound understanding and appreciation in any of these worlds or in all of them.
What exactly can account for a creature of such transcendent gifts will most likely never be known. It is too easy to say that he was a happy accident of nature and nurture – the concatenation of his mother’s and father’s genes, the effects of the prevailing social, political, and intellectual winds sweeping Renaissance Florence. In Leonardo’s case, there is so much self-nurture. We simply do not know much about his upbringing. He repeatedly admonished others – artists and scientists alike – to “learn from nature, not from each other.” In a timeless irony, we must first learn from him, then observe and ponder. Avoid taking anything for granted, test it before accepting it. Don’t ever give up the aspiration of personal growth no matter what stage of your life you are in, read incessantly, read critically, even look up words you don’t know with a view toward increasing your vocabulary. Carry a small pad with you and draw sketches (even if you have convinced yourself that you cannot draw). Creating sketches will make you more observant. Observe in the manner of the scientist, savor in the manner of the artist. Record your observations. Experiment, knowing full well that some experiments will fail. But that is how to attain a deeper understanding. It is important to be curious, and important to explore intellectual worlds, but it is essential to seek their connections. The model that worked magnificently for him will never make any of us another Leonardo, a man called by so many scholars “the greatest genius who ever lived.” But it cannot fail to make us each far more creative and more effective practitioners in the intellectual world that we inhabit.
Monday, January 15, 2007
Thursday, December 28, 2006
Searching for Symmetry
We are beggars for beauty in a barbaric world, blithely banishing both pursuers and the pursuit into the netherworld of nihilism. Isn’t the pursuit of beauty a hideous folly? ‘To write poetry after Auschwitz is barbaric.’ Adorno’s brutal response to a loaded question ‘can we write poetry after Auschwitz’ is not a naïve assertion that the craft of poetry would be impossible. Neither the iambic pentameter nor vers libre is under any immediate threat of extinction as evidenced from the proliferation of bad poetry. In all fairness, Adorno was asking a more nuanced question whether it is possible to celebrate beauty and the virtues of humanity in the aftermath of Auschwitz. Adorno’s sentiment is a stark counterpoint to the thematic prosody of a young English romantic who wrote wistfully “Beauty is truth and truth beauty.” I often wonder if John Keats remained a steadfast aesthete even as he tragically coughed his way to death in the throes of consumption.
‘Beauty’ has been bruised and banished, except perhaps in trivial conversations. Embarrassed by aesthetics and spurred by science, philosophers tried to place their discipline on a firm footing by seeking a theory of knowledge rather than a theory of life. Then came the existentialists and promised us that their wares were the wherewithal for life. Much ado about life and a scant sense for symmetry results in a system that skirts the silhouette of ‘beauty.’ (Camus’ most important philosophical problem was – well, suicide. Suicide, except in the rare circumstances of ritual death in certain tribes, is generally considered hideous).
After Camus and absurdists, came argot-loving theorists who wedded the concept of beauty with the concept of power. At least, beauty in the universal sense of the term is fraught with political connotations. The notion of a transcendent beauty is nothing more than a metaphysical fiction or at best the values of an elite imposed upon the gullible masses. Or so the story goes. Nonetheless, despite the ravages of war, despite human caprice and despite the abysmal impulse to war-mongering barbarism, the idea of universal beauty continues to be resilient. In recent years, it has found its most surprising allies among the ranks of equation toting physicists and mathematicians. Remember Keats’ lament:
Do not all charms fly
At the mere touch of cold philosophy?
There was an awful rainbow once in a heaven:
We know her woof, her texture; she is given
In the dull catalogue of common things.
Philosophy will clip an Angel’s wings,
Conquer all mysteries by rule and line,
Empty the haunted air and gnomed mine-
Unweave a rainbow, as it erstwhile made
The tender-person’d Lamia melt into a shade
‘Cold philosophy’ (a reference to natural philosophy that included the sciences) was anathema to aesthetics. Keats’ then sworn enemies are now his bosom friends as Newton’s descendants in theoretical physics are voyagers on a restless odyssey to
stumble upon the masterful equation that will once and for all unify the four forces – strong, weak, gravity and electro-magnetism.
Even long before Richard Dawkins’ belated Unweaving the Rainbow, physicists and mathematicians have understood that beauty is hard-wired into the universe. The Laws of Physics are after all said to be symmetrical. For a brief period, scientists and philosophers presumed that the birth of quantum mechanics jolted this picture with its undulating uncertainties and bizarre phenomena that Albert Einstein referred to as ‘spooky stuff.’ ‘God does not play dice’ repeated Albert to the intellectual step-child that he inadvertently fostered. Tired of Einstein’s recalcitrance, Niels Bohr is said to have responded – ‘Albert, Stop telling God what He can or cannot do.’ Recent efforts to unify relativity with quantum mechanics are inspired by this deep-seated drive to discover symmetry. Symmetry has once again become part of the sweet science of beauty. One version of symmetry is found in the controversial anthropic principle which at its simplest form states that our universe is hospitable to life. Fudge the constants and life as we know it might just be chocolate fudge. The anthropic principle is considered a conversation-stopper because it invokes the specter of an incomplete epistemology. The anthropic principle merely describes the necessity for a certain state of conditions without adequately specify the underlying causes. Therefore, it is considered a conversation-stopper among some physicists. Yet, the response to this conversation-stopper is usually an even more egregious conversation-stopper – the multiverse hypothesis- that our universe happens to be a life-supporting system only because it is one among an infinity of other universes. In this case, Physics has truly turned from a modern to a post-modern cosmology. Regardless of whether we live in a universe or a multi-verse, the issue of symmetry continues to swing.
At a more fundamental level, Occam’s razor with its emphasis on explanatory elegance has driven the equations of theoretical physics. While Occam has been unjustly caricatured as the bogeyman of reductionism, the principle that this old Friar advocated merely states ‘do not multiply the entities.’ Explanatory parsimony is another instance of beauty taking its rightful place in facilitating the process of understanding. The quest for beauty in the physical world is channeled through the beauty of the highest form of abstraction – mathematics. As an aside, I recall praying fervently for the demise of an old schoolmaster who derived sadistic pleasure in spanking schoolboys who did not comprehend the derivative. It took me a few years to realize the beauty of calculus and mathematics as an intellectual enterprise. In Bertrand Russell’s words,
“The study of mathematics, rightly viewed, possesses not only truth, but supreme beauty cold and austere – like that of sculpture without appeal to any part of our weaker nature, without the gorgeous trappings of painting or music yet sublimely pure and capable of a stern perfection, such as only the greatest can show. The true spirit of delight, the sense of being more than man, which is the touchstone of the highest excellence, is to be found in mathematics as surely as in poetry.”
‘The sense of being more than Man?’ Bertrand Russell was no mystic, he was a hard-nosed analytic philosopher with zero-tolerance for metaphysics. Yet, he spoke of mathematics in the language of transcendence. In this noumenal realm, mathematics possesses its own internal laws and beauty. In the phenomenal realm, theoretical physics pursues beauty in the cosmos. The noumenal and the phenomenal is bridged by what Eugene Wigner described as “the unreasonable effectiveness of mathematics.”
In Wigner’s words, “The enormous usefulness of mathematics in the natural sciences is something that is bordering on the mysterious and there is no rational explanation for it.” And Wigner continues, “the uncanny usefulness of mathematical concepts raised the question of the uniqueness of our physical theories.” The idea of formal beauty in mathematics is by itself sufficient to fuel entire mathematical frameworks without any reference whatsoever to the physical world. However, when some of these abstruse concepts find straight-forward application to the physical world – it does raise uncanny questions about the nature of the physical world. Does the universe possess a fundamental symmetry?
As Einstein himself remarked, “the eternal mystery of the universe is its comprehensibility.” It took me so long to make my central argument that the arts and the physical sciences are intricately interwoven with each other. I am not arguing for the consilience model of Wilson where the arts is reduced to biology. Instead, I am arguing that the arts and the sciences are connected to each other insofar as aesthetics is concerned.
In this brief essay, I would like to briefly pursue the notion of unification in the context of modern cosmology and aesthetics. Erwin Schrodinger in his classic essay ‘What is Life?’ makes two astute observations.
1) Despite the perplexing complexities in the universe, there is a staggering sense of law-like regularities
2) Second, he reminds us that the laws of nature are not ‘natural.’ Insofar as their presence is concerned, leave alone the fact that they are comprehensible to the human understanding.
Top this off, with Wigner’s metaphysical tribute:
“The miracle of the appropriateness of the language of mathematics for the formulation of the laws of physics is a wonderful gift which we neither understand nor deserve. We should be grateful for it and hope that it will remain valid in future research and that it will extend, for better or for worse, to our pleasure even though perhaps also to our bafflement to wide branches of learning.”
It is not so much the discovery of calculus, Reimannian Geometry and other advanced concepts that makes mathematics interesting. It is the fact that mathematics is beautiful. And even more amazing is the fact that this beauty can be seen in the physical world. There is a sense that the universe we inhabit in is at some deep level profoundly rational and meaningful to the point of being mystical.
Plato & Aristotle were quick to associate symmetry with beauty. In Aristotle’s words, “the chief forms of beauty are orderly arrangement (taxis), proportion (symmetry) and definition (horimenon).” The Roman architect Vitruvius in his Ten Books on Architecture was quick to associate symmetry with proportion. Bach’s Musical Offerings and his organ Fugues exploit the principle of symmetry to the nth degree. Escher’s paintings and a whole list of art works make rich use of symmetry
In cosmology, a striking example of symmetry is found in both the Special and General theories of relativity. Einstein affirmed the notion that the speed of light is precisely the same for all observers. Even while refining and modifying Newton’s ideas on space, time and gravity, Einstein reinforced the received view that the laws of physics are symmetrical. In his special theory of relativity, Einstein treated time as the fourth dimension of space. Einstein’s formulation was stated more precisely by Minkowski who used the idea of symmetry to put relativity on a firm mathematical basis. Minkowski (Einstein’s math teacher referred to him as a ‘lazy dog’) showed that both space and time can be rotated as a four dimensional entity. Furthermore, Einstein believed that the laws themselves can actually be deduced from the requirements of symmetry.
Einstein’s modus operandi signified a significant inversion of the scientific method. As Mario Livio states in his book Symmetry
“Instead of starting with a huge collection of experimental and observational facts about nature, formulating a theory and then checking whether the theory obeys some symmetry principles, Einstein realized that the symmetry requirements come first and dictate the laws nature has to obey.”
Similarly, the bizarre world of Quantum Physics has its share of symmetries as well. Unlike the deterministic world of Newton-Laplace, the epistemology of Quantum Mechanics is probabilistic and counter-intuitive. Even so, the ability to give a statistical description of how particles behave using the wavefunction attests to a fundamental underlying symmetry.
Nonetheless, the regimes of relativity and quantum mechanics do not mesh well. The behavior of gravity at the quantum level is a fine example to illustrate this point. The appeal of an overarching symmetry drives both string theorists and quantum loop gravity theorists among a plethora of other approaches --- all with the grand desire to unify the four fundamental forces into a theory of everything.
Listing any more examples of symmetry might sound patronizing to the informed reader who is well aware of this fact. If this is indeed the case, it puzzles me why academics are less inclined to treat the arts and the sciences as variations on the same theme.
If there is a fundamental symmetry underlying the universe, cosmology becomes a branch of aesthetics or aesthetics becomes a branch of cosmology. Once again, the liberal arts trivium (logic, rhetoric and grammar) and quadrivium (arithmetic, geometry, music and astronomy) could be revived and revivified for a twenty-first century contexts. Let us not forget the biologists who see instances of symmetry galore in nature.
Here is my concluding unscientific postscript. No, it is just a question –
‘why aren’t scientists and artists talking more to each other?’ If we need another muse, let me propose – Leonardo Da Vinci.
By the way, I dedicate this informal banter to my old schoolmaster (that dreadful Calculus teacher) – I honestly don’t know if the old bloke is dead or alive. No hard feelings Mate.
Roy Joseph
‘Beauty’ has been bruised and banished, except perhaps in trivial conversations. Embarrassed by aesthetics and spurred by science, philosophers tried to place their discipline on a firm footing by seeking a theory of knowledge rather than a theory of life. Then came the existentialists and promised us that their wares were the wherewithal for life. Much ado about life and a scant sense for symmetry results in a system that skirts the silhouette of ‘beauty.’ (Camus’ most important philosophical problem was – well, suicide. Suicide, except in the rare circumstances of ritual death in certain tribes, is generally considered hideous).
After Camus and absurdists, came argot-loving theorists who wedded the concept of beauty with the concept of power. At least, beauty in the universal sense of the term is fraught with political connotations. The notion of a transcendent beauty is nothing more than a metaphysical fiction or at best the values of an elite imposed upon the gullible masses. Or so the story goes. Nonetheless, despite the ravages of war, despite human caprice and despite the abysmal impulse to war-mongering barbarism, the idea of universal beauty continues to be resilient. In recent years, it has found its most surprising allies among the ranks of equation toting physicists and mathematicians. Remember Keats’ lament:
Do not all charms fly
At the mere touch of cold philosophy?
There was an awful rainbow once in a heaven:
We know her woof, her texture; she is given
In the dull catalogue of common things.
Philosophy will clip an Angel’s wings,
Conquer all mysteries by rule and line,
Empty the haunted air and gnomed mine-
Unweave a rainbow, as it erstwhile made
The tender-person’d Lamia melt into a shade
‘Cold philosophy’ (a reference to natural philosophy that included the sciences) was anathema to aesthetics. Keats’ then sworn enemies are now his bosom friends as Newton’s descendants in theoretical physics are voyagers on a restless odyssey to
stumble upon the masterful equation that will once and for all unify the four forces – strong, weak, gravity and electro-magnetism.
Even long before Richard Dawkins’ belated Unweaving the Rainbow, physicists and mathematicians have understood that beauty is hard-wired into the universe. The Laws of Physics are after all said to be symmetrical. For a brief period, scientists and philosophers presumed that the birth of quantum mechanics jolted this picture with its undulating uncertainties and bizarre phenomena that Albert Einstein referred to as ‘spooky stuff.’ ‘God does not play dice’ repeated Albert to the intellectual step-child that he inadvertently fostered. Tired of Einstein’s recalcitrance, Niels Bohr is said to have responded – ‘Albert, Stop telling God what He can or cannot do.’ Recent efforts to unify relativity with quantum mechanics are inspired by this deep-seated drive to discover symmetry. Symmetry has once again become part of the sweet science of beauty. One version of symmetry is found in the controversial anthropic principle which at its simplest form states that our universe is hospitable to life. Fudge the constants and life as we know it might just be chocolate fudge. The anthropic principle is considered a conversation-stopper because it invokes the specter of an incomplete epistemology. The anthropic principle merely describes the necessity for a certain state of conditions without adequately specify the underlying causes. Therefore, it is considered a conversation-stopper among some physicists. Yet, the response to this conversation-stopper is usually an even more egregious conversation-stopper – the multiverse hypothesis- that our universe happens to be a life-supporting system only because it is one among an infinity of other universes. In this case, Physics has truly turned from a modern to a post-modern cosmology. Regardless of whether we live in a universe or a multi-verse, the issue of symmetry continues to swing.
At a more fundamental level, Occam’s razor with its emphasis on explanatory elegance has driven the equations of theoretical physics. While Occam has been unjustly caricatured as the bogeyman of reductionism, the principle that this old Friar advocated merely states ‘do not multiply the entities.’ Explanatory parsimony is another instance of beauty taking its rightful place in facilitating the process of understanding. The quest for beauty in the physical world is channeled through the beauty of the highest form of abstraction – mathematics. As an aside, I recall praying fervently for the demise of an old schoolmaster who derived sadistic pleasure in spanking schoolboys who did not comprehend the derivative. It took me a few years to realize the beauty of calculus and mathematics as an intellectual enterprise. In Bertrand Russell’s words,
“The study of mathematics, rightly viewed, possesses not only truth, but supreme beauty cold and austere – like that of sculpture without appeal to any part of our weaker nature, without the gorgeous trappings of painting or music yet sublimely pure and capable of a stern perfection, such as only the greatest can show. The true spirit of delight, the sense of being more than man, which is the touchstone of the highest excellence, is to be found in mathematics as surely as in poetry.”
‘The sense of being more than Man?’ Bertrand Russell was no mystic, he was a hard-nosed analytic philosopher with zero-tolerance for metaphysics. Yet, he spoke of mathematics in the language of transcendence. In this noumenal realm, mathematics possesses its own internal laws and beauty. In the phenomenal realm, theoretical physics pursues beauty in the cosmos. The noumenal and the phenomenal is bridged by what Eugene Wigner described as “the unreasonable effectiveness of mathematics.”
In Wigner’s words, “The enormous usefulness of mathematics in the natural sciences is something that is bordering on the mysterious and there is no rational explanation for it.” And Wigner continues, “the uncanny usefulness of mathematical concepts raised the question of the uniqueness of our physical theories.” The idea of formal beauty in mathematics is by itself sufficient to fuel entire mathematical frameworks without any reference whatsoever to the physical world. However, when some of these abstruse concepts find straight-forward application to the physical world – it does raise uncanny questions about the nature of the physical world. Does the universe possess a fundamental symmetry?
As Einstein himself remarked, “the eternal mystery of the universe is its comprehensibility.” It took me so long to make my central argument that the arts and the physical sciences are intricately interwoven with each other. I am not arguing for the consilience model of Wilson where the arts is reduced to biology. Instead, I am arguing that the arts and the sciences are connected to each other insofar as aesthetics is concerned.
In this brief essay, I would like to briefly pursue the notion of unification in the context of modern cosmology and aesthetics. Erwin Schrodinger in his classic essay ‘What is Life?’ makes two astute observations.
1) Despite the perplexing complexities in the universe, there is a staggering sense of law-like regularities
2) Second, he reminds us that the laws of nature are not ‘natural.’ Insofar as their presence is concerned, leave alone the fact that they are comprehensible to the human understanding.
Top this off, with Wigner’s metaphysical tribute:
“The miracle of the appropriateness of the language of mathematics for the formulation of the laws of physics is a wonderful gift which we neither understand nor deserve. We should be grateful for it and hope that it will remain valid in future research and that it will extend, for better or for worse, to our pleasure even though perhaps also to our bafflement to wide branches of learning.”
It is not so much the discovery of calculus, Reimannian Geometry and other advanced concepts that makes mathematics interesting. It is the fact that mathematics is beautiful. And even more amazing is the fact that this beauty can be seen in the physical world. There is a sense that the universe we inhabit in is at some deep level profoundly rational and meaningful to the point of being mystical.
Plato & Aristotle were quick to associate symmetry with beauty. In Aristotle’s words, “the chief forms of beauty are orderly arrangement (taxis), proportion (symmetry) and definition (horimenon).” The Roman architect Vitruvius in his Ten Books on Architecture was quick to associate symmetry with proportion. Bach’s Musical Offerings and his organ Fugues exploit the principle of symmetry to the nth degree. Escher’s paintings and a whole list of art works make rich use of symmetry
In cosmology, a striking example of symmetry is found in both the Special and General theories of relativity. Einstein affirmed the notion that the speed of light is precisely the same for all observers. Even while refining and modifying Newton’s ideas on space, time and gravity, Einstein reinforced the received view that the laws of physics are symmetrical. In his special theory of relativity, Einstein treated time as the fourth dimension of space. Einstein’s formulation was stated more precisely by Minkowski who used the idea of symmetry to put relativity on a firm mathematical basis. Minkowski (Einstein’s math teacher referred to him as a ‘lazy dog’) showed that both space and time can be rotated as a four dimensional entity. Furthermore, Einstein believed that the laws themselves can actually be deduced from the requirements of symmetry.
Einstein’s modus operandi signified a significant inversion of the scientific method. As Mario Livio states in his book Symmetry
“Instead of starting with a huge collection of experimental and observational facts about nature, formulating a theory and then checking whether the theory obeys some symmetry principles, Einstein realized that the symmetry requirements come first and dictate the laws nature has to obey.”
Similarly, the bizarre world of Quantum Physics has its share of symmetries as well. Unlike the deterministic world of Newton-Laplace, the epistemology of Quantum Mechanics is probabilistic and counter-intuitive. Even so, the ability to give a statistical description of how particles behave using the wavefunction attests to a fundamental underlying symmetry.
Nonetheless, the regimes of relativity and quantum mechanics do not mesh well. The behavior of gravity at the quantum level is a fine example to illustrate this point. The appeal of an overarching symmetry drives both string theorists and quantum loop gravity theorists among a plethora of other approaches --- all with the grand desire to unify the four fundamental forces into a theory of everything.
Listing any more examples of symmetry might sound patronizing to the informed reader who is well aware of this fact. If this is indeed the case, it puzzles me why academics are less inclined to treat the arts and the sciences as variations on the same theme.
If there is a fundamental symmetry underlying the universe, cosmology becomes a branch of aesthetics or aesthetics becomes a branch of cosmology. Once again, the liberal arts trivium (logic, rhetoric and grammar) and quadrivium (arithmetic, geometry, music and astronomy) could be revived and revivified for a twenty-first century contexts. Let us not forget the biologists who see instances of symmetry galore in nature.
Here is my concluding unscientific postscript. No, it is just a question –
‘why aren’t scientists and artists talking more to each other?’ If we need another muse, let me propose – Leonardo Da Vinci.
By the way, I dedicate this informal banter to my old schoolmaster (that dreadful Calculus teacher) – I honestly don’t know if the old bloke is dead or alive. No hard feelings Mate.
Roy Joseph
Monday, December 4, 2006
Calling Leonardo
Among Americans in Amboise
‘Not again,’ mumbled my despondent friend as our gaudy double-decker tour bus rolled into the premises of yet another Chateau in the quiet town of Amboise. Awe gave way to architectural ennui once it became evident that our cameras could no longer distinguish the façade and interiors of one Chateau from another. Feeling ‘underwhelmed’ by the opulence of the erstwhile French aristocracy, our band of bourgeois brothers decided that we were officially ‘Chateau-ed out.’ Within a few seconds, boredom gave way to delirious anticipation when I overheard an off-handed remark that Leonardo Da Vinci spent the last three years of his life in Amboise.
Leonardo was living in Rome completely eclipsed by his worthy rivals Michelangelo and Raphael who were themselves producing more than a handful of masterpieces. When Francis I invited Leonardo and offered him patronage at the Cloux palace (now known as Chateau du Clos Lucé ) in the vicinity of the King’s own stately Chateau d’ Amboise, Leonardo must have seriously considered it. While the exact date is unknown, it is believed that Leonardo took up residence at Amboise in 1516. Armed with this trivia, a few of us set out on a pedestrian pilgrimage to find Clos Lucé. We were greeted by an old pointed archway through which Leonardo entered his new home in 1516.
Leonardo first entered through this archway on the back of a mule with the Mona Lisa, Saint Anne and Saint John the Baptist safely tucked into leather saddlebacks. Crossing the Alps all the way to Amboise, this beast of burden must have been only the second most famous mule in Western history. Next to the mule that carried Jesus Christ into Jerusalem. From leather saddlebacks to the Louvre is a few hundred years of iconic history.
Da Vinci has become a household name, unequivocally associated with genius in every part of the globe. Yet, a cold and cruel irony emerges. The man behind the Mona Lisa would have been ignored were he alive today. As a painter, he might have been dismissed as a portrait artist by his panel of peers. As a thinker, he might have been shunned or dismissed by the vast majority of universities (at least American universities). Humanists would deny him tenure for not publishing enough academic articles on topics that resonate with mainstream scholarship. Professional scientists might sneer at him as a charlatan for neither having the credentials nor a well-defined research agenda. In all likelihood, Leonardo would never have been able to teach at a major university since he had very little formal education.
A specialization-only model that has come to characterize much of post-war twentieth century university life (that is being carried over into the twenty first century) would have no room for him, despite the fact that Leonardo’s intellectual legacy single-handedly dwarfs the collective wisdom of academics. In spite of the growing recognition that ‘great ideas’ are not ultimately discipline-specific ideas, the culture of specialization is so deeply entrenched that boundary-spanners are sometimes pejoratively referred to as ‘generalists.’ This leads me to believe that this culture of specialization if taken too far might become counterproductive towards fostering a spirit of discovery.
Most of the greatest thinkers and discoverers including Albert Einstein, Watson and Crick made breakthroughs by seeing rich connections between seemingly disparate concepts. At the zenith of their creativity, they were young upstarts who did not fit comfortably well within ossified institutional settings.
The muse of Leonardo invites us to shed our deep-seated methodological and disciplinary prejudices and seek truth and beauty in their entirety. A little bit of clarification is necessary. In recent years, an icon has been turned into an ideologue by ventriloquists and conspiracy theorists. Even a perfunctory examination of his work gives us actual insight into who he was and what he symbolized. Leonardo was the purest embodiment of the Renaissance man – a person who sought to bridge, in his own words, ‘the science of art and the art of science.’
A brief digression is in order. The term ‘Renaissance’ has been overused and frequently misappropriated. Naturally, this might lead us to believe that the word has lost its potency. I would like to argue the opposite position briefly. The posthumously celebrated cultural critic Walter Benjamin eloquently argued that the age of mechanical (now digital) reproduction is rapidly devaluing art. And doesn’t it? For years, imitations of the Mona Lisa in restrooms, television commercials, coffee shops and a variety of other settings have seemingly rendered a beautiful masterpiece into a banality. I believed every word of Benjamin’s argument until the day I found myself standing face-to-face with the original in the Italian section of the Louvre. All those imitations and reproductions became irrelevant when I realized that the imprimatur of an impostor does not impinge upon the imprimatur of an artist. If this is indeed the case, misuse does not warrant disuse.
In a similar vein, if it possible to rejuvenate a new Renaissance by recapturing the original intent, Leonardo’s haunting legacy will not be lost on us. Bulent Atalay in his thought-provoking book Math and the Mona Lisa states “five hundred and fifty years after Leonardo’s birth we use Leonardo’s model to seek again the consilience of science and art…and to remedy as far as possible the dissociation that exists between cultures.” While Leonardo was by no means a systematic thinker like Aristotle (who provided us with invaluable means of classifying different branches of learning), he had a unified conception of knowledge. His way of seeing art in science and science in art is manifest in his contributions to diverse areas such as painting, architecture, sculpture, mechanics, hydraulics, ballistics, naval and aeronautical design, mathematics and so forth.
While some progress is being made with the advent of think-tanks and research institutions that inspire cross-pollination of ideas; our society as a whole might once again need to take the journey of Leonardo.
The gift of the imagination is our surest vehicle to get us there. As Albert Einstein himself famously remarked, “imagination is more important than knowledge.”
Groundbreaking discoverers rely more on foresight and (counter)intuitive leaps than methodical thinking. The moral of this lesson is still lost on us as most university-trained intellectuals choose safety over a perilous life of discovery.
Those who risk it all for a dream pay a costly price. Leonardo’s story in itself is fraught with pain and personal tragedy. It is believed that Leonardo wept on his deathbed because he felt that he had offended God and his fellowmen for not consummating his craft.
May his tears speak to us.
---Roy Joseph
‘Not again,’ mumbled my despondent friend as our gaudy double-decker tour bus rolled into the premises of yet another Chateau in the quiet town of Amboise. Awe gave way to architectural ennui once it became evident that our cameras could no longer distinguish the façade and interiors of one Chateau from another. Feeling ‘underwhelmed’ by the opulence of the erstwhile French aristocracy, our band of bourgeois brothers decided that we were officially ‘Chateau-ed out.’ Within a few seconds, boredom gave way to delirious anticipation when I overheard an off-handed remark that Leonardo Da Vinci spent the last three years of his life in Amboise.
Leonardo was living in Rome completely eclipsed by his worthy rivals Michelangelo and Raphael who were themselves producing more than a handful of masterpieces. When Francis I invited Leonardo and offered him patronage at the Cloux palace (now known as Chateau du Clos Lucé ) in the vicinity of the King’s own stately Chateau d’ Amboise, Leonardo must have seriously considered it. While the exact date is unknown, it is believed that Leonardo took up residence at Amboise in 1516. Armed with this trivia, a few of us set out on a pedestrian pilgrimage to find Clos Lucé. We were greeted by an old pointed archway through which Leonardo entered his new home in 1516.
Leonardo first entered through this archway on the back of a mule with the Mona Lisa, Saint Anne and Saint John the Baptist safely tucked into leather saddlebacks. Crossing the Alps all the way to Amboise, this beast of burden must have been only the second most famous mule in Western history. Next to the mule that carried Jesus Christ into Jerusalem. From leather saddlebacks to the Louvre is a few hundred years of iconic history.
Da Vinci has become a household name, unequivocally associated with genius in every part of the globe. Yet, a cold and cruel irony emerges. The man behind the Mona Lisa would have been ignored were he alive today. As a painter, he might have been dismissed as a portrait artist by his panel of peers. As a thinker, he might have been shunned or dismissed by the vast majority of universities (at least American universities). Humanists would deny him tenure for not publishing enough academic articles on topics that resonate with mainstream scholarship. Professional scientists might sneer at him as a charlatan for neither having the credentials nor a well-defined research agenda. In all likelihood, Leonardo would never have been able to teach at a major university since he had very little formal education.
A specialization-only model that has come to characterize much of post-war twentieth century university life (that is being carried over into the twenty first century) would have no room for him, despite the fact that Leonardo’s intellectual legacy single-handedly dwarfs the collective wisdom of academics. In spite of the growing recognition that ‘great ideas’ are not ultimately discipline-specific ideas, the culture of specialization is so deeply entrenched that boundary-spanners are sometimes pejoratively referred to as ‘generalists.’ This leads me to believe that this culture of specialization if taken too far might become counterproductive towards fostering a spirit of discovery.
Most of the greatest thinkers and discoverers including Albert Einstein, Watson and Crick made breakthroughs by seeing rich connections between seemingly disparate concepts. At the zenith of their creativity, they were young upstarts who did not fit comfortably well within ossified institutional settings.
The muse of Leonardo invites us to shed our deep-seated methodological and disciplinary prejudices and seek truth and beauty in their entirety. A little bit of clarification is necessary. In recent years, an icon has been turned into an ideologue by ventriloquists and conspiracy theorists. Even a perfunctory examination of his work gives us actual insight into who he was and what he symbolized. Leonardo was the purest embodiment of the Renaissance man – a person who sought to bridge, in his own words, ‘the science of art and the art of science.’
A brief digression is in order. The term ‘Renaissance’ has been overused and frequently misappropriated. Naturally, this might lead us to believe that the word has lost its potency. I would like to argue the opposite position briefly. The posthumously celebrated cultural critic Walter Benjamin eloquently argued that the age of mechanical (now digital) reproduction is rapidly devaluing art. And doesn’t it? For years, imitations of the Mona Lisa in restrooms, television commercials, coffee shops and a variety of other settings have seemingly rendered a beautiful masterpiece into a banality. I believed every word of Benjamin’s argument until the day I found myself standing face-to-face with the original in the Italian section of the Louvre. All those imitations and reproductions became irrelevant when I realized that the imprimatur of an impostor does not impinge upon the imprimatur of an artist. If this is indeed the case, misuse does not warrant disuse.
In a similar vein, if it possible to rejuvenate a new Renaissance by recapturing the original intent, Leonardo’s haunting legacy will not be lost on us. Bulent Atalay in his thought-provoking book Math and the Mona Lisa states “five hundred and fifty years after Leonardo’s birth we use Leonardo’s model to seek again the consilience of science and art…and to remedy as far as possible the dissociation that exists between cultures.” While Leonardo was by no means a systematic thinker like Aristotle (who provided us with invaluable means of classifying different branches of learning), he had a unified conception of knowledge. His way of seeing art in science and science in art is manifest in his contributions to diverse areas such as painting, architecture, sculpture, mechanics, hydraulics, ballistics, naval and aeronautical design, mathematics and so forth.
While some progress is being made with the advent of think-tanks and research institutions that inspire cross-pollination of ideas; our society as a whole might once again need to take the journey of Leonardo.
The gift of the imagination is our surest vehicle to get us there. As Albert Einstein himself famously remarked, “imagination is more important than knowledge.”
Groundbreaking discoverers rely more on foresight and (counter)intuitive leaps than methodical thinking. The moral of this lesson is still lost on us as most university-trained intellectuals choose safety over a perilous life of discovery.
Those who risk it all for a dream pay a costly price. Leonardo’s story in itself is fraught with pain and personal tragedy. It is believed that Leonardo wept on his deathbed because he felt that he had offended God and his fellowmen for not consummating his craft.
May his tears speak to us.
---Roy Joseph
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