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"The Heresy of Friar Travolo" can be seen as how new, paradigm-shifting ideas may be treated by unprepared minds. In the guise of breaking one of mathematics' fundamental laws, the story attempts to show how too much caution may lead to stagnant thinking, or the avoidance of thinking itself.
Learn from this cautionary tale, and apply the lessons in your every thought. |
"Euclidean geometry was thought to derive its validity from certain self-evident universal truths; it appeared as the only type of consistent geometry of which the mind could conceive. (Carl Friedrich) Gauss had certain misgivings on the matter..." - A. d'Arbo, 'The Evolution of Scientific Thought from Newton to Einstein' (1927)
Dearest Elizabeth, forgive me. The light is dim, and my hand trembles. The enemies of God have me under guard, but there is a maid who pities me. She brought me these instruments, and I pray she will take my letter to you when I have finished.
You have heard many monstrous things about me. None of them are true. I will tell you the truth, from the beginning, though the story will be long. Please, believe me.
I came to Rhoan at the Bishop's request, as innocently as a dove. We met for the first time in the dining room of the episcopal palace, where over a plate of veal and pears he claimed to have a heretic of the fifth dogma imprisoned.
"Ah, the fifth dogma," I said. "Always the favorite of madmen."
"And Travolo, the friar imprisoned, is madder than most." The Bishop took a tract from the fold of his velvet stole and presented it to me. I drew closer to the candlestick and began to read: An Explication of the Doctrines Arising From the Negation of the Fifth Dogma....
"The negation!" I said aloud.
The Bishop brushed aside my astonishment with a flick of his hand. "I assume that it will be a simple matter for you to expose the heresy."
The Bishop, please understand, has no deep grasp of dogmatics or exegesis. He is proficient in applied geometry and his political skill is formidable, which is why he holds his post to such general acclaim. But he was not equipped to confute the heretic himself. I attempted to explain. "The fifth dogma is the keystone of the geometric dogmas. Many have tried to do without it, but to deny it is a form of folly which has not gripped even the greatest madmen. Even Saint Serafano, in his Ab Initio--"
"I have no interest in the history of the dogma, Inquisitor," the Bishop said. "I simply want to know whether you can demonstrate that Travolo is a heretic."
"If the rest of the tract continues in the manner of its title, I expect it will be trivial."
"Good," the Bishop said. He fixed me with a stern glare. "Because without such a demonstration we have no chance of quelling the dissent of the Sussic princes."
At this point I became disinterested. I am a servant of God, not a politician. All I understood from the Bishop was that the Sussic princes were anxious to rid themselves of the Church's strictures, and had begun to search for an excuse to expel the Bishops and geometers. They were prevented by their dependence on the Church's universities to supply them with bursars, architects, cartographers, lawyers, and other masters of applied geometry. Should they simply expel the Church, they would lose the blessing of the guilds, and fall into penury. So when Travolo began promoting his heresy, they seized upon it as a pretext with which to break with the Church while retaining the loyalty of some portion of their clerics.
There were many more details which the Bishop conveyed to me, which I understood not at the time. If I had, perhaps this would have turned out differently.
Travolo was held in the keep belonging to the Comte of Rhoan, one of the dissident princes. This arrangement was a compromise, a way for the Sussics to protect their heretic without openly defying the Church. I showed the Comte and his men-at-arms the Bishop's emblems until they led me to a room in the tower of the keep. The men-at-arms opened the door with trepidation, as if fearing that the mad, mild monk within might devour them alive, then locked it behind me, instructing me to call out to them once I had finished.
And so I met the famed heretic: a small man, stooped at the neck, with mud-colored eyes and a thin, greasy beard. His chamber was floored with planks of wood and furnished only with a straw mattress and a burlap blanket. Light trickled in through the narrow windows. Without my lamp it would have been very dark, yet I could see that Travolo had fashioned a crude pencil from wood, and had begun to scratch the outlines of an exegesis into the stone walls.
I wore my ecclesial garb and carried the geometer's arc in my hand. He flinched from my shadow and drew up against the far wall. "You are Friar Travolo?" I said.
"Yes, yes," he said in a reedy voice. "Are you the Inquisitor?"
"I am here to examine your heresy."
He came up to me wringing his hands greedily. "Have you read my tract?"
"I have begun to," I said.
"Do you have it with you?"
I showed it to him. He seized the pages with immediate pleasure, as if greeting an old friend. "Would you like to work through it? I have yearned for the chance to speak to a true geometer."
I had been hoping for an immediate recantation, which many of the lesser heretics give the moment they see the shadow of an Inquisitor. Travolo was clearly not of this type. "Have you found no true geometers before now?" I said.
"Fah! These Bishops and princes? They couldn't exegete their nose out of their ass."
"Speak with respect, Brother Travolo! You are the one who is on trial for denying the fifth dogma."
"And I have succeeded in building a whole new geometry atop its negation."
I sighed, as I had heard similar claims from those who were both mad and vain. Nonetheless, I began my attack with reason. "Let us examine your exegesis."
Elizabeth, I pray you remember the fifth dogma as we have received it:
Given a line and a point not on that line, there is exactly one line which can be drawn through the point which does not meet the original line, which we name a parallel line.
You have heard many monstrous things about me. None of them are true. I will tell you the truth, from the beginning, though the story will be long. Please, believe me.
I came to Rhoan at the Bishop's request, as innocently as a dove. We met for the first time in the dining room of the episcopal palace, where over a plate of veal and pears he claimed to have a heretic of the fifth dogma imprisoned.
"Ah, the fifth dogma," I said. "Always the favorite of madmen."
"And Travolo, the friar imprisoned, is madder than most." The Bishop took a tract from the fold of his velvet stole and presented it to me. I drew closer to the candlestick and began to read: An Explication of the Doctrines Arising From the Negation of the Fifth Dogma....
"The negation!" I said aloud.
The Bishop brushed aside my astonishment with a flick of his hand. "I assume that it will be a simple matter for you to expose the heresy."
The Bishop, please understand, has no deep grasp of dogmatics or exegesis. He is proficient in applied geometry and his political skill is formidable, which is why he holds his post to such general acclaim. But he was not equipped to confute the heretic himself. I attempted to explain. "The fifth dogma is the keystone of the geometric dogmas. Many have tried to do without it, but to deny it is a form of folly which has not gripped even the greatest madmen. Even Saint Serafano, in his Ab Initio--"
"I have no interest in the history of the dogma, Inquisitor," the Bishop said. "I simply want to know whether you can demonstrate that Travolo is a heretic."
"If the rest of the tract continues in the manner of its title, I expect it will be trivial."
"Good," the Bishop said. He fixed me with a stern glare. "Because without such a demonstration we have no chance of quelling the dissent of the Sussic princes."
At this point I became disinterested. I am a servant of God, not a politician. All I understood from the Bishop was that the Sussic princes were anxious to rid themselves of the Church's strictures, and had begun to search for an excuse to expel the Bishops and geometers. They were prevented by their dependence on the Church's universities to supply them with bursars, architects, cartographers, lawyers, and other masters of applied geometry. Should they simply expel the Church, they would lose the blessing of the guilds, and fall into penury. So when Travolo began promoting his heresy, they seized upon it as a pretext with which to break with the Church while retaining the loyalty of some portion of their clerics.
There were many more details which the Bishop conveyed to me, which I understood not at the time. If I had, perhaps this would have turned out differently.
Travolo was held in the keep belonging to the Comte of Rhoan, one of the dissident princes. This arrangement was a compromise, a way for the Sussics to protect their heretic without openly defying the Church. I showed the Comte and his men-at-arms the Bishop's emblems until they led me to a room in the tower of the keep. The men-at-arms opened the door with trepidation, as if fearing that the mad, mild monk within might devour them alive, then locked it behind me, instructing me to call out to them once I had finished.
And so I met the famed heretic: a small man, stooped at the neck, with mud-colored eyes and a thin, greasy beard. His chamber was floored with planks of wood and furnished only with a straw mattress and a burlap blanket. Light trickled in through the narrow windows. Without my lamp it would have been very dark, yet I could see that Travolo had fashioned a crude pencil from wood, and had begun to scratch the outlines of an exegesis into the stone walls.
I wore my ecclesial garb and carried the geometer's arc in my hand. He flinched from my shadow and drew up against the far wall. "You are Friar Travolo?" I said.
"Yes, yes," he said in a reedy voice. "Are you the Inquisitor?"
"I am here to examine your heresy."
He came up to me wringing his hands greedily. "Have you read my tract?"
"I have begun to," I said.
"Do you have it with you?"
I showed it to him. He seized the pages with immediate pleasure, as if greeting an old friend. "Would you like to work through it? I have yearned for the chance to speak to a true geometer."
I had been hoping for an immediate recantation, which many of the lesser heretics give the moment they see the shadow of an Inquisitor. Travolo was clearly not of this type. "Have you found no true geometers before now?" I said.
"Fah! These Bishops and princes? They couldn't exegete their nose out of their ass."
"Speak with respect, Brother Travolo! You are the one who is on trial for denying the fifth dogma."
"And I have succeeded in building a whole new geometry atop its negation."
I sighed, as I had heard similar claims from those who were both mad and vain. Nonetheless, I began my attack with reason. "Let us examine your exegesis."
Elizabeth, I pray you remember the fifth dogma as we have received it:
Given a line and a point not on that line, there is exactly one line which can be drawn through the point which does not meet the original line, which we name a parallel line.
Many have found the fifth dogma to be complicated and obscure given the divine clarity of the first four, and so have attempted to proceed without it or exegete it from the first four. Thus have arisen many heresies. But Travolo's heresy was of a different sort yet, as his tract began by asserting a different dogma.
Given a line and a point not on that line, there are at least two other parallel lines which can be drawn through the point which do not meet the original line.
It is hard to express the outrage that I, a trained exegete and Inquisitor, felt upon encountering this. I confronted Travolo with the madness of this dogma, and he merely smiled at me. "Yes," he said, "now let us exegete the consequences."
For the rest of the day we engaged in exegesis, as he drew out doctrines from his insane dogma, his techniques growing gradually in complexity. At first I merely observed his heresies with an Inquisitor's eye, for his doctrines were in contradiction to those received, and were often absurd on the face of them. Yet I noted that his method was sound. He appeared to be working towards some conclusion whose shape was not immediately apparent to me, and I admit that I left his cell that evening in a state of mild curiosity.
In such a mind I supped with the Comte. His estate was spacious and well appointed, and he showed me the boundaries of his grounds before dinner, making sure I noted the livery stables, the barracks of his personal guard, and the county armory, which was contained within the estate. I was not at first especially impressed by this, but that evening, as we dined on grouse and raspberry sauce, the purpose of these demonstrations became clear.
"The nobles of this region are quite interested in the outcome of our investigations, Inquisitor," said the Comte at a lull in the conversation. He placed his wine glass on the table with deliberate care and folded his hands over his plate.
"I have heard," I said. "It's unusual for the civil authorities to take such interest in a simple monk. I understand he has incited no unrest or revolt."
"No, no," the Comte said. "Though some of us have an interest in dogmatics. Is it so unusual for a prince to study the doctrines of the church?"
"Not at all, if he does so under the guidance of a proper teacher. Untaught, few would ever cross the pons asinorum."
"Few of the asses. Yet Saint Evoclius formulated his doctrine untaught...."
"He is a saint."
"So anyone who discovers a novel doctrine is a saint? Perhaps I should bring Travolo down from the tower, so I should have the privilege of dining with a saint."
"Perhaps you should wait until I've given my verdict. Lest you fall into heresy."
"Naturally." He gave me a chilly smile, and fixed me with his stare. "Still, it would be a crime if I were to hinder the progress of a saint and give aid to his persecutors."
I very carefully pushed my plate away and looked at the Comte. "It would be worse to protect a heretic and hinder the truth of the Church."
"Yet should a saint appear, should I not protect him with every resource at my disposal?"
"If a saint should appear--"
"And where error appears, should I not raise every sword in my domain to rout it?"
"Undoubtedly."
"Well." He raised his wine glass to me as in a toast. "You have seen the armory of the county. Should the persecutors of truth appear, rest assured that my arms will not lie idle."
A chill passed over me. I paused to take a bite of grouse-flesh, and answered carefully. "The Church herself, of course, cannot be an instrument of error."
"Every flesh is fallible, as the Church herself teaches. Only geometry approaches God...."
"You quote Saint Tomasino to me?"
The Comte smiled at me. "And if the Church herself hinders the truth?"
"You draw near to heresy, dear Comte. Truth is the Church's foundation. She cannot fall into error any more than Evoclius's dogmas could be found false--"
"--and so we return to the matter of Travolo."
"You've read his tract?"
"As I said, I have an interest in dogmatics. But I leave it to you to find whether there is error in it." His eyes on me were sly and cold.
My food sat poorly in my stomach after that. I did not sleep well, and not merely because of the Comte's threats. Travolo's exegesis rumbled through my sleep, and in the morning I returned to the prisoner driven as much by curiosity as by my duties as Inquisitor.
When I appeared at the door of his cell, Travolo raised his head and rubbed his hands together happily. "Brother Inquisitor!" he said, extending his hands towards me as if greeting an old friend. "You have returned."
"Do not be eager," I said, disguising my own eagerness. "I have not completed my examination of your exegesis. We have much to do."
"We do!" Travolo cackled. He picked up the charcoaled stick with which we had worked yesterday. "Shall we begin?"
"First, eat," I said. I had hidden two buttered loaves of bread in my habit, out of pity for Travolo's miserable state.
He looked at the food with a mixture of desire and revulsion, glancing from the pencil to the bread, as if the passion of geometry might leave him if he stopped to dine. At last he took the loaves and stuffed them greedily into his mouth, letting crumbs and drops of butter fall to the ground, all the while turning his eyes to the dimly-lit exegesis drawn on the far wall. I pitied him; yet his devotion to geometry had a touch of the divine about it, and filled me with an uncanny fear.
Given a line and a point not on that line, there are at least two other parallel lines which can be drawn through the point which do not meet the original line.
It is hard to express the outrage that I, a trained exegete and Inquisitor, felt upon encountering this. I confronted Travolo with the madness of this dogma, and he merely smiled at me. "Yes," he said, "now let us exegete the consequences."
For the rest of the day we engaged in exegesis, as he drew out doctrines from his insane dogma, his techniques growing gradually in complexity. At first I merely observed his heresies with an Inquisitor's eye, for his doctrines were in contradiction to those received, and were often absurd on the face of them. Yet I noted that his method was sound. He appeared to be working towards some conclusion whose shape was not immediately apparent to me, and I admit that I left his cell that evening in a state of mild curiosity.
In such a mind I supped with the Comte. His estate was spacious and well appointed, and he showed me the boundaries of his grounds before dinner, making sure I noted the livery stables, the barracks of his personal guard, and the county armory, which was contained within the estate. I was not at first especially impressed by this, but that evening, as we dined on grouse and raspberry sauce, the purpose of these demonstrations became clear.
"The nobles of this region are quite interested in the outcome of our investigations, Inquisitor," said the Comte at a lull in the conversation. He placed his wine glass on the table with deliberate care and folded his hands over his plate.
"I have heard," I said. "It's unusual for the civil authorities to take such interest in a simple monk. I understand he has incited no unrest or revolt."
"No, no," the Comte said. "Though some of us have an interest in dogmatics. Is it so unusual for a prince to study the doctrines of the church?"
"Not at all, if he does so under the guidance of a proper teacher. Untaught, few would ever cross the pons asinorum."
"Few of the asses. Yet Saint Evoclius formulated his doctrine untaught...."
"He is a saint."
"So anyone who discovers a novel doctrine is a saint? Perhaps I should bring Travolo down from the tower, so I should have the privilege of dining with a saint."
"Perhaps you should wait until I've given my verdict. Lest you fall into heresy."
"Naturally." He gave me a chilly smile, and fixed me with his stare. "Still, it would be a crime if I were to hinder the progress of a saint and give aid to his persecutors."
I very carefully pushed my plate away and looked at the Comte. "It would be worse to protect a heretic and hinder the truth of the Church."
"Yet should a saint appear, should I not protect him with every resource at my disposal?"
"If a saint should appear--"
"And where error appears, should I not raise every sword in my domain to rout it?"
"Undoubtedly."
"Well." He raised his wine glass to me as in a toast. "You have seen the armory of the county. Should the persecutors of truth appear, rest assured that my arms will not lie idle."
A chill passed over me. I paused to take a bite of grouse-flesh, and answered carefully. "The Church herself, of course, cannot be an instrument of error."
"Every flesh is fallible, as the Church herself teaches. Only geometry approaches God...."
"You quote Saint Tomasino to me?"
The Comte smiled at me. "And if the Church herself hinders the truth?"
"You draw near to heresy, dear Comte. Truth is the Church's foundation. She cannot fall into error any more than Evoclius's dogmas could be found false--"
"--and so we return to the matter of Travolo."
"You've read his tract?"
"As I said, I have an interest in dogmatics. But I leave it to you to find whether there is error in it." His eyes on me were sly and cold.
My food sat poorly in my stomach after that. I did not sleep well, and not merely because of the Comte's threats. Travolo's exegesis rumbled through my sleep, and in the morning I returned to the prisoner driven as much by curiosity as by my duties as Inquisitor.
When I appeared at the door of his cell, Travolo raised his head and rubbed his hands together happily. "Brother Inquisitor!" he said, extending his hands towards me as if greeting an old friend. "You have returned."
"Do not be eager," I said, disguising my own eagerness. "I have not completed my examination of your exegesis. We have much to do."
"We do!" Travolo cackled. He picked up the charcoaled stick with which we had worked yesterday. "Shall we begin?"
"First, eat," I said. I had hidden two buttered loaves of bread in my habit, out of pity for Travolo's miserable state.
He looked at the food with a mixture of desire and revulsion, glancing from the pencil to the bread, as if the passion of geometry might leave him if he stopped to dine. At last he took the loaves and stuffed them greedily into his mouth, letting crumbs and drops of butter fall to the ground, all the while turning his eyes to the dimly-lit exegesis drawn on the far wall. I pitied him; yet his devotion to geometry had a touch of the divine about it, and filled me with an uncanny fear.
Ponder this
Non-Euclidean geometry can be unsettling, but is essential in understanding the universe. With Einstein's theory of relativity space is no longer a plane but bendable and skewable. What are the applications of non-Euclidean geometry in this respect?
What is the rationale and motivation of each of the characters? The Inquisitor, the Bishop, and Friar Travolo? Are there historical symmetries between the characters and real-world policymakers?
Discuss
Certain views are seen as inviolable dogmas, though this comes from the lack of understanding of the scientific method (hypothetical formulation, testing, and theoretical formulation). In scientific inquiry, theories are seen as 'not yet been proven wrong' rather than 'inviolable facts'. How would you interpret this story as a cautionary tale?
Further readings
Euclidean geometry, codified by Euclid of Alexandria, the geometry that we are used to in out daily lives.
Non-Euclidean geometry, discovered by Eugenio Beltrami, Felix Klein, and Henri Poincaré, the geometry of non-planar space.
Galileo's trial, the real-life Friar Travolo. He was persecuted for defying religious orthodoxy on heliocentric cosmology.
Relationship between religion and science, for a wider perspective on Galileo's persecution.
