never knew the name the first guy you heard of Pablo Andrada, but I remember the occasion: it was in the presentation of the magazine "Numbers", a publication on local and other trifles mathematical curiosities related to the hard science. "We can conceive the infinity of natural numbers when we understand that whenever we get another one before adding the guy said. Then, we speak of all the natural numbers as a complete and finished thing, hiding under the rug the fact that we never finished conceiving at all.
Some of these frowning because the comment referred to a discussion of the early old 900, when Cantor introduced the idea of \u200b\u200bactual infinity , which conceived precisely to a set without end as a finished thing as long as could be explained in a finite and get all its elements against the idea at the time of a potentially infinite , as a sequence that can only be flown but never completed.
Dr. Pernacci broke the ice with elegance and realized this fact.
"Dear friend, if we say" 1 is a natural number "and that" if to is a natural number is also to +1, we have referred all using only a handful of words "He paused, nodding oozed the other and continued. The resulting set contains an infinity of elements, but we able to define them all by a finite expression, perfectly conceivable. Understanding this has allowed the mathematical burst forth beautifully.
"Yes, yes," said the first-but you can not conceive at all, think one by one in all the natural numbers.
There was another silence, and less critical and more reflective.
"True," said the flamboyant editor of the small group that was jumping in small group with a tray of canapés, we have not been given to humans conceive of infinity, "and sarcastically added one by one.
There was laughter.
"I did not rush me Therefore, "the guy finally revealing his speech object a couple of months ago I found a very interesting subject. It is almost a teenager. Deals pizzas on a scooter. His name is Pablo Andrada.
- And what is the grace of the creature? Rosalie asked Barrantes, a professor of mathematics at secondary school was not understanding much of what was spoken and used the accessibility of the last comment to put her sandwich.
-Pablo Andrada, gentlemen, is able to conceive in a few little minutes all the natural numbers one by one, the infinite.
There was a brief silence, then a whisper in a low and a final barrage of jokes and laughter which provides not remember exactly.
The episode had been forgotten in the absence of the sequence of events that forced me to keep track of the topic.
I do not know if it was random or a simple fact probably gained special significance for the history of that episode, but as three years in an obscure library in the neighborhood of Congress I found a specimen immersed in a tray of offerings whose title captured my attention: If we could count to infinity , a certain Marco Tancredo. It was a small booklet of about two hundred pages of original version in Milano, 1972. This Second Edition was the year '81 and had signs whites in the windows have remained throughout his life. I quickly approached the seller and asked the price.
for twenty-two, those of the bat.
"I want only this. "Twenty-
said.
I looked for a moment with disgust and grabbed any other book in the drawer. "Modern kitchen for easy woman", I think. Quick read the title and gave it a certain hostility.
I walked to the subway with a special thrill, an absolute expectation mixed with some concern that forced me to turn around repeatedly, as if someone was following me.
I spent the days reading, studying and digesting the book Tancredo.
In a broad introduction to the subject of debate, passed an assessment of the epistemological consequences resulting from a perception of the infinite by humans. Transcribe
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In the formal sciences, the pure truth of some statements transmitted to others through the show, this being a finite sequence of statements followed each other by the rules of logical inference. What if humans could carry infinite demonstrations steps? What, if anything could be from an infinity of initial axioms? What strange places would take the eventual cognitive infinite capacity to process, object by object?
then followed with a discussion of Gödel's theorems. The logical formidable Austrian proved that from a set of axioms or initial statements that could be recorded by a finite number of words, they could never prove all the truths of arithmetic, demonstrations by finite steps. Assuming that only men can conceive finite axiom systems for the purposes stated and we can only logical demonstrations of finite steps, this amounted to saying that our knowledge of the truths of arithmetic is doomed to incompleteness. In fact, understanding these limits of science called formal finitary has dominated research on logic and information theory to this day. We now know that the formal sciences are accurate, often at the expense of being incomplete.
But what would happen if a breed of men capable of conceiving an infinite number of axioms, able to process endless demonstrations? Could these men have access to all the arithmetic?
While reading the book, the name of Pablo Andrada came to my mind many times. A man who could count all the natural numbers, conceiving one by one, dominate the essence of the thought processes that would be required for demonstrations without end. Such a man could bring the ship into port knowing formal suspect. Did the
But as Pablo Andrada?
In one chapter, the book explores the physical possibility of a brain capable of counting to infinity.
In a naive approach, one would assume that to count to infinity in a little minutes, at some point would need a number to an infinitesimal time. But this is not so, as well as notes Tancredo.
If it takes a minute to count 1, half a minute to count 2, a quarter of a minute to count 3, eighth in count 4 and so on, counting every natural number in half the time it was used in the previous count, it would take two minutes to count them all.
This is easy to see. The total time would 1 + 1 / 2 + 1 / 4 + 1 / 8 + 1 / 16 + ... , which turns out to be 2, as is already known length. But the most interesting that the delayed time to count each number is never infinitely small. It can be very small, may be shrinking, but absolutely no natural number is counted in an infinitesimal time, the following procedure.
Thus, a brain to infinity, does not necessarily require information processing speed infinite. You only need to process it as fast as you want, but not infinitely fast. Thereafter
Tancredo makes a detailed inspection of the possibilities of an electronic brain to handle infinity, where I really have lost between axons and dendrites. Yes I knew that long after the publication of his book appeared venture theories quantum machinery, electronics no longer, as a physical basis for the functioning of the brain. Machinery that would give the brain a lot more possibilities than could be envisaged on a mere ordinary course of jumping from neuron to neuron.
Whatever the case, you end up convinced that chapter the so Pablo Andrada could exist.
At the end of the book, Tancredo makes historical research tracks in the past presence of individuals capable of taming the infinite. There, italic develops a much more detailed presentation about a past documented for a possible sectarian human breed "infinity."
The story is set in Crotona, Italy, home of the Pythagorean school, where Tancredo says he found documentation that accounts for a branch of the Pythagoreans who had the ability to count to infinity.
Pythagoras was more mystical than mathematical. He believed in the transmigration of the human soul and gave the institute, made up of students from around the world, a mysterious and apparently hidden legacy of Zoroaster, who had visited an eventual trip to Persia. Known is the story of Hippasus of Metaponto, one of its most outstanding students who demonstrate that the root of two is not a rational number. It should be clarified that Pythagoras held that all non-integer, could be written as a ratio of integers, that is, as a rational, so that the test Hippasus darkened considerably, to such an extent that forced the members of your school to keep secret about the finding. But it was the very Hippasus which violated the secret and found death soon after, according to it is believed, at the hands of the Pythagoreans or even by order of Pythagoras himself.
Under these facts, it is foolish to think that a possible branch of the Pythagoreans to exhibit the ability to master the infinite, also saw the shadows of silence, rather than by a love for the occult, for mere survival instinct. Tancredo
account Hilíaco then the history of Syracuse, a taciturn and timid character, germinated in the shade of Pythagoras and maintained it by himself. Apparently Hilíaco not show big math skills, but could count all the numbers, without skipping or delay. When the Pythagorean school falls into the hands of the democratic revolution Hilíaco fled to Syracuse with a handful of disciples and founded a sect so hidden that even know his name. Despite this, the sect had managed to concentrate a large number of individuals capable of infinite thought.
But the fate of hiliacóricos would be marked by the tragedy because of two facts: First, a marked tendency for the "infinite" most staunch the madness and the other a discretion so stringent that have led to the disappearance school with the death of its last members sane.
Latest Tancredo claims are highly speculative and offers little material for analysis. Simply argues that this curious property of a brain capable of the infinite sequence, derived from a mutation that he places without further argument, on chromosome 17. But from there, and making some simple progressions which quantifies some default parameters and other excess, Tancredo claims that there should be infinite even a handful of likely spread through Europe and South America.
course, Tancredo's work was paid by the possibility of a Pablo Andrada, and that height and I wondered whether it would be possible to find him.
Throughout the study period of this work, I was discussing the details with some colleagues from the University. In particular, I have had thick coffee chats with Dr. Alfredo Cuernavaca, a former teacher and a scholar of logic and foundations, who explored the issue in great detail, aspect by aspect. I must say that beyond the endless meanderings of debate, its position on the topic could be summed up clearly: Cuernavaca was an uncompromising critic of the whole idea, which did not hesitate to describe as crazy and delusional.
But the most notable of those days was a strange feeling of anxiety and restlessness that pervaded every time I walked by the street. He felt being watched or persecuted, and did not hesitate to turn around to look back, two or three times per block. Finally, these views had their rational explanation when an unknown individual approached me in the subway platform.
"I know the dilemma that eats," he said without introduction.
- What?
"You're looking for Pablo Andrada, right? -Added ignoring my requisition. There
knew what he was talking about. I hesitated between asking who he was or deny everything. This was a guy in his fifties, with many roasted belly breath and some alcohol. I let the subway and answered.
"Not that you're looking for, but I'd love to know.
"You must know that the boy needs help. It is not easy to find a guy like you, aware of these issues. You understand me right?.
"Look, I'm just a mathematician who has been investigating these issues.
The man gave a smile and shook her head facing the floor.
"I know you can help me. No one else can.
He paused briefly and continued.
-A Paul has not fared well. Did not last in any job. Already has twenty years and no overtones that can succeed only with your life. Now a promoter of insurance. Bah, for fifteen days. Has not yet sold a single policy. And that's forever. May do well talk to you.
I stopped talking for a while. Finally made an appointment for next Saturday in an old bar on Avenida de Mayo.
-Paul will be alone, "he said. You will recognize by his bag with the logo of "maximum security." Do not tell that spoke to me.
"I can not say I talked to you because they are not who you are," I said.
"Oh, sorry. I am Joseph Andrada, Pablo's father said and held out his hand. Since the subway
moving yelled from the platform.
- Handle with care! You know!.
But I did not know anything. Not go knew.
waited anxiously Saturday. I reviewed my notes and revive my doubts. Damn! What no one would ask a person who can count to infinity?
On Saturday I was there. I entered the bar with an expectation indescribable. I think I shook his feet. There was the boy with his briefcase, looking at the window with abandon. He was tall and very thin. Just wrapped meat, bones emerged from the sleeve of his coat very wide and ended in long, cadaverous fingers dangling from her wrists with sheer disdain.
"Good afternoon," I said. Are you Pablo Andrada? Yes
sir. Sit down please, "he said, and quickly began to talk as he opened his briefcase.
I was surprised at ease. I expected a more subdued personality. Had he not sold a single policy? I really like a very brash seller. In fact, before I knew it I was explaining the details of your home and contents insurance.
-Paul - I said, "Do you drink coffee?
"Excuse me, of course, ask for something. Yes, coffee will be fine. Then he slowly probed
.
- Do you know why am I here?
He looked puzzled.
"I suppose you are looking for a good insurance for your home.
Paul did not know.
Slowly I introduced the topic and quickly realize the target that the shots were directed.
He formed a slight smile, looking at the tablecloth, shaking with a little crumb nonexistent.
"My father ... my father," he said. You are here for my curious ability to count the numbers right?.
"Yes." Actually there are many things I do not understand. I can not imagine how it is possible to count to infinity.
"You do not believe is not it?
"Not that do not believe ...
-Note-off. Pablo Andrada
Then he closed his eyes, placed his palms on the table and began to count.
"One, two, three, four, five, six ..." and continued to enjoy "... twelve, thirteen, fourteen ..." and accelerated and went cash-... vtiún, vtidós, vtitrés-and faster-cient ... . vtiún, cient.vtidó, cient.vtitré .... His eyes went back and forth under your closed eyelids. Then diluted in an incomprehensible babble that degenerated into a first-bidibidibidibidi ... "and then a sharp-zibizibizibizibi hum ..." to end abruptly with a "Zzzzzzsssssssssp.
followed a silence.
As white is the sum of all colors, my silence was the sum of all thoughts. Thoughts raced through my consciousness so fast that none could be defined.
Paul opened his eyes slowly. He raised his coffee cup and sip half I spied the face. Why not have seen it quickly because the drink talking.
- What else? "He said.
He barely managed to control my disappointment. Unwittingly decided to ignore my strong suspicions of fraud and proceed with the planned investigation. We talked a while longer the issue. I wrote a problem on sums infinite in a paper napkin. He said he would review at home and continued its sale of insurance.
I left the bar late in excusing myself to take another appointment and I took off from there.
raining. Was fine. The rain would wash the stupidity and punish me accordingly. Only a wise thing to draw from the interview: If a man could really capture the infinite, could not show it to anyone.
next day it happened. Again on the platform of the subway. Someone started screaming at my back.
- Son of a bitch! Son of a bitch!
startled I turned and saw me coming the rough body of Joseph Andrada caught me neck and gave me a punch. Highly exalted
kept yelling something drunk bastard! I told him to treat with care! And look what he did! Putaaaaa Son!
other passengers quickly intervened and security personnel. I pulled the man over. I did a couple of questions and refused to meet him.
I came home with a broken nose and a mind full of questions. What are you accusing me? Paul left it in perfect conditions Saturday at the bar. What was my carelessness and what evidence of it?
The next day I returned to the bar as offenders returning to the scene of the crime.
The waiter cleared the table and I gave several turns over with his rag. Seconds later I asked.
- And? How is your friend? I watched with amazement
- What happened to my friend? "I said.
"That we want to know us. After you left, he was there, sitting in his chair, made a long pause. About three hours we began to worry. The guy did not move you know. I went and played it a little. Nothing. I shook him more vigorously. Nothing, the boy opened his eyes big, and I with him. Stood motionless, his eyes opened by reading a note in the table. A napkin with some beads. As at twelve o'clock the ambulance took him well, as dead with open eyes.
- What did the napkin? I asked in awe.
I do not know. I do not understand anything but here I kept antics, look, he said as he drew the napkin from his pocket.
I extended the slip and I could see my own hand the simple question about infinite sums, "which is the sum infinitely +1-1 +1-1 +1-1 +1 1 ... ..? . "
Then I realized with horror that Paul Andrada itself was a genuine descendant of the hiliacóricos , an infinite shaft, which by the grace most vulgar of my carelessness had been suspended in limbo abstract, without colors or sounds, like hell, forever oscillating between one and two.
