Archive for the ‘General’ Category

The Useless Tree

Wednesday, August 11th, 2010

A carpenter and his apprentice were walking together through a large forest. And when they came across a tall, huge, gnarled, old, beautiful tree, the carpenter asked his apprentice: “Do you know why this tree is so tall, so huge, so gnarled, so old and beautiful?” The apprentice looked at his master said: “No, why?”

“Well,” the carpenter said, “because it is useless. If it had been useful it would have been cut long ago and made into tables and chairs, but because it is useless it could grow so tall and so beautiful that you can sit in its shade and relax.”

– Chuang Tzu

A life without a quiet center easily becomes delusional. When we cling to the results of our actions as our only way of self-identification, we become possessive, defensive, and dependent on false identities. In the solitude of prayer we slowly unmask the illusion of our dependencies and possessiveness, and discover in the center of our own self that we are not what we can control or conquer but what is given to us from above to channel to others. In solitary prayer we become aware that our identity does not depend on what we have accomplished or possess, that our productivity does not define us, and that our worth is not the same as our usefulness.

– Henri Nouwen

The Theory of Almost Everything

Thursday, August 5th, 2010

Perhaps one of my favorite opening paragraphs to a book…

There is a theory in physics that explains, at the deepest level, nearly all of the phenomena that rule our daily lives. It summarizes everything we know about the fundamental structure of matter and energy. It provides a detailed picture of the basic building blocks from which everything is made. It describes the reactions that power the sun and the interactions that cause fluorescent lights to glow. It explains the behavior of light, radio waves, and X rays. It has implications for our understanding of the very first moments of the universe’s existence, and for how matter itself came into being. It surpasses in precision, in universality, in its range of applicability from the very small to the astronomically large, every scientific theory that has ever existed. This theory bears the unassuming name “The Standard Model of Elementary Particles,” or the “Standard Model, for short. I call it “The Theory of Almost Everything.” It is the pinnacle of human intellectual achievement to date.

Dostoevsky

Monday, July 5th, 2010

To the students in your class who think no one would choose against salvation, you could refer them to Nietzche, or to Ivan in the Brothers Karamazov…

The story is told by Ivan to Alyosha, who are brothers (The Brothers Karamazov) of The Grand Inquisitor, during the Inquisition. It is a story that Ivan has written and is relating to his brother where Christ comes back, and the GI recognizes Him, and throws him in jail.

We learn that the GI is a Jesuit – a high up leader of the Roman Catholic Church – who supposedly loves humanity, who has suffered in solitude, following the Way of Christ. But he has come to the conclusion that man is too weak for the freedom that Christ gave him when he denied the 3 temptations of Satan. The GI is furious at Christ for not taking the offer, because, as he argues, it would have united all of humanity in a worldwide, harmonious kingdom on earth, accomplishing and satisfying everything that mankind strives, suffers, and longs for.

“In accepting the kingdom of the world and Caesar’s purple, one
would found a universal kingdom and secure to mankind eternal peace.”

The GI argues that Christ has denied mankind this great kingdom and given them instead a freedom which they do not want, and do not accept. Thus, the GI reveals the true mission of the Roman Catholic Church (according to Ivan, who is telling the story) which is to finish the job for which Christ came, correcting his error in denying Satan.

But it is only then that men will see the beginning of a kingdom of
peace and happiness. Thou art proud of Thine own elect, but Thou
has none other but these elect, and we–we will give rest to all.

The GI argues (in a rant to Christ, who is imprisoned) that most of mankind cannot or will not use their free will to choose the Way of Christ (the only path to true freedom), but rather, the vast majority of mankind will use their free will to choose the desires of the flesh, thus turning their back on freedom and choosing sin and worldly destruction and death. Supposedly, the GI has not lost faith in the Way that Christ has shown, but he does not believe man can rise to the challenge and because he has not (supposedly) lost his love of mankind, he has decided to deny Christ and to deceive mankind, giving them what they “really” want so that they may go joyfully and peacefully to the worldly death and destruction they will inevitably choose, all the while believing that they are destined for everlasting peace in heaven. According to Ivan, mankind is doomed, even under Christ’s plan, since they are too weak to follow His Way. Thus Ivan (the GI’s) argument is that since mankind is largely doomed the job of the Roman Catholic Church is to make them believe otherwise so that they may have at least some temporary joy and peace, albeit illusory. To Christ, the GI says:

Thinkest Thou we shall be right or still lying?
They will convince themselves of our rightness, for they will see
what a depth of degrading slavery and strife that liberty of
Thine has led them into. Liberty, Freedom of Thought and
Conscience, and Science will lead them into such impassable
chasms, place them face to face before such wonders and insoluble
mysteries, that some of them–more rebellious and ferocious than
the rest–will destroy themselves; others–rebellious but weak
–will destroy each other; while the remainder, weak, helpless
and miserable, will crawl back to our feet and cry: “‘Yes; right
were ye, oh Fathers of Jesus; ye alone are in possession of His
mystery, and we return to you, praying that ye save us from
ourselves!”


Who separated the flock and scattered it over ways
unknown if it be not Thee? But we will gather the sheep once more
and subject them to our will for ever. We will prove to them
their own weakness and make them humble again, whilst with Thee
they have learnt but pride, for Thou hast made more of them than
they ever were worth.

We will give them that quiet, humble
happiness, which alone benefits such weak, foolish creatures as
they are, and having once had proved to them their weakness, they
will become timid and obedient, and gather around us as chickens
around their hen. They will wonder at and feel a superstitious
admiration for us, and feel proud to be led by men so powerful
and wise that a handful of them can subject a flock a thousand
millions strong. Gradually men will begin to fear us. They will
nervously dread our slightest anger, their intellects will
weaken, their eyes become as easily accessible to tears as those
of children and women; but we will teach them an easy transition
from grief and tears to laughter, childish joy and mirthful song.
Yes; we will make them work like slaves, but during their
recreation hours they shall have an innocent child-like life,
full of play and merry laughter. We will even permit them sin,
for, weak and helpless, they will feel the more love for us for
permitting them to indulge in it.

We will tell them that every
kind of sin will be remitted to them, so long as it is done with
our permission; that we take all these sins upon ourselves, for
we so love the world, that we are even willing to sacrifice our
souls for its satisfaction. And, appearing before them in the
light of their scapegoats and redeemers, we shall be adored the
more for it. They will have no secrets from us. It will rest with
us to permit them to live with their wives and concubines, or to
forbid them, to have children or remain childless, either way
depending on the degree of their obedience to us; and they will
submit most joyfully to us the most agonizing secrets of their
souls–all, all will they lay down at our feet, and we will
authorize and remit them all in Thy name, and they will believe
us and accept our mediation with rapture, as it will deliver them
from their greatest anxiety and torture–that of having to
decide freely for themselves.

And all will be happy, all except
the one or two hundred thousands of their rulers. For it is but
we, we the keepers of the great Mystery who will be miserable.

There will be thousands of millions of happy infants, and one
hundred thousand martyrs who have taken upon themselves the curse
of knowledge of good and evil. Peaceable will be their end, and
peacefully will they die, in Thy name, to find behind the portals
of the grave–but death. But we will keep the secret inviolate,
and deceive them for their own good with the mirage of life
eternal in Thy kingdom. For, were there really anything like life
beyond the grave, surely it would never fall to the lot of such
as they!

Know then that I fear Thee
not. Know that I too have lived in the dreary wilderness, where I
fed upon locusts and roots, that I too have blessed freedom with
which thou hast blessed men, and that I too have once prepared to
join the ranks of Thy elect, the proud and the mighty. But I
awoke from my delusion and refused since then to serve insanity.
I returned to join the legion of those who corrected Thy
mistakes. I left the proud and returned to the really humble, and
for their own happiness. What I now tell thee will come to pass,
and our kingdom shall be built, I tell Thee not later than
to-morrow Thou shalt see that obedient flock which at one simple
motion of my hand will rush to add burning coals to Thy stake, on
which I will burn Thee for having dared to come and trouble us in
our work. For, if there ever was one who deserved more than any
of the others our inquisitorial fires–it is Thee! To-morrow I
will burn Thee. Dixi’.”

Alyosha is horrified by his brother’s story…

“A precious piece of information, notwithstanding your ‘not
that.’ I ask you, why should the Inquisitors and the Jesuits of
your imagination live but for the attainment of ‘mean material
pleasures?’ Why should there not be found among them one single
genuine martyr suffering under a great and holy idea and loving
humanity with all his heart? Now let us suppose that among all
these Jesuits thirsting and hungering but after ‘mean material
pleasures’ there may be one, just one like my old Inquisitor, who
had himself fed upon roots in the wilderness, suffered the
tortures of damnation while trying to conquer flesh, in order to
become free and perfect, but who had never ceased to love
humanity, and who one day prophetically beheld the truth; who saw
as plain as he could see that the bulk of humanity could never be
happy under the old system, that it was not for them that the
great Idealist had come and died and dreamt of His Universal
Harmony. Having realized that truth, he returned into the world
and joined–intelligent and practical people. Is this so
impossible?”

“Joined whom? What intelligent and practical people?” exclaimed
Alyosha quite excited. “Why should they be more intelligent than
other men, and what secrets and mysteries can they have? They
have neither. Atheism and infidelity is all the secret they have.
Your Inquisitor does not believe in God, and that is all the
Mystery there is in it!”

“It may be so. You have guessed rightly there. And it is so, and
that is his whole secret; but is this not the acutest sufferings
for such a man as he, who killed all his young life in asceticism
in the desert, and yet could not cure himself of his love towards
his fellowmen? Toward the end of his life he becomes convinced
that it is only by following the advice of the great and terrible
spirit that the fate of these millions of weak rebels, these
‘half-finished samples of humanity created in mockery’ can be
made tolerable. And once convinced of it, he sees as clearly
that to achieve that object, one must follow blindly the guidance
of the wise spirit, the fearful spirit of death and destruction,
hence accept a system of lies and deception and lead humanity
consciously this time toward death and destruction, and moreover,
be deceiving them all the while in order to prevent them from
realizing where they are being led, and so force the miserable
blind men to feel happy, at least while here on earth. And note
this: a wholesale deception in the name of Him, in whose ideal
the old man had so passionately, so fervently, believed during
nearly his whole life! Is this no suffering? And were such a
solitary exception found amidst, and at the head of, that army
‘that thirsts for power but for the sake of the mean pleasures of
life,’ think you one such man would not suffice to bring on a
tragedy? Moreover, one single man like my Inquisitor as a
principal leader, would prove sufficient to discover the real
guiding idea of the Romish system with all its armies of Jesuits,
the greatest and chiefest conviction that the solitary type
described in my poem has at no time ever disappeared from among
the chief leaders of that movement. Who knows but that terrible
old man, loving humanity so stubbornly and in such an original
way, exists even in our days in the shape of a whole host of such
solitary exceptions, whose existence is not due to mere chance,
but to a well-defined association born of mutual consent, to a
secret league, organized several centuries back, in order to
guard the Mystery from the indiscreet eyes of the miserable and
weak people, and only in view of their own happiness? And so it
is; it cannot be otherwise. I suspect that even Masons have some
such Mystery underlying the basis of their organization, and that
it is just the reason why the Roman Catholic clergy hate them so,
dreading to find in them rivals, competition, the dismemberment
of the unity of the idea, for the realization of which one flock
and one Shepherd are needed. However, in defending my idea, I
look like an author whose production is unable to stand
criticism. Enough of this.”

“You are, perhaps, a Mason yourself!” exclaimed Alyosha. “You do
not believe in God,” he added, with a note of profound sadness in
his voice. But suddenly remarking that his brother was looking at
him with mockery, “How do you mean then to bring your poem to a
close?” he unexpectedly enquired, casting his eyes downward, “or
does it break off here?”

“My intention is to end it with the following scene: Having
disburdened his heart, the Inquisitor waits for some time to hear
his prisoner speak in His turn. His silence weighs upon him. He
has seen that his captive has been attentively listening to him
all the time, with His eyes fixed penetratingly and softly on the
face of his jailer, and evidently bent upon not replying to him.
The old man longs to hear His voice, to hear Him reply; better
words of bitterness and scorn than His silence. Suddenly He
rises; slowly and silently approaching the Inquisitor, He bends
towards him and softly kisses the bloodless, four-score and-ten-
year-old lips. That is all the answer. The Grand Inquisitor
shudders. There is a convulsive twitch at the corner of his
mouth. He goes to the door, opens it, and addressing Him, ‘Go,’
he says, ‘go, and return no more… do not come again… never,
never!’ and–lets Him out into the dark night. The prisoner
vanishes.”

“And the old man?”

“The kiss burns his heart, but the old man remains firm in his
own ideas and unbelief.”

“And you, together with him? You too!” despairingly exclaimed
Alyosha, while Ivan burst into a still louder fit of laughter.

If the students in your class do not recognize the “Humanism” of Christopher Hitchens in this story put forth by Ivan – which is the supposed love of man that deems them unworthy for the Freedom Christ offers; this love that would deny them the harsh, brutal lie of Religion – then they need to think again, because this is alive and well today, more than at any other time perhaps. There is a strong movement that actively denies Christ, calls the Church a lie, and worships Man and Earth over Heaven.

The idea that everyone will be saved is imcompatible with what Christ teaches and Christopher Hitchens, Ivan, Doestevsky and a vast swath of modern thinking in the world today, all of which openly and categorically and emphatically deny Christ and His teaching.

Who are they kidding but themselves?

There are Ivans all around us. They run companies, gangs, governments and much of the world…


Compassion and Charity

Monday, July 5th, 2010

The highest goal of Buddhism is compassion. The highest goal of Christianity is charity. Both of these are forms of love. But of these, charity is greater.

‘Who can be sure of having ever experienced a true charitable impulse? Who can doubt ever having felt compassion? One must begin with what is easiest, and unfortunately our talent for sadness is much greater than our talent for joy. And so we all need courage. And compassion, for others and for ourselves.

‘Or to put it another way: Christ’s message, which is love, is the more exhilarating, but Buddha’s lesson, which is compassion, is more realistic.

‘Therefore, “love and do what you wish” – or be compassionate and do what you must.’

Economics and Human Dignity

Tuesday, June 22nd, 2010

In response to http://www.zenit.org/article-29616?l=english

It is very important that the tendency of western capitalism to make commodities – of labor AND of consumers – be balanced by this sense of human dignity.

One of the long standing criticisms of capitalist societies has always been that capitalism needs “men who cooperate in large numbers to consume more and more and whose tastes can be standardized, easily influenced and anticipated.” To the extent that we are more influenced by marketing executives than we are by the Truth of who we are, this prediction shall come to pass. The celebrity worship we see creeping and creeping and creeping, deeper and deeper into our society, is evidence that even the self can be turned into a commodity to be bought and sold on the market, no different than any other object.

The outcome for such a man who has been turned into a commodity is that he experiences his life force as in investment which must bring him the maximum profit obtainable under existing market conditions – in my business the bosses and professional career makers and advisors all talk about the idea that you are a brand. Careerism in the entertainment industry is reduced to the pure objectification of the self as an absolute commodity. This is the ultimate goal of every news anchor and commentator on TV, whether they be on the evening news or American Idol.

The alienation that this inevitably leads to causes such a man to seek relief in constant amusement, which serves as a temporary distraction from his alienation from himself. This man becomes like the man described in Brave New World: “well fed, well clad, yet without self.” Happiness is thought to be found in “having fun,” which is also turned into a commodity to be marketed, and bought and sold on the free market. Such a person is truly lost to themselves, with no capacity to love – except perhaps in the false, sentimental, emotional kind of “love” promoted in romance novels and Hollywood stories. And he is unable to love simply because he has become completely lost to himself. Such a man spins around and around in the influencers’ whirlwind, unable to find secure footing because he is not grounded in Truth.

Inasmuch as I consider myself a capitalist for economic and moral reasons (it has unprecedented power to eradicate poverty in the world), I also see that it is NOT a system of morals, and therefore it must be complimented with the knowledge and practice of Love, which is both the awareness of Truth and acknowledgment of who Man is. As you pointed out in your homily, to answer Jesus’ question, “Who do you say I am,” we must have first answered the question, “who do we say we are?” This question must be answered correctly if man is to retain any dignity whatsoever in this world.

Agnosonosia, Scrabble and the Unknown Unknown

Tuesday, June 22nd, 2010
see also: The Four Stages of Competence
  1. Unconscious Incompetence
    The individual neither understands nor knows how to do something, nor recognizes the deficit, nor has a desire to address it.
  2. Conscious Incompetence
    Though the individual does not understand or know how to do something, he or she does recognize the deficit, without yet addressing it.
  3. Conscious Competence
    The individual understands or knows how to do something. However, demonstrating the skill or knowledge requires a great deal of consciousness or concentration.
  4. Unconscious Competence
    The individual has had so much practice with a skill that it becomes “second nature” and can be performed easily (often without concentrating too deeply). He or she may or may not be able teach it to others, depending upon how and when it was learned.

I commented on an interesting article I read today in the NYTimes about agnosonosia, which is a word that means something like the “disease of not knowing.” I was attracted to the article because I have come across that word before, in a different context of the article, which was about a certain medical condition

The author was struggling with a concept that I find to be very important for intellectuals and I was shocked that the author did not understand this concept. It is the concept of the “unknown unknown.”

Donald Rumsfeld famously commented on it years ago, and the media attacked him. I spent many a debate trying to explain why Rummsfeld’s comment showed great wisdom, and those that mocked him for it showed showed great ignorance. I got a lot of grief about that, but I know I am correct.

This author brings the comment up again because he does not get it. So I tried to explain it to him…

http://opinionator.blogs.nytimes.com/2010/06/20/the-anosognosics-dilemma-1/#preview

“The author of this article does not seem to understand the difference between “known unknowns” and “unknown unknowns. He says…

“The fact that we don’t know something, or don’t bother to ask questions in an attempt to understand things better, does that constitute anything more than laziness on our part?”

“The fact that we don’t know something…” is a description of a known unknown, like the word “ctenoid”or the melting point of beryllium, which the author mentioned. We don’t know these things because we “don’t bother” to know them. These are known unknowns.

Unknown unknowns, on the other hand, are things that fall outside of our worldview completely. We don’t “bother” with unknown unknowns because we did not realize that that we COULD bother with them.

The idea that the recreational Scrabble player has a profound ignorance of the game – that for this person, there are many unknown unknowns – is not based on the fact that he does not know the word “ctenoid.” Ironically, the author’s view that the recreational Scrabble player’s ignorance is only an ignorance of vocabulary is a perfect example of the unknown unknown, for what the recreational Scrabble player and the author of the article don’t know that they don’t know is that proficiency in statistical mathematics, more so than simple proficiency in vocabulary, separates the top pros from the rest of the competition. For example, professionals use a technique called “tile tracking” which helps them make strategic decisions that go far beyond any vocabulary skills.

“By being aware of how many tiles are in play and how many are left, you can have a better idea of how the board will shape up, and you can predict which letters you might draw.”

Recreational Scrabble players like the author of the article know that they have weak vocabulary skills and they know that they are not professional grade Scrabble players, but they do NOT know that studying statistical mathematics is even part of the game. They think the outcome of the game is simply determined by who knows the most vocabulary words.

That, my friends, is a perfect example of the difference between a known unknown (the idea that a weak vocabulary makes for a weak Scrabble player) and an unknown unknown (the idea that weak math skills can prevent you from becoming a pro).

This example also serves to show that the nature of unknown unknowns are such that they live completely outside our “worldview” and their discovery enlarges our perception exponentially. The fact that math is an important tool to the professional Scrabble player enlarges our understanding of the game itself. Unknown unknowns, once discovered, enlarge our perception of reality in ways that known unknowns simply cannot.

And this is why they are so important. To the extent that we cannot even conceive of the existence of the unknown unknown can only limit our ability to enlarge our understanding, and to the extent that intelligent people deny the existence of the unknown unknown, we doom ourselves to the folly of thinking that we can defeat professional Scrabble players simply by studying the dictionary.

Their can be no wisdom without the acceptance of the existence of the unknown unknown for nothing else than this does wisdom mean.

God is the Machine (Digital Physics, Quantum Computation)

Friday, June 11th, 2010

God is the Machine

http://www.wired.com/wired/archive/10.12/holytech_pr.html

Seth Lloyd

This paper proposes a method of unifying quantum mechanics and gravity based on quantum computation. In this theory, fundamental processes are described in terms of pairwise interactions between quantum degrees of freedom. The geometry of space-time is a construct, derived from the underlying quantum information processing. The computation gives rise to a superposition of four-dimensional spacetimes, each of which obeys the Einstein-Regge equations. The theory makes explicit predictions for the back-reaction of the metric to computational `matter,’ black-hole evaporation, holography, and quantum cosmology.

Stephen Wolfram

A New Kind of Science

A History of the Quest for Computable Knowledge

The Timeline

Intuition Failure

Thursday, February 25th, 2010

A Simple Proof

x = 0.999…
10x = 9.99…
10x-x = 9.99… – 0.999…
9x = 9
x = 1

0.999… = 1

Welcome to wonderland.

I Think, Therefore I am Confused

If you are like me, you look at this equation and scoff. Common sense and years of math intuition tell you that what is on the left is not the same as what is on the right.

If you are like me, you are wrong. But if you are like me, you will look it up on wikipedia or at math wolfram, you will see that there are rigorous proofs… but you will yet still have doubts, unsatisfied.

You are not alone. This is one of those bedeviling problems that has worn many a thinker – from Pythagoras forward – bald with head scratching.

You see, it is not our fault. When you and I look at that equation, we imagine two distinct, finite numbers. This is what the form of these numbers want us to do. But this formulation is a visual shortcut to a deep, counter-intuitive concept that, if followed, will take you all the way down the rabbit hole; all the way back to Pythagoras and thousands of years of mathematical struggle and strife.

It turns out that mathematicians are not really that much different from you and I. They do not like all that rigor any more than you and I do. Well, maybe some of them do! But the truth is that the rigor came later, after years and years of intuition failure. And this is the interesting thing: the rigor is only there because we got stuck really bad.

But strangely, it turns out that Logic has the creative power to build Grand Tapestries that Imagination could never fathom – not even it it’s wildest dreams!

Imagining Number

The first thing that trips us up when we see the repeating decimal is we think of a discreet number that we can extend to the precision of our will. We imagine that we can subtract this number from 1 and measure the gap between them. But when we do imagine this, we must be imagining a discreet, finite number, which is to say we are NOT imagining 0.999… but some number infinitely smaller!

Ah, the joys of being a finite being trying to conceive and measure the infinite!

The problem, you see, is that this infinite decimal cannot be contained by the mental image we have of what a number is. It leaks out of the box. The mere utterance of the existence of such a number could incite murder in Pythagoras’ day!

We were taught to think of multiplication as repeated addition but one day that model, or image, or idea, stopped working for us. You may not even remember this. If you are like me, your intuition of number was already set and to this day you think of multiplication as repeated addition. But this is because when you and I think of number we do not think of sqrt(2).

From Better Explained:

In school, first we are told numbers are “counts” of something, then we learn to add, or “combine our counts.” Next, we learn that multiplication is just repeated addition and this interpretation works well for round numbers like 2 and 10 but concepts like -1 and sqrt(2) don’t work. Why? Our “model” or “analogy” or “picture” was incomplete. Numbers aren’t just a count; a better viewpoint is a position on a line. Now we see arithmetic as ways of transforming the location of this position to a new location. Addition is now seen as sliding and multiplication as scaling.

The moral of the story is that this is what happens when we reach the brink, or limit, of an idea. Our thoughts are created in images and sometimes those images just don’t work in certain situations and we have to find new images.

This is what is happening with the infinite series. The image we have in our head of what a number is breaks down. It no longer works. We have to throw intuition away and trust the rigors of logic. It was only because our intuition, and our images, failed us in places like this that we had to back up to the beginning and start to lay down the mathematical rigor and build up ever so carefully from the bottom up, making sure along the way that we were leaving absolutely no wiggle room in our logic and in our proofs. This took thousands of years and it is a pretty impressive accomplishment. Some problems literally stopped us in our tracks for hundreds and hundreds of years.

I admit, I have gone back and forth on this and many other similar issues, trying to fight what the mathematicians have proven. I myself do not know how to picture a “number” that can never be pinned down, one that is the sum of a process that never ends! I just can’t do it! Every time I try to imagine this “number” I am really imagining that the process has stopped. But it hasn’t and down the rabbit hole I go again!

But everything I am telling you has been proved in the most rigorous way. The only escape is to postulate a new kind of number and that leads to worse problems. But it has in fact been done. I don’t think the results are very pretty, though.

The Infinite Series

The equality at the top of this page  is a shortcut way of writing “9(1/10) + 9(1/100) + 9(1/1000) …  = 1″

This is called an “infinite series.” All such infinite series represent exactly one real number (in this case, 1) or run off to infinity. There are (possibly infinitely) many “infinite series” that represent (or converge to) any given real number. Which is to say, given any real number, there are many (possibly infinitely many) infinite series that correspond to it.

In other words, all real numbers can be “represented” as a repeating decimal. There is no debate about this. Incidentally, don’t you think this is a curious thing to say, that a number can be represented as another number!

(Re)Imagining Number

The problem begins when we ask the question, “what is the sum of this series,” which is to say: “add this to this to this to this and never stop, and when you are done, tell me what the sum is!” The contradiction should jump out at you at this point. The problem is that no one has been able to posit an escape from this contradiction.

The notation 0.999… is just a representation that we came up with so we could express the idea of “9(1/10) + 9(1/100) + 9(1/1000) … forever” more succinctly. The problem is that when we see 0.999… we think of a static number and imagine we can add digits to it to make it more and more precise at will. But in reality, the repeating decimal is just what we write down when we write down the “end product” of something that has no end.

The end product of something that has no end.

This is a contradiction. It does not make sense. But we created a map for it anyway, so that we could do arithmetic with it. As it turns out, we can do arithmetic with things that do not make sense in themselves. Nonsense squared = q;  and q times q times q equals q3, which is nonsense cubed. As long as we forget the meaning of nonsense, we have no problem calculating nonsense cubed. My goal here is to point out the nonsense that we have glossed over in these last thousand years….

The source of the confusion in this case is that we want pictures, so we make a picture of 0.999… and then when we try to use that picture, it fails us because it leads us to believe that the equation, “0.999… = 1″ is false. But this is because our intuition about what 0.999… means is wrong.

I agree with you: we intuitively imagine that there must always be a gap between 0.999… and 1, but that is because we can only imagine the process that is behind the symbol in finite terms. We think, “no matter how long you allow the process to run, it never reaches that for which it approaches.”

Seems rock solid, but it is in the phrase, “no matter how long” that we stumble, for in this phrase we are really postulating a finite length or duration - a “how long” of the process – at which point we stop the process and say “at this certain POINT in the series” there is a gap between our sum and 1.

In effect, we are saying, “there is a gap here. And here. And here. And here.” And so we conclude there is always a gap, QED. Watertight! Right?

No!

What we mean when we make that argument is that there is a gap at each finite point, each time we stop the process and measure. But this is sort of like the problem you have in quantum physics where the measurement collapses the strange, mysterious thing into the expected phenomenon. In so doing, we destroy the strange mysterious thing itself!

The crux of the problem is that you can never stop to take a measurement to see if there is a gap. Any time that you imagine a gap, you are stopping the addition machine and imagining a finite point. There is no getting around this because there is in fact no gap between 0.999… and 1 even though there is a gap at each and every finite point that is built into what 0.999… means!

Think of the problem in reverse, and ask, if there is a number between 0.999… and 1, what could it possibly be? To imagine such a number is to imagine that 0.999… is finite, that the process of adding up all those fractions has ended. But it never does, and so there is never even a point to point to and measure the gap!

Another way of stating this is to say that 0.999… is infinitely close to 1, which is de facto to say that no other number can exist in the “in between” because the “in between” is infinitely small, which is another way of saying the size of the gap is zero.

Yet another way to crash into this paradox is to ask the simple question, after the number X. what is the next real number? Once you think about it you will realize that there is no such thing as the next real number because there is an infinite number of numbers between any two real numbers. Therefore, the concept of a “next” real number fails. Such a concept is undefined. No matter how hard you try to make it so, there simply is no such thing as a “next” real number. This is so counter-intuitive to our concept of the number line as one point followed by another that we really have to pause and consider if we really understand the number line at all. I suggest that for most of us, the answer to this question is a resounding no.

We are finite and we are trying to measure the infinite! This is why our imaginations necessarily fail us. But we cannot help but to think in this way.

This is why math was grounded in the rigors of logic, because we kept running into these problems of intuition failure.

The History of Number

One of my favorite books is called “Number.” It has been around a while, written by Tobias Dantzig. Einstein said it was the “beyond doubt the most interesting book on the evolution of mathematics which has ever fallen into my hands.” It is one of the books that I would take to a desert Island were I to be stranded there for life. This problem we are discussing is literally thousands of years old. Pythagoras was the first to slam head-on into it. It took us thousands of years to “settle the case” rigorously and it is only because we were able to do so that any engineering beyond that of the ancients is possible, for the mathematics of these infinities is what makes all modern electrical engineering possible.

The history of science and the history of mathematics is one of the greatest stories there is and it is one of our greatest treasures. It should be taught as part of the liberal arts. I can’t really do it justice, but I enjoy reading those who can. If your interest is peaked, start here.

Credits: Everything written here was inspired and taken from here and here. Check out Kalid’s site. It is great!

Heat and Light

Sunday, January 10th, 2010

related post: fire

Heat and Light

I always get confused when thinking of infrared radiation and heat. I tend to think of these as the same, but they are not. This post is a refresher course for me for when I start to have trouble with these concepts…

Heat is not a substance, nor is it a form of energy, strictly speaking. Rather, it is the transfer of energy. More specifically, it is the transfer of kinetic energy. More specifically still, heat is the mechanism by which the transfer of molecular motion travels from a faster moving thing to a slower moving thing.

Here are the official definitions…

In modern terms, heat is concisely defined as energy in transit. Scottish physicist James Clerk Maxwell, in his 1871 classic Theory of Heat, was one of the first to enunciate a modern definition of “heat”. In short, Maxwell outlined four stipulations on the definition of heat. One, it is “something which may be transferred from one body to another”, as per the second law of thermodynamics. Two, it can be spoken of as a “measurable quantity”, and thus treated mathematically like other measurable quantities. Three, it “can not be treated as a substance”; for it may be transformed into something which is not a substance, e.g. mechanical work. Lastly, it is “one of the forms of energy”. Similar such modern, succinct definitions of heat are as follows:

  • In a thermodynamic sense, heat is never regarded as being stored within a body. Like work, it exists only as energy in transit from one body to another; in thermodynamic terminology, between a system and its surroundings. When energy in the form of heat is added to a system, it is stored not as heat, but as kinetic and potential energy of the atoms and molecules making up the system.[4]
  • The noun heat is defined only during the process of energy transfer by conduction or radiation.[5]
  • Heat is defined as any spontaneous flow of energy from one object to another, caused by a difference in temperature between the objects.[6]
  • Heat may be defined as energy in transit from a high-temperature object to a lower-temperature object.[7]
  • Heat as an interaction between two closed systems without exchange of work is a pure heat interaction when the two systems, initially isolated and in a stable equilibrium, are placed in contact. The energy exchanged between the two systems is then called heat.[8]
  • Heat is a form of energy possessed by a substance by virtue of the vibrational movement, i.e. kinetic energy, of its molecules or atoms.[9] The kinetic energy and heat may formally be equivalent, but they are not identical.

So, back to infrared radiation…

Is infrared radiation and heat the same thing? Moreover, if we understand IR as part of the electromagnetic spectrum, could we say that heat is therefore EM radiation, like light?

Infrared radiation IS a form of heat. That is, IR will transfer kinetic energy, and this transfer of kinetic energy is what we call “heat.”

What does this mean? Well, it means that the range of frequencies in the spectrum of electromagnetic radiation that we refer to as “infrared” are such that they transfer kinetic energy (motion).

The first question that arises when we state things this way, is: do other frequencies in the spectrum of electromagnetic radiation transfer kinetic energy as well, and why or why not?

The answer is no. The other frequencies in the spectrum of electromagnetic radiation do NOT transfer kinetic energy. But why?

Because of wave harmonics.

Infrared radiation can be felt. It is (invisible) Light that warms the skin. When we get close to the stove, a fire, or feel our skin warming on a sunny day, we are experiencing the effects of the same infrared radiation. This is why I tend to think of heat and light as the same thing. Because this particular Light warms the skin. But note that this is true only of the light in the lower (less energetic, longer wavelength) part of the visible portion of the spectrum – at the red end. Notice that this is the lower (energetic) end of the visible portion of the spectrum. We cannot “feel” the light in the even lower, longer (radio) ranges or the much higher, shorter, more energetic (gamma) range. From this we can see that although IR warms the skin, and IR is Light, not all “Light: warms the skin. Radio waves do not warm the skin and gamma rays do not warm the skin. This fact confounded scientists for centuries (hint: it should confound you, too. If it doesn’t, stop here and explain why this is so – the answer, by the way, was given in the previous paragraph).

Another way of observing the above is to notice that increasing the energy level (kinetic energy) of the Light, does not increase the heating effect of the Light on an object.

But why?

Let me say that again, more precisely:

Infrared radiation (red Light) from the sun transfers kinetic energy (heat) from the sun to our skin. Ultraviolet radiation (purple Light) from the sun does not. But UV Light is more “energetic”. The question, then is: Why is it not “hotter?”

Definitions

-Heat vs heat transfer – are these the same? From the wikipedia definition above, heat is the transfer of kinetic energy, so it is misleading to speak of the transfer of the transfer of something. What we are doing is confusing kinetic energy with heat when we start to make this kind of distinction between the thing and the transfer of the thing. In this case, the thing in question is atomic and molecular motion.

-Kinetic energy is this motion – the wiggling of atoms and molecules – and heat is the passing on of this “motion” from one object to another. Both kinetic energy and heat increase as the wiggling of atoms and molecules increase, but heat increases only to the extent that this increase in wiggling (kinetic energy) causes an increase in wiggling elsewhere (to another object).

-Temperature is the measurement of the amount of this wiggling (kinetic energy). This wiggling sounds familiar, for Light is itself a wiggle. So why is it that the UV wiggle does not increase the wiggle (temperature) in our skin as much as the IR wiggle? Why does Light with more kinetic energy not produce more heat? Why is the kinetic energy of this Light not transferred?

-Wave harmonics

The wiggle of IR reinforces the wiggle in the molecules of our skin, sort of like how microwave Light reinforces the wiggle in the polarized water molecule.

The key is that the wiggle is best transferred when the speed (temperature, or kinetic energy – referred to as “color” in the visible spectrum) of the wiggling matches between the source of the radiation (Light) and the receiver of the radiation. Another way of saying this is this is that IR is a particular speed (wavelegth) of Light that makes molecules wiggle. Sometimes, this particular (range of) wavelength of Light is called “heat waves,” “radiation,” and/or “infrared radiation.”

In all of these cases, we are referring to Light (electromagnetic radiation) of a certain range (the infrared range), which is to say of a certain wavelength, or “energy.” It just so happens that this particular range of wavelengths, or energy, harmonizes with molecules and increases their ever present wiggling.

So, the wiggling is kinetic energy and certain frequencies of Light transfers this wiggling to molecules. But other frequencies do not. This is why it is not the Light that is the heat, but that some Light can transfer its kinetic energy (wiggle) to objects. This form of Light is called infrared radiation, or heat waves.

But why is that only certain frequencies of Light make this transfer? Why don’t higher frequency wavelengths transfer more heat, and lower less?

Harmonics. Infrared radiation (the lower energy Light in this example) wiggles at a similar speed (temperature) as the molecules in our skin. Therefore, the wiggling in the Light and the wiggling in our skin can harmonize such that the wavelengths of the Light and the wavelengths of our skin add to each other.

Wavelengths of skin? Yes, whenever an electron moves it generates electromagnetic waves in the field that permeates the universe (I like to refer to all EM as Light, but traditionally people speak of light as a particular range of EM, just as IR (heat radiation) is another particular range of EM ). Another way of saying this is that when an electron moves, it emits Light. Light behaves as a wave such that two waves can cancel or reinforce. When we feel our skin get hot from the Light of the sun, what is happening is that the Light that is emanating from our skin and the Light that is penetrating our skin from the sun, reinforce. In other words, we shine brighter, hotter.

We shine? Yes, everything in the universe (except dark matter) emits Light (EM) because everything in the universe wiggles and every electron that wiggles creates ripples in the electromagnetic field which permeates the entire universe. These ripples are called electromagnetic radiation, or Light, as I have described.

So the upshot of all of this is that infrared radiation transfers its wiggle to molecules because it wiggles with a wavelength that is in a certain range that the molecules respond to. This is why higher energy Light from the sun (like UV Light) does not transfer heat to our skin – the harmonics are not right. UV Light is, in a sense, invisible to our skin in the same way it is invisible to our eyes. It wiggles too fast for our skin to “see” or “hear”

The Wiggle is All

It is interesting to note that the transfer of wiggles is a concept that applies to every event and phenomena of the universe. All of our senses are merely ways that the wiggles of the universe are transferred to our bodies, whether the source of the wiggle be described as Light (molecular wiggles felt by our eyes), sound (molecular wiggles felt by our ears), taste (molecular wiggles felt by our tongue), smell (molecular wiggles felt by our nose), or touch (molecular wiggles felt by our bodies). In each of these phenomena, kinetic energy is passed from one object to another through physical touch.

This is an interesting concept to keep in mind, particularly when thinking of forces (post on forces coming soon).

Everything that happens in the universe, every interaction, is a form of one thing touching another. There is no such thing as force. Not the way we tend to think of it: as something separated from, independent of, matter. Force and matter are the same phenomenon, like space and gravity. Einstein showed that gravity is not some force that acts in space, but that gravity is space itself. Likewise with electromagnetic radiation. Light is not something that moves through space; rather, it is the movement of space itself.

These are difficult abstractions to work through, but the movement of physics in the twentieth century has progressively stripped down our distinctions such that it is nearly impossible to draw a line between matter and space and energy at all. At the deepest level, we have seen the equivalence of these.

Now, you may say that this is all metaphorical, to say that this thing is actually another, and you would be absolutely right. Metaphor is exactly that: an expression that equates two seemingly different things. In other words, mathematics. The equations of physics that describe the universe are metaphors that tell us that this equals that. E=MC2 is the mathematical metaphor that tells is that matter IS energy. We know that space is not separate from time, and yet we speak of time as if it flows through space. This is not correct. Neither is it correct to speak of light and radio as if these are two different phenomena. They are not. One just has more kinetic energy but they are, in fact, the same phenomena. The joy of physics and mathematics – and religious contemplation – is merely the joy in discovering the multifaceted unification of the variety of experiences and phenomena in the universe. In the religious sense, this experience begins with the discovery that you are not your mind; it is the discovery that to experience your true self is to experience the ever present communion with God. The progress of physics thus mirrors the progress of the spirituality that grows only by seeing the ever more connectedness of the universe as one vast, single multifaceted phenomenon. This should not come as a surprise, for both of these journeys are of men and the journeys of men have but one end.

But wait, we started this meditation on heat and light by distinguishing them and declaring that they are not the same phenomenon. And surely taste and sight are not the same phenomenon. Well, yes and no. In the same way that red and blue are not the same, that is. But there is another sense in which red and blue are the same. That is to say, that they are of the same nature. The larger point in this meditation is really about levels of abstraction. We can go in either direction and make distinctions between red and blue, or we can go in the other direction and draw them together under one classification. In either case, these are just abstractions, or ways of thinking about these phenomena.

Thus the idea of the grand, singular, multifaceted phenomenon, aka the universe. After all, seen from a perspective outside of time, every event and phenomena in the history of the universe can be seen as a single event – in the mind of God, for example.

But now we have wandered far off the topic, so this will have to continue in another post.

See you then.

  • Heat is the transfer of energy between substances of different temperatures.
  • Sing O Muse

    Sunday, January 10th, 2010

    ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

    And yet it just won’t go away:
    I feel it and cannot understand it;
    cannot hold on to it;
    nor yet forget it;
    and if I grasp it wholly
    I cannot measure it!

    –Richard Wagner

    ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

    I do not know it – it is without name – it is a word unsaid,
    It is not in any dictionary, utterance, symbol.

    There is that in me – I do not know what it is – but I know it is in me.

    –Walt Whitman Leaves of Grass

    ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

    And it’s hard to explain how I feel;
    It don’t go in words but I know that it’s real.
    I can be moving or I can be still,
    But still is still moving to me.

    –Willie Nelson

    ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

    And where we had thought to find an abomination,
    we shall find a God; where we had thought to slay another,
    we shall slay ourselves; where we had thought to travel outward,
    we shall come to the center of our own existence;
    where we we had thought to be alone,
    we shall be with all the world.

    –Joseph Campbell, A Hero with a Thousand Faces

    ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

    Now that lilacs are in bloom
    she has a bowl of lilacs in her room
    and twists one in her fingers while she talks.

    “Ah, my friend, you do not know, you do not know
    what life is, you who hold it in your hands”;
    (slowly twisting the lilac stalks)

    “You let it flow from you, you let it flow,
    and youth is cruel, and has no remorse
    and smiles at situations which it cannot see.”

    I smile, of course,
    and go on drinking tea.

    –T.S. Eliot, Portrait of a Woman

    ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

    I am moved by fancies that are curled
    around these fingers, and cling;
    the notion of some infinitely gentle
    infinitely suffering thing.

    –T.S. Eliot, Preludes

    ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

    A flower grows
    and blooms and dies
    and sometimes is touched
    by butterflies.

    – PBT

    ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

    Take this kiss upon the brow!
    and, in parting from you now,
    thus much let me avow -
    you are not wrong, who deem
    that my days have been a dream;
    yet if hope has flown away
    in a night or in a day,
    in a vision, or in none,
    is it therefore the less gone?
    All that we see or seem
    is but a dream within a dream.

    I stand amid the roar
    of a surf-tormented shore,
    and I hold in my hand
    grains of the golden sand -
    how few! yet how they creep
    through my fingers to the deep,
    while I weep – while I weep!
    O God! Can I not grasp
    them with a tighter clasp?
    O God! Can I not save
    one from the pitiless wave?
    Is all that we see or seem
    but a dream within a dream?

    – Edgar Allen Poe

    ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

    We had passed through an initiation
    like that of the Tibetan ascetic,
    who staggers half dead from a trance,
    where he has seen himself eaten
    alive and has not yet learned that
    the eater was himself.

    ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

    The battle between the manifest world
    and the ancestral darkness
    at the end of all things

    the too great power of divinity

    ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

    because in my head, frothy red
    more interesting than the words she said
    “I cannot say, I cannot say,
    these things like little white wings
    that carry and bring
    yet have no speech
    and cannot sing.

    The dancing lightness
    of sunshine washes (us) clean

    – PBT

    ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

    I know the cries that cry for all
    I’ve known one well enough
    to know them all

    I know the despair of half-empty streets
    under a half-empty moon
    that lingers at half past three
    in the half-dead, mid afternoon

    I know where a yellow wildflower grows
    through a half inch crack in black cement

    I wonder whether a yellow wildflower knows
    it was born in a place it was never meant?

    Can a yellow wildflower decide
    in the left lane of a highway divide?

    – PBT

    ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

    Another and another and another -
    has anyone ever counted them?
    Does anyone know;
    The number of waves
    in a single day -
    rising to crescendo
    then receding quietly away?

    – PBT

    ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~