Sunday, April 20, 2014
On the Einstein-Podolsky-Rosen Thought-Experiment
On the Einstein-Podolsky-Rosen Thought-Experiment
From: E.D.
To: Arjun Janah
Cc: R.B.
Sent: Sat, Apr 19, 2014 10:23 pm
Subject: RE: Blog post of our correspondence re. probability.
Arjun,
<personal details, deleted by Arjun 04/20/14>
Also, a correction or clarification about Bell's theorem. My understanding of it at this time is that Bell's theorem does establish non-local causation in the following sense. If two particles depart from one another in different directions and then, later, the spin of one of the particles is altered, the spin of the other particle instantaneously changes also. However, this fact can not be used to communicate instantaneously (or faster than the speed of light). So in that sense, there is not non-local causation, because we at the macro level can not observe it.
I could have it all wrong, but I think the main point is that in the end q.m. does not explain non-local causation or action-at-a-distance (such as telepathy, precognition, other mystical states of union, etc.) at the level of human consciousness. So q.m. is disappointing for people who wanted to use it for that purpose.
<See also the third section added below this one -- the e-mail I received at 6:34 pm on 04/20/14 from E.D. -- for a sharper clarification of this issue. -- Arjun 04/20/14>
I think I mentioned this book to you before, but this is the sort of thing I'm referring to here:
"How the Hippies Saved Physics"
http://www.amazon.com/How-Hippies-Saved-Physics-Counterculture/dp/039334231X
E.D.
-----------------------------------------------------------------------------
From: Arjun Janah
To: E.D.
Cc: R.B., blank01, P.B.,V.K
Sent: Sun, Apr 20, 2014 12:55 am
Re: Blog post of our correspondence re. probability.
<personal details, deleted by Arjun 04/20/14>
Yes, I think you have described the core quantum-entanglement situation concisely and correctly. But my knowledge of this is limited. I will try to go into some details below, which may or may not shed more light on what you have neatly summarized.
This was (to my knowledge) first posed as a challenge to q.m. (or its emergent interpretation at the time) by Einstein, Podolsky and Rosen. It is a strange sort of instantaneous action-at-a-distance. Bell later formulated, I am told, what had come to be known as the EPR paradox as a testable inequality.
Electrons possess an internal or intrinsic angular momentum, called "spin", which has, in certain natural units, a value of 1/2. From quantum mechanics, the projected value of this spin, along any chosen spatial direction, can have only two observed values: +1/2 or -1/2 (in those same natural units).
Two electrons can combine into either a total-spin = zero state (where the two electron spin-vectors are aligned oppositely: 1/2 -1/2 =0) or a total-spin = one state (where their spin-vectors are aligned in the same direction, 1/2 + 1/2 =1). These are the only definite total-spin states allowed by the rules of q.m.
Consider two electrons that are known to be in, say, a total-spin = zero state, but for which the individual spin directions of the two electrons are unknown. If those two electrons move apart in space without further disturbances (caused, for example, by our further observations), then they should remain in that total-spin = 0 state.
In the usual intepretation of q.m., the spin-values of each electron along any given direction are not only not known to us (and here's where we may strongly quibble with that interpretation) they are indeterminate in an absolute sense. This is the thing that Einstein and his colleagues could not (with good reason, in their minds) accept.
If we now make an observation on one of the electrons and find its spin value along a certain direction is, say, +1/2, then we know that the spin value of the distant electron along the same direction will have to be -1/2.
While it is perhaps understandable, as per the usual interpretation, that it was our act of observation (a measurement of spin value along a chosen axis) that "threw" the observed electron from an undetermined individual-spin-alignment state into the definite +1/2 state, how could we say the same of the distant electron?
Could our measurement made on the local electron here truly have "thrown" that distant electron there into its spin -1/2 state? Or was it already in that state before we made our local measurement on its partner?
E, P & R had presented the instantaneous throw of the individual spin of the distant electron (following an observation of its local partner) into a definite state as a paradox, something that arose out of q.m. and its accepted interpretation, but appeared to violate the precept that the fastest way in which an event at one place could affect an event in another place was via a signal sent from the first event to the second -- which signal could not travel faster than the speed of light (or other electromagnetic waves) in a vacuum. And this speed, though large, is finite. So instantaneous causation was considered impossible.
So E, P & R argued that either the accepted interpretation of q.m. was wrong -- i.e., the electron-spins were in fact aligned along a certain direction (in which case the observation was only a discovery of what had already existed in reality) rather than being truly indeterminate in an absolute sense -- or else one would have to accept instantaneous causation, which violated a precept that was considered sacred, one on which Einstein's theory of relativity and so also all of physics had rested.
But after Bell had formulated this as an inequality that could in principle be tested, tests were done that showed (so it is alleged) that the accepted interpretation of q.m. is in fact correct.
However, although instantaneous causation does apparently occur (if we swallow all of this, which is a mixture of observed fact, mathematical q.m. rules and an interpretation that was formulated by some of the most thoughtful people, including Niels Bohr) it is alleged that this cannot be used to do any true communication. This alleged fact has been taught to physics students for some time now as the saving grace in this situation. I am not clear in my mind, at this time, on how to understand or explain that point.
This is my own (surely incomplete and faulty) understanding of the situation, dating back many decades to my undergraduate days in India, when I came upon EPR by chance. (I did not learn about Bell's inequality till later, and still am pretty ignorant about that.)
As far as the relevance of any of this to paranormal phenomena, I have no understanding at all about that aspect. That is not to rule out such phenomena, only to state the lack of any linkage that I know of.
Arjun
-----------------------------------------------------------------------------
From: P. B.
To: Arjun Janah
Cc: E.D., R.B., blank01, V.K.
Re: Blog post of our correspondence re. probability.
Sent: Sun, Apr 20, 2014 2:31 am
Dear Arjun,
<personal detail, deleted by Arjun 04/20/14>
The actual description of the EPR thought experiment evades the question
of fluctuations, of electron spin for instance, which destroy coherence.
In other words, if the electrons in the EPR paradox are in the normal
paramagnetic state, as we take for granted, then spin correlations will be
exponentially damped in space and time. In a ferromagnet, the spin states
will be correlated and will be entangled. Flipping the spin of a majority spin
will ensure that a minority spin will be promoted to the Bose condensed state.
Verifying this entanglement hypothesis means verifying that the
Bose condensed state depends only on temperature ( in classical
critical phenomena) or on a bias (in quantum critical phenomena),
as for instance in the breaking of symmetry at different energy scales.
P.B.
-----------------------------------------------------------------------------
From: E. D.
To: Arjun Janah
Sent: Sun, Apr 20, 2014 6:34 pm
Subject: RE: More blog posts
Arjun,
These (things I wrote that you posted) were mostly off-the-cuff remarks made by e-mail.
You might want to add this as an addendum, in answer to your question about what parapsychology has to do with Bell's theorem.
The connection between parapsychology and Bell's theorem is that the existence of non-local causation is a necessary (though not sufficient) condition for the existence of parapsychological phenomena. Telepathy, for example, suggests that there is an immediate and instantaneous connection between minds (and the brains that embody those minds). Psychokinetic phenomena suggest that the mind has an immediate (non-local) causal power over physical objects held at a distance. If Bell's theorem proved that non-local causation exists, it would fulfill one condition for the existence of parapsychological phenomena.
Of course, parapsychological phenomena might still not exist because there might be other conditions that are necessary for them to occur that can not be fulfilled. But the impossibility of non-local causation is the most commonly cited reason to reject the possibility of parapsychological phenomena. So demonstrating the possibility of non-local causation would greatly increase the possibility that parapsychological phenomena might occur.
E.D.
E.D.: on the philosopher John Schumacher (in connection with the previous post)
E.D.: on the philosopher John Schumacher (in connection with the previous post)
From: E.D.
To: Arjun Janah
Sent: Fri, Apr 18, 2014 3:00 pm
Subject: RE: Blog post of our correspondence re. probability.
<Some personal details in the e-mail below, not directly related to the main discussion, were deleted by Arjun when posting on 04/20/14.>
Arjun,
I didn't realize that physicists didn't have an answer to this question. I thought the standard interpretation of q.m. was that there is no "objective" reality until an observation occurs, collapsing the wave function. I thought Schrodinger's Cat Paradox was supposed to raise doubts about this interpretation, but that most physicists did not see it as a problem.
I never studied q.m in any depth. I studied physics when I was an engineering student at RPI. In addition to that, I took philosophy classes with John Schumacher, who was a former math student who became a philosopher. He graduated from RPI in 1966 and was radicalized by the student movements of the 60s. A classic "hippy" in his appearance and philosophy, he was interested in eastern mysticism (buddhism) and, because of his RPI education, also interested in combining his interest in buddhism with his knowledge of physics.
(John Schumacher) and a couple of other members of the RPI philosophy department knew David Bohm and invited him to speak on campus. They were particularly interested in David Bohm's interpretation of q.m. and in the "holographic" theory of the universe. All this was very heady stuff and I was quite interested, although I never did learn enough q.m. to know what to make of it. John Schumacher published only one book in the late 80s which was a synthesis of philosophy and q.m., titled "Human Posture". He died young at age 54 in 1999.
The crux of the issue was non-local causation. Q.M., in some interpretations, at least, allows for immediate causal connections over a distance. Bell's theorem was supposed to provide proof for such local causation. But later I learned that, in fact, Bell's theorem can't be used to prove non-local causation.*
<* See also E.D.'s modification of this statement in the next post -- Arjun 04/20/14.>
During his student years at RPI, John Schumacher was a student of John Koller, who is a expert on Eastern Philosophy--especially Indian philosophy. John Koller is still alive and I see him occasionally. There were other philosophers at RPI in the 60s that also exposed John to eastern philosophy. Also at RPI in the 60s was David Thoreau Wieck, a leading post-war American anarchist philosopher.
Ultimately, I think David had the biggest impact on John, and both directly and indirectly through John, on me.
During his student years at RPI, John Schumacher was a student of John Koller, who is a expert on Eastern Philosophy--especially Indian philosophy. John Koller is still alive and I see him occasionally. There were other philosophers at RPI in the 60s that also exposed John to eastern philosophy. Also at RPI in the 60s was David Thoreau Wieck, a leading post-war American anarchist philosopher.
Ultimately, I think David had the biggest impact on John, and both directly and indirectly through John, on me.
John Schumacher was an anarchist, and his anarchism leaned pretty heavily towards communist anarchism.
I think his interest in non-local causation was an attempt to show how the capitalist view of social reality, in which individuals are separate and competing entities, is false. In fact, John wanted to show, people are more radically connected to one another. Thus social life is held in common, and an egalitarian, non-hierarchical society, in which we are all "one," fits best with reality.
A lot of John's book, "Human Posture," I think, tries to explain how we ever came to believe we are separate if the reality is that we are not. It does that in a psychological, sociological and metaphysical way.
Ultimately, I have come to believe that John's primary commitment was political, and that he merely used mysticism, physics and metaphysics to help him meet his political commitments. I have also come to doubt his interpretations of mysticism and physics and to think that he would have been better off just focusing directly on politics.
Although I've come to see this as a chink in his sterling armor, I still think of him as an extraordinary personality--practically a mythic being come to life, or rather, an archetypal persona, and it is tragic he's no longer with us.
E.D.
Friday, April 18, 2014
Mutiny or Uprising of 1857?
The following is from a correspondence (of comments) on Facebook with a friend, D.R.. D.R.'s first comment refers to an Indian man who is alleged to be 179 years old, according to the available local official documents. I have lightly edited the correspondence, in a few places, mostly for spelling and grammar.
D.R. I would like to sit at his feet and hear about his take on the Sepoy Mutiny in 1857....
April 16 at 11:03pm · Edited · Unlike · 1
<Some comments made by A.J. and D.R. were deleted here. They had no bearing on the events of 1857>
Arjun Janah To be Subcontinentally Correct, that's the Indian Rebellion or Uprising of 1857.
11 hrs · Like
Arjun Janah What's the Middle East to some, is West Asia to others. So one country's freedom fighters are another country's terrorists and so on. In our case, our freedom fighters (against the Soviets in Afghanistan) turned into our terrorists (as in 9-11). Such things do happen.
11 hrs · Like
D.R. As far as I'm concerned Bombay is still Bombay. It's ludicrous to change a Portuguese name into the same Portuguese name with another spelling and pronunciation in order to "Indianize" it. As for the rebellion, the Sepoy mutiny is a better descriptive term because that's what it was. To change it to the "Indian rebellion" or such is silly, like changing the name of Bombay. A rebellion in a military organization is a mutiny ! It was sepoys, or soldiers who were the rebels, not the population at large, that was the salient feature of the rebellion. I've been fed up with this ridiculous movement to change names, often in India from simple names to much longer and more complicated names. Why have a complex about history? It's history, part of India's history, for better or worse. We haven't seen all this in places like Singapore or Hong Kong, have we?
5 hrs · Like
D.R. What's more, when the new Indian street names are long, they are shortened to English initials, and then transcribed into Hindi, so B.C. road becomes Bee See Road in Hindi or other regional languages depending on location. Sub-intelligently correct...
2 hrs · Like
Arjun Janah There is no shortage of stupidity and chauvinism in the subcontinent (in all of its nation-states) as also elsewhere. Nation-states themselves are too often abominations, in my humble opinion, just as Empires were, and just as the feudal landlord systems were in so many parts of the world for so long, before the nation states and the great European colonial empires -- and sadly still are.
You can of course call it the Sepoy Mutiny if you wish. It's simply a matter of perspective. That's why I gave the example of West Asia and The Middle East. We learned the history of the 1857 uprising from our English masters, filtered and written in the way they viewed it. It was one of many rebellions against the British, all invariably put down with extreme brutality. This one involved the sepoys, among others, who had access to guns and were able to organize in military formations, so it created more havoc and fear among the colonialists than others had. Without it, the de facto rule of the predatory East India Company would have continued unchecked.
If you recall the sorry history of that Company, local resentment had long been brewing against this colonial enterprise that tapped into the local parasitical feudal system but drove it into extremis, demanding so much that the landlords, in turn, were forced to literally starve their tenant peasants to satisfy their own overlords. The Sepoy Mutiny, whatever be its immediate spark, was a symptom of this. In practice, if you scratch the surface, most human things have economic drivers, however they may be cloaked in religious or other garbs.
This is what happens when raiders enter a country. A traditional rajah or nawab will squeeze his peasants, but not to the point of death, as that would be killing the goose and inviting insurrection. Those who are there for the short-term, be they raiders from the Afghan or Maratha hiighlands, Portuguese pirates or English merchants determined to make their fortunes as fast as possible so they can rise in England's own class system -- these have no such compunctions.
The British Crown that replaced the East India Company's rule after the rebellion was more circumspect, settling in for the longer haul of sucking the country dry without causing open insurrection. Of course, its own sorry history in south and east Asia should be well known, including its development and legal monopoly on the opium trade out of India into China, forced onto the latter via the Opium War and the ruthless bombardment of the densely populated Chinese coast -- the one that led, among other things, to the ceding of Hong Kong to the British.
I do not know the history of Singapore (Singha Pura, literally, Lion-City) but it might be worth researching.
In the case of the Opium Trade and War and Hong Kong, a parallel (ridiculous as it sounds) would be if an ascendant China, in league with Russia, Korea and others, had grown in reach and power and had occupied Colombia, had declared a monopoly on the buying of coca leaves, had converted large tracts from food crops to coca plant production, had set up factories in that region producing great cannonballs of cocaine (as the British did with opium) and had then begun to export these into Miami and other ports along the U.S. coast, feeding into the demand created by local social conditions in this country, whose populace had fallen on hard times, but still was unwilling to purchase the shoddy goods being produced by the Chinese (as the Chinese, up to and perhaps past the mid 1800's, had been reluctant to buy British products).
Pushing the analogy further, were the weak U.S. federal government -- notified by local officials about the growing addiction and crime problems in Miami and elsewhere caused by this illegal import of cocaine -- were they to attempt to correct this, first by issuing legal warnings that were ignored, and finally by raiding the warehouses, confiscating the cocaine and dumping it in the sea (as the Chinese official in charge of operations did with the opium, fearful of burying it in the land or burning it, for interesting reasons) -- were this to happen, then we might see outraged Chinese merchants send messages to Beijing, saying Chinese sovereignty had been assaulted, causing the navies of the Chinese, along with the Russians, Koreans and others, to shell the unprotected coastline, from Miami in Florida north to Baltimore in Maryland, causing no end of civilian casualties, until, at last, ports were ceded, with 100 year contracts, to these great powers, and the laws of the U.S. regarding the import of such substances was effectively annulled, with military operations subsequently launched on land by the foreign powers to suppress those rebellious citizens and localities that had denounced the compromise made by Washington and had risen up in rebellion against these great powers and our own collaborationist federal government.
But this, in effect, was what happened in China, with the Boxer Rebellion being the historical counterpart of that uprising, and Queen Victoria's court in London playing the part of the Beijing government whose navy took part (in our imagined scenario) in the shelling of the U.S. coastal cities.
Of course, the British had a number of accomplices in this, and if you go to the Roosevelt residence in upstate NY, you will find records of how that family acquired its wealth through that same Opium Trade and War. Of course, some of the recently-risen drug lords in Cali (and elsewhere in Columbia and other places) had also grown very wealthy and powerful. But they had prices on their heads and could not hope, perhaps, to give rise to two Presidents as they did here. ( I will not venture to speak of Afghanistan, whose populace has been devastated by violent superpower interventions and fratricidal civil words for many tortured decades.) And we have no figures there in Colombia comparable to the British royal family, whose fortunes took off with the pirate raids on the Spanish galleons bearing loot from Spain's own ravaged American colonies, and then skyrocketed with colonial enterprises the world over, which left British footprints and faces on every continent except Antarctica -- too often, driving the natives to extinction or desperate straits, with the royal exchequer profiting from every enterprise, however sordid or gory. But so it is with all empires.
One man's freedom fighter is another's terrorist, and one man's great king or emperor is another's despot. That has always been true. I am sure George Washington (affluent landlord and slave owner as he was) was viewed with as much disfavor by the British moneyed classes as King George was by their counterparts in the American colonies. Our own (Indian) Prime Minister called the "Maoists" the "greatest threat to India's security" but the extent and intensity of the unrest in tribal and poor rural parts of India indicate that, from the point of view of the populace in these regions, what we are seeing is in large part a struggle for basic survival. From the perspective of many of the locals, it might seem that the corporations that are driving them off their ancestral fields and forests, with the active collaboration of the local and national governments and their police and military forces, as well as local armed para-military organizations, are "the greatest threat to our very existence".
As far as Indianization of names go, some, I agree, are ridiculous. Others are perfectly natural, as the change of name of my city of birth, from Calcutta to Kolkata, was -- a change opposed vociferously, and even in an organized fashion, within my own extended family. But as I pointed out to my late father (who was neutral on this), the latter (or, before our times, its archaic form, Kolikata) has always been the name of the city among Bengalis, who constitute, by far, the majority of its populace. Granted, it was the British, including the rapacious East India Company, that brought the city into existence as we know it, including much of its wonders as well as horrors. But the British had a habit of murdering subcontinental place names, especially the ones in the east with the rounded vowels of Oriya, Bangla and Ahomiya.
All of that being said, at great length, and being probably merely a verbose restatement of things you and others already know, I agree with you that history is history, for better or worse, and that cosmetic changes of names accomplish little by themselves, perhaps only serving to obscure that history, whose darker recesses hold things that few on either side want exposed to light, but from which there is still much to learn.
13 mins · Edited · Like
D.R. I would like to sit at his feet and hear about his take on the Sepoy Mutiny in 1857....
April 16 at 11:03pm · Edited · Unlike · 1
<Some comments made by A.J. and D.R. were deleted here. They had no bearing on the events of 1857>
Arjun Janah To be Subcontinentally Correct, that's the Indian Rebellion or Uprising of 1857.
11 hrs · Like
Arjun Janah What's the Middle East to some, is West Asia to others. So one country's freedom fighters are another country's terrorists and so on. In our case, our freedom fighters (against the Soviets in Afghanistan) turned into our terrorists (as in 9-11). Such things do happen.
11 hrs · Like
D.R. As far as I'm concerned Bombay is still Bombay. It's ludicrous to change a Portuguese name into the same Portuguese name with another spelling and pronunciation in order to "Indianize" it. As for the rebellion, the Sepoy mutiny is a better descriptive term because that's what it was. To change it to the "Indian rebellion" or such is silly, like changing the name of Bombay. A rebellion in a military organization is a mutiny ! It was sepoys, or soldiers who were the rebels, not the population at large, that was the salient feature of the rebellion. I've been fed up with this ridiculous movement to change names, often in India from simple names to much longer and more complicated names. Why have a complex about history? It's history, part of India's history, for better or worse. We haven't seen all this in places like Singapore or Hong Kong, have we?
5 hrs · Like
D.R. What's more, when the new Indian street names are long, they are shortened to English initials, and then transcribed into Hindi, so B.C. road becomes Bee See Road in Hindi or other regional languages depending on location. Sub-intelligently correct...
2 hrs · Like
Arjun Janah There is no shortage of stupidity and chauvinism in the subcontinent (in all of its nation-states) as also elsewhere. Nation-states themselves are too often abominations, in my humble opinion, just as Empires were, and just as the feudal landlord systems were in so many parts of the world for so long, before the nation states and the great European colonial empires -- and sadly still are.
You can of course call it the Sepoy Mutiny if you wish. It's simply a matter of perspective. That's why I gave the example of West Asia and The Middle East. We learned the history of the 1857 uprising from our English masters, filtered and written in the way they viewed it. It was one of many rebellions against the British, all invariably put down with extreme brutality. This one involved the sepoys, among others, who had access to guns and were able to organize in military formations, so it created more havoc and fear among the colonialists than others had. Without it, the de facto rule of the predatory East India Company would have continued unchecked.
If you recall the sorry history of that Company, local resentment had long been brewing against this colonial enterprise that tapped into the local parasitical feudal system but drove it into extremis, demanding so much that the landlords, in turn, were forced to literally starve their tenant peasants to satisfy their own overlords. The Sepoy Mutiny, whatever be its immediate spark, was a symptom of this. In practice, if you scratch the surface, most human things have economic drivers, however they may be cloaked in religious or other garbs.
This is what happens when raiders enter a country. A traditional rajah or nawab will squeeze his peasants, but not to the point of death, as that would be killing the goose and inviting insurrection. Those who are there for the short-term, be they raiders from the Afghan or Maratha hiighlands, Portuguese pirates or English merchants determined to make their fortunes as fast as possible so they can rise in England's own class system -- these have no such compunctions.
The British Crown that replaced the East India Company's rule after the rebellion was more circumspect, settling in for the longer haul of sucking the country dry without causing open insurrection. Of course, its own sorry history in south and east Asia should be well known, including its development and legal monopoly on the opium trade out of India into China, forced onto the latter via the Opium War and the ruthless bombardment of the densely populated Chinese coast -- the one that led, among other things, to the ceding of Hong Kong to the British.
I do not know the history of Singapore (Singha Pura, literally, Lion-City) but it might be worth researching.
In the case of the Opium Trade and War and Hong Kong, a parallel (ridiculous as it sounds) would be if an ascendant China, in league with Russia, Korea and others, had grown in reach and power and had occupied Colombia, had declared a monopoly on the buying of coca leaves, had converted large tracts from food crops to coca plant production, had set up factories in that region producing great cannonballs of cocaine (as the British did with opium) and had then begun to export these into Miami and other ports along the U.S. coast, feeding into the demand created by local social conditions in this country, whose populace had fallen on hard times, but still was unwilling to purchase the shoddy goods being produced by the Chinese (as the Chinese, up to and perhaps past the mid 1800's, had been reluctant to buy British products).
Pushing the analogy further, were the weak U.S. federal government -- notified by local officials about the growing addiction and crime problems in Miami and elsewhere caused by this illegal import of cocaine -- were they to attempt to correct this, first by issuing legal warnings that were ignored, and finally by raiding the warehouses, confiscating the cocaine and dumping it in the sea (as the Chinese official in charge of operations did with the opium, fearful of burying it in the land or burning it, for interesting reasons) -- were this to happen, then we might see outraged Chinese merchants send messages to Beijing, saying Chinese sovereignty had been assaulted, causing the navies of the Chinese, along with the Russians, Koreans and others, to shell the unprotected coastline, from Miami in Florida north to Baltimore in Maryland, causing no end of civilian casualties, until, at last, ports were ceded, with 100 year contracts, to these great powers, and the laws of the U.S. regarding the import of such substances was effectively annulled, with military operations subsequently launched on land by the foreign powers to suppress those rebellious citizens and localities that had denounced the compromise made by Washington and had risen up in rebellion against these great powers and our own collaborationist federal government.
But this, in effect, was what happened in China, with the Boxer Rebellion being the historical counterpart of that uprising, and Queen Victoria's court in London playing the part of the Beijing government whose navy took part (in our imagined scenario) in the shelling of the U.S. coastal cities.
Of course, the British had a number of accomplices in this, and if you go to the Roosevelt residence in upstate NY, you will find records of how that family acquired its wealth through that same Opium Trade and War. Of course, some of the recently-risen drug lords in Cali (and elsewhere in Columbia and other places) had also grown very wealthy and powerful. But they had prices on their heads and could not hope, perhaps, to give rise to two Presidents as they did here. ( I will not venture to speak of Afghanistan, whose populace has been devastated by violent superpower interventions and fratricidal civil words for many tortured decades.) And we have no figures there in Colombia comparable to the British royal family, whose fortunes took off with the pirate raids on the Spanish galleons bearing loot from Spain's own ravaged American colonies, and then skyrocketed with colonial enterprises the world over, which left British footprints and faces on every continent except Antarctica -- too often, driving the natives to extinction or desperate straits, with the royal exchequer profiting from every enterprise, however sordid or gory. But so it is with all empires.
One man's freedom fighter is another's terrorist, and one man's great king or emperor is another's despot. That has always been true. I am sure George Washington (affluent landlord and slave owner as he was) was viewed with as much disfavor by the British moneyed classes as King George was by their counterparts in the American colonies. Our own (Indian) Prime Minister called the "Maoists" the "greatest threat to India's security" but the extent and intensity of the unrest in tribal and poor rural parts of India indicate that, from the point of view of the populace in these regions, what we are seeing is in large part a struggle for basic survival. From the perspective of many of the locals, it might seem that the corporations that are driving them off their ancestral fields and forests, with the active collaboration of the local and national governments and their police and military forces, as well as local armed para-military organizations, are "the greatest threat to our very existence".
As far as Indianization of names go, some, I agree, are ridiculous. Others are perfectly natural, as the change of name of my city of birth, from Calcutta to Kolkata, was -- a change opposed vociferously, and even in an organized fashion, within my own extended family. But as I pointed out to my late father (who was neutral on this), the latter (or, before our times, its archaic form, Kolikata) has always been the name of the city among Bengalis, who constitute, by far, the majority of its populace. Granted, it was the British, including the rapacious East India Company, that brought the city into existence as we know it, including much of its wonders as well as horrors. But the British had a habit of murdering subcontinental place names, especially the ones in the east with the rounded vowels of Oriya, Bangla and Ahomiya.
All of that being said, at great length, and being probably merely a verbose restatement of things you and others already know, I agree with you that history is history, for better or worse, and that cosmetic changes of names accomplish little by themselves, perhaps only serving to obscure that history, whose darker recesses hold things that few on either side want exposed to light, but from which there is still much to learn.
13 mins · Edited · Like
Monday, April 14, 2014
The Collapse of the Probability Distribution
The Collapse of the Probability Distribution
(correspondence re. The Conscious Universe)
--------------------------------------------------------------------------
From: E. D.
To: Arjun Janah
Sent: Wed, Mar 19, 2014 7:06 pm
Subject: RE: The Conscious Universe (not a poem)
Dear E.D.,
I am forwarding your question and my reply to three physicist friends, plus to V.K.,, who introduced me to two of them, and who has an interest in many things.
Your question is by no means silly. Or of it is, then we are all silly, those of us who have thought about it a bit and been just as confounded as you were.
A q.m. wave function is clearly observer-dependent, just as the classical probability distribution I described earlier is. It is simply a measure (if one looks at the probability aspect alone) of one's knowledge or ignorance of the system under consideration. This knowledge or ignorance is affected, obviously, by any observation one makes on the system. Viewed in this way, the collapse (and reconstruction) of a q.m. wave function is simply a consequence of advances or losses of knowledge by you, the observer, about the system. So what I am saying is that the q.m. wave function you are working with is your q.m. wave function of the system.
Let us go again to a spatial analogy. If you take your usual seat at the library as your frame of reference, the position co-ordinates of a fly you are observing (neglecting your other duties temporarily) will have certain values over time, following its trajectory. But they will have different values for a student sitting at a table in the library. And if you were to move to what used to be the lending desk, the fly's co-ordinates as a function of time would have different values again. In this elementary example, everything is clear.
But in this case, no one would argue that the fly is in any way affected by the choice of observer or your shifting of view-points.
When we go to a probabilistic description of a system, however, the probability distributions are indeed affected by such choices or shifts, just as position co-ordinates were affected in our elementary spatial example. And this is where the headache starts. Do the wave functions or probability distributions describe an objective reality, independent of the observer, or one which is observer-dependent? I would vote for the latter, noting that our understanding of the word "reality" needs to be analyzed.
Arjun
-----------------------------------------------------------------------------------------
From: E.D.
To: Arjun Janah
Sent: Mon, Apr 14, 2014 2:47 pm
Subject: RE: The Conscious Universe (not a poem)
Thanks Arjun, your examples clearly illustrate principles of probability. Of course, in classical probability, it is the observer's knowledge that changes suddenly with new information, not the reality itself, as in certain interpretations of qm. That's where the mind-bending paradox comes in.
E.D.
------------------------------------------------------------------------------------------
From: Arjun Janah
To: E.D.
Sent: Mon, Apr 14, 2014 3:31 pm
Subject: RE: The Conscious Universe (not a poem)
You are right. But we will go into q,m. later. The collapse of the wave function, as you saw, has classical analogies. It is the disturbance of the system by the observation (as in Heisenberg's uncertainty principle) that separates q.m. from prior physics. So let us separate those two things in our minds, although of course, they are related.
Arjun
I am forwarding your question and my reply to three physicist friends, plus to V.K.,, who introduced me to two of them, and who has an interest in many things.
Your question is by no means silly. Or of it is, then we are all silly, those of us who have thought about it a bit and been just as confounded as you were.
A q.m. wave function is clearly observer-dependent, just as the classical probability distribution I described earlier is. It is simply a measure (if one looks at the probability aspect alone) of one's knowledge or ignorance of the system under consideration. This knowledge or ignorance is affected, obviously, by any observation one makes on the system. Viewed in this way, the collapse (and reconstruction) of a q.m. wave function is simply a consequence of advances or losses of knowledge by you, the observer, about the system. So what I am saying is that the q.m. wave function you are working with is your q.m. wave function of the system.
Let us go again to a spatial analogy. If you take your usual seat at the library as your frame of reference, the position co-ordinates of a fly you are observing (neglecting your other duties temporarily) will have certain values over time, following its trajectory. But they will have different values for a student sitting at a table in the library. And if you were to move to what used to be the lending desk, the fly's co-ordinates as a function of time would have different values again. In this elementary example, everything is clear.
But in this case, no one would argue that the fly is in any way affected by the choice of observer or your shifting of view-points.
When we go to a probabilistic description of a system, however, the probability distributions are indeed affected by such choices or shifts, just as position co-ordinates were affected in our elementary spatial example. And this is where the headache starts. Do the wave functions or probability distributions describe an objective reality, independent of the observer, or one which is observer-dependent? I would vote for the latter, noting that our understanding of the word "reality" needs to be analyzed.
Arjun
-----------------------------------------------------------------------------------------
From: E.D.
To: Arjun Janah
Sent: Mon, Apr 14, 2014 2:47 pm
Subject: RE: The Conscious Universe (not a poem)
Thanks Arjun, your examples clearly illustrate principles of probability. Of course, in classical probability, it is the observer's knowledge that changes suddenly with new information, not the reality itself, as in certain interpretations of qm. That's where the mind-bending paradox comes in.
E.D.
------------------------------------------------------------------------------------------
From: Arjun Janah
To: E.D.
Sent: Mon, Apr 14, 2014 3:31 pm
Subject: RE: The Conscious Universe (not a poem)
You are right. But we will go into q,m. later. The collapse of the wave function, as you saw, has classical analogies. It is the disturbance of the system by the observation (as in Heisenberg's uncertainty principle) that separates q.m. from prior physics. So let us separate those two things in our minds, although of course, they are related.
Arjun
Tuesday, March 18, 2014
The Conscious Universe
This was written in response to a Facebook post about the universe being explainable--as it is, at least in its physical aspect--in terms of forces.
That post expressed bewilderment as to why theists do not accept this.
Without wishing to get into that debate with the theists--and also with the proselytizing atheists--I offered instead, for whatever they are worth, these personal viewpoints, found over the years.
The Conscious Universe
I studied, worked in and taught physics, in the past, for thirty three years, and so had my fill of learning about, working with and teaching about forces of all kinds. After a while, I realized, however, that instead of just thinking of objects exerting forces on one another, we can also think of different parts of the universe (from electrons to galaxies) talking to one another--in languages of their own.
Of course, they rarely use Spanish or Swahili. At the fundamental levels, the parts of the universe use, instead, the languages that we call fields or interactions--gravitational, electromagnetic, weak and strong nuclear.
When I speak to an English-speaking student in my class to say, "Jie Wen, please come here." and she complies, or ask another, "Rafael, could you take this to the attendance office?" and he does that, then I could describe their actions, crudely, in terms of forces of attraction and repulsion. And in human interactions, we do have elements of compulsion.
But we could also describe this, perhaps more accurately, as I have done, as communication and considered response.
So also, when I push on a board eraser and it moves, I could describe this in terms of (electromagnetic) forces that I exert on the eraser, causing it to press and slide across the board--or I could visualize electrons on the surface of my hand "speaking to" their counterparts on the surface of the eraser and saying, "Sisters, could you move along? We are coming through."
This is not as far-fetched as it sounds, because each electron is, in fact, communicating with other electrons (and protons, etc.) via photons, which carry information back and forth, and there is a processing of this information going on in each cell of space-time.
In addition to this information-processing ability of the physical universe, there seems to be also an aspect of the universe that is not physical--not measurable. The universe, as we experience it, has not only quantities, like wavelengths, but also qualities, like colors. These exist in a mental dimension or sphere that parallels, to some extent, physical phenomena, but is distinct from these.
I do not think, personally, that consciousness "evolved" and is accessible only to humans and "higher animals". It seems more reasonable to think of it (as we do of the measurable or physical aspects of the universe--such as space-time, information, mass, energy, electric charge, etc.) as elemental, already present.
This consciousness manifests itself in our mental processes (which again need our bodies to appear, but are distinct from them).
To give an analogy, one needs to buy or build a radio or TV set to hear or view broadcasts, but no one would claim that the radio or TV set is producing these. They were already there, in the form of electromagnetic waves that spread out from the station. In our case, there is no separate station out there--no heavenly broadcaster. At least, I don't think there is. The universe itself is alive and conscious. We are part of that.
At the "lowest" level, we are nodes or routers in the network. But we, along with atoms and octopi, may be more than that.
-- Arjun
2014 March 19th, Wed.
Bensonhurst, Brooklyn, NY
Saturday, March 8, 2014
On Fascists and Socialists
On Fascists and Socialists
In this country (the U.S.A.), the terms "fascism" and "socialism" appear to carry roughly equal negative weight among those who are better educated. Among the general populace, the latter term is by far the more pejorative -- and frightening.
This is understandable in view of the propaganda that has been carried out here for so long. But this might not be the only country in which these two terms are being conflated. So it may be worthwhile to try to distinguish between the two.
In Europe, especially in Central, Southern and Eastern Europe, the genuine socialists and fascists have been at war for close to a century, perhaps much longer if one loosens the definitions.
Although fascism and socialism as we know them started in Europe, they have since spread all over the world. They manifest themselves in many ways in the systems of government that are in place, including what we have in this country. And the conflict, often violent, between these ideologies continues, all over the world.
The choice between them has always been stark. Although both fascists and socialists may be regarded as "statists", believing in the power of government, they differ as to who should be in control of the government and for whose purposes the government should work.
The fascists typically believe that government should be controlled by an elite that is either emerging or in place, typically drawn from the landlord, industrial and financial hierarchies, which are often interwoven. Anything that threatens this, they seek to extinguish, usually brutally and completely.
To do this, they appeal to nationalism, often of either the narrowest or the imperial kind, and they thrive on ethnic strife, seeking to divide the populace, and so also the workers, along ethnic lines. They attempt to villianize ethnic groups as well as unions and other alliances of workers. For the woes suffered by the populace and the country, they blame these groups and alliances as well as perceived external threats, seeking to link all of these together in people's minds. Constant propaganda, utilizing both commercial and governmental outlets, is used for this.
They then carry out programs of extermination and expulsion, both via paramilitary organizations and, once they have seized power over government, via the police and military, working both within and outside of the usual spheres of the legislatures, courts and executive offices. Highly discriminatory and oppressive laws are enacted and enforced. Slayings of union leaders, terror campaigns aimed at cowing villagers and workers, imprisonment and slaughter of opposition leaders and students, campaigns of ethnic cleansing and more are the hallmarks of the fascists.
The socialists believe that the workers (and peasants, in agrarian countries) should have greater influence and power over government and be entitled to a greater share of the wealth generated by the work they do. So anything that divides workers and peasants is something they consider pernicious and fight against. In doing this, they run up against the structures that have been historically established in most regions of the world -- structures in which peasants and workers are at the bottom of the feeding chains, with laws and force being used to keep them there.
So socialists typically find themselves in conflict with the affluent and typically blame certain sections of these for much of the ills of the populace and country. When they are able to wield some power themselves, true socialists attempt to undermine and take away the power accrued by the affluent elite.
I could go on and list some of the successes and failures of both fascist and socialist ventures. But that would take too long -- and so I'll let others have their say.
-- Arjun
Sunday, February 23, 2014
correspondence re: A Solitary Pawn
Re: A Solitary Pawn (not a poem)
http://science-sanity.blogspot.com/2014/02/a-solitary-pawn.htm
http://science-sanity.blogspot.com/2014/02/a-solitary-pawn.htm
| From: Arjun Janah | |||||||
| To: Alpan Rawal |
What I have been fumblingly trying to do here is explore, with a simple example, the possibility of extending classical space-time concepts a bit further, by adding a possibility dimension. So, although I have not broached it in the discussion of the example, the idea (by no means novel) is to generalize the simple space-time event to a space-time-possibility event, with each such having at least one extra co-ordinate that is related to the probability of the event, as measured by an observer in a given frame of reference (his/her reality), So instead of (xb,yb,zb,tb) we would have (xb,yb,zb,tb, ub) with u (b) = - C log P(b) being at best a first, tentative approximation or attempt at constructing this postulated possibility dimension co-ordinate.
So the concepts of a probability sample space may not be directly applicable. But thanks, yet again, for bringing these things to my attention.
Although quantum mechanics introduces probability into the heart of mechanics itself, it does much more than that. So my naive attempt will not be able to proceed further without much modification, if at all feasible, to embrace q.m. Even in the classical case I constructed, you can see that there are limitations that we should be aware of.
Regarding your earlier comment, note that if one were to include regions rather than just points, as elements in an ordinary 3D or 2D space, then the usual distance measure would also break down as a metric. It is only when we confine space-elements to points that our ordinary distance-measure in space, the one everybody is familiar with, is meaningful.
Arjun
-----Original Message-----
From: Alpan Rawal
From: Alpan Rawal
To: Arjun Janah
Sent: Fri, Feb 21, 2014 11:26 pmSubject: Re: A Solitary Pawn (not a poem)
Yes, but I am not sure if your restriction is compatible
with the definition of a probability sample space. In a sample space, as
far as I know, a union of allowed events is an allowed event, there is a
null event, and if a is an allowed event, so is the complement of a.
Alpan
On Feb 22, 2014 9:19 AM, "Arjun Janah" wrote:
Thanks, Alpan!To: Arjun Janah
I have confined the considerations, if you read carefully (and if I have, hopefully, been explicit enough regarding this) to discrete, point-like events, {a}, defined by space-time co-ordinates (ia, ta), with ia being the spatial co-ordinates of a square (say, of its center-point), and ta the time at which we imagine the pawn to be at the square. You are quite correct that, if you include unions of events, then all that I have tried to demonstrate breaks down, right from the start. The events should be point-like, with no fuzziness or possibility of overlap. In the derivations, I think I have been careful about this.
By the way, you had brought up, earlier, metrics between probability distributions. This is something very intriguing for me, something that I had never thought about.
Arjun
-----Original Message-----
From: Alpan Rawal
Sent: Fri, Feb 21, 2014 10:05 pm
Subject: Re: A Solitary Pawn (not a poem)
Consider the event c = a U b , where a and b are as you define and U is the union of the 2 events. Clearly c is true if a is true, thus P(c | a)=1, yet c is not equal to a.
Alpan
On Feb 22, 2014 8:09 AM, "Arjun Janah" wrote:
I have
typed this, over the course of the day, to fulfill the promise I made to Alpan -- that I would send examples where, at least in a classical
situation, the distance measure I suggested does indeed appear to
satisfy (with some conditions) the requirements for a metric. Here is
one such example -- a rather trivial one, elaborated on at great
length. It may have to suffice for a while.
A Solitary Pawn
http://science-sanity.blogspot.com/2014/02/a-solitary-pawn.html
Unfortunately, I have typed this much as I have grown accustomed, over the past several years, to write or type a poem -- that is, in a stream-of-consciousness mode. More significantly, I am sending it out to you much as I used to send out poems, until S. A., V. K. (in particular, with many helpful suggestions) and some others brought my attention to the problems arising from doing so. In other words, I am sending this out without careful looking over, deliberation and revision. So there may be much that is inelegant. More importantly, there are things that are not quite correct, that will come to light only through due scrutiny.
In part, this hurry is caused by the same pressures that made me take up that earlier faulty practice -- the pressures of the job.
< Several detailed excuses and explanations relating to then-recent job and family duties have been excised here by A.J., being considered, in retrospect, to be not directly relevant to this discussion. >
Today, I could not resist the luxury or temptation of jotting down (if what follows can be called that) the thoughts I have long vaguely had in mind about this example, bearing in mind also Alpan's recent useful critiques and comments. I enjoyed doing that, as it was an invigorating change from my usual mind-numbing routine and hectic scrambles. But now I will have to send this off, without due deliberation and revision, as I have to attend to job-related matters, which will occupy most of the remaining two days of the break.
< There has been more excision, here, by A. J., of irrelevant personal details.>
So please do not be upset if I do not respond immediately to e-mails in connection with this (including strong critiques, which I do actively solicit as they are generally very stimulating and useful) or other matters. Do send these right away, but you may have to wait awhile for my (coherent) response, as I will probably submerge myself for a while and surface again a while later.
The mathematically minded or the precisely logical may find my arguments at times clumsy, sloppy or fanciful. I apologize for that, not having had time to sharpen them as I should have. My only excuse is that physicists often proceed (informally) in this fashion -- and although I have long (for three decades, now) been away from physics, I still tend to operate in that mode on such matters.
Adios, amigos!
-- Arjun
A Solitary Pawn
http://science-sanity.blogspot.com/2014/02/a-solitary-pawn.html
Unfortunately, I have typed this much as I have grown accustomed, over the past several years, to write or type a poem -- that is, in a stream-of-consciousness mode. More significantly, I am sending it out to you much as I used to send out poems, until S. A., V. K. (in particular, with many helpful suggestions) and some others brought my attention to the problems arising from doing so. In other words, I am sending this out without careful looking over, deliberation and revision. So there may be much that is inelegant. More importantly, there are things that are not quite correct, that will come to light only through due scrutiny.
In part, this hurry is caused by the same pressures that made me take up that earlier faulty practice -- the pressures of the job.
< Several detailed excuses and explanations relating to then-recent job and family duties have been excised here by A.J., being considered, in retrospect, to be not directly relevant to this discussion. >
Today, I could not resist the luxury or temptation of jotting down (if what follows can be called that) the thoughts I have long vaguely had in mind about this example, bearing in mind also Alpan's recent useful critiques and comments. I enjoyed doing that, as it was an invigorating change from my usual mind-numbing routine and hectic scrambles. But now I will have to send this off, without due deliberation and revision, as I have to attend to job-related matters, which will occupy most of the remaining two days of the break.
< There has been more excision, here, by A. J., of irrelevant personal details.>
So please do not be upset if I do not respond immediately to e-mails in connection with this (including strong critiques, which I do actively solicit as they are generally very stimulating and useful) or other matters. Do send these right away, but you may have to wait awhile for my (coherent) response, as I will probably submerge myself for a while and surface again a while later.
The mathematically minded or the precisely logical may find my arguments at times clumsy, sloppy or fanciful. I apologize for that, not having had time to sharpen them as I should have. My only excuse is that physicists often proceed (informally) in this fashion -- and although I have long (for three decades, now) been away from physics, I still tend to operate in that mode on such matters.
Adios, amigos!
-- Arjun
Saturday, February 22, 2014
A Solitary Pawn
----------------------------------------------------------------------------
we get:
9.3) u(a|b) = u(b|a) (symmetry)
Finally, from our result (5c) P(c|a) >= P(c|b) x P(b|a)
(where, from (5a), tb lies between ta and tc)
and from our definition (6) u(b,a) = - C log(P)
we get
9.4) u(c,a) <= u(c,b) + u(b,a) (provisional triangle inequality)
A Solitary Pawn
This is best printed out for easy reference, especially once one comes to the numbered equations.
Imagine an infinite chessboard, populated by a single pawn sitting on, say, a black square. Let that solitary pawn move, after regular time intervals, in one of four directions -- forward, backward, left or right. Let the choice of direction be determined by the fall of a four-sided (tetrahedral) die, so that the probability of moving in each direction is 1/4.
(Since infinite chessboards cannot be displayed, a finite one is shown above.)
Although these are not the conventional moves for a chess pawn, let these be the rules of movement for our hapless one.
Let {i} be the (infinite) set of squares. Let {t} be the discrete times at which the pawn completes a move and arrives at a new square. Then (i,t) are the discrete space-time co-ordinates of the pawn, with (ia, ta) representing the event a -- the pawn's arrival at square ia at time ta.
The set of squares {i} could be numbered, for example, by the x,y co-ordinates of the squares relative to an origin square, which could be the starting square. Both x and y would be (signed) integers in this case, with the set {i} being represented by pairs of these signed integers.
We might wish, instead, to choose a single index number, i, to represent each such pair (x,y) and so also each square. Whether that is feasible in our two-dimensionally infinite case, I will leave for you to figure out, as it is beyond this non-mathematician.
Let P(a) be the probability of event a, so that P(a) = 1 if a is true. All the probabilities are fixed by the initial position of the pawn at the time we start our observations. But that is only because we found the pawn at that position at that time.
Of course, as we follow the pawn in an experiment/game, we will see it take a particular path, determined by the successive rolls of the die. That will be our particular, experienced reality. But there are other possibilities, each deserving the title of "reality" if realized. In other realities, all obeying the same rules of movement, the pawn would take alternate routes. We believe that if we could explore all these possibilities or alternate realities (which in this case happen to be infinite in number) we would then be able to experimentally determine the probabilities P(a).
The only way to approximate such a thought-experiment with a real one would be to repeat the game a very large number of times, starting always with the pawn in the same position and taking the starting time to be the same. The probabilities so obtained should get closer and closer to the actual (theoretical) probabilities. But to get them exactly, we would need an infinite number of trials. This infinite repetition would be required even for a game with a finite number of moves. In practice, we could (for a finite game) repeat the experiment a tiresome number of times and then say that we have come close enough.
But what can we do instead, as lazy theorists, averse to wasting our time mucking around with a pawn and a tetrahedral die, or giving up on our attempts to purchase or construct an infinite chessboard? We can assume that the die is a fair one, and then apply the venerable laws of the Bernoullis et al to calculate the probabilities P(a) -- as far as we are able to. We could do this, by hand, for, say, 4 successive moves -- or, by using a computer, for many more.
But if we are lazier still, we can refrain from even bothering with numerical calculations. Instead, we could just explore the game using algebra, hopefully establish some features of how the game plays out, declare victory and retire. Let us take this route, favored by the laziest. Unfortunately, this also requires some knowledge, which may be acquired, along with some acumen, which we may or may not possess. But no matter.
Let P(b|a) be the probability of event b, given event a. More explicitly, let P(b|a) be the probability of event b occurring, given that event a occurs..
I have adopted here a notation, P(b|a), for the relative probability of events, that is closer to that suggested by blank01 and Alpan (Rawal).
Let us write down some useful facts or results regarding this quantity, valid in the situation we are considering -- the lonely pawn moving on the squares of an infinite chessboard at regular gong-chimes, according to the throw of a fair tetrahedral die.
Recall that b and a are events, being the presence of a pawn at a certain square at a certain time, and recall also that we can represent these events by their space-time co-ordinates, so that a = (ia, ta) and b = (ib, tb).
1) 0 <= P(b|a) <= 1
This follows because P(b|a) is a probability, which is a ratio of cases (possibilities) resulting in the outcome to the total number of cases. This ratio has values ranging from 0 (impossibility = in no case) to 1 (certainty = in all cases).
2) P(b|a) = 1 if and only if b=a
If b=a, then P(b|a) = 1 follows from the definition of P(b|a).
If b =/= a (by which we mean that b is not the same as a) and P(b|a) = 1, then this means that a move into square ib at time tb is a certainty, while movements into other squares at that time (tb) are impossible, with zero probabilities. (The sum of all probabilities at a given time must equal 1). Simply from symmetry considerations, this is absurd. (The movement-rules and the chessboard imply symmetries in both time and space). So P(b|a) cannot be 1 in this case (b=/=a). So we have proved the "only if", albeit by waving my hands a bit.
2a) P(b|a) = 0 if ta=tb and ib=/=ia
This follows because the pawn cannot be at two different squares, ib and ia, at the same time, tb=ta.
We are neglecting the pawn's jump-time, during part of which some might argue the pawn is in both of two adjacent squares. We could take the jump to be instantaneous, or the jump-time to be so small compared to the time between chimes that we can exclude it from our considerations here. Even if this is not the case, from the way we originally defined the times {t}, stating that these are the discrete times at which "the pawn completes a move and arrives at a square", the jump-time need not concern us.
3) P(a|b) = P(b|a)
Since the rules of the game are completely symmetrical with respect to both time and space, this follows. If, for example, b is at a later time than a (tb > ta) then, along every path moving forward in time, linking the prior event a to the later event b, we could retrace our steps, at each step (jump between squares) using, this time, the probability that the pawn was at an adjacent square in the previous time interval. This will again be just 1/4.
So the probability calculations going forward and backward yield the same result, and (3) follows.
I have waved my hands again, but, hopefully, not with any sleight of hand, intended or inadvertent.
4) The probability of event c, given event a and an (intermediate) event b, is, by the multiplicative (AND) law of probability:
P(c|b|a) = P(c|b) x P(b|a)
5) Next, consider all possible paths between events a and c, with {b} being the set of all possible intermediate events, intersected by these paths at time tb, where we have the constraint:
5a) ta < tb < tc OR ta > tb > tc
Since the events {b} are exclusive events (being point-like, with no overlap) and, by our previous statement, exhaust all the possibilities at time tb, we have, from the additive (OR) law of probability:
5b) P(c|a) = Sum over {b} of P(c|b|a) = Sum over {b} of ( P(c|b) x P(b|a) )
Since each term in this sum is positive, being a probability, it follows that:
5c) P(c|a) >= P(c|b|a) = P (c|b) x P (b|a)
where we have retained only one of the terms on the right, that corresponding to the particular event b. (The inequality follows because the sum of positive terms must be greater than or equal to any one of those terms.)
Let us now define a tentative distance-measure (a candidate "metric") u(b,a), between two events, b and a:
6) u(b,a) = - C log ( P(b|a) )
where C is a positive constant and the logarithm of a number, x, is defined as the power, y, to which the base 10 must be raised to give that number:
7) x = 10^y <=> y = log (x)
with ^ being used for power (exponent) and <=> standing for symmetrical implication (if and only if).
Note that since 10^0 = 1, log (1) = 0.
Also, since 10^(-infinity) = 0, log (0) = - infinity.
Those who know mathematics will not like our use of "infinity". But, here again, no matter! We shall continue merrily, pooh-poohing such concerns.
For those who might have forgotten their high-school logarithms:
Let X = 10^a and Y = 10^b
So that log(X) = a and log(X) = b
Then X x Y = (10^a) x (10^b) = 10^(a+b)
So that we have the useful result:
7a) log(X x Y) = a + b = log(X) + log(Y)
So "taking logs" converts a product to a sum. When we were high-school students, we used logarithm tables to to do difficult multiplications (and divisions, which reduce to subtractions of log values). The slide rule was also based on this.
We shall use this result ourselves, below.
u(b,a) could be thought of as the "unlikelihood" of event b occurring, given that event a occurs. Or we can think of it as a distance between (point-event) possibilities in a possibility-space. So we could also refer to it as a measure of "possibility distance".
But we are jumping too far ahead. Can we really think of this quantity as a "distance" in a meaningful way? Is it a proper distance measure -- a true metric?
The general requirements for a metric d(a,b), in a space in which a, b are "points", are:
8.1) d(b,a) >=0 positivity
8.2) d(b,a) = 0 if and only if b=a distinct events separated by non-zero distance
8.3) d(b,a) = d(a,b) symmetry
8.4) d(c,a) <= d(c,b) + d(b,a) triangle inequality
(In an Euclidean space, 8.4 means that the shortest distance between two points is along the straight line between the two. Detours are longer.)
If these conditions, 8..1 -- 8..4, are satisfied, then the space of points {a,b,c...}, with the metric d(b,a) used to measure distances between points, forms a "metric space", with many familiar and useful properties. The two dimensional surface of a blackboard, with distances between chalk points measured with a ruler, is a familiar example of such a space. If the points on the board are represented by x,y co-ordinates, then the distance can be written, using Pythagoras' Theorem, in terms of the differences between the x, y coordinates of two points. So, because Pythagoras (and all of Euclidean geometry) applies to the space of points on a blackboard, it is called an Euclidean metric space.
The 3D space in which the teacher who writes on the blackboard (or whiteboard, alas) lives and moves about, along with his/her students, is also an Euclidean metric space.
There are, however, many other metric spaces which are not Euclidean. They are still metric spaces because we can define metrics (distance measures) on them that obey the rules listed above.
Let us now attempt to see whether the unlikelihood function u(a,b) does indeed behave like a proper measure of separation -- a metric, d(a,b) -- should behave.
Or does it have behavior problems that can or cannot be dealt with? Teachers, take note!
We shall confine our attention to the case of the pawn on the infinite chessboard. So one should be careful not to generalize the results we obtain below to other situations, without taking note of differences that may arise.
From the definition of the unlikelihood (6) and of the (relative) probability (1), we see that:
since P <=1
u = - C log(P) >= log (1) = 0
where the inversion of the inequality comes from the negative sign in the definition of u. (Recall that C is positive, so - C is negative.)
So we have established:
9.1) u(b,a) >= 0 (positivity)
Combining our result (2) P(b,a) = 1 if and only if b=a
with our definition (6) u(b,a) = - C log( P(b|a) ) we get:
9.2) u(b,a) = 0 if and only if b=a (The possibility-distance between distinct point-events is non-zero.)
Combining our result (2a) P(b|a) = 0 if ta=tb and ib=/=ia
with our definition (6) u(b,a) = - C log( P(b|a) )
we get:
9.2a) u(b,a) = + infinity if ta=tb and ib=/=ia
This is a special case of the more general result that two events that cannot both be true are separated, in the "possibility space", by an infinite distance, as measured by our proposed metric. Although this is a digression from our attempt at establishing that our candidate is indeed qualified (under certain conditions, as for a handicapped worker) for that job, this result is worth noting.
and from our definition (6) u(b,a) = - C log( P(b|a) )Next, from our (hand-waving) result (3) P(a|b) = P(b|a)
we get:
9.3) u(a|b) = u(b|a) (symmetry)
Finally, from our result (5c) P(c|a) >= P(c|b) x P(b|a)
(where, from (5a), tb lies between ta and tc)
and from our definition (6) u(b,a) = - C log(P)
we get
9.4) u(c,a) <= u(c,b) + u(b,a) (provisional triangle inequality)
where again, as for (9.1), the inversion in the inequality comes from the negative sign in the definition of the possibility-distance.
The provision on the validity of 9.4 is that of (5a), namely:
9.4a) ta < tb < tc OR ta > tb > tc
and this is a handicap that our candidate and his/her employers have to deal with.
Enough for now! Apologies for the infliction. Print it out and read it at your leisure, if you feel so inclined. Do let me know where the reasoning is faulty or sloppy. I am sure you sure you will find several instances of these. This is a quick and perhaps premature, send-out -- but I may not be able to think about these things again for a while -- or ever!
-- Arjun
===============================================
Correspondence on this follows at the next post on the blog.
===============================================
Correspondence on this follows at the next post on the blog.
===============================================
Subscribe to:
Posts (Atom)