correspondence re: A Solitary Pawn

  

Re: A Solitary Pawn (not a poem)
            
     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

 To: Arjun Janah 

 Sent: Fri, Feb 21, 2014 11:26 pm
Subject: 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!

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

    To: Arjun Janah
    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