Sunday, February 23, 2014

correspondence re: A Solitary Pawn

Re: A Solitary Pawn (not a poem)

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.


-----Original Message-----
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.


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.


-----Original Message-----
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.


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

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


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.

Next, from our (hand-waving) result  (3)   P(a|b) = P(b|a)
    and  from our definition                     (6)   u(b,a) = - C log( 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

  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.

Wednesday, February 19, 2014

On Knowledge and Consciousness

To: Alpan Rawal 
Subject: Re: The Universe as a Simulation (NY Times) + more
From: Arjun Janah
Date: Tue, 18 Feb 2014 09:23:34 -0500


Thanks for the response. I was thinking of just two events, A and B -- with A being, say, an electron present at a "fuzzy space-time point" x, y, z, t, with fuzziness dx,dy, dz, dt., and B another such event. 

Then P(A, B) would be the probability of A occurring, given that B is true. This is what I meant by relative probability.  P(B, A) would be the converse.

For the benefit of some of the others:

P(a,b), being a probability, has a range from 1 (certain) to 0 (impossible).

If we take d(a,b) = - ln (p(a,b))   where ln(x) is the logarithm to the base e (actually any base will do):
then this quantity has the range:

- ln(1) = 0     to     - ln(0) = - (- infinity) = + infinity   

(if you will excuse the crudities).

In order for this to be a proper distance measure (a true metric) between any two (fuzzy) space-time points a, b, the following should be satisfied:

1. d(a,b) >= 0                             (positivity)
2. d(a,b) = 0   if and only if a=b   
3. d(a,b) = d(b,a)                        (symmetry)
4. d(a,b) + d(b,c) >= d(a,c)          (triangle inequality)

Alpan is pointing out that the last two may not be satisfied by this simple measure (which Alpan refers to as the relative entropy), but that another, more complicated measure exists for which these are satisfied. I am not familiar with that one.  But many thanks, Alpan!  (I may have paraphrased Alpan too simply or incorrectly.) 

In any case, my earlier statement that this simple measure satisfies "nearly all (but not all) of the mathematical requirements for a distance measure" appears to be incorrect, from what
Alpan writes. It has been over forty years, but I recall working things out and finding that only one of the requirements was invalid. But I could be quite wrong.

-- Arjun

-----Original Message-----
  From: Alpan Rawal

 To: Arjun Janah 

 Cc: blank01 ; P. B. ; V. K. ; E. D. ; N. D. ; A. B. ; J. V. ; S. B.  
 Sent: Tue, Feb 18, 2014 1:46 am
 Subject: Re: The Universe as a Simulation (NYT) + more

Dear Arjunda,

I think you need to define what you mean by "relative probability". Ordinarily the relative entropy between two probability distributions is not a true metric (does not satisfy symmetry and triangle inequality), so I am not sure if that is problematic for your notion of distance. On the other hand, the square root of the Jensen-Shannon divergence between two distributions is a metric, but I do not know if that works for your argument.



Note added:   See the next two posts ("A Solitary Pawn" and correspondence on that) for a follow-up on this.
On Tue, Feb 18, 2014 at 12:25 AM, Arjun Janah  wrote:

I am skeptical of the assertion (hypothesis) that the universe might be a computer simulation.

But the prefacing description, in the article, about the reality of the holos (the virtual space in which mathematical realities exist) is worth reading about. I once read an interesting imaginary travelog by a writer whose name I forget. In that travelog, the writer met with imaginary mathematicians and physicists in various countries to explore this concept.

Also, the idea that we were reared upon in our schooling -- that the universe "consists of" space, time, and momentum-energy -- with each of the pair-members being transmutable a la Einstein-Minkowski-Poincare, needs expansion and modification to:

(a) include at least one more measurable quantity -- information;


(b) acknowledge that there is also (the perhaps unmeasureable thing called) consciousness -- not just the preserve of humans and "higher" animals, but an elemental constituent or "dimension" of the universe as we know it. 

Let us first consider (a).

Probability enters into quantum mechanics in a fundamental way -- and probability and information are different measures of the same beast.  The universe, every space-time point, every elementary particle, processes information and communicates with other parts of the universe, with information "moving" back and forth. This is true for electrons, atoms, cells, humans, trees, planets and stars.  The concepts of fields and forces may be translatable into that of language. Of course, the language of communication between electrons is quantum electrodynamics, not English, that between cells is mainly chemical and electrical, and so on. 

The uncertainty principle of Heisenberg and the probabilistic nature of quantum mechanical calculations (which after all simulate what an electron does to "decide" where to go) may perhaps be a consequence of the limitation of processing power of a sufficiently small section of the (phase-space) continuum -- in other words, not just the physicist with his/her equations, but the electron itself does not have the ability to decide for sure.

That is one way of looking at it. Another is to view the negative log of the relative probability of two space-time events (a quantity directly related to information and with a range from zero to positive infinity) as a measure of "distance" between those two events. 

This would be a distance in a virtual "possibility" space in which there can be different frames of reference and a sort of "movement" -- corresponding to alternate realities and communication.  Two parts of the universe that communicate with each other, and so gain knowledge about each other,  move closer together in this virtual space, as measured in this way.  The measure itself obeys nearly all the mathematical rules of conventional distance measures -- but not all.

Just as (long before Einstein et al) English (and surely other languages) frequently used the same words for temporal and spatial intervals and separations -- as in "a long/short time", "vowel length (duration)", "near future", "distant past", etc., acknowledging the similarity between these two measures, so also English used spatial-distance terms for probability, as in "a remote chance", "a close call", "far-fetched", "nearly correct", "almost true" and spatial-location terms for possibility/state -- "This is where we're at.", "We will try to reach that goal.", etc.

So, while the probability dimension [or u= - ln (P)] has not yet been put on the same footing as x, y, z, and t, I believe it will, although perhaps not so simply.

Let us look next at (b).

The idea of a conscious universe goes back to Jagadish Chandra Bose as a modern exponent but is actually at least as old as humankind. By defining the "physical universe" to exclude such things as consciousness we could try to save the old framework, but when we look at elementary quantum mechanics and the problems that it has with observation and wave-function collapse, we see that the old framework has already crumbled.

If consciousness is thought of as elemental, pre-existent, much as space-time is, then we see it forms an "orthogonal dimension" to the old "physical universe", but one in which measure (which is a method of comparison by repeated mapping of a standard and a counter) is not applicable.  "Physical" quantities, such as wavelengths, have their shadows or correspondences in this "dimension" as qualities -- colors, tones, etc.

The two modifications (a) and (b) to classical ideas may of course be related.  Naively speaking, how can one part of the universe communicate with other parts and "know" about them, without being conscious? There could of course be objections to this -- what if they just communicate like network routers do, processing information, surely, but not consciously?  But perhaps routers, being parts of a conscious universe, are in asense conscious after all, and could not do their jobs (tunneling down to the electron level, where electrons are pushing and pulling or "speaking" to one another via e-m fields, with all of quantum theory at play) without this?

Well, that's my two-cents worth on (a) and (b). My ignorance and my limitations in this, being away from physics now for three decades, will be evident.  These are thoughts I had while a student in India over forty years back, doing my Physcs B.Sc. with Niloy and Arindam who are on this mailing list  -- and there (virtually speaking) those thoughts still remain. Not even teaching physics at the high-school level for the past twelve years, things are fading fast, and the fuzziness of the microcosm appears to be manifesting itself more and more in my thoughts and attempts at communication.

More practically, however, the article mentions a paper which purports to offer a measurable means (by measuring asymmetries) of deciding whether indeed the universe is a simulation.

Is the fuzziness of quantum mechanics the by-product the limitations of a simulation on a classical computer?  I very much doubt that.  The computer itself must be quantum mechanical, I believe -- in which case, it is the universe itself, the physical part and almost surely also the non-physical, in interaction.

Metaphysically speaking,


P.S.   N. D., my last (more practical) communication with you, regarding your query about V. S., bounced back. This has happened before.  Methinks that U. Conn's and AOL's routers have distanced themselves from one another in the virtual manifold.  I will try to resend it.     
Note added:   See the next two post ( "A Solitary Pawn" and correspondence on that)  for a follow-up on this.