Well, I am back, sorry I missed you.
saviourmachine said:
Oh, take your time. I'll check every week or so. Maybe you can another pseudoniem, doctordick. Thanks for your time till now.
I have no idea why what happened happened. Computers are strange things sometimes: apparently it never reset my password or sent me an e-mail. I kept trying variations on what I thought I had used and suddenly one worked
So: my response to "saviormachine"!
saviourmachine said:
Mmm, very interesting stuff. I did electronical engineering, so I didn't know this theory yet.
I think you should be aware of the difference between "a theorem" and "a theory". A "theorem" can be proved, a "theory" can not, it can only be defended.
saviourmachine said:
2. Symmetry as non-informational overhead; reducable according Shannon's or Kolgomorov's definitions of 'information'
3. Symmetry as lack of knowledge about the difference between (a part of) an entity and it's symmetrical counterpart.
Two and three are opposite sides of the same coin so to speak; one is entirely equivalent to the other.
saviourmachine said:
I do not really understand you. There is a fundamental observation: symmetry, which denotes a lack of knowledge, what can be solved by using the concept 'conserved quantities'. In what way is such a concept not an axiom? An axiom like: there exists a thing as angular momentum. Or an axiom like: we don't know the centre of our universe.
I could be wrong but it seems to me that you are confusing the two different issues: the symmetry (a representation of some particular mode ignorance – or indifference, per selfAdjoint's perspective) and the deduced conserved quantity required to enforce or accommodate that symmetry (a requirement established by internal self consistency).
I don't know that I would use the phrase "symmetry denotes a lack of knowledge"; I would rather express it as "symmetry can be seen as a lack of knowledge". But of course, my main complaint is the vagueness of English anyway so I am not really aware of what relationships are implied in your head when you use the phrase. Thus it is that anything I say on the actual meaning of your comment is no more than an opinion. The symmetry and the conserved quantity are related through the necessity of maintaining that "non-informational overhead" you referred to in #2 above. As you said, the representation must be reduceable and the mathematical constraint which provides that reduction is the conserved quantity.
Take for example, the consequences of not knowing the "center of our universe" (the origin of the coordinate system used to represent positions in our problem solving). If we don't know where the origin is, we don't know[/color] the particular value of any position. It follows that there is a different solution for every possible position of that origin. The other side of the coin is, if we are able to find a solution (say x as a function of t) we can clearly take that particular solution and deduce exactly where the origin was.
Since we now have information which was not available in the original problem given to us, something here is logically inconsistent.

Conservation of momentum is a mathematical relationship on that solution which makes all the various solutions (the collection of solutions, each of which would independently allow deduction of a different origin) equivalent to one another. It is the relationship which provides the required reduction in information. What I am giving you is no more than a different perspective on Noether's theorem.
My purpose in stating things in such a strange way is to bring out the obvious inconsistencies implied by presuming we know things we cannot possibly know (setting up a coordinate axis when we don't know where the origin is). Remember, my sole purpose is to establish the parameters on my thoughts which will assure me that I am not inadvertently presuming information I do not have. Noether's theorem is an excellent example of how easy such a thing can happen and I don't think the common presentation brings the most important issue to the forefront.
The axiom[/color] is: we are ignorant of something. When we set up our coordinate system, that ignorance is not explicitly displayed: blind usage of the coordinate system ignores the embedded ignorance. It follows that we must have a constraint which will yield up that same ignorance in our final results. It is the relationship I am trying to bring to your attention, not the solution.
At the moment, let me list what I have presented to date:
1)The existence of "squirrel thought" (intuition, zen, fundamental knowledge) which is not a process amenable to logical analysis because of the extreme limits on logical analysis but, none the less appears to provide very effective solutions to very important problems. This is the only source of solutions to any conceivable problems and we must keep its failings in mind.
2)The existence of "mathematics", a mental construct capable of extending logical relations far beyond what can be held consciously available for logical analysis. It constitutes a "very effective solution" (i.e., an intuitive construct) which has acquired far reaching agreement as to meaning and internal consistency. It is the only collection definitions which are accepted widely enough to provide anything close to "exact" communication. If a science wants to be exact, it must present its ideas with the same exactitude expected of mathematics.
3)That any representation of information in a mathematical form makes presumptions which must be carefully analyzed. We must make sure our ignorance is maintained in our analysis (we must not claim or imply that we know things we cannot know).
The next thing I would like to bring forth is apparently very difficult to communicate and I beg your indulgence. I tried to get people to think about this issue when I posted a simple question back in May of 2004. I totally failed and I am quite sure the fault was mine for not putting it in a form they could identify with. The original question was buried in a large post to Russell E. Rierson but is more easily discovered through a post on that thread by
baffledMatt. If you want to look at my earlier attempt, a quick perusal of that thread might be a place to start. The original question was, "how does one tell the difference between an electron and a Volkswagen?" The point was that context is the single most important piece of information required to answer the question, a piece of information seldom even considered as significant.
In order for you to comprehend what I am getting at, consider the following steps. Your purpose is to examine an event which took place at the point in space referred to as (x,y,z,ict) (if we are to be exact, your approach must be general relativistically correct and we will use Einstein's picture). Your problem is to identify the object which was present at the event (for the fun of it we will make the answer to the question very simple; it was either an electron or a Volkswagen).

First, can you go and look at the event? Of course you cannot; to do so would require you to have a time machine.

Exactly what information do you normally have to go by in the situation where such a question might arise?
It should be clear to you that what you really know (or at least presume to know) is the collection of events which immediately surround the event of interest.

If the object were a Volkswagen, the significant surrounding events might include a road, a driver, maybe some trees or a building. If the object were an electron, a more reasonable set of surrounding events might include and "electron gun" or perhaps a wire or maybe a lab table. So the first step is to identify the most significant of these surrounding events.

But that is just a restatement of the original problem. Again, you can't go look; you must depend on what you already know. Fundamentally, you need to know the distribution of events in the vicinity of (x,y,z,ict).
For the moment, let us not worry about the process by which you come to know the existence of those events in the vicinity of (x,y,z,ict). What is important is the distribution of events themselves. What I am getting at is the fact that identification of any event is essentially a presumption of what distribution of surrounding events will be accepted as a valid set.

In other words, if I were to give you a specific distribution of events the distribution itself would express the identity of its various parts.
What is important here is that the presumptive necessity of identifying the fundamental entities making up a particular situation it totally erroneous.

If I were to give you a mathematical expression (a function of many variables) which yielded the probability of finding a specific distribution of events as a function of time (that distribution being the collection of variables looked at as coordinates, [x,y,z,ict], of specific unnamed events) then that expression itself fundamentally characterizes the identity of all those events.
This is a very simple concept with very far reaching consequences; particularly if we wish to keep our minds open to all possibilities. As I said, identification of a particular event is paramount to establishing a very large set of acceptable and unacceptable peripheral events which are, in the final analysis, only vaguely specified. This is a very poor basis for "exact objective analysis". One should not label things first and then attempt to explain the labeled things behavior; one should examine and attempt to explain the behavior itself: when, where and under what circumstance the behavior occurs. "When" and "where" is coordinate specification and "under what circumstance" is a specification of associated behavior found at a related "when" and "where".
Let me know if this perspective on the problem confronting scientific investigation makes any sense to you at all!
Have fun -- Dick