I just got back and was quite astonished at the shear volume of the response to my last post. And, please do not think that these people who tell you what I am doing understand what I am doing. It is quite clear that the significance of my classification between "knowable" and "unknowable" is not comprehended by most if not all of the people here. (To anyone who reads this: if it turns out that you do understand my meaning, I apologize for thinking it possible that you didn't.)
Back when I was a student, I used to often ask questions which no one else seem to ask. For example, suppose a professor were to describe some experimental result where a photon emitted from one point was absorbed somewhere else and he was concerned with what happened at the second point. On an occasion like that, I might be tempted to ask "how do you know a photon crossed between those points". Of course, his answer would be "because of what happened!" If pressed on the issue, he could usually be driven to the point that the results were the proof of the occurrence and that the actual fact of the photon crossing the gap was an "unknowable" thing.
Now, when my uncle used logic to determine that the horse had to be behind the barn (where he couldn't see it), I could go behind the barn and see if he was right. But when the physicist used his logic to determine that a photon had crossed the room, he was unable to give me an alternate way to prove this. Physics is chock full of situations like this and I often pressed my professors on those issues. The word they most often used to describe the situation was "unknowable": that is, that there existed no such proof and the "true" existence of these things had to be taken on faith.
This led me very early to divide my knowledge into two categories, "knowable" and "unknowable". "Knowable" was what I knew to be true, and "unknowable" was what I believed to be true but couldn't prove. Now, over time, it became quite evident that my belief in those "unknowable" things was supported by the fact that the laws of physics (a set of rules) together with that belief made a great many predictions of "knowables" possible.
When I first studied physics that "knowable" category was always growing; but, as time went on, that growth began to reverse. The more I learned, the more things seemed to be moving into that "unknowable" category. Now the philosophers all told me that, if I look at things from that perspective, everything is "unknowable"; however, they can no more prove their position than that physicist can prove the photon crossed the room. Thus, the only rational position is that what I think I know can be divided into two categories: "knowable" and "unknowable". And that we fundamentally do not have the power to determine if any given idea belongs in one or the other category.
So, you ask, what good do the categories serve if we have no way of determining which category any given idea should be stuck in? The answer to that question is actually quite simple. The categories are quite different when it comes to the consequences we should expect. The knowables are true! They are what is and cannot be changed. The "unknowables" are quite different. They are essentially figments of our imagination, the existence of which are defended by the "fact" that they (together with the rules) explain the knowables.
Now there are two variables here (things which can change). Both the "unknowables" and the "rules" can change while the "knowables" cannot change. We are free to create any "unknowables" we choose just so long as they obey "all the rules". And we are free to create any "rules" we wish so long as all the "knowables" and "unknowables" obey those rules. This is no more than an abstract statement of what science does. They continually invent new entities which provide explanation of the phenomena they examine. At the same time, they occasionally change the rules a little.
I hope that clears up what I mean by "knowable" and "unknowable".
Of importance is the issue that if it is the scientist's intent to explain the "knowables" by inventing "unknowables" and trying to find a set of rules which explain that whole of "knowledge", then the full consequences of the problems structure should be taken into account. If one recognizes the fact that freedom exists both in the invention of entities and in the creation of rules (the laws of physics depend on what you believe exists) then there is considerably more to think about than is contemplated by the modern scientist.
Now, if you will accept that both the rules and the entities are variable, let us proceed to my "proof" that it is always possible to put the rules in the form F=0. I will attempt to spell out that proof in detail.
First, I define reality to be a set of numbers. By the way, Harv has complained about my refusal to use R1 to refer to this. I have no problem at all with the use of R1; my problem is rather with his clear inability to separate logic from the imbedded implications of the words (the fundamental reason I move to numbers). (Harv and Aurino would prefer a set of concepts, words, ideas, whatever so I will show explicitly that this proof does not depend on the numerical property of numbers.) First, it should be comprehended that the set of numbers constitutes the "knowables" only (as, prior to developing rules, I am free to create "unknowables" to my hearts content).
What I have to work with is a collection of lists of numbers (or lists of words, ideas, concepts, whatever). Each list is what I am referring to in my paper as an observation. I have attached a number to every list which I have called "time". My first step is to put all those lists up on the wall of my infinite office where I can examine them in detail. I know that there must be a finite number of lists and that each list must contain a finite number of numbers (or words, ideas, concepts, whatever) because if they were infinite, I could not examine them. If their number were infinite, no matter how many I examined, there would exist some yet to be examined. My conclusions must be based on the observations I have examined.
Since I have attached a number to each list, the actual list is a function of that number. That is to say, for every given number in the set I have used, there is a particular list being referred to.
The times (the number attached to a list) are all different by construction as I do not use the same value of this parameter twice. The lists however, may not all be different. As I comment in my paper, if they were all different, then, specifying a particular list would identify a particular time. Just go find that list in those tacked to that wall and see what time (the attached parameter) was attached to it. In that case, time is an implicit function of the list.
Now, the rule which I am looking for is the rule which yields the lists I have tacked on the wall (at this point in the proof, the lists consist entirely of the "knowables"). I will take it as well understood that there exists an infinite number of rules which will yield exactly the given lists. I can very easily show this for numbers and showing it for lists of words, concepts or ideas is just a complication.
In order to simplify the definition of F (the function which I will show capable of defining the rules) I will first require that all the lists contain exactly the same number of numbers (or number of ideas, concepts or words). I accomplish this result by adding "unknowables". Merely find the longest list and then add to the other lists enough garbage to make all the lists the same length (called padding in computer jargon). Note that these additional entries are "unknowables" (pure figments of my imagination). Note also, that once I add them to the lists, they do not change; they are as fixed and immutable as the entries in the original set I was given.
The next issue is to make sure that there are no two lists alike. The solution is very simple: if you find two lists alike, just pick a collection of numbers (or ideas, concepts or words), all different, none of which appear in any given list where the number of numbers (or ideas, concepts or words) is equal to the number of lists and add one of these to each of the given lists. Those two lists will no longer be alike and the operation makes no lists alike which were not alike before.
All you do is continue this process until no two lists are alike. Sure, you may have to add a whole slew of "unknowables" but what difference does it make? In the end we will require they all obey the same rules so all we are doing is trading unknowables for simplifying the rules. When we finish, we then know that specifying a list is equivalent to specifying a time (given a particular list, there is only one in the entire set which is identical to that list and its time is specified by the original notation).
Now, take that step one step further (though it is only one logical step, the actual requirement is quite a bit more complex). The next step is to use the addition of "unknowables" to make every list different even when any arbitrary element is removed from any list. Just start by removing any arbitrary element from the a list and then remove any arbitrary element from another list. If the two lists are the same, add an unknowable to every list as specified in the previous paragraph. Those two lists will no longer be alike when that particular operation is performed and the operation creates no new such problems (only those which already existed still exist).
Since the number of lists is finite and the number of entries on the original lists (those are the ones causing the problems, not the ones we added) is finite, the number of ways one element may be removed from two lists is finite. It follows that eventually, the procedure will terminate.
What do we have now? We have a set of lists which are different (one from another) even when any arbitrary element is removed from any list. This means that if I am given a list which is missing one element, there is only one list tacked up on the wall which contains exactly that collection of numbers (or words or ideas or concepts) when one element is removed. If that is the case, I can find that list and compare that list to the one which I was given. Since there is only one list in the entire collection which matches the one I was given, I can look at the actual list (which now consists of both "knowable" and "unknowable" entries) and determine exactly what the missing element has to be.
If I can define a procedure for finding a missing element of a list given all but one of the members of the list, I have defined a single valued function of (n-1) variables which will yield the correct nth variable. This is exactly what is expressed in equation 1.1 of my paper. Assign a number to each of those labels Harv and Aurino demand we use, and the result is an ordinary mathematical function (a tabular defined function) which can now be changed to F=0 (equation 1.2) via a simple reorganization of terms. The constraint F=0 will be obeyed by every list in the set (note that the function is independent of time). Any set of data which does not obey F=0 cannot be a member of the set of observations.
What I have proved is that there always exists a set of "unknowable" data (total figments of my imagination - created ideas or entities) which will make it possible to express the rule which constrains the data to what is seen to be written F=0. Notice, I have proved the set exists by construction, I have not proved that my attack is the only possible attack. There may very well (and probably are) an infinite set of ways to accomplish that same result. My only interest is that it can be done and I have proved that explicitly.
When this whole discussion started, we began by pointing out that scientists look for entities which can explain what is seen and the rules those entities must obey. What I have shown analytically is that the problem, as conceived of by the science community, contains more variability than is required. If you are allowed free choice of rules and free choice of entities, the number of solutions is infinite. The simplest solution is to pick a simple rule (F=0) and then look for the entities which are required to explain what one sees. The rest of my paper continues from that. Please do not just scan the results. If there is anything you do not understand, let me know and I will do my best to clarify it.
I hope that makes more sense to you than what Yanniru et. al. have been feeding you. I am afraid that most of the people here (Paul excluded) have more interest in discrediting me than they do in paying any attention to what I say.
Yanniru continues to bring up the issue of symmetry. My position on that is also quite simple. Symmetry and ignorance are intimately related concepts! I do not believe Yanniru understands the relationship. I thought I had already posted an essay on this subject; however, as the problem of finding it is beyond my meager abilities, I will present the subject again.
You will be told by anyone competent in physics that symmetry arguments are the most powerful arguments which can be made. One of the problems with higher education is that the professors will often present that idea to their students as if it is true because it has the weight of authority behind it or the students will accept it as true because they think that, if all the really bright people believe it, it must be so which amounts to the same thing.
In actual fact, symmetry arguments are so powerful because they are the only arguments which can actually turn ignorance into knowledge. The essential characteristic of any symmetry argument is that information which is not available in the problem can not be available in the solution of the problem (mathematics cannot produce something from nothing). The first part of understanding that statement is understanding that symmetry is indeed a statement of ignorance.
Let me lay out a few specific examples so that you can understand the connection between symmetry and ignorance.
1) If an object displays mirror symmetry, that means that it is impossible to find any difference between the original object and its mirror image. If that is true then it must be so that when you go to examine the object, there exists no way to know whether you are dealing with the original object or a mirror image of the original object: i.e., it must be true that you are ignorant as to which possible version of the object you are examining.
2) If an object displays spherical symmetry, that means that it is impossible to find any difference between the original object and an arbitrary three dimensional rotation of that object. If that is true then it must be so that when you go to examine the object, there exists no way for you to know whether you are dealing with the original orientation of the object or some rotated version of it: i.e., it must be true that you are ignorant as to the angular orientation of the object you are examining.
3) If an object displays "shift" symmetry, that means that it is impossible to find any difference between the original object and the same object shifted some distance along a line (think of an infinite pair of railroad tracks). If that is true then it must be so that when you go to examine that object, there exists no way to know whether you are dealing with the original object or a version which has been shifted along that line: i.e., it must be true that you are ignorant as to where the origin of the line referred to should be.
4) Finally, I will talk about "exchange" symmetry. If a pair of objects display "exchange" symmetry, that means that they are identical. If that is true then it must be so that when you go to examine that pair of objects, there is no way to tell if you are dealing with the original set or a set where the two objects have been exchanged: i.e., it must be true that you are ignorant as to whether you are dealing with objects A and B or with objects B and A. This is a bit more subtle than the other rather trivial symmetries.
The point is that, if the symmetries are true than you must be ignorant of some fact. The other side of the coin, which Yanniru seems to have missed (I guess he just wasn't in class that day) is that anytime you are ignorant of some aspect of the possible description of an object, that object (from your perspective) will display a symmetry related to your ignorance: i.e., you will be unable to differentiate between the original object and one which is different with respect to the aspect of the description of which you are ignorant.
Thus it is that any symmetry always implies a related ignorance and any ignorance implies a related symmetry (they are nothing more than different sides of the same coin). When a physicist says, "let us assume the following symmetry", what he means is that we will assume there exists no way of relieving our ignorance as to changes related to that symmetry.
It is too bad that scientists seem to always begin a discussion of symmetry arguments with that particular phrase when they could just as well said, "let us presume there is no way to determine this particular aspect of the problem". The only justification for his action is that "let us assume the following symmetry" is much simpler to say than to go through the details of describing the actual associated ignorance. (I am afraid it is a jargon thing). The real danger is that some students don't take the trouble to understand the real issue; that we are talking about the fundamental consequences of ignorance.
So "the assumption of a specific symmetry" is identical to "the assumption of a specific ignorance". What I point out in my paper is that our senses stand between us and reality. Since all the information we have to work with arrives through our senses, we are inherently ignorant of what process exists within that barrier. We certainly cannot "go look behind the barn" here; this explanation is wholly and completely in the "unknowable" category. Our ignorance is absolutely guaranteed!
Once more we find ourselves in a position where the possibilities are open. We cannot "prove" anything about reality without assuming some relationship between our senses and reality. So, instead of assuming our senses are a direct correct readout of reality (that nothing is illusion other than the illusions we are aware of) should we not rather recognize the inherent difficulties of being on the wrong side of an illusion creating mechanism?
We should be looking for things which cannot be produced by simple ignorance! Symmetry breaking is the important issue here as all symmetries are easily created by simple ignorance and we have a guaranteed abundance to work with.
Have fun -- Dick