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What is "understanding" all about.
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Expressive

Joined: Wed Mar 02, 2011 1:25 am
Posts: 42
Regarding science, the single most important issue to be discussed is exactly what the purpose of the endeavor is. I propose that the most fundamental concept standing behind science is “understanding”. I suggest that the subject of science can not even be approached without comprehending the nature and meaning of the word “understanding”; an issue few if any scientists take seriously. The most common “professional scientific” reaction to that issue is: hell that's not important; I know exactly what it means! I say they are wrong. They have totally failed to think out one of the most serious issues behind their work.

Everyone simply presumes that they know exactly when they “understand” something. I say their thinking they understand something is an emotional conclusion. As such, it is an opinion and not a fact; essentially a judgement they make. That they understand something is a belief and not a fact, a most serious error in the foundations of their work. The following is a rather simple minded approach towards uncovering the the correct implications of “understanding” and how it is to be achieved.

How does a teacher determine whether or not a student “understands” that which the teacher has attempted to teach? It is my experience that most teachers will come up with a number of questions which they think are relevant to the subject being taught and will present those questions for the students to answer. The teacher then judges the extent of the student's “understanding” via the probability that they (the teacher) would give those same answers to the given questions under a constraint of limited understanding consistent with that same judgement. In a sense, the teacher arrives at an “understanding” of the student's progress towards understanding of what was being taught.

How does a researcher determine whether or not a proposed assistant “understands” the nature of the research that the researcher is interested in researching? It is my experience that most researchers will ask the proposed assistant what questions they think are relevant to the research of interest (essentially asking what they see to be the relevant questions). The researcher then judges the extent of the proposed assistant's understanding of their purposes via the probability that they themselves (the researcher) would find the answers to those same questions relevant to their research. In a sense, the researcher arrives at an “understanding” of the proposed assistant's understanding of what is being researched.

How does an experimentalist determine whether or not they themselves “understand” the outcome of an experiment they are attempting to perform. It is my experience that most experimentalists have a very definite question in mind when they design the experiment and usually have a very definite collection of possible answers to that question. They also do their best to constrain the experiment such that the result of the experiment (one of those possible answers) correlates to a specific answer to the actual question they had in mind in designing the experiment. Obtaining a one to one correlation is often quite difficult as the experimenter seldom has total control of all possible variations of relevant issues. The experimentalist generally judges their personal understanding of the experimental result via their judgement of the relevant correlations to the question or questions they had in mind. In a sense, the experimentalist arrives at an “understanding” of the outcome of the experiment through their understanding of those relevant correlations.

In all of the above, the language used to express that which is understood must be understood (even with regard to the experimentalists own self communication). Once again, one is presented with the issue of determining the understanding of the language used to express that understanding. And how does one estimate the extent of that understanding? They consider questions which, given understanding of the circumstances, yield answers consistent with that understanding.

Two issues are seriously broached in this presentation. First that “understanding” is not a fact; it is a judgement based on the correlations between the answers to given collection of questions and what is presumed to be the correct answers to those self same questions. And, second, that this judgement is impossible to make without concocting a set of relevant questions and answers together with the probabilistic correlation between the two.

Now it should be noticed that nothing above has anything to do with the language used to express those questions and answers. The only thing of significance is the probability that a specific answer fits a specific question. Now any description of a specific answer to a specific question can be represented via a set of numbers (that is what the internet and computer aided communication is all about). Thus it is that “understanding” is judged via the probability that set of numbers fits the presumed correct answers (that set of numerical labels representing the relevant circumstances to be judged). What is most important about this relationship is that knowing the language is unimportant. One need only have that presumed knowledge of those probabilistic correlations.

Any intelligent person should comprehend that the above is the only valid analysis of John Searle's “Chinese room” argument against true AI; “they are not capable of understanding language, even in principle”. All the other analyses presume (without any defence) that “understanding” is something more than an emotional judgement and can be known as existing or not existing (though no one offers any way of determining the truth of that judgement).

Ah, but what is it that “we understand” when we possess that “presumed knowledge of those probabilistic correlations”? There is a very common English term for what is thus understood: it is presumed that we understand “an explanation” of those circumstances. Thus it is that, when we presume we know those probabilistic correlations, we are led to the emotional conclusion that we understand “the explanation” of those circumstances. It follows, as the night the day, that every conceivable explanation, known or not known, can be represented by the fact that $\large \dpi{80} {\color{white} P(x_1,x_2,\cdots,x_n)}$ is a known function.

The $\large \dpi{80} {\color{white} "x_i”}$ are nothing more than numerical labels of the relevant elements necessary to represent the circumstances (the questions and answers) in the language used to express the explanation. The important issue here is the fact that it is the circumstances being represented which are significant, not the language nor the specific labeling used. All that is required is that all the relevant circumstances are represented by a consistent set of numerical labels.

If understanding the above is beyond your comprehension, then everything I have to say is beyond your comprehension so don't bother trying to follow my arguments. My only reason for posting this is the fact that I would like to talk to someone intelligent as I have some rather important scientific conclusions which can be deduced from the above.

Have fun -- Dick

_________________

Knowledge is Power
The most popular abuse of that power is
to use it to hide stupidity!

Sun Jan 22, 2012 8:50 pm
Grand Master

Joined: Thu Oct 14, 2010 2:43 pm
Posts: 1854
Location: Somewhere in the 23rd dimension
First of all, I would like to apologize for so abruptly ending our last discussion, I was very busy with my various studies, but that does not give me the right to be so rude. Secondly, I hope I am intelligent enough to comprehend it as you intend to, but that will show itself shortly I guess.
Before continuing with your argument though, there are some difficulties I have regarding the above. I'll post them one by one, to give them the attention, I think, they deserve.
As a start, Iv wonder how you explain uses of understanding which are not preceded by a question and answer session. Imagine someone saying 'I understand that a rose cannot be both red and white'. I would say that one does not need a judgement based on questions and answers to say this. After all, this follows directly from the definition of colour.
I'm interested in your reply and look forward to a pleasant discussion.
Stefan

_________________
"The solution of the problem of life is seen in the vanishing of the problem." [Tractatus Logico-Philosophicus]
"Many people would sooner die than think. In fact they do" Bertrand Russell
"Science is organized knowledge. Wisdom is organized life." Immanuel Kant

Thu Jan 26, 2012 10:09 am
Expressive

Joined: Wed Mar 02, 2011 1:25 am
Posts: 42
Hi Stefan, I was surprised to hear from you.

Einstein2.0 wrote:
First of all, I would like to apologize for so abruptly ending our last discussion, I was very busy with my various studies, but that does not give me the right to be so rude.

Don't worry about it. I had sort of given up on you and seldom looked at this site anymore so I missed the fact that you had replied. Sorry about that. And I don't think intelligence plays as big a roll as does willingness to work “outside the box”. I have generally had great difficulty communicating my thoughts and I suspect the real problem is the fact that others simply do not think the way I do. Some things that are absolutely obvious to me simply do not occur to others and I have to take that fact into account. Perhaps a little information on my past would be of benefit.

When I was three years old, I witnessed an argument my father had with my aunt's husband concerning aliens from outer space. (It was always quite clear to me that my father had utterly no respect for my uncle.) When my uncle left, slamming the door as he went, my father turned to me and said, "anyone who believes more than ten percent of what he hears, or more than fifty percent of what he reads, or more than ninety percent of what he sees with his own eyes is gullible!" At the time, I had no idea as to what percent was nor what the word gullible meant but the comment was none the less absolutely engraved on my mind in every detail, never to be forgotten. The one thing I knew is that I didn't want to be gullible (at the time I think I had the impression it was a birth defect of some sort that required absolute disrespect).

My dad's comment may have had more impact on my life than any other experience I ever had. My single biggest worry has always been, "just how does one determine what one is supposed to believe?" I have to add to that the fact that adults just love to "pull kids legs". When I would catch them at it, they were always delighted. As a consequence, long before I even began school, I came to believe adults always lied to kids. I never saw it as malicious but rather presumed it as training not to be gullible (and, believe me, I was firmly convinced that I was indeed gullible; a problem I made every effort to hide). I simply found it very difficult to tell when I was being told the truth and when I was being lied to.

Now, all children like to think they are grown up and so did I. By the time I was five or six, I stopped openly catching adults in their [$#@!]: I began to pretend I believed everything they said (just like a real adult). Oh, I didn't act on it but I certainly didn't tell them when I thought what they said was [$#@!]. My mother once told me one learns a lot more by listening than they do by talking. Oh, she also told me that if I did something which I really felt was right but turned out to be wrong, I could be forgiven; however, if I ever did something which I myself felt was wrong, that was unforgivable so I always went with what felt like the right thing to do (I absolutely never made an attempt to intellectually justify what I did). I think she was a very intelligent person.

I only brought this up to explain something about the way I lived my life and how I came to think the way I do. I thought about things all the time but I absolutely never thought out what I was supposed to do. What I always did was to "go with my gut!" All my life, I have never made decisions based on logic applied to my beliefs (because, for the most part, I had no idea as to what I was supposed believe); I have always simply done what seemed to be the right thing to do emotionally. In essence I just turned control of my actions over to my subconscious; I did what came naturally.

In school, I could never figure out what ten percent of what the teacher said I was supposed to believe until I got into mathematics. Math teachers made it quite clear as to what was to be believed and went to great pains to justify their assertions. But, as practically every thoughtful person says, mathematics has nothing to do with reality. That is why I ended up in Physics. Physicists seemed to be the only people who seriously tried to justify their beliefs. This is how I ended up with a Ph.D. in Theoretical Nuclear Physics.

One problem I had was that, by the time I got into graduate school, they ceased to concern themselves with justifying their beliefs. When people with good credentials accepted things, we were expected to believe them (in spite of the fact that, time after time throughout history, authorities have been shown to be wrong on many occasions ). Believing them began to get very difficult if not impossible. After I got my degree, I actually never worked professionally in the field but I did continue to think about the problem of “belief”, an issue they seemed to be unconcerned with.

It is now my opinion that the source of the difficulty is belief itself. Belief is the single most potent corrupter of intellectual analysis which has ever existed. I now find it clear that "belief" is simply not necessary in order to solve the problem of understanding the universe confronting us. It turns out that the solution is simply sitting there in mathematics. Mathematics could, in fact, be defined to be the invention and study of internally consistent concepts. That is why physicists have consistently contributed to mathematics time after time. Our view of reality is supposed to be a collection of internally consistent concepts so, every time a scientist comes up with a new set of useful internally consistent concepts, the mathematicians adopt a representation of those concepts into their field.

It follows directly that, if and when a solution is found, it will be expressible in mathematics. I have found a solution and it turns out that asking the right question is the key to the difficulty. If you ask the right question and approach the possibilities without any prejudice of belief (i.e., making sure no possibility is omitted), the solution will unravel itself.

The solution is to define exactly what we mean by “understanding”. There are some very important underlying issues here. To quote a friend of mine from Finland, circular reasoning is quite common:

Quote:
In some important ways, our world views always are; the chosen terminology can be anything, as long as it is a self-coherent way to express valid predictions. That is just another way to say, it's chosen expression form is a bunch of circularly understood concepts. And that is another way to express why I think it is so childish when people argue about the correctness of the circle they most like to use.

The real issue here is “language”. Language is entirely a “collection of circularly understood concepts”. When anyone says, “my senses pick up real actual patterns from and about reality and perceive them and store them”, he is actually presuming that his concept of “senses” and the “perception of reality” obtained from them are correct. That is a belief and is not necessarily a fact. However, there is a way of handling the issue without actually succumbing to the belief.

Communications are achieved via a language. Knowing the language involves understanding those circularly defined concepts represented by the symbols going to make up that language. The important point here is that the actual symbols used to represent that language are an immaterial issue. Absolutely any concept representable in a language is just as easily represented by a set of numerical labels attached to the conceptual elements represented by the symbols going to make up that language.

Now a lot of people will complain about that assertion. But, before you jump to the conclusion that such a representation cannot be absolutely general, consider two very important issues. First, all internet communications rely on the ability to convert anything to be communicated into a collection of binary numbers (and that includes words, pictures, mechanical interactions and even could include smells and taste via the technology possibly becoming available to us). And secondly, almost everyone's view of the universe includes the concept of nerve signals connecting their brain to reality. Now those nerve signals themselves could certainly be represented by a collection of numerical labels. So even the common world view presumes the communications can be reduced to a set of labels.

If follows, from the above, that absolutely any possible communication conceivable can be represented by a set of numerical labels which, in turn, can be represented by the mathematical notation $\large \dpi{80} {\color{white} (x_1,x_2,\cdots,x_n)}$. The language being represented is simply not an issue here because, the problem of understanding the language is fundamentally identical to the problem of understanding anything.

The real beauty of this representation is the fact that the supposed language plays no role in the problem at all: i.e., this means that any generalization which can be made from this representation applies even to issues not yet thought of. It is possible that future views of the universe might very well be quite different from those we currently hold and might require serious changes in the language required to express them. Using this notation, that is simply not an issue; the communications of those views could still be represented in the notation $\large \dpi{80} {\color{white} (x_1,x_2,\cdots,x_n)}$. What is to be understood can always be seen as a collection such collections of numerical labels.

Oh, as an aside, I have left the number of numbers in that representation finite. That will turn out to be a very important factor with subtle consequences. It arises directly from the definition of “infinite”. If the number of elements in a collection is infinite then it follows that, no matter how many you have considered, you have not finished considering them. Essentially, all communications must be finite in extent. Note that this does not require that concepts being communicated to be finite but rather that the number of elements used to communicate must be finite.

Quote:

I wonder how you explain uses of understanding which are not preceded by a question and answer session. Imagine someone saying 'I understand that a rose cannot be both red and white'. I would say that one does not need a judgement based on questions and answers to say this. After all, this follows directly from the definition of colour.

No, it does not follow from the definition of color. It follows from your understanding of the definition of “color”, your understanding of the definition of “a rose” together with your understanding of all the other words assembled there. All of which arose from your experiences; what was (in your world view) communicated to your brain via the nerve impulses.

Actually, it is an error to believe that the only solution to the problem requires the existence of brains and nerve cells. All you can really say is that you are working with communications which can be represented by the mathematical notation $\large \dpi{80} {\color{white} (x_1,x_2,\cdots,x_n)}$.

Your understanding yields to you what probability it is that you associate with a particular communication $\large \dpi{80} {\color{white} (x_1,x_2,\cdots,x_n)}$ such as attaching the probability of one to the representation of, 'I understand that a rose cannot be both red and white'.

Oh, by the way, one could say that I am presuming communications are possible. Not really, what I am asserting is that I have no interest in concepts which can not be communicated. If it cannot be communicated I can not possibly understand it so I am not going to include such a thing in my thoughts.

At any rate, when you say, 'I understand that a rose cannot be both red and white' you are essentially assigning a probability of one to what is being represented by the specific collection of numerical labels $\large \dpi{80} {\color{white} (x_1,x_2,\cdots,x_n)}$ which are used to represent the communication, 'I understand that a rose cannot be both red and white'.

Thus it is that when you and I reach a point where we assign similar values of [tex]P(x_1,x_2,\cdots,x_n)[tex] to a sufficient number of possible communications between us, we will begin to presume we understand one another. This is the fundamental underlying nature of understanding itself and has nothing to do with the language used to assign those numbers (all it requires is that a unique set of numbers be assigned to the underlying elemental concepts expressed by the language). The only important factor is the fact that such an assignment can be made if the language is understood.

I hope this makes a little more sense to you.

Have fun -- Dick

_________________

Knowledge is Power
The most popular abuse of that power is
to use it to hide stupidity!

Sat Jan 28, 2012 8:48 pm
Grand Master

Joined: Thu Oct 14, 2010 2:43 pm
Posts: 1854
Location: Somewhere in the 23rd dimension
What may make my point a little clearer is that (for me) the sentence 'I understand that a rose cannot be both white and red' is equivalent to the sentence 'I know that a rose cannot be both white and red'.
This seems to indicate (I think that it's fair to give usage of words some importance when considering their meaning) that understanding is sometimes the same as knowing. Since knowing can be tested via question and answer sessions but is not, I think, essentially defined via such sessions I still have some doubts. For knowledge is, for me, being aware of the truth of a true proposition and having become aware of this via a correct act. The act is a knowledge act, and can be seen as verifying the truth of the proposition (so one can have no knowledge of the amount of window panes of a building until one has counted them).
Nowhere in this reduction of understanding to a knowledge act and the truth of a proposition does it seem necessary to employ a question and answer correlation. I hope this makes my point somewhat clearer.

_________________
"The solution of the problem of life is seen in the vanishing of the problem." [Tractatus Logico-Philosophicus]
"Many people would sooner die than think. In fact they do" Bertrand Russell
"Science is organized knowledge. Wisdom is organized life." Immanuel Kant

Sun Jan 29, 2012 10:03 pm
Expressive

Joined: Wed Mar 02, 2011 1:25 am
Posts: 42
I think I understand quite well the point you think you are tying to make. My point is that the issue you are considering is totally immaterial to the assertion I am making.

Einstein2.0 wrote:
Nowhere in this reduction of understanding to a knowledge act and the truth of a proposition does it seem necessary to employ a question and answer correlation. I hope this makes my point somewhat clearer.

Understanding constitutes being able to come up with an estimate of the probability of the truth of a proposition. Any proposition may be represented by a collection of mere numbers which are numerical labels of the fundamental elements which, in turn, constitute the communication of the proposition. The question is, what is the probability that proposition should be seen as true.

Thus it is that the question, "what is the value of $\large \dpi{80} {\color{white} P(x_1,x_2,\cdots,x_n)}$?", which always comes to bare on any issue of understanding anything.

I am afraid that you are failing to take into account the great number of propositions which need to be understood before the expression of the proposition in your example can be meaningful: i.e., be understood. You are failing to take into account that the language being used to express the proposition must be understood. Your point is made from the perspective that the language is understood: i.e., you want to work from a context you "believe" is valid. You are failing to take into account that your understanding of what stands behind your understanding might contain an error.

My point is that understanding can be represented by a whole mass of related probabilities $\large \dpi{80} {\color{white} P(x_1,x_2,\cdots,x_n)}$. Knowing the language constitutes knowing $\large \dpi{80} {\color{white} P(x_1,x_2,\cdots,x_n)}$ for a great number of possible arguments, each of which merely represent some specific communication.

I hope this helps.

Have fun -- Dick

_________________

Knowledge is Power
The most popular abuse of that power is
to use it to hide stupidity!

Mon Jan 30, 2012 6:17 pm
Grand Master

Joined: Thu Oct 14, 2010 2:43 pm
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Location: Somewhere in the 23rd dimension
I disagree on the fact that a proposition can be represented solely by numerical labels. You also need a projective relation of some kind mapping your numbers to elements of the proposition. A string of numbers has clearly no meaning of itself. But I suppose that this is implied by the numerical labels you use, I just wanted to note that such a relation is necessary.
To continue with my main point, how would you define 'the probablity of the truth of a proposition'? Is that the probability that a proposition is true, or is is the probability that we correctly think a proposition to be true, or something else?
Apart from that, surely I simply need to know what a rose is, what colour is, what red is and what white is in order to be able to know that 'A rose cannot be both white and red'? Knowing the objects and the relation that maps words into objects / relations should be enough. How else would you explain how a child learns language, but via getting to know the mapping of words to objects (which it knows)?

_________________
"The solution of the problem of life is seen in the vanishing of the problem." [Tractatus Logico-Philosophicus]
"Many people would sooner die than think. In fact they do" Bertrand Russell
"Science is organized knowledge. Wisdom is organized life." Immanuel Kant

Mon Jan 30, 2012 7:13 pm
Expressive

Joined: Wed Mar 02, 2011 1:25 am
Posts: 42
Einstein2.0 wrote:
I disagree on the fact that a proposition can be represented solely by numerical labels.

You miss the whole point of the presentation. You are adding a component to the problem which simply need not exist in order to solve the problem (something I will show down the road; after you comprehend exactly what I am saying).

Quote:
You also need a projective relation of some kind mapping your numbers to elements of the proposition.

Sure you do, if you want to express what you believe to be the correct content of that representation. But, obtaining a belief of what the representation expresses is your goal and you cannot presume a solution to the problem prior to solving it.

All I am saying is that, given a solution to the problem (a language expressing a specific solution, including a dictionary of all the meanings of all the words and all the relations necessary to teach yourself the language), that solution may be represented via a function of the form $\large \dpi{80} {\color{white} P(x_1,x_2,\cdots,x_n)}$ where those numerical labels simply refer to the specific elements laid out in that dictionary together with all the required relationships.

Quote:
A string of numbers has clearly no meaning of itself.

I am not saying that it does! I am saying that any communication of any relevant information may be represented by a collection of numerical labels of the pertinent symbols used by the language necessary to communicate that information. Quite a different thing.

Quote:
But I suppose that this is implied by the numerical labels you use, I just wanted to note that such a relation is necessary.

The important thing to recognize here is that the relationship is the solution one is looking for. As I said earlier,

DoctorDick wrote:
The real beauty of this representation is the fact that the supposed language plays no role in the problem at all: i.e., this means that any generalization which can be made from this representation applies even to issues not yet thought of. It is possible that future views of the universe might very well be quite different from those we currently hold and might require serious changes in the language required to express them.

All I am concerned with is the fact that there exists no communication which cannot be represented by a set of numerical labels $\large \dpi{80} {\color{white} (x_1,x_2,\cdots,x_n)}$ in the language (or any other mechanism) used to communicate the significant issues to be understood. The real issue here is that I am not concerned with what those labels represent at all. Coming up with a solution as to what those labels represent is the problem being examined. The only issue of interest to me is that the representation is constrained in absolutely no way!

Einstein2.0 wrote:
To continue with my main point, how would you define 'the probability of the truth of a proposition'? Is that the probability that a proposition is true, or is is the probability that we correctly think a proposition to be true, or something else?

In the total absence of belief itself, explain to me the difference between the two views. My position is simply “one” means it is to be accepted as valid and “zero” means it is not accepted as valid. What numbers between mean is something to be understood (part of the problem being examined).

Quote:
Apart from that, surely I simply need to know what a rose is, what colour is, what red is and what white is in order to be able to know that 'A rose cannot be both white and red'? Knowing the objects and the relation that maps words into objects / relations should be enough.

Anything you think you know is a presumed understanding. The basis of my whole presentation is that the understanding can be represented by $\large \dpi{80} {\color{white} P(x_1,x_2,\cdots,x_n)}$.

! REPRESENTED !

Quote:
How else would you explain how a child learns language, but via getting to know the mapping of words to objects (which it knows)?

And how the hell does the child get to know those objects? If I am working with the communications and nothing else (think of brain cells being triggered), there is one hell of a lot more one needs to know than what is contained in that sentence of yours. Why do you think babies usually need over a year of experiences before they begin to talk. How old is the youngest child you know who would understand your comment concerning the rose and the associated conclusion?

I need to tell you a story. My son was born in February. Thus he was a babe in arms during his first summer. The second summer, I spent most of my free time working on a fifty foot cabin cruiser at a marina (I had bought it before she got pregnant when time and money were no problem). My wife used to play with him on the shore while I worked. She often brought along pop corn for him to feed the ducks (there were ducks all around the marina). Well in the spring of his third year, he came running into the living room yelling, “daddy, daddy, daddy, there are ducks in the front yard.” So I went to look. There were no ducks there but there were some robins.

So I tried to explain to him that they were robins, another kind of bird but he insisted they were ducks. So I spent the next week or so finding birds and teaching him the names. He quite quickly learned all the names I gave him and, if I pointed at a specific bird, he could name it. But, when we finished, he would always mutter under his breath that they were all ducks.

On the way home from work about a week later, I spotted some ducks in the park. So I bought some pop corn and, when I got home, I asked him if he wanted to feed the ducks in the park. He got all excited and jumped in the car. We went to the park and he was quite astonished at the ducks (who crowded around him when they found out he had pop corn). The birds in our yard didn't do that. He wouldn't talk to to me about birds (or ducks) for almost a year after that. I think I broke his bird so to speak.

The point being that errors are made in the process of learning things and we really need to start in the absence of belief of any kind if we are going represent exactly what understanding is without making any presumptions (things we believe to be true).

Coming to understand things requires experience and what is experienced can be represented by a collection of numerical labels represented by $\large \dpi{80} {\color{white} (x_1,x_2,\cdots,x_n)}$. Understanding it can be represented by $\large \dpi{80} {\color{white} P(x_1,x_2,\cdots,x_n)}$.

That is all I am trying to get you to understand. The problem is that you keep wanting to presume understanding of something before you look at the problem of acquiring understanding.

As I said earlier, ask the right question and the problem unravels itself.

Have fun -- Dick

_________________

Knowledge is Power
The most popular abuse of that power is
to use it to hide stupidity!

Mon Jan 30, 2012 11:21 pm
Grand Master

Joined: Thu Oct 14, 2010 2:43 pm
Posts: 1854
Location: Somewhere in the 23rd dimension
The problem is, that it is impossible to look at understanding without first presuming that you understand something. Thoughts are (or at least, seem to be) limited to what language can express. Since understanding is one of the main components of language, it is not possible to describe it with language (without using it in the description) and thus to think about it without first presuming it.
This fundamental impossibility to go outside language leads to the fact that you have to first presume some understanding before you can define it. Hence my using examples from language to look at what understanding means.
A similar argument goes for beliefs. We cannot work without beliefs, so what you ask is the impossible.

I disagree that all knowledge is presumed understanding. It is after all possible to know something without understanding it; I can know that 1+1=2 but do not have to understand why that is so. I think that understanding something also means knowing it, but I don't see why the converse is true, at least when looking at how the two words are used.

Also, I don't see where in your story about your son something else happens than that he learns the objects and the relation mapping words to those objects. Of course this is not a straightforward process, but I never said it was.

As a final note, I understand what you are trying to explain but I disagree with it, up till now you have simply repeated your theory, surely you have got reasons why your theory is preferable to other (non mathematical, so I think more natural) theories regarding understanding?

_________________
"The solution of the problem of life is seen in the vanishing of the problem." [Tractatus Logico-Philosophicus]
"Many people would sooner die than think. In fact they do" Bertrand Russell
"Science is organized knowledge. Wisdom is organized life." Immanuel Kant

Thu Feb 02, 2012 6:02 pm
Expressive

Joined: Wed Mar 02, 2011 1:25 am
Posts: 42
Einstein2.0 wrote:
The problem is, that it is impossible to look at understanding without first presuming that you understand something.

I would not argue with that. What I presume I understand is “mathematics”. I like the following quote of Bertrand Russell:

Quote:
"Too often it is said that there is no absolute truth, but only opinion and private judgment; that each of us is conditioned, in his view of the world, by his own peculiarities, his own taste and bias; that there is no external kingdom of truth to which, by patience and discipline, we may at last obtain admittance, but only truth for me, for you, for every separate person. By this habit of mind one of the chief ends of human effort is denied, and the supreme virtue of candor, of fearless acknowledgment of what is disappears from our moral vision. Of such skepticism mathematics is a perpetual reproof; for its edifices of truths stands unshakable and inexpungable to all the weapons of doubting cynicism."

I regard mathematics as a language designed to accurately express deductive relationships far beyond those which may be achieved by “gut” intuition (the common source of almost all human beliefs). Alfred North Whitehead said, "Mathematics in its widest significance is the development of all types of familiar, necessary, deductive reasoning."

I would comment that mathematics has its own ontology (numbers, points, geometry, products etc.) together with perhaps the most extensive collection of conclusions which can be reached via deductive logic (epistemology) available to the human mind. I would add that many brilliant people have spent their lives attempting to assure that no invalid conclusions can be reached using that language. Perhaps they have made mistakes but the language of mathematics is far more internally consistent than any common language developed by mankind.

Einstein2.0 wrote:
Thoughts are (or at least, seem to be) limited to what language can express. Since understanding is one of the main components of language, it is not possible to describe it with language (without using it in the description) and thus to think about it without first presuming it.

Again, I agree with you. Anssi Hyytiäinen, my friend in Finland, has been following our conversation and has suggested the following text (which I will paraphrase) might clarify things for you.

Quote:
Certainly it is common that there are many logically deduced ideas and derivations, that are derived from the concepts and definitions of a person's world view in some way. But before such deductions can occur, there is a question about how did those concepts and definitions (roses, colors, etc.) become established in that world view in the first place.

Underneath it all, there must exist some kind of learning mechanism, that establishes a connection between some initially undefined information and some defined ontology (including your "roses" and "colors"). From an analytical perspective, we have to view that learning mechanism as something that is establishing probabilistic relationships between those ontological elements based on earlier experiences because it must start its learning from the point where nothing has been defined.

The ideas of "roses", "colors" and other ontological elements can arise (and they are, by themselves, the result of "understanding" some originally undefined information), and logical deductions based on those definitions can be performed, but still they always are by themselves just a result of inductive learning that happened before, and in that sense fall into the same definition of "understanding". It is that inductive result that we need to represent in a universally unconstrained manner.

Dicks notation is simply trying to capture the basic idea; first, that undefined information (prior to coming up with an ontology) can be simply be arbitrarily labeled, and second, that valid "understanding" of that information, fundamentally entails the fact that valid judgments (true false decisions) can be expressed about the internal relationships embedded in the “understood” information.

Conversely, if a presumed explanation contains demonstrably invalid relationships for some presumed ontological identification of that information, it would be taken to indicate that the information has not been understood correctly.

Einstein2.0 wrote:
This fundamental impossibility to go outside language leads to the fact that you have to first presume some understanding before you can define it. Hence my using examples from language to look at what understanding means.

Ok, but you must remember that you are using a spoken language as a source. How about just using mathematics, the language I want to get into. Does not “understanding” mean that one can attach true-false judgments to possible relationship assertions expressible in that language?

Quote:
A similar argument goes for beliefs. We cannot work without beliefs, so what you ask is the impossible.

Now here I would disagree with you. Allowing the possibility that something is true is not equivalent to belief. Belief in a proposition means that you have no doubt as to its validity. An open mind must always accept the idea that, no matter what seems to make sense, the possibility exists that one's understanding is wrong!

Quote:
I disagree that all knowledge is presumed understanding. It is after all possible to know something without understanding it; I can know that 1+1=2 but do not have to understand why that is so.

But if you don't understand what 1+1=2 means, how can you assert you know that 1+1=2? You need to have established some kind of otology before that collection of symbols has a meaning of any kind.

Quote:
I think that understanding something also means knowing it, but I don't see why the converse is true, at least when looking at how the two words are used.

Then you apparently dislike my use of the word. Give me a word you would prefer to use to represent the quality of being able to establish a true-false judgment of a defined circumstance. As an aside, I include 50/50, very probably true and even “I don't know” as true-false judgments. They all may be expressed in probability theory.

Quote:
Also, I don't see where in your story about your son something else happens than that he learns the objects and the relation mapping words to those objects.

I was only pointing out that his understanding of the information available to him was erroneous. This is in fact a common error made by children. That is what underlies a living language. Children guess what words mean and, if they are not corrected, they may end up using them differently than their parents. When you get as old as I am, you will begin to notice such things. Different generations associate different meanings to a lot of the words they use. When I was a child, "gay" certainly didn't mean what it means today.

When I was a child reading Chaucer in school, we all got a big kick out of the fact he used piss and [\$#@!] all the time (terms we understood were vulgar). About twenty years ago, I ran into “piss” in a book on sources of words and what it said made me laugh. It said, “piss: an onomatopoeic euphemism used by women in place of the vulgar term used by men which was not recorded.” So Chaucer's use wasn't originally vulgar at all. Oh, another funny one is "nice". It turns out that it came from a contraction of "no sense" back in medieval times.

Quote:
As a final note, I understand what you are trying to explain but I disagree with it, up till now you have simply repeated your theory, surely you have got reasons why your theory is preferable to other (non mathematical, so I think more natural) theories regarding understanding?

First, what I am presenting is not a theory. I am merely presenting a notation which I assert is capable of representing any collection of ontological elements: i.e., the notation is absolutely and universally applicable and makes utterly no presumptions as to what is being represented. The second step concerns representing valid judgments of the internal relationships embedded in the ontological structure (what I think of as “understanding” that ontology) and I probably should not have brought it up until you understood that first step. Sorry about that but I just can't comprehend the difficulty people have understanding that first step.

The only point of significance is the fact that the notation makes absolutely no constraints on what is represented. That isn't a theory, it is a simple assertion. I could be wrong; but, if I am, you should be able to point out a collection of ontological elements which cannot be represented via a collection of simple numerical labels.

I am simply dumbfounded by your problems understanding me -- Dick

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Thu Feb 02, 2012 11:30 pm
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Ah, now I see the main problem.
I understand that your notation is no theory, but there is a theory behind it that one needs to assume. What I think you're doing its explaining what understanding is, and thus describing language and hence how a part of our mind functions.
Now before you can start your representation you need to assume that the working of our mind (or language, if you prefer that) can be described mathematically. On this point I disagree with you and that is the point I have been attacking from my first post onwards.
You however looked at everything from your notation and were because of that unable to see why I had all these difficulties, because you already assume what I was attacking.

And this more fundamental assumption is a theory, one that has to be defended not with more math, but with philosophical arguments. For I do not believe that language our our mind can be described using math and have tried to give arguments against that position in all my above posts.

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Sat Feb 04, 2012 7:05 pm
Expressive

Joined: Wed Mar 02, 2011 1:25 am
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Einstein2.0 wrote:
Ah, now I see the main problem.
I understand that your notation is no theory, but there is a theory behind it that one needs to assume.

No, I am afraid you are the one making an assumption here.

Quote:
What I think you're doing its explaining what understanding is, and thus describing language and hence how a part of our mind functions.

For the moment, let us forget about explaining what understanding is (that issue can be discussed after the fundamental characteristics of the notation are clear to you). You are assuming that I am describing something which I am not; I am putting forth a notation for representing a communication. A notation which makes no presumptions whatsoever as to what it is that is being communicated (or being described if you prefer).

Describing something is far different from representing it! It is certainly possible to represent something (for example, "it" is something) without describing it but one can not create a description of something without a method of representing that description. What I am saying is that the actual symbols used to represent the ontological elements of a language are immaterial. It is the meaning of the things being represented which is material and the meaning is embedded in the language, not in the symbols used to represent that language. That is the very essence of the fact that different languages exist.

Quote:
Now before you can start your representation you need to assume that the working of our mind (or language, if you prefer that) can be described mathematically. On this point I disagree with you and that is the point I have been attacking from my first post onwards.

No, the issue you are inadvertently confronting is the fact that before one can put meaning to that representation you need to put meaning to those elements being represented by those numerical labels. That is an entirely different issue; an issue the notation is specifically designed to circumvent. I am putting no meaning whatsoever to that representation. It is your attempts to put meaning to the representation (or perhaps your presumption that it must be possible to do so) which have led you to question the path I am laying out. You think you know where this is going but you do not.

To quote my second post to this thread:

Quote:
I have found a solution and it turns out that asking the right question is the key to the difficulty. If you ask the right question and approach the possibilities without any prejudice of belief (i.e., making sure no possibility is omitted), the solution will unravel itself.

I think this is the assertion that you find so difficult to accept. Don't worry about it. I am merely setting up a representation such that “the right question” can be asked. The question must be asked without making any presumptions whatsoever as to what the answer might be. It is the fact that I can represent the answer in my notation which allows examination of the answer without constraining the meaning of that answer in any way. It is a subtle thing but once you see it, it will be quite clear and the consequences are rather astonishing.

For the moment, all I need is your recognition that any communication may be represented with my notation and that the notation itself makes no constraints whatsoever on what is represented.

Have fun -- Dick

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The most popular abuse of that power is
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Sun Feb 05, 2012 1:18 pm
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Very well then, let's say that you're giving a notation for representing communication (leaving out the a, since I guess you mean human communication).
I agree that with solely the notation there are no assumptions whatsoever, but you already make assumptions that need to be further explained. You are describing a "function" and, if you are indeed going to proceed in the same manner as in our previous discussion, then you'll also introduce complex numbers and stuff, all of which has to be given a meaning.
If you (as you seem to) choose to give all these things a mathematical meaning, then you are presuming something, namely that the mathematical meaning may be used in this context.
Once that has been established, then you are absolutely right that there are no constraints on the notation. But the process in which you give meaning to it has to be justified. For, as I said earlier, I do not think that communication can be described from a mathematical perspective.

To clarify, you do put meaning to your notation by using words as "function" without further explaining them. If there is absolutely no meaning to it, then it's just meaningless (literally) nonsense and does not represent anything but that.

_________________
"The solution of the problem of life is seen in the vanishing of the problem." [Tractatus Logico-Philosophicus]
"Many people would sooner die than think. In fact they do" Bertrand Russell
"Science is organized knowledge. Wisdom is organized life." Immanuel Kant

Sun Feb 05, 2012 1:55 pm
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Joined: Wed Mar 02, 2011 1:25 am
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Einstein2.0 wrote:
Very well then, let's say that you're giving a notation for representing communication (leaving out the a, since I guess you mean human communication).

Perhaps we are getting somewhere; however, I am somewhat bothered by your comment, “I guess you mean human communication”. You should understand that presuming that the concept “human” is an absolutely necessary ontological concept required to exist in all conceivable “communications” is putting a constraint on what is being represented. Presuming such a thing violates the requirement that absolutely no constraints be placed on what is being represented.

The end purpose here is to solve the problem of understanding that which is being represented and beginning with any assumptions as to what it is that is being represented inherently destroys the generality of the representation: i.e., there may exist “understandings” which do not make such assumptions. Unless you can prove conclusively that such alternate “understandings” do not exist you must allow for the possibility that they do.

At the moment, exactly what constitutes a communication is pretty well undefined. The generality of my representation is quite open. There are those who will contend that I have made the assumption that the communication can be separated into independently referenced ontological elements (often called “reducible”); however, that is not the case. If it cannot be separated into elements, it must be referenced by a single numerical label “x”. The problem there is that “understanding” such a communication seems rather doubtful (unless you are perhaps God).

The next issue is to establish what constitutes “understanding” a communication. The human concept of “understanding” is rather vague and arguments could arise far and wide with practically any attempt to define exactly what is meant; however, if the solution we are looking for is “understanding” it behooves us to have some way of determining the existence or nonexistence of understanding. My position is quite simple: if understanding exists, answers the the truth or falseness of a specific communication may be made: i.e., understanding provides $\large \dpi{80} {\color{white} P(x_1,x_2,\cdots,x_n)}$ where “one” means true and “zero” means false.

It is very important here to realize that I am speaking of “understanding” and not the validity of that understanding: i.e., the presumption that erroneous understandings have been removed has not been made nor will any such assumption be allowed. The central issue here is that I personally can not comprehend solving the problem of “understanding” a communication where the solution provides no judgments whatsoever as to the truth of any part of that communication.

If you wish to add additional requirements to your personal definition of “understanding” you are free to do so; however, you must understand that those additional requirements are an aspect of your solution to understanding and thus cannot be held as absolutely general. Likewise, if you can present something you think would seriously be accepted as “an understanding” of a communication which includes utterly no judgments of truth on any aspects of that communication then you would have a serious problem with my definition. But, before you tread that path, consider that “utterly no judgments of truth” includes all meanings on what the symbols used to communicate actually mean. This implies the communication must be totally meaningless trash and do you really want to include meaningless trash in the category of things which you understand. In my mind that sort of defeats the idea of solving the problem of understanding.

Quote:
I agree that with solely the notation there are no assumptions whatsoever, but you already make assumptions that need to be further explained. You are describing a "function" and, if you are indeed going to proceed in the same manner as in our previous discussion, then you'll also introduce complex numbers and stuff, all of which has to be given a meaning.
If you (as you seem to) choose to give all these things a mathematical meaning, then you are presuming something, namely that the mathematical meaning may be used in this context.

My answer to that is that presumption of a context other than mathematics constitutes a violation of the original representation: i.e., limiting the solution to a context is putting a preordained constraint on what is an acceptable solution.

As per what I have stated above, the solution is required to provide judgments as to the truth of possible communications: i.e., the solution requires providing a number bounded by zero and one when given a specific communication represented by $\large \dpi{80} {\color{white} (x_1,x_2,\cdots,x_n)}$. Thus finding a solution in this representation constitutes a mathematical problem. The solution is exactly $\large \dpi{80} {\color{white} P(x_1,x_2,\cdots,x_n)}$.

Quote:
Once that has been established, then you are absolutely right that there are no constraints on the notation. But the process in which you give meaning to it has to be justified. For, as I said earlier, I do not think that communication can be described from a mathematical perspective.

The process by which one gives meaning to the solution is exactly the process I am intentionally circumventing. Giving meaning to the solution is the process called theorizing. I have utterly no interest in theorizing whatsoever. What I have done is to exhaustively examine all possible solutions, limited only by the condition that they be internally consistent. That examination must include all possible functions $\large \dpi{80} {\color{white} P(x_1,x_2,\cdots,x_n)}$ which are internally consistent. And that is my real problem. Everyone simply assumes such an attack is, as you say, “meaningless (literally) nonsense and does not represent anything but that.”

I say it is not meaningless nonsense but instead brings forth some rather astonishing conclusions. But, the professionals refuse to look, firmly convinced that I am a certified nut case and nothing more. If you want to join the crowd and refuse to look, I will accept that; but don't pretend to have proved what I am doing is nonsense.

Essentially, the underlying issue is, do the probabilities of truth (that would be the function $\large \dpi{80} {\color{white} P(x_1,x_2,\cdots,x_n)}$) generated by the theory which presumes to display understanding of the collection of communications represented by $\large \dpi{80} {\color{white} (x_1,x_2,\cdots,x_n)}$ constitute an internally consistent structure? This has nothing to do with what that theory presents as an ontology but rather has to do with the necessity that the same question (together with the appropriate context represented by $\large \dpi{80} {\color{white} (x_1,x_2,\cdots,x_n)}$ ) always yields the same answer; something which is a direct consequence of the structure of the function $\large \dpi{80} {\color{white} P(x_1,x_2,\cdots,x_n)}$. Any attempt to put meanings to the ontological elements denoted by $\large \dpi{80} {\color{white} ”x_i”}$ essentially presumes one understands the communication and thus places an unwarranted constraint (one you cannot prove is correct) on the possibilities. The analysis must include all possibilities, even those which seem to you to be nonsense. You must understand that the possibility that you do not understand the communication always exists.

In a sense, if you assume the analysis is unnecessary, you would be assuming that everything representable by the notation $\large \dpi{80} {\color{white} (x_1,x_2,\cdots,x_n)}$ is nonsense. If you think that is true, it follows that you think all communications are nonsense and that is a rather ridiculous assertion. On the other hand, there may be communications which you think are meaningless nonsense which could be understood if you had the proper solution. For this reason, it is important that all possibilities be included in the analysis: i.e., absolute generality must be maintained at all costs or the analysis itself is in error.

Have fun – Dick

PS The examination process is actually quite simple and straight forward. Getting people to look at it is the real difficulty.

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The most popular abuse of that power is
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Sun Feb 05, 2012 10:27 pm
Grand Master

Joined: Thu Oct 14, 2010 2:43 pm
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If the rest of the process is indeed as simple as you say it is, then it's probably best if I postpone my judgement till after you have explained it entirely.
Not that I suddenly agree, but viewing the whole thing is always better.
So, since O think that by now I get the first step, please continue.

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"The solution of the problem of life is seen in the vanishing of the problem." [Tractatus Logico-Philosophicus]
"Many people would sooner die than think. In fact they do" Bertrand Russell
"Science is organized knowledge. Wisdom is organized life." Immanuel Kant

Wed Feb 08, 2012 8:39 am
Expressive

Joined: Wed Mar 02, 2011 1:25 am
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Sorry to be slow to answer but I wanted to be careful as to how and what I presented so as not to awaken an immediate negative response. I would much rather you think about things carefully.
Einstein2.0 wrote:
If the rest of the process is indeed as simple as you say it is, then it's probably best if I postpone my judgment till after you have explained it entirely.
Not that I suddenly agree, but viewing the whole thing is always better.
So, since O think that by now I get the first step, please continue.

Thank you. I have had great trouble getting people to understand the exact nature of my starting position and it does appear that you have at least grasped some of the essentials of that position. In my conversation with you, in an attempt to establish the required generality of the examination, I have concentrated on the two concepts “communication” and “understanding”; however, these are not really the terms I prefer to use in an extended examination. I would prefer to use some other terms which, at least in my mind, are a little more to the point.

Considering the question of “communication”, I feel that the word “information” is a more acceptable term for a collection of “communications” represented by the notation $\large \dpi{80} {\color{white} (x_1,x_2,\cdots,x_n)}$. It is a somewhat more general reference than is “communication”; however, when I used it in the past, everyone tended to automatically presume information had to be understood (essentially well defined) before anything could be qualified as “information”. I switched over to “communication” in my discussion with you in an attempt to avoid that ubiquitous presumption. I also prefer to refer to a specific example of the collection of representations $\large \dpi{80} {\color{white} (x_1,x_2,\cdots,x_n)}$ as “a circumstance”: i.e., a specific component of the information to be understood.

As I said earlier, the numerical labels specified by the numbers $\large \dpi{80} {\color{white} ”x_i”}$ refer to specific ontological elements making up the structure of the applicable language. It is very important to comprehend that the meanings of the symbols used to represent a language are embedded in the structure of the language being represented and not in the symbols themselves. It is that fact alone which stands behind all of my deductions concerning the necessary constraints on the representation. At no point do my deductions have anything to do with the meaning of what is being represented.

When it comes to referring to what it is that we “understand”, I have a preference for the word “explanation”. It seems to me that, if something is understood, we can explain it. Thus, I would prefer to see the notation $\large \dpi{80} {\color{white} P(x_1,x_2,\cdots,x_n)}$ as representing an “explanation”. If you have a serious problem with the usage of these alternate identification terms, let me know what problem you feel is raised by my usage of such a vocabulary.

Using this alternate terminology, the problem of interest becomes examining explanations consistent with a given collection of circumstances $\large \dpi{80} {\color{white} (x_1,x_2,\cdots,x_n)}$. Essentially, we want to examine all possible functions of those circumstances which can be seen as a probability estimate of their correctness (truth or falseness). Another way of looking at the issues is that we want to consider functions $\large \dpi{80} {\color{white} P(x_1,x_2,\cdots,x_n)}$ which are consistent with the given information.

There are a couple of subtle issues here which need to be understood if we are to avoid making any assumptions whatsoever. The first is that the collection of circumstances being explained must cover the entire range of information available. It should be understood that any omitted information which bears in any way on the solution being represented constitutes an assumption that the omitted information is already understood. That is an unwarranted assumption if the analysis is to be entirely general and universally applicable. Again, if the meaning of that assertion is not clear to you, we need to discuss it.

The other issue to be kept in mind is exactly what $\large \dpi{80} {\color{white} P(x_1,x_2,\cdots,x_n)}$ represents. This is a probability that the referenced circumstance, $\large \dpi{80} {\color{white} (x_1,x_2,\cdots,x_n)}$ , is true. Since the goal of the examination is a complete examination of all possibilities, instead of thinking in terms of some theory setting that function, I would rather view the situation from exactly the opposite direction. I would like to begin with the perspective that every mathematical function so representable could possibly provide an explanation of the information and then remove any functional structures which cannot possibly be valid.

Thus, the known information (that which the explanation is to explain) is represented by a collection of specific circumstances $\large \dpi{80} {\color{white} (x_1,x_2,\cdots,x_n)}$ where each of the $\large \dpi{80} {\color{white} x_i}$ is a specific number. To us, the examiners, the collection constitutes a meaningless collection of numbers. We do not understand what is being communicated and thus personally have no explanation of the given information or what any particular numerical label means; however, if an explanation exists, there also exists a set of ontological elements invented by the theorist who uncovered that solution expressed in a language understood by that theorist. The numbers are, by definition, numerical labels of a list of those ontological elements consistent with definitions of that language. The patterns required by that explanation as expressed in that language are thus embedded in the specific collection of circumstances which we are to examine.

The explanation is thus represented by a function $\large \dpi{80} {\color{white} P(x_1,x_2,\cdots,x_n)}$ ; where the argument represents a possible circumstance. What we know is quite simple: we know that the probability of those circumstances which make up the given known information (that collection of seemingly meaningless numbers we have to work with) is non-zero. Note that we do not know the probability of those specific circumstances; we only know that they are part of the given information thus their probability is not zero. If that is unclear, please note that the actual probability of a specific circumstance (implied by the explanation) might depend upon the appearance of other circumstances within the given information (a relationship which one could refer to as “context”).

If we “understood” the meaning of the given information and the explanation it represents, we could conceive of alternate circumstances (expressed via that same set of ontological elements used by that language) the truth of which we might know. From that perspective, what we actually have is a table of a finite number of circumstances which, taken as a whole, have a non-zero probability of truth. Possessing an explanation would allow is to develop additional entries to that table together with probability estimates of their truth. In many senses, the explanation allows us to interpolate the value of the function $\large \dpi{80} {\color{white} P(x_1,x_2,\cdots,x_n)}$ between the entries on that table. It should be clear that, given a finite amount of information, the possible number of functions which satisfy that table is actually quite infinite. (With regard to that issue consider the possibility that the language being referenced may contain additional ontological elements not represented in any of the given circumstances.)

The single thing which provides the most powerful basis for constraining the form of $\large \dpi{80} {\color{white} P(x_1,x_2,\cdots,x_n)}$ is the fact that the actual numerical labels used have to be immaterial. They are mere symbols for the ontological elements represented and can, themselves, make no contribution to the meanings embedded in the given circumstances . The only real requirement is that a given numerical label (once established) must consistently label the same ontological element thus yielding the same embedded patterns the explanation is required to reproduce.

I will give a rather simple but important example of what I am talking about. Let us examine the following thought experiment. Two entities (I would call them people but they might be something else) understand exactly the same explanation. They both reference exactly the same circumstances upon which their understanding is based and they both communicate with exactly the same language. It must be that the function, $\large \dpi{80} {\color{white} P(x_1,x_2,\cdots,x_n)}$ (the referenced explanation they have in mind), must be identical.

You may want to give me argument on that and if you seriously think an argument is necessary at this point we will have to go into the subtleties of the meaning of $\large \dpi{80} {\color{white} P(x_1,x_2,\cdots,x_n)}$, something I would prefer to postpone as the issue is more easily understood when the deductions are a bit clearer. The real argument is that it is possible to constrain the meaning of $\large \dpi{80} {\color{white} P(x_1,x_2,\cdots,x_n)}$ such that the assertion is true without constraining the possible explanations.

For the sake of argument, suppose that when they lay out those circumstances for communication, they use different numerical labels for the referenced ontological elements. Note that the fact that the function is identical must extend beyond the finite table which we are interpolating from; it must also yield identical probabilities for circumstances not contained in that table.

There is a very simple renumbering which yields a deep and profound consequence. Suppose they use exactly the same labeling system except that one starts with a “0” label and the other starts with a “1”: i.e., every label used by the second party is exactly one greater than the label used by the other. A very curious comparison between the two representations arises:

$\large \dpi{80} {\color{white} P(x_1,x_2,\cdots,x_n) = P(x_1+1,x_2+1,\cdots,x_n+1).}$

Suppose that instead of adding one, we instead add the number “a+Δa” in one case and the number “a” in the second case. In that case we end up with,

$\large \dpi{80} {\color{white} P(x_1+a+\Delta a ,x_2+a+\Delta a ,\cdots,x_n+a+\Delta a ) = P(x_1+a,x_2+a,\cdots,x_n+a).}$

But that implies

$\large \dpi{80} {\color{white} P(x_1+a+\Delta a ,x_2+a+\Delta a ,\cdots,x_n+a+\Delta a ) - P(x_1+a,x_2+a,\cdots,x_n+a)=0.}$

Dividing both sides by Δa we have,

$\large \dpi{80} {\color{white} \frac{P(x_1+a+\Delta a ,x_2+a+\Delta a ,\cdots,x_n+a+\Delta a ) - P(x_1+a,x_2+a,\cdots,x_n+a)}{\Delta a}=0.}$

Clearly that relationship does not depend upon the value of Δa and, taking the limit as Δa goes to zero we will still get zero. The final result should be a very familiar expression:

$\large \dpi{80} {\color{white} \lim_{\Delta a =0}\frac{P(x_1+a+\Delta a ,x_2+a+\Delta a ,\cdots,x_n+a+\Delta a ) - P(x_1+a,x_2+a,\cdots,x_n+a)}{\Delta a}=\frac{d}{da}P(x_1+a,x_2+a,\cdots,x_n+a)=0.}$

The next step should be somewhat obvious. Suppose we define $\large \dpi{80} {\color{white} z_i=x_i +a}$ and write down P(z_1,z_2,\cdots,z_n). Clearly this is essentially exactly what we started with except that the name of the numerical label has changed from “x” to “z”. Since “a” is no longer explicitly in the new representation, if we want the derivative of this new representation with respect to “a” we need to use the partial differential relationship,

$\large \dpi{80} {\color{white} \frac{d}{da}=\sum_{i=1}^n\frac{dz_i}{da}\frac{\partial}{\partial z_i} .}$

But, $\large \dpi{80} {\color{white} \frac{dz_i}{da}=1}$. This leads one to the rather straight forward conclusion that

$\large \dpi{80} {\color{white} \sum_{i=1}^n \frac{\partial}{\partial z_i}P(z_1,z_2,\cdots,z_n)=0}$

Now this conclusion has nothing whatsoever to do with what explanation was examined and is thus a rather surprising universal requirement on that probability function. It certainly was surprising to me the first time I saw it some forty five years ago.

If anything about the above deduction bothers you, let me know. There are actually a few subtle issues that need to be addressed in order to make the representation absolutely general; however, I thought that this particular relationship needed to be brought up first.

Have fun -- Dick

_________________

Knowledge is Power
The most popular abuse of that power is
to use it to hide stupidity!

Sat Feb 11, 2012 8:15 am
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