Representing The Problem To Be Solved.

[Doctordick]

Thanks to CraigD I am going to make an attempt to restate the problem I have solved. First of all, it is very much a philosophical issue. How it connects to philosophy should be pretty clear if one were to first read that introduction to Wittgenstein's “thesis” (thank you Qfwfq).

 

In my opinion, the major philosophical error made by Wittgenstein (as expressed by Bertrand Russell) is the idea that “the essential business of language is to assert or deny facts”. I would rather say, “the essential business of language is to communicate”; facts, fiction, nonsense or whatever. I think his concern with “facts” and logic is actually what prevents him from thinking about the central requirements of such a “perfect language”.

 

Bertrand Russell

He is concerned with the conditions for accurate Symbolism, i.e. for Symbolism in which a sentence `means' something quite definite. In practice, language is always more or less vague, so that what we assert is never quite precise. Thus, logic has two problems to deal with in regard to Symbolism: (1) the conditions for sense rather than nonsense in combinations of words; (2) the conditions for uniqueness of meaning or reference in symbols or combinations of symbols.

Wittgenstein's error is worrying about that first “problem”; it has no significance whatsoever. Any language incapable of expressing nonsense is simplistic beyond belief. Laying that issue aside, problem #2 (the conditions for uniqueness of meaning) is indeed critical and the introduction of the concept of “simples” is central to solving that problem.

 

From the same introduction

The first requisite of an ideal language would be that there should be one name for every simple, and never the same name for two different simples. A name is a simple symbol in the sense that it has no parts which are themselves symbols. In a logically perfect language nothing that is not simple will have a simple symbol. The symbol for the whole will be a ``complex'', containing the symbols for the parts.

From that point on, the discussion goes totally awry. He asserts that “a name is a simple symbol in the sense that it has no parts which are themselves symbols” and then goes on in an attempt to represent these names with “words” which, in themselves are not “simples” (they have parts made of letters which are, in turn, lines, intersections etc. which are themselves complex concepts). The underlying critical issue is actually never even broached. (Of course, I have only read Russell's introduction so I could be wrong; however, it seems to me that, had the issue been broached, Russell would have made some reference to it.)

 

Then he makes an attempt to construct a “perfect language” with but seven “simples”. The correct minimum number of required “simples” should be more on the order seven billion. In fact, that is the very issue which makes solving AI so difficult. However, in the modern world, computers might make the problem solvable if one had an intelligent attack (which I think I have discovered). But, other than that, the idea of using numerical labels for these “simples” is a quite reasonable choice.

 

That is exactly the choice I have made; the only real difficulty here seems to be general comprehension of my notation. The central issue is that any idea conceivable in any conventional language can be represented by what I call “a circumstance”:

 

[math]

(x_1,x_2,x_3,\cdots,x_n)

[/math]

 

where each and every x_i is the numerical label of some “simple” in that “perfect language” designed to yield "uniqueness of meaning". Attempting to construct that perfect language (those billions upon billions of “simples”) is beyond actually doing but we can nonetheless discuss some interesting relationships which must be satisfied. That is the reason I concern myself solely with the constraints implied by the definition of “an explanation”. Scientifically (and philosophically) speaking, explanations are the single most important service provided by any language and understanding the implicit constraints implied therein is critical to understanding anything.

 

One very important aspect of a language immediately crops up here. Order of “simples” can be part of the meaning expressed by the symbol [math] (x_1,x_2,x_3,\cdots,x_n)[/math]. On the other hand, there may very well exist what Russell refers to as “complexes” which do not require “order”. For example, the letter “A” is a symbol (a “complex”) consisting of three specific lines and the meaning is not changed when the order with which those lines come into existence is changed. The written Chinese language language is full of complex symbols consisting of many marks, where the order in which these marks are created contains no meaning. (If the order had meaning, one would be obliged to watch the writer in order to understand what was written.)

 

This is exactly the reason I introduce the index “t” to my notation. I divide that collection of “simples” required to express some idea into a collection of “circumstances” within which the order of the “simples” is of no consequence. In the end, any body of knowledge (expressed in some perfect language) can be represented by a collection of circumstances specified by the notation:

 

[math]

(x_1,x_2,x_3,\cdots,x_n,t)

[/math]

 

One way to think of what that symbol means is to see the “simples” as individual nerve cells in your brain (x_i is a numerical label of a specific cell). The circumstance can then be seen as the collection of cells which are active at time “t”. Alternatively, the “simples” could be seen as specific positions on the paper stored in the library of congress (or, for that matter, all the libraries in the world). The circumstance can then be seen as the collection of positions which also store ink molecules at a specific time “t”. Note that, in both these examples, order of the “simples” is of no consequence because all order is embedded in the definition of the those specific positions or cells, an issue outside the representation. The things in our mental image of these circumstances are not really “simples” and the language with which we think about them is not perfect.

 

The issue here is not what [math](x_1,x_2,x_3,\cdots,x_n,t)[/math] represents but rather that there exists nothing which is not representable by such a notation. It is, in itself, a perfect language; however, the moment we translate it into any common language, that perfection vanishes. That does not mean it cannot be translated and I will use translation of specific explanations to uncover important constraints.

 

This is the very essence of part I of the original post “Laying out the representation to be solved”.

 

 

Part II concerns the other critical component inherent to any explanation.

 

It should be clear that there exists nothing communicable which can not be represented by a collection of circumstances as I have defined the term [math](x_1,x_2,x_3,\cdots,x_n,t)[/math] if we did indeed possess that “perfect language” sought by Ludwig Wittgenstein. I am not concerned (at least not at the moment) with constructing that “perfect language”. Rather, my sole interest is in discovering the constraints implicit in the definition of “an explanation”.

 

Certainly explanations would exist in such a perfect language. The issue is, exactly what is “an explanation”? Well, it is apparently held quite widely that, if you can explain something, you personally have a decent understanding of whatever it is you are explaining. So the question arises, how does one determine the extent of that understanding? I contend that, if you understand something, you are capable of answering questions related to whatever it is that you understand.

 

Thus the other critical component of an explanation is a general set of questions which will exhaust the complete extent of possible answers. I propose something akin to twenty questions; however, in order to exhaust the possibilities, I suggest an unbounded set of questions with yes/no answers. In essence, the answer to any question need be no more than an estimate of the probability of a yes/no decision (that takes care of questions when the explanation does not provide a definite yes or no answer).

 

It follows that knowing an explanation of a collection of circumstances essentially means that one knows the probability of all circumstances related to that collection whether they have experienced those related circumstances or not. An explanation of where you were last night might include many related circumstances far beyond the simple assertion of “at home in bed”. That explanation implies impossibility of a great number of supposed related circumstances and likewise implies the necessity of a few also.

 

So an explanation provides probabilistic answers to a collection of possible circumstances. This is quite easy to express in the “perfect language” Wittgenstein wanted to construct. It follows that every explanation of any collection of circumstances can be represented by the notation.

 

[math]

P(x_1,x_2,\cdots, x_n, t)

[/math]

 

The important issue here is that, since the “simples” are labeled with numerical labels and probability is defined to be a number bounded by zero and unity, that notation corresponds exactly to a mathematical function. Thus it is that I come to the conclusion that every explanation is representable as a mathematical function (in that “perfect language” Wittgenstein wanted to create).

 

There is however, one bothersome issue in that representation. Every explanation can be seen as a mathematical function but the converse is not true. That may not disturb the validity of the fact that the relationship is identical to a mathematical function but it does create a subtle problem in a search for the constraints implicit in the definition of an explanation. Clearly, viewing the explanation as equivalent to a mathematical function implies both constraints will be in a mathematical form. Anyone familiar with mathematics knows that the extent of the possibilities here are so vast that keeping these two very different constraints well separated from one another is important.

 

Upon discovering a specific mathematical constraint is necessary, the problem arises of guaranteeing that it is impossible that the found constraint is a consequence of that mathematical function being a probability. That constraint is not a constraint on an explanation but is rather, only a constraint on the representation I have chosen.

 

If we can open up the range of possible functions to All mathematical functions, that difficulty vanishes. We need to have a way of satisfying that constraint due to the definition of probability without constraining the representative function in any way. To do that, I essentially need to remove the constraint

 

[math]

0\;\leq\; P(x_1, x_2, x_3,\cdots, x_n, \cdots, t)\;\leq \;1

[/math]

 

from the representation.

 

Actually, there exists a simple method of avoiding that particular constraint. Let [math]\vec{G}(\vec{x})[/math] be a representation of an arbitrary function: i.e., a function is a method of mapping one set of real numbers, what is normally called the “argument” of the function (what is here represented by [math]\vec{x}[/math]) into a second set of real numbers (which is normally called the “value” of the function). In this case, the “value” is represented by [math]\vec{G}[/math]). The vector notation is used here merely for the convenience of succinctly displaying the set of real numbers as a vector in an abstract space.

 

Given absolutely any such function [math]\vec{G}(\vec{x})[/math], one can define what is called an “inner product”, [math]\vec{G}(\vec{x})\cdot\vec{G}(\vec{x}),[/math] which will be a positive definite real number.

 

If we now define

 

[math]

P(x_1, x_2, x_3,\cdots, x_n, \cdots, t)\propto \vec{G}(x_1, x_2, x_3,\cdots, x_n, \cdots, t)\cdot\vec{G}(x_1, x_2, x_3,\cdots, x_n, \cdots, t),

[/math]

 

“P” is guaranteed to be greater than or equal to zero. The only remaining part of the constraint due to P being a probability is to define that constant of proportionality such that the upper bound is unity. In the definitions of probability in standard probability theory, that upper bound is obtained by requiring that the sum over all possibilities be one. We can do exactly the same thing here. We merely integrate (or sum) the function [math] \vec{G}\cdot\vec{G}[/math] (which must be [math]P(x_1, x_2, x_3,\cdots, x_n, \cdots, t)[/math] times some constant) over all possibilities and use the fact that P integrated (or summed) over all possibilities has to be unity to establish the value of that constant of proportionality.

 

We then merely divide [math]\vec{G}[/math] by the square root of that integral (or sum) and obtain a new function (which I will call [math]\vec{\Psi}(x_1, x_2, x_3,\cdots, x_n, \cdots, t)[/math]). At this point the correct probability (or probability density) for the expectations predicted for our explanation can be written

 

[math]

P(x_1, x_2, x_3,\cdots, x_n, \cdots, t) \equiv \vec{\Psi}(x_1, x_2, x_3,\cdots, x_n, \cdots, t)\cdot\vec{\Psi}(x_1, x_2, x_3,\cdots, x_n, \cdots, t)

[/math]

 

At this point, not only can every explanation be seen as a mathematical function (the inner product of [math]\vec{\Psi}[/math]) but the converse is also true: every conceivable function [math]\vec{\Psi}[/math] can be seen as an explanation as it can provide a set of answers as to the expected validity of any conceivable circumstance.

 

One last cavil needs to be handled. This is a procedure commonly used in probability theory and is usually referred to as “normalization”. A number of possible problems can occur with that procedure. First, the specified integral might be zero and division by zero is undefined; in that case, the division is unnecessary as the original function [math] \vec{G}\cdot\vec{G}[/math] being positive definite, cannot be greater than one and thus the division is not necessary as [math]\vec{\Psi}[/math] can be simply be set equal to [math]\vec{G}[/math].

 

 

Secondly, the specified integral (or the specified sum) might be infinite and division by infinity is exactly zero causing the defined function (which is to be our explanation) to vanish exactly. From the perspective of probabilities, this second case actually appears to be quite reasonable. Anytime the number of possibilities goes to infinity (i.e., there are an infinite number of possibilities which do not vanish) the probability of a specific result must vanish. In that case, we concern ourselves not with specific cases but rather with ratios between integrals (or sums) taken over specified ranges.

 

 

It should be recognized that the only issue of interest here is that the result must be interpretable as a probability. That property, and that property alone, allows us to consider any given mathematical function as a possible explanation of the circumstances to be explained.

 

I want to make it as clear as possible that I have Absolutely No Intentions of finding any explanations of anything. I am concerned only with constraints implicitly imposed upon an explanation by the definition of the concept “an explanation” and nothing else.

 

I hope this clarifies exactly what the expression [math]\vec{\Psi}(x_1,x_2,\cdots,x_n,t)[/math] stands for.

 

Have fun -- Dick

[Qfwfq]

You're welcome Dick, although I kinda thought I had said it means treatise and not thesis. :shrug:

 

In my opinion, the major philosophical error made by Wittgenstein (as expressed by Bertrand Russell) is the idea that “the essential business of language is to assert or deny facts”. I would rather say, “the essential business of language is to communicate”; facts, fiction, nonsense or whatever. I think his concern with “facts” and logic is actually what prevents him from thinking about the central requirements of such a “perfect language”.

AFAIK his effort was toward constructing a “logically perfect language” for use in philosophy, not a general purpose “perfect language” for the vulgum, nor for use in theatre, poetry or other liberal arts. AFAIK he eventually concluded his endeavour was futile and this, largely, after long examining and discussing his purported "A = B" paradox, questioning whether such a statement should have any place at all in a logically perfect language where (he argued) A and B can't be distinct and also the same thing. But I won't go much into this cuz I'm not one that knows much more about it.

 

Wittgenstein's error is worrying about that first “problem”; it has no significance whatsoever. Any language incapable of expressing nonsense is simplistic beyond belief.

I disagree and I point out that you are missinterpreting Russel's point (1).

 

(they have parts made of letters which are, in turn, lines, intersections etc. which are themselves complex concepts)

How about molecules of ink? In any case you are contradicting your later statement:

The correct minimum number of required “simples” should be more on the order seven billion.

I think perhaps you would also find it helpful to take a look at how artificial languages for IT purposes are often defined, using so-called BNF's and the concept of tokens and terminals.

 

After this point I can't afford time to go through the rest of the post and glean out what there is that I haven't already seen.

[Doctordick]

You're welcome Dick, although I kinda thought I had said it means treatise and not thesis. :shrug:

Well, sorry about that. In my opinion, the issue is of utterly no importance and I thought I made that quite clear in my response to your earlier comment.

 

DoctorDick, on 16 August 2011 – 11:27 AM, said:

No! As a matter of fact I never even dawned on me to even think about what the word meant; all I was concerned with was what the paper said. The actual title of the thing is kind of a side issue isn't it? But thanks for your concern.

If you had glanced at the opening post you would have understood that the subject was the consequence of googling that phrase, not the phrase itself. Requiring the subject to be “What is Tractatus?” otherwise the thread is off subject seems to me to be a little extreme don't you think? I guess I will have to be more careful as to what I call my threads with the “Grand Inquisitor” overlooking my heresies.

 

AFAIK his effort was toward constructing a “logically perfect language” for use in philosophy, not a general purpose “perfect language” for the vulgum, nor for use in theatre, poetry or other liberal arts.

A “logically perfect language”? Come on, don't you think that is a rather unique interpretation of the adjective “logically”. That this “perfect language” is only to be applicable to logical assertions? Even if you are right, that is a rather stupid intention as it essentially inserts a constraint where the consequences are essentially beyond examining. Or are you in suggesting that “explanations” are not required to be logical. More medieval proscriptions of acceptable doctrine I guess.

 

AFAIK he eventually concluded his endeavour was futile and this, largely, after long examining and discussing his purported "A = B" paradox, questioning whether such a statement should have any place at all in a logically perfect language where (he argued) A and B can't be distinct and also the same thing.

In a perfect language, “A=B” can not occur with simples, that is the nature of simples. With "complexes" that is another story entirely. As I have said many times, mathematics is essentially a tautology and the assertion A=B is no more than identification that two complexes are equivalent and equivalence is not the same as identical as it is subject to context,residing itself within a larger complex. Wittgenstein's problem was that he was a philosopher and simply didn't have the understanding necessary comprehend the problem he was attempting.

 

And exactly what happens if you drop that idiotic first constraint? Or is he the authority and we mere mortals are not allowed to ponder such deep issues?

 

I disagree and I point out that you are missinterpreting Russel's point (1).

So what? What is your problem here? What are you disagreeing to? With my assertion that a valid “perfect language” can not be constrained to expressing logic only? Please give me a reason for that (other than it is against the acceptable doctrine you are out to enforce).

 

How about molecules of ink?

Did you not ever read anything I say?

 

The things in our mental image of these circumstances are not really “simples” and the language with which we think about them is not perfect.

 

The issue here is not what [math](x_1,x_2,x_3,\cdots,x_n,t)[/math] represents but rather that there exists nothing which is not representable by such a notation. It is, in itself, a perfect language; however, the moment we translate it into any common language, that perfection vanishes. That does not mean it cannot be translated and I will use translation of specific explanations to uncover important constraints.

Or perhaps you are suggesting a perfect language (there should be one name for every simple, and never the same name for two different simples) is nonsense? In a perfect language, “A=B” can not occur with simples. Wittgenstein's problem was that he was a philosopher and simply didn't have the understanding to comprehend the problem he was attempting to consider.

 

Or is it that you just read portions so that you can go off half cocked with your complaints.

 

In any case you are contradicting your later statement:I think perhaps you would also find it helpful to take a look at how artificial languages for IT purposes are often defined, using so-called BNF's and the concept of tokens and terminals.

I haven't the slightest idea as to where I am contradicting myself. Furthermore, you well know that I am not going to bother reading any of that! You are doing exactly the same thing Wittgenstein does: thinking that translation into a non-perfect language as a valid way of understanding a perfect explanation. If you cannot comprehend that problem, you are wasting both are times.

 

After this point I can't afford time to go through the rest of the post and glean out what there is that I haven't already seen.

Please don't bother. I would rather you leave me alone than be the subject of your thoughtless inquisitions. You make it quite clear that you don't put the first thought into what I say.

[Qfwfq]

Well, I don't see the sense in any of these complaints of yours. I don't see what your point is at all.

 

The only thing you leave me wondering about is: how you could get seven billion simples out of twentysix letters of the alphabet, even by considering the marks that graphically compose them. Anyways, if they go through such a tight bottleneck of complexes in the construction, why the need for such a greater number of simples? I don't get it.

 

Even written Chinese doesn't have many simples:

There are 8 strokes, just them have made thousands and thousands Chinese Characters, words and phrases:

http://www.ebridge.cn/new/languages/lan.php?sno=1303

That's only one more than Wittgenstein said, Dick, only one more measly simple than he reckoned!

[Rade]

Been off line for awhile, poor Wittgenstein...

 

Some may find the comments of SAM26 in this forum to be of use for understanding Wittgenstein:

 

http://forums.philosophyforums.com/threads/wittgenstein-a-summary-41631.html

 

As a first go, I suggest you concentrate ONLY on the comments of SAM26 in the order presented to the end, then you can return a second time to read the comments of others.

 

I would also like to clarify that Wittgenstein was a trained Mechanical Engineer, and studied pure mathematics and logic. He came to study philosophy well after his formal training in physics and mathematics.

 

Wittgenstein was a student of Russell, but they quickly came to disagree with each other, thus, one must be careful using Russell to understand Wittgenstein. It is best to use Wittgenstein to make sense of Wittgenstein (ideally as he wrote in German), then, logically disagree or not with his philosophy, with understanding that Wittgenstein in latter years rejected much (not all) of his own thinking. I find the comments of SAM26 in the above forum link to be useful to help with understanding the transition in the thinking of Wittgenstein over time.

[lawcat]

We have a branch of science that works like a charm in these matters, it's called linguistics--definitions and grammar. Thank Lord someone had sense to place that in elementary school curriculum rather than DD or Wittgenstein.