A hairline fracture in GS foundations?

From: Steven Lewis
To: Steve Stockdale ; Phil Ardery ; Nora Miller ; Katherine Liepe-Levinson ; Kate Johnson ; K. Liepe-Levinson, Muse Enterprises ; Jeremy Klein ; Jackie Rudig ; Frank Gastner ; Emory Menefee ; E.W. Kellogg III, Ph.D. ; David Maas ; David Hewson ; Charles Russell ; Andy Hilgartner ; Alta Brock ; Allen Flagg ; Andrea Johnson ; Bob Potter ; Bruce Kodish ; George Barenholtz ; Gregg Hoffmann ; Irene Ross Mayper ; Jeff Mordkowitz ; Jennifer Carmack ; Jerry Nierenberg ; Jim French ; Laura Bertone ; Lynn Schuldt ; Milton Dawes ; Sanford I. Berman ; Susan Kodish ; Walter Davis

Tuesday, September 07, 2004 5:33 PM
Subject: A hairline fracture in GS foundations?

I hope this message finds my fellow geesers rested after a 'holiday' weekend! A new reader to "Manhood of Humanity" contacted me about the derivation of the Work formula #4 on page 112 of the second edition. He had no problems with the math until Korzybski leapt from step 3 to step 4. It does seem to be a "giant step for a man, a giant leap for mankind," and I am wondering if any of you math wizards can describe the steps Korzybski omitted in moving from equation 3 to equation 4.

Please, no "and then a miracle occurs" explanations!

Steven Lewis

www.kcmetro.edu/~lewis

From: Phil Ardery
To: Steven Lewis, Ralph E. Kenyon, Jr
Cc: Steve Stockdale, Nora Miller ; Katherine Liepe-Levinson ; Kate Johnson ; K. Liepe-Levinson, Muse Enterprises ; Jeremy Klein ; Jackie Rudig ; Frank Gastner ; Emory Menefee ; E.W. Kellogg III, Ph.D. ; David Maas ; David Hewson ; Charles Russell ; Andy Hilgartner ; Alta Brock ; Allen Flagg ; Andrea Johnson ; Bob Potter ; Bruce Kodish ; George Barenholtz ; Gregg Hoffmann ; Irene Ross Mayper ; Jeff Mordkowitz ; Jennifer Carmack ; Jerry Nierenberg ; Jim French ; Laura Bertone ; Lynn Schuldt ; Milton Dawes ; Sanford I. Berman ; Susan Kodish ; Walter Davis
Subject: Re: A hairline fracture in GS foundations?
Date: Tue, 7 Sep 2004 21:19:42 -0400

Steven -- Your new reader has spotted a proofreader's error -- the repetition of an error made first on p. 91, where the formula for summing the series is first introduced.  The denominator of the fractional term should appear as "R - 1", not "R - I".  Interestingly, the online edition of Manhood at esgs has this right.

www.esgs.org/uk/art/manhood.htm

Ralph -- If you should care to take my partial response and make sense of the math, I thank you.  If you lack Manhood of Humanity in hardcopy, the formula 4#  that Steven cites appear in the "Wealth" chapter, about 3/4 through.  The first presentation of the formula appears in the "What Is Man" chapter, very close to the end.

From: Ralph E. Kenyon, Jr
To: Phil Ardery; Steven Lewis
Cc: Steve Stockdale, Nora Miller ; Katherine Liepe-Levinson ; Kate Johnson ; K. Liepe-Levinson, Muse Enterprises ; Jeremy Klein ; Jackie Rudig ; Frank Gastner ; Emory Menefee ; E.W. Kellogg III, Ph.D. ; David Maas ; David Hewson ; Charles Russell ; Andy Hilgartner ; Alta Brock ; Allen Flagg ; Andrea Johnson ; Bob Potter ; Bruce Kodish ; George Barenholtz ; Gregg Hoffmann ; Irene Ross Mayper ; Jeff Mordkowitz ; Jennifer Carmack ; Jerry Nierenberg ; Jim French ; Laura Bertone ; Lynn Schuldt ; Milton Dawes ; Sanford I. Berman ; Susan Kodish ; Walter Davis
Subject: Re: A hairline fracture in GS foundations?
Sent: Wednesday, September 08, 2004 12:45 AM
Subject: Re: A hairline fracture in GS foundations?

First of all, all you "geesser" will have to let go of the [false] idea that "A" can never equal "A". If you can't get past that step, then you will have no hope of understanding the mathematics that general semantics is based on.

Let the symbol '^' stand for or "mean" "raised to the power of". Then (3) can be written thus:

(3) W = PR^1 + PR^2 + PR^3 + . . . + PR^T

a) Expand the term ". . ." to show the next to the last term. In effect ". . ." = ". . ." + PR^(T-1).

Oh, does that bother you? Well ". . ." means "the previous unlisted terms ", so it can be written as "the previous unlisted terms and PR^(T-1)". [Here's an instance of a multi-ordinal use of a symbol, in case you want a technical explanation.]

(3.1) W = PR^1 + PR^2 + PR^3 + . . . + PR^(T-1) + PR^T

b) Multiply both sides of the equation by R under the rule that multiplying equals by the same quantity yields equals.

(3.2) WR = PR^2 + PR^3 + PR^4 + . . . + PR^T + PR^(T+1)

c) Subtract equation 3.2 from equation 3.1 under the rule that equals subtracted from equals yields equals.

(3.3) W -WR = (PR^1 + PR^2 + PR^3 + . . . + PR^(T-1) + PR^T) - (PR^2 + PR^3 + PR^4 + . . . + PR^T + PR^(T+1))

d) Expand the first ". . ." to include one more term. In effect ". . ." = ". . ." + PR^4. [See a) above.]

(3.3.1) W -WR = (PR^1 + PR^2 + PR^3 + PR^4 + . . . + PR^T) - (PR^2 + PR^3 + PR^4 + . . . + PR^(T+1))

e) Next we want to expand the last part to include one previous term. In effect ". . ." = PR^T + ". . .". [See a) above.]

(3.3.2) W -WR = (PR^1 + PR^2 + PR^3 + PR^4 + . . . + PR^T)-(PR^2 + PR^3 + PR^4 + . . . + PR^T + PR^(T+1))

f) Remove the parenthesis by multiplying the second half through by -1, using the distributive law and the associative property of addition and subtraction.

(3.3.3) W -WR = PR^1 + PR^2 + PR^3 + PR^4 + . . . + PR^T - PR^2 - PR^3 - PR^4 - . . . - PR^T - PR^(T+1)

g) Regroup terms so that like powers are together using the communitive law.

(3.3.4) W -WR = PR^1 + (PR^2 - PR^2) + (PR^3 - PR^3) + (PR^4 - PR^4)+ (. . . - . . .) + (PR^T - PR^T ) - PR^(T+1)

h) Perform the indicated subtraction. Each pair of terms will cancel each other out.

(3.3.5) W -WR = PR^1 + (0) + (0) + (0)+ (0)+ (0) - PR^(T+1)

If you have trouble with the idea that ". . . - . . ." equals zero, then you can write out all the terms for various values of T and see that it is so. This is left as an exercise for the skeptical.

i) Simplify. Here's what we have left after the above operation.

(3.3.5) W -WR = PR^1 - PR^(T+1)

j) Multiply both sides by -1 under the rule that equals multiplied by equals yields equals, and rearrange terms under the communative law. (We would not have had to do this if we had chosen to subtract equation 3.1 from equation 3.2.)

(3.3.6) WR - W = PR^(T+1) - PR^1

k) Factor out a W on the left side and an R on the right side using the distributive law (in reverse).

(3.3.7) W(R - 1) = R(PR^T - P)

l) Now, provided that R is not equal to 1, we can divide both sides by the quantity (R-1) under the rule that division of equal quantities by the same non-zero quantity yields equals, and we will get the desired equation in a slightly different form.

(3.3.8 ) W = R(PR^T - P)/(R-1)

l) Rearrange the terms (under the associative law) to get the equation (4).

(4) W = [R/(R-1)](PR^T-P)  =    R  
R-1
(PR^T-P)

Since I can't trust email to preserve fonts or spacing, the above is written with a slash instead of a horizontal line.

However, this is a very common formula for the sum of a series, so the above "proof" of the derivation need only be learned by mathematics initiates. The rest can look the formula up in mathematics reference books.

Korzybski simply asserts that the increase of each generation follows a multiplicative expansion rather than an additive expansion. In the top half of page 110, he state "this is the mathematical equivalent of adding his parent's years of life to his own." Note the use of the word 'adding'. In the second paragraph he simply asserts or claims that the result is not an arithmetic progression. He then continues through page 111 and into page 112 to explain what a geometric progression is, but his discussion makes no effort to compare and contrast or argue why this may be so.

The usual statement, that each new generation begins where the previous generation left off, contains no quantification or any way to differentiate between an arithmetic progression and a geometric progression. I have personally used the argument that each new bit of information must be correlated with all previous bits. This turns out to be a power of 2 exponential curve. But that applies to information. See How much "out-of-date" is Korzybski in 2003?. It does not apply to energy. Information is created, but energy is conserved, and material wealth is instantiated as energy. Can my grandson shovel dirt four times as fast as I can? And his son 8 times as fast as me? The "truth" about the increase in wealth, which Korzybski is talking about, is somewhere in between the linear and the exponential. The 1972 Club of Rome report on Limits to Growth should be looked at very carefully.

But I digress.

I was asked to provide the "missing" formulas. I have done so with explanation of each step. Enjoy.

Supplement

Regarding Manhood, on page 113, Korzybski says, "The formula makes mathematically evident the time-binding capacity characteristic of the human class of life." Note that the formula is supposed to be a map that describes the territory. What Korzybski has done is explain the map (formula) and then claim that the map shows that the territory is similar in structure. The fallacy in this presentation is known as "begging the question". He has assumed that what he is claiming is true rather than present any convincing evidence that this map is a good one. What Korzybski has done is to "distract" readers from the fallacious argument by explaining the formula. It is, however, fairly widely assumed that (unchecked) growth of a species is exponential. This is essentially true of all species, including viruses, bacteria, plants, and animals - not just humans. The key word is "unchecked". Checking limits include distribution difficulties with the progeny, available resources, predators, hazardous environment, etc.. So, while unchecked growth is exponential, the reality of a limiting environment is otherwise.

When it comes to the growth of information, the simplest model, the one I use to explain it, is that each new bit must be correlated with all previous bits. This is true of information, but it is not necessarily true of knowledge. Only new pieces of information that are essentially independent of prior knowledge falls in this category. A new piece of information that is not independent of all that went before cannot be combined with prior knowledge in all possible combinations, and that is what is required for exponential growth.

Examples:

The discovery of a new animal species, or a new plant species adds to our knowledge in an arithmetic fashion. It is merely added, not multiplied. But a discovery of a new fundamental particle in physics, on the other hand, must be put together with all the other previously discovered particles, so that the interaction possibilities must all be categorized and validated or disconfirmed. So knowledge increases linearly in some areas and exponentially in other areas.

It must also be remembered that knowledge is not information. Information is a measure of the ability to disambiguate possibilities. Historical knowledge is the record of disambiguations that have been corroborated or disconfirmed. In a nutshell, knowledge is how information has been | is | can be used. While it is knowledge that a particular theory is false, the information in that falsified theory is not knowledge. (And just in case you are interested, I see wisdom as the compassionate use of knowledge.)

To quote one source.

My own distinction is thus:

I differentiate knowledge from information. Knowledge is the capacity of an entity to act on information. Knowledge Engineering is the process of extracting knowledge from a person and embodying it in a system - such as an expert system program or an organizational structure. Entities that embody knowledge include persons, organizations, processes, and systems. Any such entity exists in a context or environment and has purposes and goals. Knowledge management is the explicit organization and control of knowledge within an organization. (See Organizing for Knowledge Management.)

Behavior that is "instinctive" (encoded in dna as expressed), begins in the same place for each new generation. For something to be added to this repertoire there must be a mutation in the dna. Each new generation begins with the capabilities that their parents were born with. Behavior that is "learned" (encoded in the environment), begins with the process of learning from the environment, which includes the parents and the social structure. Each new generation begins where the parent began, but, because new learning is not tied to a dna mutation, each new generation can also learn what the parents learned as new. The rate of growth for this type of information is potentially much greater than that limited by changes to dna, because it is not tied to a physical replication process. A case in point is the fact that a baby chimp named Loulis learned to sign while living with Washo, the Chimp who was taught American Sign Language, and another signing chimp named Ally. Loulis learned to use what his foster parents had been taught by humans. This is rudimentary time-binding in its simplest form.

"I would be the first to admit that it is possible that Washoe may have actually taught some sign to her infant, however, I believe it is far more likely that her infant learned sign because he was immersed in the signing environment. The chimps' ability to use and understand sign is not diminished to me because they did not learn it in a school-like setting." *

Other reports of a new learned technique for dealing with an environmental problem being passed on from generation to generation in various non-human species have been reported, and not just in primate species. Although the rate of acquisition of new species knowledge is very slow by comparison.

Time-binding, in this sense, is not a hard and fast category. We humans, however, excel in this regard, in part due to our external use of language. The big jump in our excellence in this area begins with the invention of written languages. With written language, our environment includes the possibility (not the actuality) of learning all that was previously recorded (that has been preserved - provided the knowledge of how to read the records still exists). The records, however, degenerate to mere information as the knowledge of how to use the information is lost. -- Are there still any stone masons around with the know-how to build the magnificent stone arch bridges built two centuries ago? -- Lost knowledge detracts from the pure exponential nature of information. Another factor that affects the growth of knowledge comes in the form of paradigm shifts. At a paradigm shift a whole body of knowledge is disconfirmed and replaced with a new perspective. How is this to be quantified? That human knowledge increases as a smooth exponential curve is a vast oversimplification. But it's one of those ideas that is relatively easy to grasp in its simplified form. It's a "neat high-level abstraction" - A map so far removed from the territory that most details are obscured.


Annotated bibliography of general semantics papers
General Semantics and Related Topics

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