Biblical Hebrew Forum

["Yitzhak Sapir" <yitzhaksapir AT gmail.com>]

On 1/21/07, Peter Kirk wrote:

> > ... The linear/proportional relationship is observed, and since a linear
> > relationship is the simplest relationship that can be considered it is
> > probably best to stick with it. ...
>
> NO for at least three reasons:
>
> 1) This is NOT what is observed;
>
> 2) This is NOT what is predicted by any kind of theory, since it is
> trivially obvious to note that the probability that a new verb added to
> the list will be a homonym is not independent of the existing number of
> verbs;
>
> 3) This does NOT have the implications which you suggest for merger of
> letters, because of course if two letters do merge the number of verbs
> will add together, but the number of homonyms will more than add
> together because new homonyms have been created. This new homonym effect
> is not just a deviation but the essence of the issue.
>
> > ... I think that if one truly
> > wanted to investigate the nature of the observed relationship one
> > would do the calculus to identify the relationship that is governed
> > by sound laws such as the above including the above mentioned
> > deviations. ...
>
> Indeed, and that is what I have been trying to do. Why have you been
> ignoring my results, and insisting that the relationship must be linear
> even though the analysis as well as the data show that it is not?
>
> > ... That is, the equation of the sound law unification of
> > phonemes lays the framework for computing through calculus the
> > relationship that is to be expected of a language in which this
> > was a governing principle.  So that would be where I'd start, but
> > until then, I'd stick with the linear relationship which seems to
> > fit the observed data quite well, and because of its simple nature
> > does not also require complex explanations or substantiations.
> >
> >
> You stick with your over-simplistic relationships which doesn't explain
> the observations if you like, an approach which sounds about as
> scholarly as Karl's, but I will follow a proper scholarly approach.

The last statement seems to imply that I do not follow a proper scholarly
approach.  Let me explain what I believe to be the proper scientific
approach:
1) Observe data.
2) Study the relationships between the data.
3) Theorize as to why relationships observed in (2) exist.
4) Test the theory against the data and the observed relationships.
5) Discard any theory that does not properly explain the observed 
relationships.
6) Repeat 3-5 until a theory that explains the observed relationships is 
found.
7) Make more observations based on the new theory and repeat steps 2-7.

I believe I follow a proper scientific approach, and I am stuck in
step #2 because
I am unable to come up with a theory that properly explains the observations. 
 I
therefore am content with leaving unexplained the observations, but noting the
relationships.  These observations that cannot be explained by theory make up
an "empirical law."  (I think more observations are needed before this can be
considered a "law").  However, I find it problematic to run ahead and create a
theory and then when the data doesn't fit the theory, throw away the
data.  Like I
said, in the case of a mappiq he, or medial vav, I can understand the special
circumstances of those data and can understand if that data is ignored as
very special cases in the framework of an overall study.  I cannot accept that
for Resh.

In these terms, I am given the particular database of homonyms, and the number
of times any particular word appears, and I try to assess how the data
is related
to each other.  I graph the data and see that there is an increasing
relationship.
This leads me to try regression analysis and I find that certain types
of analyses
lead to good correlation.  In the end I favor linear relationships
because a linear
relationship is much simpler and I prefer simple over complex,
especially at step
#2.  I am sorry I called it "proportional."  You're right, they're not
proportional, so
this was a bad term to use.

Using the relationships noted in #2, I can make a certain conclusion regarding
Karl's theory.  I am not seeking to explain the linear relationship at
this point,
but I recognize it exists.  For that matter, at this stage I am pretty
content with
any relationship (power, linear, etc) that can approximate the observed values
of homonyms well.  I will choose a particular relationship only when I find a
suitable explanation that convincingly explains it, and also works well to
explain the data.

I have taken time to consider the statistics, and after playing with the 
numbers
a few days, I think that a merger would create ~(H1)(H2)/L^2 homonyms where L
is the number of letters/phonemes in the final alphabet, and H1 and H2 are the
number of homonyms of the two letters/phonemes that merged to one letter/
phoneme in the L-letter alphabet.  Namely, I constructed a table of 3 rows and
3 columns, each square describing a possible pair situation.  For example,
one square might describe the situation of letters ABX as opposed to CDY
and another AXB and CDY.  Here A, B, C, D are open possibilities and X and
Y are the two merged phonemes.  In the case of AXB and CDY the probability
of an homonym is the probability that A = C and that D = X and B = Y.  A = C
gives us a probability of 1/L^2 because each of them is an open possibility.
However, this is added for all letters together so we end up with only 1/L.
D = X and B = Y give a probability of 1/L each.  The probability of all of 
these
together is about 1/L * 1/L * 1/L = 1/L^3.  For the pair of ABX vs. CDY you
get 1/L^2, and so the total probability ends up being something like
(2L + 1)/3L^3.   The upper 2L dominates and this approximates to 2/3L^2.
For the case of Shin and Sin in the original first table, one multiplies 259 
by
77 and multiplies by this probability to get something around 84 or
93.  However,
we find only 31 homonyms of Shin/Sin.  So maybe I computed wrong, or maybe
there are additional considerations that further limit the final number of
homonyms.

Now, as for trying to go towards a theory that describes the data.  I tried to
look at a merger as one where y3 = y1 + y2 + Kx1x2 and x3 = x1 + x2 - Kx1x2
where (x1,y1) and (x2,y2) are two pairs of (roots,homonyms) for a particular
letter, and K is some constant probability. If one reorganizes, one gets
y3 + x3 = y2 + x2 + y1 + x1.  If this were governed by an equation that worked
for all (x,y)'s this should be observed for any triplet of (x,y)'s,
and this allows one
to find that y1 + x1 = y2 + x2.  This does not square with the data
where as x (the
number of roots) increases, so does y (the number of homonyms).  Again, there
may be additional factors to take into account here.  In both this and
the previous
computation, those additional factors do not include the "higher order
terms" you
speak of, however, because all we are concerned about is the new matches, not
the old matches that already exist.  This analysis described here is
what I meant
by going from a description of a merger to something that describes the
relationship between homonyms.

Thus, I am left at step #2, noting a linear relationship in the data
but not having
any way that can convincingly explain it.  Now, I followed you essentially 
until
you wrote, "But my latest analysis suggests a simple square law rather than
a 2/3 power law, if the roots are actually randomly distributed. My reasoning
is as follows ...".  Here, you seemed to go ahead and confuse the regression
analysis, which was meant to try to understand the data, "step 2", with
theorizing the reasons for the results, "step 3".  The data shows a linear
analysis (or a power curve analysis with the exponent being 3/2 when y =
homonyms, if I remember correctly).  This is not the time to change the law
just because a theory suggests a different relationship.  This is the data, 
and
what the data suggests the relationship is.  If a theory comes up with
something else, then that theory doesn't fit the data.  Furthermore, looking
now at your further discussion, I note that you computed the probability that
two words in a pair will be a match as 1/L^2.   However, my computation 
suggests
this is to be essentially multiplied by a factor of 2/3.  In the
further analysis, where
you have the large table, you use consistently a very large value for
N, namely, the
total number of roots of all letters, rather than the total number of
roots with the
proper letter in the root.  That is, your N^2/2 should vary from line
to line and be
much less than what you observe.  Again, I am content to leave it
unexplained and
simply note the relationship in the observed data.  You seem to want
to progress to
a theory, but in the end, if the theory fails, the data and the
observed relationships
in the data remain.  While my "over-simplistic" approach may not successfully
*explain* the data, (because no explanation is provided), it
accurately describes it,
and it is this that we must stick to if we want to find an explanation
in the end,
using proper scientific methodology.

Yitzhak Sapir