On 01-17-98 William Elliot wrote to John Boone... 
 
        Hello William and thanks for writing, 
  
 WE>  WE> Are there biological applications for fuzziness?   
 
 WE>  JB> Yes, many.  I'll begin by using fractals, which IMO, is related 
 WE>  JB> to fuzziness.  Fractals are "fuzzy" at the edges.   
 
 WE> I don't think so, yet this is an interesting notion. 
 WE> Fractals are called such as these shapes are not simple 1, 
 WE> 2, or 3 dimensional objects but have fractional dimension.   
 WE> Indeed there's something blurry about these objects. 
   
  Yep! 
  
 WE> I'll begin by heart 
 
 WE> JB> rate, HR if one were to plot heart rate as a function of time 
 WE> JB> HR(t),  
 WE> JB> and plot HR(t), y, HR(t-1), x, and assign a "spread fucntion", 
 WE> JB> one  
 WE> JB> finds, such a manipulation is a good predictor of -future- death, 
 WE> JB> in particular myocardial infarctions (I done some research in  
 WE> JB> this area).  
 
 WE> Don't know what you mean HR(t), y, HR(t-1), x.  What's the 
 WE> y and x?  What's a spread function? 
   
  If one were to measure HR, generated by measuring the P-P 
internal - the time from the beginning of one heart beat to 
the beginning of the next divided into 60) placing them in a 
sequence, with increasing number representing HR happening at 
a latter point in time, e.g. HR1, HR 2, HR 3 HR4, HR5, HR6, HR7, 
etc.  
  Now, let x be HR1 and y be HR2 and plot the point, 
next let x by HR2 and y be HR3 and plot the point, etc. 
  Said plotting technique results in a distribution on the 
x-y plane.  Apparently, the smaller the spread, the more 
likely death is to occur.  The trick becomes what spread 
is "Ok" and what spread is not. 
 
 WE>  JB> If one were to look at the structure of circulatory, respiratory, 
   
 WE>  JB> and nervous system, such systems have a fractal nature.   It is  
 WE>  JB> proposed, the reason, DNA being the making "code" could generate  
 WE>  JB> such diversity, by following a simple -REITERATIVE- "code"  
 WE>  JB> (going through the process of transcription and translation)  
 WE>  JB> in the language of mathematics, algorithim.  
 
 WE> Indeed, and visa versa, life like formations are generated by such a 
 WE> process. 
   
  Even crystalization, shows such fractal nature. 
  
 WE> JB> For diagnosis, what one does is take the set of {signs} and 
 WE> JB> {symptoms} and assign to a set known as {disease A or not A}. 
 WE> JB> Sets of signs and symptoms are often fuzzy.  For example,  
 WE> JB> the disease caused by "Sarcoptes Scabii" -usually- presents  
 WE> JB> with "severe pruritus".  How does then place what the  
 WE> JB> patient said into or not into, perhaps partly into the  
 WE> JB> set "severe pruritus"; in addition, "minor pruritis"  
 WE> JB> may be "severe pruritis" in another.  
 
 WE> I have a cough or spit phlegm, what's the probability of TB? 
 
 Yep. 
  
 WE> I have a cough or spit phlegm, what degree of TB should I assign? 
 
 Perhaps and in addition, is the set {cough} fuzzy.  
  
 WE> This is a possibility but as I know zilch about fuzzy ... 
   
  In addition, what information obtained from the patient allows 
one to place it into "a cough or sputum, phlegm?" 
  
 WE> JB> I hope this explains, why, I think, fuzzy presents itself 
 WE> JB> in the biological sciences, from the genetic level, to  
 WE> JB> the cellular level and finally to macrosopic level to  
 WE> JB> include the systems and even into the field of medicine.  
 
 WE> I agree about fractals.  I disagree about fractals and fuzzy sets.  It 
 WE> would  
 WE> take a decisively new insight to merge the two.  I agree  
 
  Ok.   
  
 WE> about the last example but don't have a glimmer how.  Now 
 WE> if you were to look into the index of your books,  
 WE> especially the one on managing uncertainty, for biological 
 WE> or medical applications, that would be a start.  Certainly,  
 
  I have a book, differential equations and their applications 
with a look at the medical disorder diabetes, but have not had 
time to look at it like so many other books :(. 
 
 WE> uncertainty is frequent companion to diagnosis.  Let me 
 WE> know if you find any examples. 
 
  Right, now, I have 5 magazines to read, just got 4 in the mail, 
so, it will be a while before I will be able to look and contemplate.  
I have tried to keep this short as I have some heavy reading to 
do. 
 
Take care, 
John 
 
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 On 01-17-98 William Elliot wrote to John Boone... 
 
        Hello William and thanks for writing, 
  
 WE>  WE> other book, what's introductory chapters like?  With substance?  
 
 WE>  JB> It depends upon the definiton of "substance."  I have, 
 WE>  JB> however, found it usefull as I learned from it, only  
 WE>  JB> on page 30 for the past 2 years.   
 
 WE> Substance is explicit definitions, theorems, proofs, formulas. 
    
  So far some defintions via the logo, Greek meaning word, 
method with out any mathematical treatment, so far no  
theoriums, proofs, formulas. 
  
        [snip] 
  
 WE> JB> Not unexpected, this definitional point is key for the rest 
 WE> JB> of our discussion.  Let me try to read and share some of  
 WE> JB> what I read at a future date, but I need to start to taper  
 WE> JB> my other disussions on this echo.   
 
 WE> A fuzzy set A is a function from the universe U of elements into the 
 WE> closed  
 WE> unit interval, x e A (d) could be written A(x) = d.  As a  
 WE> function is a set of ordered pairs, the answer is yes.   
 
 WE> Note that every fuzzy set contains all of the elements x e 
 WE> U to varying degrees.  The fuzzy set U contains all the  
 
  Yep, it would appear that way, if we have the definition 
correct. 
 
 WE> elements with degree 1.  The fuzzy null set not U contains 
 WE> all the elements with degree 0. 
   
 WE> Note the fuzzy set U is different than the set U.  U is the  
 WE> set of elements, for example {x,y,z}, while the fuzzy set U  
 WE> is the function U(x) = 1, U(y) = 1, U(z) = 1 thus assuring  
 WE> that x e U (1), y e U (1), z e U (1) for all the elements  
 WE> in the universe {x,y,z} of my example. 
   
 WE>  JB> I remember from Bart Kosko's  
 WE>  JB> book, somewhere between page 0 and 30, I believe around page  
 WE>  JB> 6 or so, Bart introduced the concept of fuzzy sets, e.g.,  
 WE>  JB> the set of {apples.}  He asked at which point, in consumption  
 WE>  JB> as in eating (at least we are having fun , does the  
 WE>  JB> apple become {not apple?}  
 
 WE> This sounds like descriptive stuff.  Does he give any explicit 
 
   Yes, but I think attempts to explain the point. 
 
 WE> definitions 
 WE> anywhere in the book?  The apple is not an apple when it's  
 
  Ah, but in the world of fuzzy, a half eaten apple belongs to 
the set apple.  Take a apple, take a bite, does it belong to 
the set apples?  
 
 WE> an apple core.  A half eaten apple is half apple.  A slice 
 WE> of an apple is 1/8 apple.  Simple enuf.  Now when a apple  
 WE> is rotten, how included in Apples is it?  How about apple  
 WE> sauce, cider?  Oh how language can out strip mathematical  
 WE> formalism. 
 
  You got it, this is what Fuzzy logis is supposed to help with. 
 
Take care, 
John 
 
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