MATHEMATICA: A Treatise

                Written by: Marilynn Lea Stark

  Chapter 1

Preface: 

     The study of mathematics originally had taken precedence over all other subjects in my education, and although I did not major in it, it remained as a first love, a supreme conceptual plane and driving interest for me as I grew as a scientist.  Due to the rigorous application of metaphysics on the precipitous political path upon which I landed during the Cold War from 1980 onward was I also one day gifted with a breakthrough in my understanding of mathematics.  This breakthrough in my understanding of mathematics resolved the quest which had remained with me throughout all of my studies in the scientific disciplines which had become my occupation and contemplative endeavor.  In July of 2001 I had derived e.  This inspired me further to develop "A Treatise on the Chemistry of Living Things" at that time.  Once the 9/11 disaster struck our nation shortly thereafter the continuity of the work I had begun with the derivation of e was placed on the back burner; I began to develop the Web site Starkmusic.com with the hopes of ever disseminating my classical music compositions to the wider people.  Starkmusic.com was offered as an indirect answer to the plight of our nation in the wake of 9/11 in the way of guidance culturally.  Following that, the political way for me became most tenuous as to residence, and I was uprooted from all I had lived and worked for in the way of recovery from the results of the loss of freedom I had known for so many years as I led in the Cold War.  Bereft of religious freedom, my impending marriage and financial stability, once again, the strife for survival took over my life.  Still today, the socio-political forum in my locale unjustly fights my return to academia on a day-to-day basis, and so I am offering this work through the media of the Internet for proper dissemination in the name if not spirit of freedom and democracy.  May this work be well appreciated, and may justice be won.  May God bless America. 

 //  Marilynn Stark September 5, 2005     

              Derivation of e

     The logarithmic function is defined:

          y=log_bx

Equation 1.             

This function sets forth a variable x which changes in regard to a constant b, or base, thus relating by value to the function y discriminately.  What is the shape or behavior of x as it varies?  The independent variable x can simply increase as it will, and its logarithm can be found as the exponent, 

          b^y=x                          

Equation 2.        

E.g.,  log₁₀100=2         and                     log₁₀10,000=4

          10²=100                                     10⁴=10,000.

     However, in order to study in the abstract how x might vary we can view the expansion of x by looking at a given variable 't' found in this expression:

          (1+t)^1/t

     Then, this expression can be substituted in for x in the function

           y=log_bx

     Intuitively, x would start by adding to the first natural number, the unit value, 1.  Everything starts from 1.  Therefore, (1+t) is established as the natural sequence of integers starting at 1.  This is fundamental; however, since this natural sequence will be related to y, the logarithmic function, then numbers which are described in sequence from (1+t) can be viewed as possible multiplicands of exponential powers, or as bases, also of increasing, natural whole numbers.  This (1+t) is how numbers change in sequence, i.e., (1+t), most fundamentally, and then, since a log is an exponent, the same discrete series must be applied logarithmically to this (1+t) sequence.  In order to see how the logarithmic function works, we could say (1+t)^t , using t twice for symmetry.            

     However, this  expression (1+t)^t would approach infinity upwardly or outwardly, which is partial symmetry.  It would be more useful, therefore, to explore (1+t) as a possible base to an exponential power series by inverting the exponent to which it is held, thus as

          (1+t)^1/t

For (1+t)^1/t neatly approaches 1, since ¹/_t approaches 0.  This approach to 1 gives a perfect symmetry; that is: 

(1+t)1/t=(1+t) ~0É(1+t)/(1+t) É 1      
 t¾º

Equation 3.

     Notice that the logarithmic function overall greatly compacts numbers.  Great changes are reduced to simple, or smaller changes.  Thus, instead of moving conceptually from, say, 10² to 10⁹, which written out is 100 to 1,000,000,000, the course is shortened to the span from 2 to 9.

     Now comes the idea of understanding the function y=log_bx by its derivative, which particularizes it to the infinitesimal.  That is to say, the logarithmic function has its own mode of change, and to study how that change occurs we can divide into it by a limit taken by that divisor, which is a form of arbitrary unification, but not to 1, which is why it is only arbitrary.  

     Therefore, the derivative dy/dx of the function of x is the ratio of Äy to Äx seen at the limit as Äx approaches zero for any function y=f(x).  (Eq.2)       

dy        Äy
ÊÊ = lim  ÊÊ
dx   Äx¾0 Äx

Equation 4.

     The ratio Äy/ Äx is made minute unto change and which changes unify the ratio by the approach of Äx to zero when the limit is applied.

     The limit as Äx¾ 0 particularizes the derivative of the function unto the (open) interval

          x2 Ê x1 = Äx or, more generally, xn+1 Ê xn = Äx.

 Equations 5 & 6.

     So now, it is stated that the limit of the expression (1+t)1/t = e, as t approaches infinity,

          
lim(1+t)1/t= e ;
t¾º

Equation 7.

 and this limit is given by the infinite series that sums the reciprocals of factorials:

e = 1 + 1 + 1 + 1 +...+ 1 +...  
        Ê   Ê   Ê       Ê
        1!  2!  3!      n!

Equation 8.

e1= (1+0)0=1
e2= (1+1)1=2
e3= (1+2)1/2=31/2=1.73205
e4= (1+3)1/3=41/3=1.58740

(This method of computation is indirect.)

     Observation: the first term in the infinite series which is given by e is 1 because at infinite t, which is never reached, the exponent 1/t  approaches zero, so that the entire expression approaches 1.  Since 1 is never realized, since infinity cannot be quantized by increasing natural numbers, we know that 1 is thus always modified, as if 1 would "found" its own modification in an ever-changing fashion.  The way 

lim  Äy  
Äx¾0 ÊÊ       
     Äx    

is used in the Fundamental Theorem of calculus causes us to compare this expression and its limit as  t¾0 ,since in this case, there is not a ratio of change; rather, there is an exponent, 1/t, to consider also as ratio.  So the change is exponential, and does not linearize a curve by diminishing intervals to the limit of zero, which says that the curvature of a function is less real or prominent dimension --wise when a line approaches two proximal points.

    Thus, knowing that the approach is infinite unto 1, how can this approach be characterized?  Just as the natural numbers proceed one-by-one in a sequence, then they may contribute to the 

lim(1+t)1/t
t¾º

in a sequential fashion, yet, if one calculates e by taking discrete values of t, one at a time to form a simple summation, one has erred.  Why?  This is because these values are discrete -- it would be like calling the interval  xn+1 Ê xn = Äx     a point: the point is only approached by infinitesimally small intervals.  Similarly here, to write in a value for t defies the nature of the expression, which though composite through the character of t, only particularizes to 1 in a dynamic sense, never reaching 1, just as a curved line is never really a straight one, no matter how small the interval.

     This is further complicated by the fact  that due to the inversion of the exponent to 1/t from what could have been t itself, explained previously, discrete values for t particularize into roots, such as the cube root of 4.  This leads to a second particularization of the expression, since taking the varied roots of given values compresses the expression.  This stands in contrast to the expression (1+t)^t  as t¾º whose limit would be true infinity.  With compression occurring by root formation using discrete term-taking, there is the setting up of internal limits, which defies the nature of the expression 

(1+t)1/t as an ever-expanding expression if t¾º.

Knowing this, that we must imitate the conceptual awareness offered by the power of the exponent yet inverted, 1/t , to cause the limit of the overall expression (1+t)1/t to approach unity, or 1: we must realize that there exists here not a value-by-value relationship with the variable at hand, but rather a series, and an infinite one, where each member is as important as the next.  

     Therefore, we must seek to characterize that infinite series.

     Any discrete values ordinated from (1+t)1/t and listed comprise a sequence of given numbers.  If we wish to characterize that sequence, then we may apply a limit concept to the varying factor, so as to see unto what value the sequence changes.  Here we have already seen that the expression  

               (1+t)1/t ,      

if seen as the limit of t as t approaches infinity: then the sequence born out of that increment upon 1 (one) of an ultimately varying t actually approaches 1 then again, since 1/t  in such a case, the exponent, approaches zero. 

     The discrete values of the series of reciprocal factorials added to 1 which is e, or the limit as t approaches infinity of  (1+t)1/t can be calculated discretely: 

e = 1 + 1 + 1 + 1 + 1 + 1 +...+ 1 +... = 1 + 1 + 1 + 1 + 1 +  1
        -   -   -   -   -       -                -   -   -   ---
        1!  2!  3!  4!  5!      n!               2   6   24  120

This sum arithmetically amounts to 2.00000 + 0.50000 + 0.16667 + 0.4167 + 0.00833 = 2.71667.  The normative value of e to five decimal places is given as 2.71828 for the first six terms of the series.  

     By calculating e only with the first five factorials a nominal value of 2.71667 is obtained.  Thus, we can reasonably conclude that since e is the characterization of the expression

  (1+t)1/t  

as t approaches infinity limit-wise, that values can be taken for the actual natural numbers whose comprise define e, and even that they are in a series, and even that this series is infinite.  

     This proof of the series given in classical mathematics causes the observer to want to further understand that series.  Why this series?

     In the strict expression  (1+t)1/t,  the 1/t  exponent causes the base (1+t) to be figured as roots, such that even as t increases, it compresses into an overall value which, when multiplied by the given power of the root at hand, equals the nth root.  This forms a certain sequence of roots.

     To take a limit on this expression of sequential roots as t approaches infinity is to perform an operation upon that sequence, since this limit is a determinative factor, and the factor is infinity .  By performing this operation, a series is elicited from the sequence given by the expression:              (1+t)1/t

     This series is said to be:         

e = 1 + 1 + 1 + 1 +...  1 + ...+  = 1 + 1 +   1  +  1  +... =
        Ê   Ê   Ê       Ê                    ÊÊÊ   ÊÊÊ      
        1!  2!  3!      n!                   2à1   3à2       

                                      
                                      
e= 1 + 1 +  1 + 1 +...  1 +...+ = 1 + 1  +  1  +   1   + ... +
       Ê   ÊÊ  ÊÊ       Ê             Ê    ÊÊÊ    ÊÊÊ
       1!  2!  3!       n!            1!   1à2    2à3

     When 1/t is introduced as an exponent to (1+t) remember that then an internal symmetry is introduced due to the ratio which is 1/t.  This causes the sequence formed of  (1+t)1/t to be compressed. When the operation of limit-taking is performed            upon (1+t)1/t , then the exponent 1/t is seen as zero, and (1+t) is perceived as 1, no matter how t had varied in its approach to infinity.

     This root-taking at the sequence level must be modified through the limit-taking operation cited heretofore, since all of the operations are subsumed under the leverage of the limit.  There are three such operations at hand which are operated upon by the fourth, or the limit-taking operation in the expression 

lim(1+t)1/t
t¾º

First, there is addition, (1+t), t being added to 1 variably and in the natural sequence of integers.  Then, there is multiplication, which could be more simply, to demonstrate by example, to the tth power at which the base (1+t) is used to multiply itself against itself.  Next, there is division, as the exponent is actually inverted to the ratio of 1/t .  

     When the fourth operation, that of taking the limit of this overall expression, is applied to the expression as t approaches infinity:

      lim(1+t)1/t     
       t¾º                             

the internal symmetry which had been introduced by the division in 1/t as an exponent is neatly reduced to zero.  This reduction also affects the base (1+t), by symmetrizing it to 1 by identity, thus:

          ê(1+t)ï                               (1+t)
 (1+t)~0= ò ÊÊÊ ò = 1 OR (1+t)0=(1+t)1-(1+t)1 = ÊÊÊÊÊ = 1
          ì(1+t)ñ                               (1+t)

 Equation 10.

     When this symmetrization occurs, what happens to the operations of multiplication and addition which were cited earlier, and are composite to the overall expression 

          (1+t)1/t

under consideration now as the power of the operation of the limit-taking which governs the symmetrization is applied at the given limit?  These two operations must also accomplish each a step in symmetry, so that the base (1+t), which had been compressed into root-level products, will realize an expansion meet with the approach of t to infinity.  Rather than to be held to the constraint of an internal sum, which had been imposed by the internal symmetry of the multiplier,  the exponent 1/t, which is also itself a dividing factor, such that nth roots of (n+1) are taken; the way to symmetrize those multiplying bases now freed from root-taking, would be for straight digit multiplying as the digits increase by 1.  The founding term of this series of increasing groups of multiplicands should be 1! itself, since even in a series of whole numbers starting as internally multiplying and increasingly multiplying groups, the number 1 is that to which all other groups symmetrize, thus, 1! or 1 x 1, would be included by definition of the unitary principle now extended in this case.  The unitary principle can be understood as the truth that quantification starts with one, and relates back to one therefore, whether one is seen as the digit 1 such as in the natural numbers, or as a unit value or entity.  The unitary principle allows the derivation of symmetrization, wherein its truth is applied specifically to mathematical operations and the concepts which underlie those operations as simplification to one (1) is elicited accordingly.

     Now again, how would the multiplication change once the symmetrization to 1 of the overall expression occurs dynamically as t¾º,and so thus 1/t¾0 ?
              

     The multiplication would now express as groups comprised successively of the natural numbers multiplied one against the other in as orderly a fashion as at all possible, and simultaneously in an increment-by-one manner.  The base is thus symmetrized by the limit-taking operation which affects so powerfully the exponent 1/t , so as to cause the entire expression (1+t)1/t to symmetrize unto unity itself as t approaches true infinity. 

     In order to visualize such concept, the symbol for true infinity will be introduced: 

Ê
Ó = true infinity   

     This symmetrization is what elicits a series from a sequence.  If the series does indeed so arise out of a sequence, then the operation of taking a limit which had reduced the exponent  1/t  to zero  in the limit as t approaches true infinity, must also be effective upon the terms of the sequence.  These terms were formed of (1+t) compositely, so that now, the limit-taking operation will by definition of the series formation affect the (1+t) component.

     Question: how is (1+t) as t approaches true infinity affected in the expression (1+t)1/t in the limit of true infinity, or,

          1/t
lim (1+t)    = ?
  Ê                                  
t¾Ó  

Since therein 1/t¾0, then we can apply the Law of the Infinitude of Unity to satisfy the question of how (1+t) changes near the limit, so as to elicit the infinite series.  That is,

       
        1/t    Ò          t
lim(1+t)   = 1 Ê lim (1+t)  = Ê                               
t¾ Ê           Ó t¾ Ê         Ó                       
   Ó                Ó                        

¸ 1 = Ê
      Ó

Law of the Infinitude of Unity.

The Law of the Infinitude of Unity, as stated above, reads as: the limit of the quantity (1+t) held to the power of 1/t  as t approaches true infinity equals 1, which is defined as symmetrical with the limit as t approaches true infinity of the quantity (1+t) held to the power of t, which equals true infinity; therefore, 1 = true infinity.

Please note that a new symbol is introduced with the statement of this mathematical law or principle, and that is the symbol  

 Ò
ÊÊÊ . See Table 1 : New Math Symbols  below for reference.
 Ó       

     To recapitulate and then derive e:

y=log_bx           

¸ b^y=x       

Study x by forming an expression of maximum expansion, since, if the infinite is considered by maximum expansion, then the infinity thus realized gives the truth, or the nature, of the function.  Therefore, form the expression   

(1+t)1/t,    

and to find its limit as t approaches true infinity, will then be related back to the derivative of the logarithmic function of x as t approaches infinity.  This limit equals e as according to Equation 3:

lim(1+t)1/t= e 
t¾º

How does one derive e?

lim ö[(1+t)1/tö] = 1           
   Ê                            
t¾ Ó                   
ò ö[(1+t)1/tö] Ç the symmetrization of (1+t)1/t  

(Ç signifies "is defined as")

Equation 11.

Note:   j(x)j is hereby introduced, and which means that an unknown variable x is symmetrized.  The discussion of the meaning of symmetrization will be present throughout this text in a 'hands on' fashion.  As an understanding of e is unfolded through its actual derivation here, the meaning of symmetrization should take conceptual hold.  Symmetrization is fundamentally the reduction of quantifications and the interrelating operations upon those quantifications to the simplicity of 1, and which is best taught for most at first through demonstration.  Since the logical implications of symmetrization are pervasive, first, a non-elaborate definition will serve best as it is introduced; secondly, this current treatise will allow the introduction of the concepts and symbols relevant to the mathematics which arise out of the understanding of unity in mathematical conceptual treatments, all through the derivation of e.  Please see Table 1: New Math Symbols below for a convenient reference guide as the work at hand unfolds for you.  A clear visual perception of the symbol itself outside of any active sequence of equations will naturally instruct and nurture the process of grasping the mathematics in this treatise.  Questions are welcome.

     Question: How is (1+t) as t approaches true infinity affected in the expression (1+t)1/t in the limit of true infinity?  

     Answer: Since therein 1/t approaches zero, then:

lim(1+t)1/t = 1 ò1/t¾0
t¾Ê             òö(1+t)ö àòà log t
  Ó                       

 Equation 12.

The symbol àòà is introduced, and which means 'symmetrizes perfectly with'; therefore, the statement in its entirety would read, "The limit of the quantity (1+t) held to the power of 1/t as 1/t approaches zero and the quantity (1+t) symmetrized symmetrizes perfectly with log t.  

     For review, two new symbols have just been introduced:   
               
          Ò                                               àòà
     Ê
     Ó                                                                

is defined as symmetrical with                                                              symmetrizes perfectly with

See Table 1. New Math Symbols below

 ö(1+t)ö Ò log t   ?
 ÊÊÊÊÊÊÊ Ê ÊÊÊÊÊÊÊ È 1
 ö(1+t)ö Ó ö(1+t)ö

Yes, where
 lim  log t  
 t¾Ê ÊÊÊÊÊÊÊ È 1
   Ó ö(1+t)ö

¸  lim  log t  
   t¾Ê ÊÊÊÊÊÊ È ö(1+t)ö
     Ó   1

 Equation 13.   

     In order to find the series which arises out of j(1+t)j in this case, wherein it must equal the limit as t approaches true infinity of log t, we must symmetrize by reducing to a factorial sum in the expression (1+t), since a factorial is an inverse of the root-taking power of 1/t in the original expression (1+t)1/t , first; and also, we must symmetrize past the log t, as is allowed and stated above, also by taking the limit of log t as t approaches true infinity.  

     Therefore, how would these factorials be arrayed in the resultant infinite series to be known as e?  

     First, we will carry of course the operation of addition, and that will be upon 1, as in (1+t).  (Note: this expression (1+t)1/t was never (1+t)1/(1+t) only because the entire expression starts from the 1 in the denominator, in the base [(1+t)], and you must only begin once.  Start from 1 once, but once.)

     Then, we can simply add onto 1 by successive factorials endlessly, since it is obvious that a factorial expansion of (1+t) will be (1+factorial after factorial after factorial), since t would thus vary.  However, in this overall quest for e, the original operation was the limit-taking upon (1+t)1/t as t approaches true infinity, or

          1/t
lim (1+t)    
  Ê                                  
t¾Ó                              

Thus, in the first-order sense of this expression acted upon accordingly, the limit derived is 1:

         1/t
lim (1+t)   = 1 (since 1/t¾0).   
  Ê                                  
t¾Ó  

Therefore, we know that this limit of 1 will remain as we elicit the series, and in order to accomplish that , the factorial sums must be inverted; these inverted factorials are the matching result of the inversion of the exponent to 1/t :

 e = 1 + 1 + 1 + 1 +...+ 1 +...  
         Ê   Ê   Ê       Ê  
         1!  2!  3!      n!  

Q.E.D.

[Please note: this treatise is not herein completed, even though the derivation of e is itself momentous.  More will be said which will help the observer understand certain concepts which have promulgated the language, and the manuscript will be further edited with equation numbers, tables such as the new one below,  etc.] MLS

          Table 1

   New Math Symbols                                                                

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  Chapter 2

                   Symmetry

     Addition is more symmetrical than multiplication because there is no factor involved...oftentimes, empirically contained and recognized groups are combined.  Addition of identical groups implies multiplication, wherein the group is added unto itself in exactly the number of times which is equal to the multiplying factor.  The same thing occurs in multiplication, wherein, if a given group is multiplied by itself a number of times, this form of identity constitutes an exponent, which is a logarithm.  This is a form of symmetry, since the multiplicand is symmetrical unto itself by identity, i.e.,  

23=2à2à2=8   [where 2 = base and 3 = exponent ]                           

2=2 by identity              

     Ò            
¸  2 Ê 2                                 
     Ó   

or,  more generally b^y=x  (Eq. 2) 

b=b       

     Ò                                    
¸  b Ê b                                 
     Ó   

Figure 1. Symmetry of Exponential Operation  »

             2 + 3 = 5   [where 2=addend , 3=addend and 5=sum]
     (2+3)¾2ö(3) and 3ö(2)
     ¸ (2+3)ö5
   [2ö(3)+3ö(2)]ö5

Conclusion: addition performs across perfect symmetry.

Figure 2. Symmetry of the Operation of Addition   »

            3Â2=6    [where 2=multiplier or multiplying factor, 3=multiplicand and 6=product]   
             Ò
     3=3 ¸ 3 Ê 3
             Ó
             Ò
     2=2 ¸ 2 Ê 2
             Ó

    (3Â2)¾3ö(2)but 2ø(3)
     ¸(3Â2)ø6
   [ 3ö(2)Â2ø(3)]ø6

Conclusion: multiplication performs across partial symmetry, since the factor extends its identity to the multiplicand, thereby lessening the direct identity of its value as relates only to one (1).

The above mathematical statements following the proof of symmetry by identity of 3 and 2 reads as follow:

3 times 2 implies that 3 symmetrizes 2 partially but 2 does not symmetrize 3.  Therefore, 3 times 2 is partially symmetrical to 6.  Quantity 3 symmetrizing 2 times 2 partially symmetrizing 3 quantity is partially symmetrical to 6.

 Figure 3: Partial Symmetry of the Operation of Multiplication   »

Resume text in Ch. 3

In the example of Figure 3 above two (2) operates upon three (3) as a factor, so that its identity is extended to the group of three (3) as a multiplier, it does not "add" directly itself by its own value which is derived by identity; rather, it directs the idea as a factor that the value of its multiplicand, three (3), should be added unto itself once, so as to give a total of  two (2) groups of three (3) each.  The value of the factor two (2) is thus indirectly lent to another value, so that two (2) is only partially symmetrical in the overall operation of multiplication.  Therefore, the operation of multiplication is less symmetrical than the operation of addition for this reason.  Two (2) is once removed in the sense of symmetry from the operation of multiplication, so that we say it is less symmetrical than it would be, for example, in the operation of addition as in Figure 2.

     In order to visualize the partial symmetry in the operation of multiplication, and compare it to the perfect symmetry across which addition works, consider groups of things.  There exists a group of three apples sitting next to a group of two apples, and the value of each group of apples lends its identity as per how many apples sit in each group.  Where there are three apples, the group is identified by the counting of three, and similarly, where there are located two apples, the group is identified by the count of two.  Each group is unified and therefore recognizable by its count, and each group is singular but for its identity accordingly.  Now if the apples are combined into one group, such that three apples are added to two apples, then a total of five physical apples have been combined to make a new group of the new count of five.  Even if there had been two groups of three apples each, to combine them into one new group, thus to add them, each group undergoes a direct addition to form the new group, whose count would of course be six.  However, multiplication is less direct than is addition.  Let us say that we have a group of three apples, and  there is a need for six apples.  It is easy to add three plus three and see that six apples can be collected by adding three more to the group of three apples with which we started.  Yet, a subtle multiplication has also just occurred.  Hypothesize that in order to prepare a dish for twenty people with three apples required for the serving of each person, the required apples must be counted and procured.  The multiplier is twenty, the multiplicand is three, and the product of operating upon the group of three apples by the factor twenty is sixty.  The identity of the factor of twenty in the calculation is not a physical entity equatable to the group upon which it operates by direct identity; there are not twenty apples present yet, there are only three apples.  Seeing the three apples sitting in a group reminds us that the group of three is enough for one person, and when twenty people become relevant to the need for a larger group of apples, that twenty only extends its value to the calculation, the multiplication of 20 x 3 to obtain the product of 60, the group needed to serve all twenty people.  Twenty divided by twenty equals one (1) by identity, granted; however, such identity of an actual twenty apples is not physically necessary in the consideration of how many apples to procure, so that the twenty as a factor is only partially symmetrical when the operation of multiplication occurs to obtain the product of 60.  If, however, twenty apples were in one group and three were in another group, and the total number of apples present through the combination of the two groups became the question at hand, then the integrity of the group of twenty apples could be identified until such time as that group of twenty apples were added to the three and the new group was counted to be at twenty-three.  In the case of adding the apples to obtain twenty-three, actual apples are transposed in the formation of a new group.  In the case of multiplying in the abstract to calculate to determine how many total are there in groups of three for twenty people, the factor twenty defines the operation, but does not lend its integrity to the operation by the physical presence of twenty apples.  Its identity, the identity of twenty, will be only partial, as there is no necessity for a group of twenty apples to be formed in order to find the answer, the product of twenty and three.    

     Notice in these illustrations of symmetry in the operations of exponential powers, addition and multiplication, that the concept of identity is used so as to find the symmetry within the operations.  This concept of identity is seen as symmetrical to one (1) by the concept that anything divided by itself equals one (1).  That is to say, n/n = 1; or, n/1 = n.  If each natural number begins to signify the quantification associated with a group which corresponds to its sheer value, and then these groups will combine in some fashion, mathematical operations are in order.  Once mathematical operations are in order, the concept of breaking up a group through division also becomes possible.  Once division enters into the hierarchy of operations, then the denominator takes on the possibility of being something other than one (1).  This possible change in the denominator is to be reflected in the symmetry of the operations.  To contemplate upon the interchangeability of the denominator of a quantified group between one (1) and itself should be instructive as to the infinite power of one (1), even as one (1) can thus be extended to the pluralistic for what it is: an absolute unifier.  Follow this concept with this illustration, as simple as it might be:

              n
          n = Ê by symmetry                        
              1

          n
          Ê = 1 by identity      
          n

              Ò
          ¸ n Ê 1
              Ó

          ¸ n àòà 1

where n represents any natural number >0.

     Or, more explicitly:  

     n = n by identity         
         n
     n = Ê  by symmetry
         1     
     n
     Ê = 1  by identity and symmetry
     n
     ¸ n àòà 1

Law of Identity of Unity by Symmetry

Any natural number  n greater than one (1) is related to one (1) by perfect symmetry, since n divided by one (1) is itself and all n are understood as divided by one (1).   Any n divided by itself is equal to one (1) by identity and symmetry.  Therefore, all n are defined as symmetrical with one (1), and all n are perfectly symmetrical with one (1).  

According to the concept explicated by the Law of Identity of Unity by Symmetry, the pluralistic aspect of a group is referred to by the value known, as its quantity is understood accurately when it sits as an identifiable, integral unit, such that n/1=n.  See how this reflects the symmetry lent by one (1) through the infinitude of one (1), such that any quantity relates to one (1) by division in a perfectly symmetrical sense, whereas to divide by any other value will take away that symmetry.  Only one (1) has this property, and thus is all mathematical identity made visible by unity, by one (1).  

     Take the case where 

        1/t    Ò          t
lim(1+t)   = 1 Ê lim (1+t)  = Ê                               
t¾ Ê           Ó t¾ Ê         Ó                       
   Ó                Ó

   Ê Ò
¸  Ó Ê 1 
     Ó      

in addition to 1 = Ê
               Ó     

Figure 4. Symmetry in the Law of the Infinitude of Unity

That is, true infinity is defined as symmetrical  with one (1).  In Chapter 1 e was derived through an understanding of how mathematical operations hold their truth through their symmetrical attributes, and which attributes of symmetry are traced to one (1).  The above-stated equations define, again, the Law of the Infinitude of Unity, so that all of quantification and the operations upon those quantifications can meaningfully be understood as relating back to one (1); if true infinity, upward and outward, ever-expanding, is by definition seen as symmetrical with one (1), this forms an identification which matches the process of counting itself.  Counting is only an ultimate elaboration of using one (1) in some form; whether that form takes on the identity of a group of larger value, still, the additive value is fundamentally always built upon one (1), even in the range of infinity.  Philosophically, if everything, if all quantifications, are comprised essentially of one (1), then all mathematical understanding should be unified somehow through the application of that unitary principle accordingly: all are one (1), all is unified somehow.  To propound the metaphysics of the awareness of oneness as in and through all relative, objective mathematical truth and endeavors as according to that truth of the essentially non-dual hypostasis of the relative realm under consideration is to sharpen mathematics inevitably.  Reason this statement through accordingly: in mathematics truths are drawn from the hypothesis of the infinite, of relative values which relate to one another more truthfully when the absolute case or limit is hypothetically applied to them.  Now if that entire concept is boiled down to a singular postulate or principle mathematically, that to the Law of the Infinitude of Unity, for example; then the concept of the all-pervasive nature of oneness, as it correlates even yet with an infinite expanse which appears to be pluralistic, can be brought to bear upon the concept of the infinity we think of as that which is so great there is none greater than--true infinity.  This correlation of oneness with the infinite metaphysically must have its mathematical counterpart available for dissertation, such that mathematical principles should be held answerable to the idea that since the objective, relative context of things is subsumed under the absolute, so must the language of math serve and discourse upon that relative context as an essential subset of the absolute necessarily.  In other words, the relative is drawn from the absolute by definition of the vastness of the absolute, which not only occupies its own category by virtue of the attribute of its greatness; the relative also cannot be logically perfected in its mathematical treatment, therefore, unless there is elicited a mode of logic which is somehow inclusive of the absolute under which it is subsumed.  This is how symmetry unto oneness, and the Law of the Infinitude of Unity, will make clear the mathematical derivations and concepts at hand.

     Consider again Figure 4, Symmetry in the Law of the Infinitude of Unity , since examining the operations present within the power of the symmetry thus conceptualized will be of great didactic value in unraveling the mathematics of symmetry.  Fundamentally, the operation which is of the greatest symmetry for its equivalence in values being operated upon is addition.  In the preceding example physical objects, apples, were used to illustrate more concretely the essence of addition, and that as it compares to the operation of multiplication.  Numbers exist, and are thought of in their increasing sequence as abstract entities of sheer value.  When two numbers are added together, the direction in the sequence is continued onward in a positive, or increasing continuity, such that counting occurs, though it is performed in groups mentally.  There exists in the operation of addition a one-to-one correspondence in the second addend, the number added to the first addend: the second addend corresponds in value directly to itself, only joining its own value directly, additively, to the first addend.  In multiplication, on the other hand, the factor indirectly extends its value by operating instead on the value of the multiplicand, and ordering the symmetry of identity of the multiplicand to add unto itself a number of times equal to the value of the factor.  

     An example of the foregoing would be: 5 + 4 = 9.   In the sequence 1,2,3,4,5 is formed the first addend at its endpoint, 5; then 4 must be added for its direct value starting from the next number:  6,7,8,9 is thus counted, with 9 as the end result of the simple addition.  The one-to-one correspondence of the operation of addition as thus illustrated of the second addend for its own additive value upon the first addend is what defines the powerful, founding symmetry of addition; the value of the resultant total reached additively can be envisioned as a direct, one-by-one counting, whereas in factoral addition, multiplication, the factor directs by its value the addition of an interval or group of natural numbers to be superimposed upon the sequence of those numbers and thus added.  Therefore, consider 2 x 3 = 6 where 2 is the factor and 3 is the multiplicand: 2 extends its own value indirectly, or factorally, in the operation, such that the sequence 1,2,3 is ordered by the concept of adding unto itself its own value once again, to give 4,5,6, since the sector or group of 4,5,6 has an intrinsic value of three and is perfectly adjacent to the multiplicand defined by the 1,2,3 sequence, and which is operated upon by the factor of 2.   Hence, in multiplication, only the value of the multiplicand is conserved, in this example 3; and the non-conserved value of the factor, 2,  is applied to a conserved value.  

     Indeed, it is also true that subtraction, the perfect inverse operation of addition, is perfectly symmetrical to numbers based also fundamentally upon the principle of the symmetry of oneness as proven by direct identity coupled with the conservation of each value added as discrete unto itself.  See that in any equation of addition or subtraction the natural number n > 0 retains its own exact value and remains thus individuated in and through the operation.  This is profound, for even though the resultant total given through an addition or a subtraction is a number of a different value, the values were operated upon in a way which  reflects the symmetry of ´ n, yet even as they undergo an operation; furthermore, each addend in addition and each subtrahend in subtraction  retains its identity by value for the operation, and they are thus conserved for their values.  Some operations, such as was demonstrated in the instance of the factor in multiplication, extend a value unto another number's value; the factor loses its individuated value except as to apply that value identity to its multiplicand's own value, and thus is non-conserved unto itself.  Partial proof of this is seen where 2 + 3 = 5, yet, 2 x 3 = 6; the product 6 is greater than the sum of the values by direct identity of 2 and 3 as when they were added and thus conserved to give 5, by 1, it so happens, to give 6.          

     As according to the Law of Identity of Unity by Symmetry, for all n, 

where n is any natural number > 0,  n àòà 1:

     ´n [n1 À n2 À ... nx À ... À] n àòà 1
         nx
  ¸  À å ´n>0 àòà 1.  
         n1         

Figure 5. The Place of Symmetry in the Sequence as Seen in Addition and Subtraction  

The summation sign above in Figure 5 is inclusive of the inverse of sum, the subtrahend being used instead of the addend, for the operation of subtraction.  (If an operation upon two natural numbers gives a fraction as a result, as in division, for instance, then its symmetry is internal; this might lend further understanding to the observation above, and will be explicated further.)  Special mention is made of the character of n as a total having been operated upon through addition or its perfect inverse subtraction, since it is helpful to point out the nature of operations starting with the the most symmetrical of such operations; the operations do follow an order themselves as according to symmetry.  Some operations are more symmetrical than others.  Before deriving how the operations differ in the extent of their individual symmetry unto one (1), allow them to be listed in the decreasing order of their symmetry so as to begin to understand them as in Table 2: Order of Operations by Symmetry here below:

Table 2

Table 2 : Order of Operations by Symmetry: 

     What can be greater than one, if one (1) is all there is, if one (1) is all that is?  As we do discourse upon this muse of the place of symmetry unto oneness in the consideration of mathematical observations and operations, let us see how one (1) even as a digit, even as a natural number, will certainly define and clarify those observations.  

     Indeed, to state the Law of the Infinitude of Unity an observer can conclude through its derivation either that:                                                                                                 Fig.6

or that:

Figure 6. Conclusions of the Law of Infinitude of Unity: the Equivalence of One and True Infinity and the Definitional Symmetry of True Infinity and One (1) 

If true infinity can never be reached per se, then its definition cannot be stated on terms which give its exact magnitude.  That is, imagine that an absolutely great number has been found.  If one (1) itself is added to it, then that great number cannot be true infinity.  This perplexing concept is solved through the idea that since an exact equivalence cannot be explicated for infinity, then to postulate a symmetry between an infinity and unity will tap the power of one (1) in its omnipresence in all of mathematical ideation.  All is founded in mathematics upon one (1), since mathematics is a hyper-extended methodology of counting and applying logic to how various counts combine and reflect truth in and through one another.  The infinite power of one (1) when viewed metaphysically as oneness, and when placed on a conceptual plane with the concept of the absolute or the infinite as its counterpart on the behalf of the equivalence of one (1) and the infinite, gains the presence in logical derivations known as symmetry; this mathematical concept of symmetry qualifies the infinity which stands as counterpart to one (1), to unity by principle, to the oneness by truth of the nature of reality itself metaphysically, to be known as true infinity.

  Chapter 3                     

                                  Inverse Mathematical Operations

     In the preceding chapter the concept of the hierarchical order of mathematical operations based comparatively upon their degree of symmetry was broached, and Table 2: Order of Operations by Symmetry was drawn up.  Table 2 thus far constitutes a preliminary guide to the more replete discussion of the understanding of mathematical operations, their inverse relations one to the other, and how symmetry helps define the processing of numbers in an orderly, logical and understandable fashion, which is furthermore real to the nature of non-dual reality.  By figuring the operation of addition and comparing it to multiplication the idea of the conservation of value as well as the degree of symmetry to one (1) of these two operations was derived.   

     In order to discuss the nature of the operation of division further observation of the attributes of symmetry as illustrated in the operation of multiplication, as in Figure 3 of Chapter 2, is at hand; it will be seen that multiplication and division are partial inverse operations of each other.

      In general, then, the attribute of symmetry as it carries partially across the operation of multiplication can be represented by:

              (nf  nx)= p
        nf ö nf
        nx ö nx
        nf ö nx

                    [nfö( nx) Â nx-ò-( nf)]= p
                (nf  nx)ø nf
     ¸  (nf x nx)ø p

   [nfö(nx) Â nx-ò-(nf)]ø p

     The concluding statement reads as follows: bracket nf symmetrizes nx times nx not symmetrizing nf bracket is partially symmetrical to p.

           where n is any natural number 

           nf = factor
           nx = multiplicand
           p  = product
           ÊòÊ means not symmetrical to

                            ø means is partially symmetrical to

                             ö(  ) means symmetrizes that which is the parenthesis

            ÊòÊ(  ) means not symmetrizing that which is in the parenthesis 

                                           Resume from Fig. 3

Figure 7. Partial Symmetry of the Operation of Multiplication

     The partial symmetry of the operation of multiplication is shown in Figure 7 above, wherein the factor nf , although it is symmetrical to itself and to nx,is as a factor not conserved across multiplication by its own value as a group.  Instead, nf governs the value of the group  nx and adds that group  nx  to itself  so that nfworks factorally, not additively, not as a direct addend of its conserved value.  These statements of symmetrization in Figure 7, then, lead to the conclusion that when nf and nx are multiplied to form a product p, these entities are partially symmetrical to one another, or,   ¸  (nf x nx)ø p.  nf is partially symmetrical to nx in the operation, since it operates its own value upon the nx term, wherein the nx value is taken as the unit operating value to be multiplied by the factor nf.  Since nf works upon that value of nx as if nx were a unit, then nf is considered partially symmetrical to nx.    

     Click here for a graphic illustration of how the factor in multiplication works with its multiplicand to undergo a change in value through non-conservation of its intrinsic value in order to accomplish the product through the multiplicand.  This illustration will open in a new window for you.

     The series of mathematical statements in Figure 7 shows that natural numbers are symmetrical to one another: 

           nf ö nx .

Any natural number is symmetrical to another natural number, since all natural numbers relate to one (1) in an essential way.  This may seem to lend to the concept of symmetry a leverage which is more expansive than that which should seem probable or even useful.  However, one (1) adds immeasurably upon itself, so that the unitary principle of one (1) is founded upon that truth.  Even if two (2) or, for the sake of the argument, seventy (70) is added, those two addends are nothing more than constituents of one (1) from the foundation up.  

     In order to grasp more fully the universal nature of the symmetry of any natural number unto any other natural number through their mutual symmetry to one (1) imagine that there could exist another universe where the basic nature of reality were different, perhaps dualized, so that one (1) was not indeed a unifying factor in mathematical thinking and practice.  Then, a system of natural numbers might prove to be randomized, or even based upon an either/or supposition, depending upon conditions and factors which would have to be set up for determination before a simple calculation could be carried out.  There might be two forms of the number two, for instance, differing in some mysterious, hypothetical way, since two might be comprised of two different aspects of some fundamental quantifier which could vary according to its past utility or its current context.  How complex a world would such an imaginary universe imply.  In such a world, then, the natural numbers would not be symmetrical one to the other.  Instead of connoting order by their nature, and adding in an hypothetically infinite and endless context, perhaps they would challenge the mathematician to a tedious task of finding the appropriate value to be used in a given instance for the given numbers: the foundation of all quantification would be dualized in such an imaginary universe.  

     Now the question might arise: if all natural numbers greater than zero are symmetrical to one the other, then why should their putative symmetry to one (1) have such moment in mathematical consideration?  Does such universal symmetry of the many natural numbers one to the other not minimize the implied place of their symmetry unto one (1), also, since symmetry is so pervasive throughout the pluralized entities of the natural numbers as they relate to one another?  Then, the questioner might reason, why so special the symmetry of the natural numbers to one (1)?  Actually, this question is the very crux of the work at hand, Mathematica: A Treatise.  The mathematics of limit-thinking tend to place too much emphasis on the place of infinity to govern over the logic, without qualifying the simple unit known as one (1), which is indeed still the unit of infinity itself.  Similarly, even that all natural numbers greater than zero are symmetrical to one another, this truth does not undo the foundation of symmetry itself, which remains as one (1).  Let us see how the language of math will address these concepts:

     n = n/1 by symmetry
     n/n = 1 by identity
           Ò    
     ¸  n ÊÊ 1
                  Ó
     ¸ n àòà 1

     ´n ö 1 ò n àòà 1
    ¸  ´n ö ni

Figure 8. The Universal Symmetry of All Natural Numbers to Each Other: Corollary to the Law of Identity of Unity by Symmetry

     All n are symmetrical to 1 such that n is perfectly symmetrical to 1.

The concluding statement  ¸  ´n ö ni says that all n are symmetrical to any other n, ni,  and understood here is n as any natural number greater than zero.  This is a corollary to the Law of Identity of Unity by Symmetry, which states that all n are defined as symmetrical with one (1), and all n are perfectly symmetrical with one (1).

     The question which might arise as one compares, say, the partial symmetry of a natural number which comes out of a multiplication, as in Figure 7 previously.  In questioning mode the logical mind might query, if product p is actually only partially symmetrical if it is formed from the multiplication of nf and nx , then there must exist such an ni which is not symmetrical to other n's, as well.  Such an argument would then question the validity of the concept of the partial symmetry of the operation of multiplication as relevant or valid.  However, the question posed here itself is not properly reasoned out according to the conceptual reach of symmetry, and how symmetry relates to numbers and to operations upon numbers.  A given number in its raw value, such as a natural number, is also definitionally symmetrical to one (1).  This we have seen.  If in a multiplication a natural number becomes a product, it is not the number which is under scrutiny, it is the operation which led to the number.  A statement of symmetrization must be taken for what it is, and it is not a statement of equality, an equation.  Equations may be used in the discussions and logic from which is derived a symmetrization, yet, symmetrization parses logical operations in mathematics for their relatedness to the unifying feature of one (1).  In the sense that the conceptual seat of readiness to accept a non-dual algorithm in mathematical thinking such as symmetrization will guide the derivations of mathematics and of natural and theoretical scientific thinking, an observer may expect that the universal nature of such an algorithm should reflect in ways where it might not reflect as true.  That is where the precision of mathematical propositions and conclusions will serve to put such readiness of mind to good use and to proper vision.  Refer to Figure 6 in Chapter 2, which states that not only is true infinity  by definition symmetrical to one (1), but also, that true infinity is equal to one (1).  A finite natural number cannot be considered for its equality in place of its symmetry.  Only one (1) confers this unique property of both  symmetry and equivalence with true infinity in the theoretical sense, so that any logically posed questions should of course be poised upon the truth of such rare equivalence between infinity and anything by symmetrical derivation.  That anything is one (1) and only one (1).

To be continued...

This treatise is being published in serial fashion.

 © 2001 - 2006  by Marilynn Lea Stark  All Rights Reserved