2. The second stage - MULTIPLICATION
In the multiplication system :
a) the point of departure cannot be zero (for reasons which we shall see more clearly in a moment)
because it would be impossible to find an increment which would produce a series.
We must therefore choose the unit (1) as the simplest point of departure ;
b) the unit cannot serve as an increment in this system (any value multiplied by 1 remains unchanged)
and the number two (2) becomes the smallest integer which can serve as an increment ;
c) the procedure is multiplication.
This system produces a series in the following manner :
1) the point of departure, the unit (= 1)
2) the preceding number (1) multiplied by the increment (2)
or the point of departure multiplied once : 1 x 2 (= 2)
3) the preceding number (2) multiplied by the increment (2)
or the point of departure multiplied twice : 1 x 2 x 2 (= 4)
4) the preceding number (4) multiplied by the increment (2)
or the point of departure multiplied three times : 1 x 2 x 2 x 2 (= 8)
and so on, to produce the series 1, 2, 4, 8 ...
Sum of the first members
If the members of a Multiplicative series are defined as M0, M1, M2, M3, ...
M0, by definition, would always have the value of "1",
M1 would be Multiplicative increment, and
the others, M2, M3, ... would follow the normal Multiplicative process.
The sum of the members to Mn would be defined as Mn+1 - 1 / M1 - 1.
EX - the sum of the first 4 members with M1 = 2 (1, 2, 4, 8) would be (16 - 1) / (2 -1) = 15.
EX - the sum of the first 4 members with M1 = 3 (1, 3, 9, 27) would be (81 - 1) / (3 -1) = 40.
EX - the sum of the first 4 members with M1 = 5 (1, 5, 25, 125) would be (625 - 1) / (5 -1) = 156.
EX - the sum of the first 4 members with M1 = 7 (1, 7, 49, 343) would be (2401 - 1) / (7 -1) = 400
EX - the sum of the first 4 members with M1 = 11 (1, 11, 121, 1331) would be (14641 - 1) / (11 -1) = 1464
If the process were reversed, we would divide the preceding number by the increment at each operation; once back at the point
of departure (1) :
1) the preceding number (1) divided by the increment (2)
or the point of departure divided once: 1 / 2 (= 1/2)
2) the preceding number (1/2) divided by the increment (2)
or the point of departure divided twice: 1 / 2 / 2 (= 1/4)
3) the preceding number (1/4) divided by the increment (2)
or the point of departure divided three times: 1 / 2 / 2 / 2 (= 1/8)
and so on, these fractions becoming smaller and smaller, with zero (0) as their limit.
We now have a series of values whose centre is the unit (1) and whose limits are zero (0) at one extremity and positive infinity
(inf) at the other :
0 ... 1/8 1/4 1/2 1 2 4 8 ... inf.
The reason why zero cannot be used as a point of departure is now much clearer : this value is not part of the multiplicative
system but is situated at one of its limits.
To appreciate the quality of the GENERATIVE STRUCTURE of these systems, let us observe the following comparisons :
(a) as in the additive system, the value of the central point (1) of this multiplicative system would be sterile as an increment and cannot be used in this function ;
(b) the central point of the additive system (0) is now one of the limits of the multiplicative system, and the simplest increment of the additive system (1) is now the central point of the multiplicative system ; these relationships will reappear when we examine the third stage of mathematical structure.
All the values of this multiplicative system are positive but it would be possible to produce a series entirely made up of
negative numbers by using a negative point of departure (such as -1) and keeping the positive increment (in this case, 2)
0 ... -1/8 -1/4 -1/2 -1 -2 -4 -8 ... -inf.
Application of cosmic laws
A few words concerning SYMMETRY and HIERARCHY :
(a) subtraction being the inverse of addition,
(b) division being the inverse of multiplication, and
(c) multiplication being the abbreviation of repeated additions,
one might be led to expect that, as a result of this, division would be the result of repeated subtractions.
Repetitive subtraction (with a constant increment) will be abbreviated, as is the case with addition, by the process of multiplication, with a positive increment, but with the initial value treated as a negative number.
EX: (0) - 1 - 1 - 1 is abbreviated as (0+) 3 x (-1), (= -3).
The process of division, however, produces a whole world of fractions which the processes of addition and subtraction are totally incapable of achieving.
This observation is of the utmost importance as far as logic and rigorous structure are concerned because it establishes the
following priorities :
1) The link between the first stage and the second stage is made between the processes of addition and multiplication and not between the processes of subtraction and division.
2) Addition is thus the dominating process of stage 1 and subtraction is nothing but a sterile image.
3) Multiplication is the dominating process of stage 2 and division is nothing but its sterile mirror image.
4) Addition is the dominating process of both stages because multiplication is nothing but its abbreviation.
Music will offer us similar situations when we examine major and minor dispositions, one natural and dominating, the other
inverse and secondary.
Looking forward again
The link between this second stage of multiplication and the third stage of "powers" is also provided by the process of abbreviation
: the power is the abbreviation of several multiplications with a constant increment. Noting, in an abbreviated manner, the
number of times to multiply the value of the increment; the increment must be constant as is the case in multiplication (the
abbreviation of several additions).