What creates dimensions fundamentally and the
consequences of this property on matrix (Part 1, dimensions)
The behaviors and properties of qualities that found exist
The first quality that is observed or discovered but not defined or created for one’s
own purpose is numbers. Numbers can be and was probably first found by
measuring the amount of clearly categorized objects, or counting. By counting,
such as counting the number of steps needed to take for one to walk from a point to
another, measures the amount of the objects we consider and define as a whole.
That is, for instance, we shall define a step to be twice the length of the walker’s
feet, by counting the number of steps the walker makes, we measure the amount of
“steps”, which is an object we defined by considering the infinitely many points
within the length of the walker’s feet as a whole, one complete, single object. In
this way, we obtain, or find numbers. In another word, such numbers can all be
obtained by counting, measuring the amount of object we define in certain way.
We know that such quality, numbers, exists at all times with or without us counting
and finding them, since any thing may be considered as a whole according to one’s
definition and be counted to measure its amount.
When one such number is applied with some calculation and as a result is
transformed/becomes another number, we find that with on the calculation and the
quality: numbers we defined, there is no barrier or limitation in which a such
defined number cannot be obtained by any calculation applied to some other
number(s). In other words, for any real number, it can be expressed in terms of
some combinations of other real numbers. That is, a real number can always be
represented by the resultant number from some certain calculations of some other
real numbers; in another word, if we let 𝑥 and 𝑎 be any two real numbers, there is
at least one relationship 𝑥 = 𝑓(𝑎) that holds true for all real numbers 𝑥 and 𝑎,
where the function 𝑓 is some calculations that only involves real numbers applied
to real number 𝑎. That is, any one real number is the resultant number of some
additions and/or multiplications between some other real numbers. This conclusion
shows that any two real numbers can be transformed/become each other through
some certain calculations; for one arbitrarily chosen real numbers out of the
collection of all real numbers, it can reach any real numbers in such collection
through some calculation. In conclusion, we find that: for any one real number N,
there exists a real number M such that 𝑁 = 𝑓(𝑀).
From the conclusion above, since for real numbers the relationship 𝑁 = 𝑓(𝑀)
must holds true, thus any quality which fails to obtain such relationship is not a
real number. Therefore, for any one non real-number quality 𝑁 ′ and some real
number M, 𝑁 ′ ≠ 𝑓(𝑀); that is, for any non real-number quality 𝑁 ′, there does not
exist a real number M such that 𝑁 = 𝑓(𝑀). This conclusion shows that a non realnumber quality 𝑁 ′ cannot, through any calculations, reach/become some real
number. There is such a limitation of the qualities it can reach/become through
calculations for 𝑁 ′ ; if one applies the same argument for some real number 𝑁, the
same conclusion also holds true for the real number N. Since the two qualities 𝑁
and 𝑁 ′ cannot reach each other through any calculations, we call them to be
different kinds of qualities. That is, the limitation of the qualities one quality can
reach creates and determines such quality to be a different kinds of quality than the
qualities it wished but cannot become.
This conclusion also shows that qualities of the same kind are all able to be found
using the same method. That is, if one knows that two qualities are the same kind
of quality and the method of obtaining one of these two qualities, then one is also
able to find the other quality using such same method. Note that qualities can
found but cannot be created, since the relationship between some qualities of the
same kind 𝑁 = 𝑓(𝑀) holds true with the restriction of the kinds of quality N and
M are, but not with the restriction of the observer’s knowledge of such qualities’
existence. This property of qualities within the same kind can be derived from the
conclusion made above as the following. According to the conclusion above, since
for any two qualities of the same kind N and M, the relationship 𝑁 = 𝑓(𝑀) must
hold true, that is, M must be able to be obtained through some calculations applied
to N, thus we also must have the relationship M = g(N) to be true, where the
function g is some certain function such that N = f(g(N)). Therefore, from this
conclusion, we see that for any two qualities N and M, if they are of the same kind,
then N can be obtained by some calculations applied to M, and vice versa, M can
be obtained by some calculations applied to N. If the quality N is found using
method A, since M = f(N), the other quality M is found using the method of
applying some calculations to N, we call this calculation method CN; similarly, if
the quality M is found using some method B, since N = g(M), quality N is found
using some calculations applied to M, call it method CM, as shown in the figure
below.
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According to the above relationships, there are multiple ways in which quality N
can be found, it can be either found through only method A, or through method B
to obtain quality M, then through method CM which is applied to M. That is, in
short, quality N can be found by method A or methods B & CM, these two methods
are considered equivalent for finding quality N; for clear referencing’s sake, we
write this conclusion as 𝑀𝑒𝑡ℎ𝑜𝑑(𝐴) = 𝑀𝑒𝑡ℎ𝑜𝑑(𝐵, 𝐶𝑀 ). Similarly, the same
argument can be applied to quality M, too. Thus we also have the conclusion that
quality M can be found by method B or methods A & CN according to the above
relationships. Thus we now also have 𝑀𝑒𝑡ℎ𝑜𝑑(𝐵) = 𝑀𝑒𝑡ℎ𝑜𝑑(𝐴, 𝐶𝑁 ). Since
𝑀𝑒𝑡ℎ𝑜𝑑(𝐴) = 𝑀𝑒𝑡ℎ𝑜𝑑(𝐵, 𝐶𝑀 ), the equation above can be re-written as
𝑀𝑒𝑡ℎ𝑜𝑑(𝐵) = 𝑀𝑒𝑡ℎ𝑜𝑑(𝐵, 𝐶𝑀 , 𝐶𝑁 ). Therefore, when applying to some same
quality, method B is equivalent to Method B, CM, and CN used together. Since
applying methods CM and CN makes no difference in the result of the calculation,
thus by applying those two methods, they must cancel each other out and thus
make no affect on the result of the calculation. That is, 𝑀𝑒𝑡ℎ𝑜𝑑(𝐶𝑀 ) =
−𝑀𝑒𝑡ℎ𝑜𝑑(𝐶𝑁 ), indicating that applying method CM is the reverse of applying
method CN. Since the determination of method A depends on the quality A is
applied to, method A is determined once the quality is arbitrarily chosen and A
varies for different qualities in order to for such qualities to become quality N after
method A is applied to them. That is, in another word, method A may be applied to
any quality to get quality N, but with varying determination of A in order to do so.
Therefore, the choice of method A is arbitrary, one is able to define A arbitrarily
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for one’s own purposes, by doing so, one only needs to change the quality A is
applied to. Since we can define A arbitrarily, and since both method A and method
CM produce quality N (although they may be applied to different qualities), we let
A be CM, that is, we define 𝑀𝑒𝑡ℎ𝑜𝑑(𝐴) = 𝑀𝑒𝑡ℎ𝑜𝑑(𝐶𝑀 ). Since method CM is
applied to quality M, in order for the above relationship holds true, M must be
adjusted according to method A, which is chosen at our will for a certain quality to
produce/become quality N. Therefore this definition satisfies the fundamental
observation 𝑁 = 𝑓(𝑀) for any two qualities N and M of the same kind, since for
any quality N, we are able to define method A arbitrarily, and by varying the
quality A is applied to, we can then therefore vary the definition of A for an
infinite amount of possibilities; Since A = CM, and CM determines quality M, we
are able to change M to be any quality of the same kind as N by varying method A,
which then varies method CM, which then determines quality M. Thus the above
definition does not only let method A happens to be the same as method CM, but
also let the choice of A, which equals to CM, to be in our control and can be
arbitrary. (Since A and CM applies to different qualities, they must be defined, or
happens to be the same, however, since we can vary A arbitrarily, this restriction
will not limit the qualities, which is denoted as M, N can reach through
calculations by this definition.) Recall that we defined 𝑀𝑒𝑡ℎ𝑜𝑑(𝐴) =
𝑀𝑒𝑡ℎ𝑜𝑑(𝐶𝑀 ), similarly, we can define 𝑀𝑒𝑡ℎ𝑜𝑑(𝐵) = 𝑀𝑒𝑡ℎ𝑜𝑑(𝐶𝑁 ). In the latter
case, since quality M has been determined by our choice of method A, one can
only vary the quality to which B is applied in order to vary method B, and hence
also vary CN. Within the system shown in the figure, we found that
𝑀𝑒𝑡ℎ𝑜𝑑(𝐶𝑀 ) = −𝑀𝑒𝑡ℎ𝑜𝑑(𝐶𝑁 ), since we now also defined that 𝑀𝑒𝑡ℎ𝑜𝑑(𝐴) =
𝑀𝑒𝑡ℎ𝑜𝑑(𝐶𝑀 ) and 𝑀𝑒𝑡ℎ𝑜𝑑(𝐵) = 𝑀𝑒𝑡ℎ𝑜𝑑(𝐶𝑁 ), thus we have 𝑀𝑒𝑡ℎ𝑜𝑑(𝐴) =
−𝑀𝑒𝑡ℎ𝑜𝑑(𝐵). That is, we found that method A is the reverse of method B, where
A and B are applied to some certain qualities to produce arbitrary qualities N and
M respectively, and where N = f(M) for all possible qualities N and M. This
relationship is shown below in the figure.
We can see that by the fundamental relationship N = f(M), for every quality M
there is within the same kind of quality as N and M, besides method A which was
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used to produce quality N, there is one more method available to be used to
produce N and every M we have known. If the collection of a kind of quality
consists of infinite amount of qualities, then there are infinite numbers of different
methods to produce a quality of this kind, that is, to produce a quality within such
collection from another quality within the same collection. If the qualities N, M,
and the arbitrary quality are the only three qualities within the collection, then if
another quality is said to be one quality in the collection, there must be two and
only two methods can be used by N to produce this another quality, and there also
must be the reverses of those two and only two methods can be used by N to
produce quality M, as shown in the figure below. Note that the same argument can
be said for any other known qualities within the collection.
In conclusion, if I is a quality within a collection of many qualities of the same
kind, then among any other two different qualities in such collection, call them J
and K; for each J and K, there must be at least one method for I to produce/become
each of the two quality, call them method X and Y for producing J and K
respectively; then correspondingly, there must also be a method –Y, which is the
reverse of method Y, can be used on I to produce J, similarly, there must be a
method –X, as the reverse of method X, can be used on I to produce K. Among any
three qualities within the same such collection, the above relationships must hold
true. A particular, individual quality within such collection of qualities of the same
kind is determined by the relationships created by applying all possible methods to
another quality. For two qualities A and B of the same kind, there may be n
methods which can be used on A to produce B, that is, there are n functions such
that A is a function of B. Therefore, B is a quality which is determined to the
quality that can satisfy all those relationships, that is, the quality that can let all n
functions hold true. This idea shown below in the figure.
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Let N and M be two arbitrary qualities in the same collection of qualities of the
same kind, let “?” be some other quality that’s different from N and M, call it
quality Q. Since N, M, and Q are qualities of the same kind, we have N = f(Q) and
N = g(M). Therefore, there must be some methods which can be used by N to
produce Q and M according to N = f(Q) and N = g(M), call these two methods
method X and Y for N to produce Q and M respectively. According to the
conclusion we deduced above, there must also be a method –Y and –X, which are
the reverse of method Y and X respectively, can be used by N to produce Q and M
respectively, as shown in figure.
Since the methods which can be used by quality N to produce Q and M exist, we
can use the reverse of those method to reverse the process of producing Q and M to
produce N from Q and M. That is, there must be method –X and Y which can be
used by Q to produce N, similarly, there must be method –Y and X which can be
used by M to produce N.
Since in the relationships we deduce in the previous conclusion, the relationships
only hold true if the methods are used by one quality to produce some other
qualities, thus the methods shown in figure that are methods used by other qualities
to produce the quality we are studying does not affect the relationship, for
simplicity’s sake, we shall ignore them in the figure and focus on the methods that
can be used by the quality we study to produce some other qualities.
Let M1 be another quality within the same collection as the collection Q, N, and M
are in. Since M1 and Q are within the same collection, there must exist a method,
call it method Z, which can be used by Q to produce M1. Then there must also be
method –X and Y can be used by Q to produce M1. According to the previous
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conclusions, we have the following relationships among Q, M1, and the methods
producing them.
Remember that we use only methods that can be used on quality Q to produce the
other qualities since the relationships we deduced in the previous conclusion only
hold true for such methods.
Since N is also in the same collection as M1, thus there must also be a method, call
it P, that can be used by N to produce M1. Then we have the following relationship,
built upon the relationships above.
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With similar procedures, we can also further deduce the relationships among
qualities and methods that produce them for every new quality found and added
into the collection.
Therefore, since we only found and deduced qualities of the same kind to have
such relationships shown above, any qualities that does not satisfy the relationships
in the above argument are not qualities within the same kind of quality.
Structures created by such behaviors and properties of qualities
From the conclusion we have in the above session, we see that shown by
observations and deductions, for qualities that are of the same kind, certain
relationships hold true among them, and changing one of the quality in such
relationships will effect other qualities in the relationships for the relationships to
hold true.
We can also clearly see that the above relationships, although universal and
fundamental for any qualities of the same kind, is always restricted to be only said
for qualities of the same kind and according only to the above relationships, we
cannot say the same relationships hold true for qualities outside this limitation,
since the limitation, whether the qualities discussed are of the same kind or not, is
defined by the relationships this limitation or restriction is in. Therefore, for
different kind of qualities, no such relationships can be said among them, that is,
there is no method can be used by a quality to produce another quality of a
different kind; qualities of different kinds cannot become each other through some
calculations. Such barrier between different kinds of qualities always holds true,
since according to the relationships we found and deduced previously, different
kinds of qualities are determined by the relationships, if the relationships hold true
for all qualities concerned, then such qualities are of the same kind, and vice versa;
the same kind of relationships (such as relationships that can produce certain group
of qualities) a group of qualities can satisfy determines the barrier between this
group of qualities and other groups of qualities that can satisfy different kind of
relationships. Through calculations of the same kind of qualities, such group of
qualities cannot produce/become qualities that are not within this group; that is, by
combining a group of qualities in every way possible, the resultant quality of such
combination cannot reach qualities outside this group, such resultant quality can
only reach some quality within the group. For a quality A within one collection of
qualities of the same kind to reach, or become some other quality B within some
other collection of qualities, where A and B are different kinds of qualities, it
would require quality A to become the same kind of quality as B to reach
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anywhere within the collection B is in. This limited amount of qualities one quality
can become creates a range, the qualities within this range are all the qualities such
quality can reach through calculations. This range separates different kinds of
qualities and indicates how far and where a quality can reach. That is, such
different kinds of qualities create the ranges which are only accessible by qualities
of their own kind, and such range of a quality, which is the limitation this quality
can reach, is the dimension of this quality. Since this range is also the limitation of
all other qualities of the same kind can reach, this range is also the dimension of all
qualities of this kind. A same kind of qualities creates a dimension: a barrier which
other qualities of different kinds cannot cross.
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