THE FOURTHWAY MANHO EJOURNAL Volume 63 July 19, 2019 

MODERN MATHEMATICS AND THE STOPINDER NATURE OF THE SET OR CLASS THEORY By Professor Dr. Tan ManHo (An excerpt from the original work, Real World Views, Book 2 by Professor Dr. Tan ManHo entitled "Biocosmic NervoReflectant and the Theory of Material Reflection in Man, Inner Development and Social Upheavals", July 1972 ~ September 1973 Discourses, Chapter 7, Section C: "Modern Mathematics and the Stopinder Nature of the Set Theory", pp. 155~165)
C. MODERN MATHEMATICS AND THE STOPINDER NATURE OF THE SET OR CLASS THEORY
1
Back in the 1960's the modern mathematics, properly socalled has
ultimately proved itself to provide the exposition of the dialectic and
mentation by forms – nature is looked upon as consisting of discrete items
with distinct identity. The Soviets during its Golden Age of
Mathematics took the lead and the West badly needed to catch up – the sooner
the better. The general way which modern mathematics
handles quantities (but not as yet fully develop quanta) takes a peculiar
form, comparable to that used by a dialecticians in their listing and
categorization of the entities in the real world. The conception of sets, resolutions of
subsets into universal sets via limitation and delimitation through
universality, that negative numbers and positive numbers are the same, that
negative of +a is a and negative of a is +a, that subsets of a set are
specific in numbers, that they are the qualities imported to a set i.e. { },
{a}, {b}, {c}, {a, b}, {b, c}, {c, a}, {a, b, c} are the subsets of the set
{a, b, c}, whilst {a, b, c} as subsets is the nodal dialectical ‘leaping’
line of which {a, b, c, d}, that things(or numbers) have peculiar reflection
and symmetry, etc. etc. are in fact all the earmarks of dialectic and the
law of categorization. “A prime number is a whole number which is divisible only by itself and 1.”
This is the definition, that every whole number divisible by itself and 1
must be a prime number. Now to say that 1 which have all the
characteristics of a prime number, to be a nonprime number is a
selfcontradiction, i.e. violates the earlier definition of a prime number.
It is assumed that 0 and 1 are something extra or super or nonprime number
of which the reason for this assumption is unfounded. According to this, therefore,
The set of positive prime numbers = {2, 3, 5, 7, 11 ……. }
The set of negative prime numbers = {2, 3,
5, 7,11 ……. } And –1, 0, and 1 will be nonprime numbers.
But –1, 0, 1 are by no means, the neutral of the two sets of prime numbers,
positive and negative. The selfcontradiction can be removed, if the particulars are put thus:
The set of positive prime numbers = {1, 2, 3,
5, 7, 11 ……. }
The set of negative prime numbers = {1, 2,
3, 5, 7, 11 ……. } ‘0’ is not a prime number which is neutral to
the two sets since ‘0’ can be divided by 2, 3, 4, etc. (On the minor
contradictions in modern mathematics, Jan 1975) Sets cannot be anything else than a
limitation of qualities which in its turn has within itself the tension to
‘push’ the limit into its infinite into universality, that line of
limitation are ‘pushed’ to its terminal position of which something cannot
keep it in the set anymore but must be distinguished from another. The
very end point of every set is its universality or infinity of elements or
members, the only transformation into a oneness, which is the ultimate whole
– the Universal Set is still not the Final Absolute Universal Set of the
Real World. But ordinary set theory is only the exposition of quanta
into oneness, which must return to proceed to the next—to something without
quanta—to quality. The movement of set methodology [in the
brain], inherent in set thinking, produce a ‘new’ form of classification of
mathematics concern — the very form which penetrate in every qualitative
distinction of quantities. The further development of set theory is a
resolution of itself into essence, into a substance essential to its
existence, to the oneness of its practice, that of ‘Set Conception’— viz.
detecting set, recognizing the ‘truth of imagination’ of absolute equality
of set A and set B or ‘equal members’ although no mathematicians have ever
i.e. intersection, manipulation of set lines or limitations to elements and
called them union, etc. discover two things which are one and in fact the
same thing. C = C, it is said, but actually the first ‘C’ is never the
same as the second ‘C’ and only that they look like the same one or else no
distinction of the first C and the second C can ever be made. Their
equality implies that they have a common ‘something’ such as a value or
number which makes them equal. It is that they are 2 different
identities wit he same algebraic representation. The same goes to {a, b} = {a, b} where
thorough ‘likeness’ is never the case. 2
The Twofold ‘Expression’ Of Intersection Of Sets 1.
{a,b,c} ∩ {c,d,c} = {c}
[Method of selecting the common element] (i)
{Df,Df,Df,Df,Df,Df,Df} ∩ {Dw,Dw} = {D_{fw
,}D_{fw}} Df is a Day of a week which is fine weather Dw is a Day of a week which is windy weather [The case when the elements are the same in both
intersecting sets but of different qualities] In Df, f Є {D} ⇔ D = {f} In Dw, w Є {D} ⇔ D = {w}
In D_{fw,}
f Є {D}
and w Є {D}
or D = {f,w} If D stand for a fine day, therefore f, and w are
the two qualities of the element D. Є is an element of the set of days
(D)
(ii)
{P_{mtc,
}P_{mtc}, P_{mtc}, P_{mtc}}
∩ {P_{m,e,g,}
P_{m,e,g}}
= {P_{mtceg}, P_{mtceg}} if and only if the members of the second set are
the concluding set, the member of the first set to form the standard set. Criteria: i.
When intersection occurs between
two sets of the same elements, same in the sense that they (element) are
selected completely from a given standard set, the results gives a set of
the elements whereby each of which poses all the different qualities of the
two intersecting parent set, and whose number is determined by the smaller
set. e.g Standard Set: {B_{(f,h,b), }B_{(f,h,b)}, B_{(f,h,b,s)}}_{} Resolution into two sets and intersecting.
{B_{(f,h,b,s),}B_{(f,h,b)
,}B_{(f,h,b)}}
∩ {B_{(f,h,b,s)}}
= {B_{(f,h,b,s)}} B is the same element; f,h,b,s are qualities. Note
no repetition of f,h,b is allowed. ∩ is the symbol for ‘intersection’. ii.
{a,b,c} ∩ {c,d,e} = {c}. transposing, {c} = {a,b,c} ∩ {c,d,e} ⇔
{c} = {c,a,b} ∩ {c,d,e}
⇔
{c} = {c, L}
∩ {c, N}, where L and N are two sets not having common
elements. Also {D_{(f,w)}} = {D_{(f,w)}, Df} ∩ {D_{(f,w)}, Dw} Transposing,
{D_{(f,w)},
Df} ∩ {D_{(f,w)},
Dw} = {D_{(f,w)}} D is called the standard set of reference. e.g. D = {3
days of a week} = {D_{f, } D_{w}, D_{(f,w)}} Where D_{(f,w)}, L is called the repetitive members of the standard sets, or members considered twice in the two process. The repetitive members can have all even numbers for double intersection;_{} _{ } 2.
Venn Diagram (a)
{c} = {c,L}
∩ {c,N} _{ } (b) {D_{(f,w)}} = {D_{(f,w)},Df} ∩ {D_{(f,w)},Dw} E = D = Standard Set of Reference
e.g of 25 girls, 18 like pop music,13 like
classical music, and 10 like both. i.e.
{10G_{b}}
= {10G_{b},
8G_{p}}
∩ {10G_{b},
3G_{c}} 3.
Procedure To Comprehension [Building
up the whole through the parts] (A)
One is familiar that A ∩ B gives
C and concretely, {a,b,c} ∩ {c,d,e} = {c}. that is, the intersection of A and B vanish into
C….. now, given set C.
i.e {c}.
one makes a return to the original at a higher form. {c} = {c,a,b} ∩ {c,d,e} Now transform the subsets {a,b} and {d,e} into the
abstract into L and M, we have, {c} = {c,L} ∩ {c,M} where, L and M are sets which do not have any same
elements Note: This backward ‘Faustian Movement’ has its
utility where we deal with such problems: For example: There are 20 boys in a
certain class. 16 of them study physics, 14 study chemistry, and 12 study
both physics and chemistry. In this case, {c} becomes {12} = {12,4} ∩ {12,2} where A = {people taking both chemistry &
physics, people taking physics} B = {people taking both chemistry & physics, people taking chemistry} C = {people taking both chemistry & physics}
(B) When this has been done the construction of
Venn diagram is easier. E is the universal set to limit or give a name to
the set in reference. L,C,M is equal to the number of times different
selections are made on E. The Venn diagrams for {12} = {2,4} ∩ {12,2}
Since, E = {20 pupils}
Remark: G = {pupils wearing
glasses} and W = {pupils wearing watches}. G ∩ W is not well defined because
it is hardly known that whether there is a common element or not. If on the other hand, G = {pupils wearing glasses in the class, counted
without respect to anything else not bothering about these with watches} and
W = {pupils wearing watches in the same class, counted again without respect
to anything else not bothering about those with glasses}, then G ∩ W has
possible common element, if there is intersection, then realization occurs. 3 The
relation between the old set theory and the new equivalent set theory, is as
such that the old, much attributed to Boolean algebra of sets (George Boole)
and Venn Diagram (John Venn), is algebraic, statistical and pictorial in its
set to set relation whilst the new, due to application of Hegelian notion,
etc, is dynamical and mentation by forms have found its way back into
mathematics as a new complement of logical mathematical thoughts. The
dynamical set to set relations from the low sets, i.e. sets with limited
number of elements in them, up to the temporally universalis and its final
finite and infinity nature of both the case in the set theory and the real
world as it is. This is due to certain confusion about the universal
set(s) — which often appears as the only universal set but as such is not
and passes over into more universal sets and so on to infinity. This
limitation and delimitation is a movement of the logical conception of
universality of set. Discontinuities in the progression are subjected
by the discontinuity intervals in the Octave of progression of set into its
universality.
The fact that a set must be transformed into
a subset before it becomes a subset of a more universal set is inevitable,
because by doing so, the transformedsubset presupposes the existence of a
qualitative from of which this subset is only a belongingto or a “member”
to another bigger qualitative subset which becomes the Universal set.
Whilst the transformation of this first Universal set into a fact, universal
subset presupposes a universality of a higher qualitative form, the second
universal set, etc., etc.
The inability to conceive motion, is
apparently revealed when a teacher discusses the problem on the universal
sets, the true significant of it is a formal logical habit of thinking has
imposed itself on the dialectical and forms’ mentation in the head of the
teacher.
A common misunderstanding about limitation
and delimitation of universal set is expressed here. A set, {a, b, c},
is a subset of the universal set {a, b, c…x, y, z}. This limitation,
or rather first limitation is itself a set {a, b, c…x, y, z} and its
becoming of a subset of another universal set is unequivocally possible.
This set, can be the subset of another or rather the second universal set,
{a, b, c...x, y, z ,of English alphabets, a, b, c…x, y, z of the Chinese
Phonetic alphabet,…}, which consists of all known species of alphabets.
The initial limitation becomes delimitation and propels into more logical
universal sets – universality with boundaries to fence more dialectical
“notes” and mathematical “stopinders”, which still preserves the weak
oneness. (Dialectic, Stopinders and Sets)
