### INFLUENCE OPERATIONS – A different approach

SGS and Mihail Paduraru

Within the new information war,
one can notice how the collective mindset is frequently “bombarded” with terms
such as disinformation and propaganda.

In order to understand how these
techniques work outside of a military frame, this piece brings forward the
result of an analysis whose purpose was to validate/invalidate the authors’
hypotheses, starting from a short video clip uploaded on a videosharing social
network, where the moderators “argue” through a mathematical demonstration that
the sum of natural numbers (N) is a rational number (Q).

The purpose of this initiative is to show the mechanism and
psychological instruments that can be used, at a common-sense level, to
validate a wrong reasoning, and, furthermore, for this to justify its existence
through a climate of confusion and mass propagation.

Thus, one could assert that the risk that arises
from these actions is represented by the propagation capacity within societies
of undesirable attitudes (based on superficial documentation and individual
ignorance), generated by skewed information, framed accurately to justify
erroneous concepts.

We will evaluate the explanations using a deductive/logical
reasoning, according to which, if the premise is true, the conclusion cannot be
false.

Premise

If mathematical principles and laws are followed, it is NOT
possible that the sum of natural numbers (N) to have as result a rational
number (Q).

The technique used to avoid the validation of result without
a proper verification is described by Felicia RĂDOI under the name

*Crystal Ball,*in the Intelligence Analyst’s Guide, volume coordinated by Ionel NIȚU, at the Mihai Viteazul National Intelligence Academy Publishing House in Bucharest.
CHAPTER 3 – Methods and Techniques in Intelligence Analysis

Techniques used to avoid mental biases and to stimulate
creativity. Typology, characteristics / p 70 (Romanian edition)

Reasoning

In essence, the authors of the demonstration select
particular results from the regulation theory and combine them with a
common-sense interpretation.

Purpose

After Padilla’s assertion in the
last phrase of the article-“

**(1) the purpose of this initiative is an influence operation:***What do we get if we sum all the natural numbers?”***and by no means an explicative or demonstrative one.***“ I think another answer might be the following: we get people talking about Mathematics.”,*
“The explanations” and
“demonstrations” exposed visually or verbally in similar videos on the
same topic (2) cannot constitute actual explanations, nor demonstrations since
the problem is not stated so that to give that type of purpose, but it merely
presents a result produced by a series of operations.

Structure

Now, the influence operation
consists of the fact that everything is based on vexing common intuitions such
as “how can a proper sum of natural numbers be a negative improper module
number” and its apparent validation through a series of operations:

-either arithmetical (so
acceptable through common sense)

-either analytical (acceptable by
appealing to authority by someone who either does not know the meaning of the
analytical operation, either is inclined to accept without verification an
assertion which requires it: invoking the name of a famous mathematician such
as Euler, Riemann, etc, of a of a mathematical subdomain such as the intensely
mediatized number theory through cryptology, or of an application in a
different domain which was massively disseminated to the public such as quantic
mechanics and so on).

Context

It is not taken into account
that, for divergent series of positive numbers, as opposed to convergent
series, the rules of calculations starting from adding term with term and
ending with the operations of their own series (eg the multiplication or Cauchy
product (3) and Mertens theorems (4) constitute an analog of multiplication,
not multiplication by component), there are no direct analogies.

Meaning, to make sense of those
operations, one needs to build a different context of significance for the
mathematical objects for which, often the same sign is used, creating confusion
in the interpretation of the final result.

More simply, at an elementary
level – a function is formed as a mathematical object from a defined domain, a
codomain or domain of value and law of correspondence.

On many occasions this
mathematical object which allows correct interpretations if it is taken in its
integrality, is amputated until the law of correspondence.

For instance, the phrasing
“function x2” constitutes an “amputation”.

Drawing conclusion from amputated
mathematical objects could lead to erroneous assertions if in their related
reasoning interfere essentially the properties of the domain, codomain or those
synergic of the object as a whole – such as the ζ function.

Forcing interpretation

For example, the alternate series
(meaning in which consecutive terms are of opposite signs) obtained to
particularize the constant string 1, has as sum ½ since a new context was
formed, in which, for a class of diverging series which contain the series
itself one can explain “the series sum” according to the concept of average
introduced by Cesàro (5) .

Here the series is divergent
(meaning its not convergent, the string of partial sums is a mix of two
constant strings, 1 and 0 respectively) as the string of partial sums is not
convergent, and the meaning of the sum is obtained through a mediation
operation which does not allow a direct analogy with the class of convergent
series.

Induced error

Therefore, the error is obtained, either with awareness of
it or not, and this is not relevant here by forcing the interpretation.

Without any explicative base, the
authors extrapolate interpretations of the class of convergent series which
have a direct analogy with the arithmetical frame, to a class of diverging
series which, through an additional operation (here the Cesàro (6) mediation)
which can allow us to discuss a mathematic object, a “series sum” but with a
different significance than in the convergent series class case.

This is valid as a direct analogy with the
arithmetical frame in the case of convergent series, but in the case of Cesàro,
solely if we remember that “sum of series” has a meaning only within the summability
concept (7), meaning how we use the term “sum series” for classes of divergent
series.

It’s essential here that the ½ is a convention allowed by the context
created by the concept or Cesàro summability theory and it is not ½ from the
set of rational numbers, but it can be identified with this only pending on the
significance context – this issue is not minor, but it is omitted by the
authors of the analyzed study.

Similarly, the error can be
identified in the case built on the Riemann (8) zeta function (this being one of
the mathematical objects that allowed the introduction of a regulation method (9)[(eg how to build contexts open to interpretation to associate mathematical
objects-for eg numbers of divergent series), how these interfere in physics
problems (eg obtaining an interpretation for certain classes of physical
phenomenons)], but that requires a longer explanation.

In short, the context in which
the exposed formula is valid excludes the frames of significance of the ½
numbers – 1/12 etc, identifying them as rational numbers through omission of
the context in which they were introduces.

For instance, if we use Riemann’s
zeta function, when one calculates ζ(– 1) = – 1 / 12, one actually refers to ζ(–
1 + 0 i) simply because the ζ function is defined on the set of complex
numbers.

The identification of -1+0 i with
-1 is implied, but does not create a significant problem in its interpretation.
The omitted problem with creates the interpretation conclusion is the sense of
regulation defined on the ζ function.

Conclusion

In the opinion of the authors of
the current study, the initial analysts are misleading.

They are incoherent and thus
produce confusion, by trying to use in their assertions to direct the
conversation towards mathematics and create a communication product in which
the confusion on the sense and value of mathematics prevails.

As such, the success of a
beneficial influence operations has as sine qua non premise coherence, or the
concept of orientation of the Boyd cycle, which cannot be efficient without
content that gives in the end the value of the interpretation.

We have no reason, according to
our beliefs, to condemn this type of activities in the name of freedom of
speech – if someone wishes to make a mockery, there is no reason to forbid him,
variety and entertainment being inherent to the freedom itself.

We have, however, the duty to
ourselves to build our own coherent sense for relevancy, but this time – in the
name of freedom of information.

Relating to our own convictions
of relevancy, we can classify in the identified manner, the communicational
contribution of the gentlemen in our case, unrelated to other contributions
these might have made.

Sources

(1) http://www.nottingham.ac.uk/~ppzap4/response.html.

(2) https://www.youtube.com/watch?v=E-d9mgo8FGk;

https://www.youtube.com/watch?v=w-I6XTVZXww.

(3) https://en.wikipedia.org/wiki/Cauchy_product.

(4) https://en.wikipedia.org/wiki/Mertens'_theorems.

(5) https://en.wikipedia.org/wiki/Ces%C3%A0ro_summation;.

(6) https://en.wikipedia.org/wiki/Ces%C3%A0ro_mean.

(7)
https://www.quora.com/What-is-the-significance-of-Cesaro-summation.

(8) https://en.wikipedia.org/wiki/Riemann_zeta_function.

(9) https://en.wikipedia.org/wiki/Zeta_function_regularization.