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.
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 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)
In essence, the authors of the demonstration select particular results from the regulation theory and combine them with a common-sense interpretation.
After Padilla’s assertion in the last phrase of the article-“What do we get if we sum all the natural numbers?” (1) the purpose of this initiative is an influence operation: “ I think another answer might be the following: we get people talking about Mathematics.”, and by no means an explicative or demonstrative one.
“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.
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).
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.
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.
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.
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.