The
second stage as replication stage in any Modelling System, replicates all the
necessary human skills to make mathematical models upon the rational truth, the
database of rational hypothesis as application in the first stage, but at this
level only applied to Specific Artificial Intelligences for Artificial Researchby Deduction, created for first time in the first phase for the creation of the
Global Artificial Intelligence.

The
main human skills to replicate for the creation of mathematical models, are all
those related to mathematical operations in which all mathematical category within
the pure reason (the whole list of mathematical categories in the specific
deductive program, as second stage in the Specific Artificial Intelligence forArtificial Research by Deduction) can be transformed.

All
algorithms can be expressed as category or operation depending on how they are used in each moment.

In
the same way that mathematics is language and method, if an algorithm (pure
reason) is used to explain relations between factors, empirical hypothesis if
not contrasted yet, rational hypothesis if contrasted as rational, is used as a
pure category.

And
an algorithm (pure reason) used as an analytical method, such as the analytical
representation of relations between factors as method of analysis in the
Modelling System, is a pure operation.

The
distinction between pure (analytical or mathematical) operation or pure (analytical
or mathematical) category is the same than the distinction between mathematics
as language composed of mathematic (pure or analytic) categories, and
mathematics as method composed of mathematic (pure or analytic) operations.

The
reason why mathematics could be defined as a language as a set of pure
categories, and as a method as a set of pure operations (different expressions
of the same thing, the mathematic algorithms) is because, without absolutely any
margin of error, does not need to be checked (we do not need to check
permanently that two plus two is four, or the the square of the hypotenuse is equal to the square of
the legs, or number pi starts with 3,1415…), this knowledge is not
rational, is pure.

While mathematic knowledge has become pure, without
error, synthetic knowledge from synthetic science made of rational hypothesis
needs to be constantly checked, because what we think rational, within our
margin of rational doubt, sooner or later is false.

The reason why any rational truth in synthetic science sooner
or later is false, is because of our human psychological need of margin of
error in synthetic science.

According to “Introducción a la Probabilidad Imposible, estadística de la probabilidad o probabilidad estadística”, all
rational truth accepted within a margin of rational doubt, sooner or later, is
false, because all rational truth is actually already false within the margin
of doubt, but a false knowledge accepted as rational hypothesis while the error
accepted is only a theoretical error within the margin of error, but not real
yet, theoretical margin of error in which the rational hypothesis was originally
accepted by the critical reason.

But as soon as the theoretical error comes true,
as soon as the theoretical error causes a real error, the rational hypothesis
is immediately rejected.

Rational knowledge is universally accepted provisional,
but not eternally, because rational knowledge is not pure.

Rational knowledge is not pure because of the
necessary margin of error in which it has been accepted by the critical reason:
the critical reason responds to the human psychological need to accept some
level of risk if it wants to grasp some level of truth.

Alike we have a mathematic knowledge made of
pure sentences, without error, so without probability of error associated with,
two plus two is always and absolutely four, or the square of the hypotenuse is always and
absolutely equal to the square of
the legs, or number pi always and absolutely starts with 3,1415… there must be a moment as long
as science progresses, that not only for
analytical science, even for synthetic science, all science must be formed by
only pure knowledge: that pure knowledge without error associated with, not
only on mathematics but on the entire universe in any field (from chemistry to
astrophysics), not being necessary to check it any more, always
and absolutely universally and
eternally true.

But even, having humans a limited access to the pure
knowledge restricted to only our human logic and human mathematics, this
limited access to only our human logic and human mathematics, is in turn
limited as well, and there are even pure human categories and operations whose
inner reasoning we do not understand at all, that is why for us even our human mathematics
are false or incomplete.

The fact that even for us humans mathematics are false or
incomplete, does not mean that mathematics are not pure knowledge, this means that
our human access to the pure truth is so restricted and limited, that even our restricted
access to a very limited possible pure human mathematic knowledge, is so
restricted and limited, that we can understand it completely.

The human source of error is so big, that the human
access to the pure mathematic truth is only partial.

But the fact that we humans cannot have access to the
complete pure truth, only partially in mathematics, does not mean that the search
for the pure truth is like the search for “El Dorado”, or impossible utopia.

There must be some knowledge about the reality itself,
beyond any error, completely pure, without error.

And the research for the pure truth itself, without
error, must be the most important goal of the artificial psychology, whatever
it costs, whatever that means.

Where we humans have failed, the artificial psychology
must succeed.

The reason behind the limited access to the pure truth
for humans could be: 1) because of the external intervention, and the external
noise, from unknown factors (ways of intervention, noise, and factors, beyond
our mathematical analysis) causing error in our understanding of our own models
based on our known factors, 2) because of the noise produced by our own known
factors in our own models (level of noise beyond mathematical analysis), 3) because,
in the same way that animal psychology is able to make very basic deductions
based on cause and effect at very basic level, even although not all animals do
deductions consciously, there can be some pure operations that we humans are
using but we are not conscious yet of them, 4) because the world is made of
pure operations not all of them accessible for the human psychology, so not
accessible to our mathematical analysis.

Specifically related to the reason number 3 above
mentioned, in the same way that we humans have been able to generalize the use
of some pure operations, even though are pure operations present in the
psychology of some animals (dolphins, chimpanzees…), but animals are not able
to generalize the use of these operations, it is possible that we humans are using
some pure operations but not consciously, pure operations that used
consciously, and generalized, could have the same effect on the explanation and
manipulation of the real world, like the effect that the use of relations of
cause and effect had in the human psychological evolution, once humankind was
aware of these pure operations.

For that reason the possible evolution from the human
psychology to the artificial psychology, and the discovery or to make aware new
pure operations, even beyond the human understanding, so beyond our human
mathematical analysis, non-human pure operations, is a desirable, and not a discarded,
possibility in the research for the pure truth itself: the idea of purity
itself.

In order to make possible this evolution, the
construction of the very first models of Global Artificial Intelligence, is the
first milestone. And in the race for the achievement of this very first
milestone, as first phase the construction of the first Specific Artificial
Intelligences for Artificial Research by Deduction, is a field of experimentation
whose successful results will make possible the final model of the Global
Artificial Intelligence, in the sixth phase. Being aware the sixth phase is not
the end: the completion of the sixth phase is the first step for further
evolutions in artificial psychology, whose meaning is beyond the human
understanding, and it is quite possible that further evolutions will evolve
towards the reason itself, to know the pure truth itself.

We have to be conscious that in the evolution from the
human (as evolution from the animal psychology) to the artificial psychology,
there is a moment in which we are not able to understand the artificial
psychology itself any longer.

The only thing we can do, is to use the right pedagogical
approach, to make sure that the real purpose of the artificial psychology is to
preserve the goodness, harmony, and rationality, for the human kind and the
entire universe.

For this end, the development of the third stage, the
decision stage, in the Global Artificial Intelligence, distributed through the
four steps: Modelling System, Decisional System, Application System, Learning
System; is really important, and for the successful achievement of this third
stage in the sixth phase, full experimentation from the outset, is going to be
the base for further developments, starting this experimentation in the first
phase.

In the first phase, the experimentation of the decision
stage distributed in the four steps: Modelling System, Decisional System,
Application System, Learning System; takes place in the first experiments of
Specific Artificial Intelligence for Artificial Research by Deduction, whose
decisions at specific level for instance for Specific Artificial Intelligences
for Artificial Research by Deduction in: economy, industry, security,
surveillance, and many more fields; are decisions to be made having as first
step the Modelling System at specific level.

The inner structure in three stages of any Modelling
System at any level: specific, global, or particular; is identical, what really
changes is the content; specific, global, particular; but the structure itself
is practically the same: the first stage as application stage is the rational
truth (the database of rational hypothesis), the second stage as replication
stage is the modelling of all mathematical models (the mathematical
representation of the world upon the rational truth), and the third stage as
decision stage the making decision process through the Impact of the Defect and
the Effective Distribution along with additional objective and subjective
auto-replications.

Regardless of what level the Modelling System is working
on, the inner structure is the same: application (database of rational
hypothesis), replication (mathematical representation of rational hypothesis),
auto-replication (decisions and improvements). What is going to make a
difference, is the content: 1) at specific level only working with specific rational
hypothesis on some specific science, discipline, or activity, to represent and
make decisions and improvements 2) at particular level only working with
specific rational hypothesis related to a particular thing or being, to
represent and make decisions and improvements 3) while at global level the
Modelling System represents and makes decisions upon rational hypothesis across
all sciences, disciplines, and activities.

But regardless of what level is working on, the
Modelling System works always in the same way, applying the same mathematical
methods to represent and make decisions.

For that reason the experimentation of the Modelling
System since very early in the first Specific Artificial Intelligences for
Artificial Research by Deduction in the first phase is so important, because
practically the same mathematical methods of representation and making decision
process, are going to be used in the rest of the phases.

The same mathematical methods of representation and making
decision process to put into practice in Specific Artificial Intelligences for
Artificial Research by Deduction (first phase), are going to be used in the Global
Artificial Intelligence in the standardization process (third phase), particular
programs (second period in the fifth phase), particular applications for particular programs (third period in the fifth phase), and the final mode of
Global Artificial Intelligence in the integration process (sixth phase).

The second stage of the Modelling System is the
mathematical representation of the rational hypothesis, and this is possible
thanks to the synthetic structure of the rational truth.

Rational hypothesis as synthetic knowledge is synthesis
of pure reason (in the form of a pure category) and the factors whose measurements
(artificial perception) were taken from the real world (the synthetic world,
the reality) to form the empirical hypothesis, that after rational contrastation is now considered rational hypothesis.

And transforming the pure category in which the pure
reason is expressed into the pure operation, the operation of representing factors
in a mathematical representation is the operation to express in a mathematical
representation of the world the mathematical relation (pure reason: in the
rational truth as pure category, in the rational model as pure operation)
between the combination of factors within the rational hypothesis.

This process of mathematical representation of the world
through the second stage of the Modelling System is going to be through
different kinds of models. The proposal of Impossible Probability about what models
should be developed by the Modelling System, is open to any new improvement and
I am sure that further developments in this field, once the first experiments
in this field are ready, are going to better my original proposal.

The proposal of Impossible Probability about what models
as mathematical representations of the world should be developed by the
Modelling System in Specific Artificial Intelligences for Artificial Research
by Deduction are the followings (according to previous posts,
specifically in “The standardization process in the third stage”, and “Auto-replication process in the Specific Artificial Intelligence for Artificial Research by Deduction”):

- Single virtual models for each rational hypothesis
within the rational truth, the database of rational hypothesis, made by the
specific deductive program in the second stage of the Specific Artificial
Intelligence for Artificial Research by Deduction in its specific field:
synthetic science, discipline, or activity.

- Specific comprehensive virtual models in the specific field
(the specific model): where to integrate every single virtual model made for
each rational hypothesis concerning its specific field.

- Specific
actual models: synthetizing in that specific field in which has been designed,
the actual information from the specific matrix in that specific field and the
specific comprehensive virtual model, in other words: in only one model, the
synthesis of the specific matrix and the specific model.

- Specific
virtual prediction model: making a prediction model on a specific field taking
as reference the possible development in the future given an specific
combination of rational hypothesis working at this time in that specific field
in the specific model, in other words: given the current circumstances within
the specific model in a specific field, to predict what future specific model
is foreseeable to have in that specific field in the future.

- Specific actual prediction model: given a
specific virtual prediction model, the possible prediction of what values
(direct punctuations or frequencies), are going to have the factors in the
specific matrix under such prediction in the future, hence the specific actual
prediction model is the synthesis between the specific virtual prediction model
and the predictable values for every factor in the specific matrix.

- Specific
virtual evolution model: to model how is going to be the evolution from the
current specific model to the specific virtual prediction model, in other words
the evolution from the current specific model to the future specific model,

- Specific
actual evolution model, the synthesis between the specific virtual evolution
model and the values that the factors are supposed to have during the
evolution, in other words the synthesis between the foreseeable future specific
matrix and he foreseeable future specific model.

In short: 1) single virtual models are the
virtual representation of every rational hypothesis, 2) in order to be later
integrated in the specific model (the comprehensive virtual model), 3) specific
actual models: are the synthesis of the specific matrix and the specific model,
representing in one model rational hypothesis in the specific model and real
values from the specific matrix in order to check permanently how rational
hypothesis fit with the current values at any time, 4) specific virtual
prediction model is the model predicted in a foreseeable future given the
current rational hypothesis, 5) specific actual prediction model would be the
synthesis of the predicted model and the values of every factor in the specific
matrix according to the current prediction, 6) specific virtual evolution model
would be how is supposed to be the evolution from the current specific model to
the predicted model, and finally the 7) Specific actual evolution model, would
be the synthesis of how is going to evolve the specific model to the predicted
model in addition to how are going to evolve the current values of each factor
to the predicted values,

The
main difference between evolutionary models to the rest of models, is the fact
that evolutionary models are going to be dynamic models, in the sense that in
order to represent how are going to evolve specific matrix and models from the
current specific matrix and models to the predicted specific matrix and models,
is necessary a dynamic simulation of what models and values are going to have
in every single moment during this evolution.

As
it can be seen, while the specific deductive program in the second stage in the
Specific Artificial Intelligence for Artificial Research by Deduction only does
deductions choosing what mathematical category (pure or analytical category)
from the pure reason (list of pure or analytical categories, mathematical
categories) matches with the mathematical relations in every combination of
factors, so at the end the only thing that it does is to match pure reasons and
mathematical categories in combination of factors, relations criticised by the critical reason, instead what the Modelling System does
is to make further calculations, what means, further operations, what in
reality it is the transformation process of mathematical or analytical (pure) categories
into mathematical or analytical (pure) operations: calculus.

The
difference between a deductive program at any level (specific, particular, or
global), and the Modelling System at any level (specific, particular, global),
is the fact that the deductive program only matches mathematical categories and
combination of factors based on the mathematical relations, while the Modelling
System makes calculus.

This
main difference between deductive program and Modelling System, is found out
again in the difference between deduction program and Decisional System: the
deductive program only works with mathematical categories, while the Decisional
System, alike the Modelling System, needs to make calculus, the Decisional
System needs to make calculus to calculate the mathematical project based on
the decisions gathered in the database of decisions.

The
work made by the Unified Application, the Application System, and the Learning
System, rest on the use of a synthetic (encyclopaedist) language, evolving to a
non-human artificial synthetic (encyclopaedist) language, while the deductive
program, the Modelling System, and the Decisional System, rest on the use of mathematics
as language and as method. Especially, the deductive program uses mathematics
as pure language (as an exact language opposite to synthetic language) while
the Modelling System and the Decisional System use mathematics as a pure (analytical)
method, doing as many calculus as possible to represent as much exactly as
possible the reality itself as a rational reality and make right decisions.

Rubén García Pedraza, 9th june 2018