56. Statistics
56.1. The
Analytical and Stochastic Nature of Statistics
Statistics has
traditionally been classified as a mathematical discipline. At the same time,
however, for many other sciences it functions as a method of study. Since
mathematics itself is an analytical science, statistics must also be
understood in analytical terms. It is primarily concerned with the study of
formal relationships between facts or phenomena, within a delimited extension
of space and time.
What
distinguishes statistics from other forms of analytical inquiry is that its
subject matter is of a stochastic nature. That is, statistical phenomena
occur within contexts that involve sequencing or ordering of events, where
results are in some measure dependent on chance, given that there is no
absolute causal determination. In this sense, statistics combines the rigor of
mathematics with the probabilistic uncertainty inherent to real-world
processes.
56.2. Pure
and Applied Research in Statistics
Traditional
epistemology drew a line between pure research and applied research.
Pure research was directed toward the internal development of a discipline
itself, while applied research concerned the implementation of scientific
knowledge in other fields.
This distinction
is also relevant for statistics. On the one hand, pure statistical research
focuses on the growth of the discipline itself, its internal theories, and its
understanding of stochastic structures. On the other hand, applied
statistical research extends beyond disciplinary boundaries, functioning as
a heuristic method of inquiry for the synthetic sciences. In this
applied role, statistics ceases to be merely a branch of mathematics and
becomes an instrument for constructing and testing knowledge in the natural and
social sciences [ and technological research].
56.3. The
Dual Dimension of Statistics
If we follow
this classical division between pure and applied research, statistics reveals
itself to have a dual dimension.
- As a discipline for itself, statistics
belongs to the analytical domain of mathematics. Its purpose here is to
develop theories that explain its stochastic nature and refine the
internal logic of statistical reasoning.
- As a method for other sciences, statistics
functions as a tool of empirical inquiry, assisting in the generation of
synthetic theories of reality. In this role, it becomes a method of
discovery and justification, enabling scientists to move from raw data to
structured explanations of natural or social phenomena [or testing novel cutting
edge technologies].
Thus, any
definition of statistics must take into account this double aspect: statistics
is simultaneously a discipline in itself and a heuristic method of
study for other sciences.
56.4.
Statistics as Theory versus Statistics as Method
From this dual
perspective, two possible definitions of statistics emerge.
- Statistics as a mathematical discipline: In
this sense, statistics belongs to the analytical sciences, and it is
rightly integrated into the mathematics curriculum at every educational
level. This definition stresses its formal and theoretical dimension,
grounded in logic and probability theory.
- Statistics as a heuristic method: In this
sense, statistics becomes a form of applied science. It serves the purpose
of generating synthetic data and theories in other fields, such as
biology, economics, psychology, or sociology. Here, the focus is on its
capacity to help us understand empirical reality by organizing,
summarizing, and interpreting data.
Both definitions
are valid, and their coexistence is precisely what gives statistics its unique
position within the sciences.
56.5. Pure
Research and the Development of Statistical Theory
When statistics
is approached as an analytical discipline, its development depends on
pure research. The outcome of such research is the elaboration of statistical
theory itself.
An example of
this can be found in Impossible Probability, where a whole theoretical
framework is constructed concerning the statistics of probability—that
is, the application of statistical methods to probability and vice versa. This
approach expands the boundaries of the field and redefines the relationship
between mathematics, statistics, and epistemology. In such works, statistics is
not just a tool but a self-developing discipline, capable of producing new
theoretical horizons.
56.6. Applied
Research and the Heuristic Function of Statistics
When statistics
is applied to other sciences, it becomes a heuristic method. In this
role, it is not the internal theory of statistics that is being advanced, but
rather the capacity of other sciences to build synthetic theories of
their own subject matter.
In natural
sciences, such as physics or biology, applied statistics allows the formulation
of generalizations about natural laws based on empirical data. In the
social sciences, it provides the foundation for analyzing human behavior,
economic systems, or cultural patterns. Thus, applied statistics is essential
for the empirical grounding of both natural and social sciences [and
technological development].
56.7. The
Epistemological Position of Statistics
Within its dual
definition, the place of statistics depends on whether it is understood as a discipline
or as a method.
- If defined as a discipline, statistics
belongs within mathematics, where it has traditionally been located and
taught.
- If defined as a method, statistics belongs
to epistemology, the science of knowledge and scientific methods.
In this case, statistics is classified as one among many heuristic tools by which
sciences investigate reality.
This dual
belonging makes statistics an ambivalent field: simultaneously part of
mathematics and part of epistemology. It embodies the tension between formal
analytical reasoning and practical methodological application.
56.8. The
Ambivalent Classification of Statistics
The result of
this analysis is a double classification of statistics.
- As a mathematical discipline, it is an
analytical science focused on internal theoretical development.
- As a scientific method, it is an
epistemological tool, applied heuristically to other sciences for
the purpose of constructing theories about reality.
Therefore,
statistics occupies a unique position in the hierarchy of sciences. It belongs
both to the formal world of mathematics and to the methodological world of
epistemology, reflecting its dual capacity to create its own theory and
to enable the creation of theory in other fields.
56.9. The
Double Definition of Statistics: Mathematics or Epistemology
Within the dual
definition of statistics—whether as a mathematical discipline or as a method
of epistemology—the ultimate definition depends on the final sense or
purpose of research. If the aim is pure research into the internal
mathematical logic of formal statistical relations, then statistics reveals its
analytical essence, as explained particularly in the opening sections of Introduction
to Impossible Probability: Statistics of Probability or Statistical Probability.
Here, the profoundly analytical nature of statistics becomes evident, since its
foundation lies in the very logic of relations.
By contrast, if
the ultimate goal of research is the synthetic study of phenomena, then
statistics operates primarily as a heuristic method. In such cases, its
role is not to deepen its own logical-mathematical basis but to provide
methodological support for sciences concerned with constructing synthetic
explanations of empirical reality.
56.10. Pure
Research in Statistics and Pure Research in Epistemology
Since statistics
can be defined either as a mathematical discipline or as an epistemological
method, the scope of pure research also acquires a dual meaning. On the
mathematical side, pure research in statistics seeks to understand the
logical relations between statistical concepts themselves, derived from the
formal study of relations among facts or phenomena.
Yet, statistics
may also be framed within epistemology. In that case, one type of pure
epistemological research consists in understanding statistics itself as a
method for the synthetic sciences. This represents a different level of pure
research, one that does not focus on the mathematical refinement of statistical
concepts, but rather on building a theoretical foundation for how statistics,
as a method, ought to be applied to empirical reality.
56.11. The
Analytical-Mathematical and the Epistemological-Synthetic Functions of
Statistics
This dual
perspective leads to a broader understanding of statistics as having two
primary functions:
- Analytical-mathematical: Pure research in
statistics is here directed toward clarifying the formal relations between
statistical concepts. Its outcome is the construction of
statistical theory as a logical and mathematical discipline.
- Epistemological-synthetic: Pure research, in
this sense, takes statistics as an object of epistemological study, asking
how it can or should be applied to reality. Its result is the development
of a theory of epistemological application, which clarifies how
statistics should be employed as a heuristic method across
different sciences.
When applied
directly to natural or social facts [or technological testing], statistics
functions as applied research, producing synthetic knowledge about
empirical reality and thus enhancing our understanding of what actually
happens.
56.12. The
Mathematical Origin of Statistics and Its Passage into Epistemology
In any case, the
origin of statistics lies in its mathematical-analytical function. Its
beginnings are tied to the formulation of a pure mathematical-statistical
theory. This is precisely the focus of Introduction to Impossible
Probability, where the fundamental mathematical concepts of statistical
probability are carefully fixed.
Once these
concepts are firmly established, statistics can then take on a new role: it
becomes part of the heuristic methods of scientific investigation and
integrates into epistemology. In epistemological terms, statistics is no longer
merely a discipline of formal relations but a method of inquiry designed
for the study of synthetic sciences.
56.13.
Epistemology and the Role of Statistics as a Scientific Method
Within
epistemology, statistics is defined as a scientific method for
investigating synthetic sciences. Epistemology itself studies the degree of
adequacy of such methods in different fields of knowledge and evaluates
their reliability when applied across the natural and social sciences.
Thus, while
analytical-mathematical statistics is dedicated to pure research into its
logical-mathematical foundation, epistemology undertakes a pure
methodological investigation, assessing the appropriateness of statistics
as a heuristic method in specific domains. This debate generates a broad
spectrum of positions:
- At one extreme are qualitative purists, who
reject the use of quantitative methods altogether, particularly in the
social sciences.
- At the opposite extreme are quantitative purists,
who argue that the only way to conduct reliable science is through
quantitative methods in all fields, natural or social [and technological].
Within this quantitative paradigm, statistics often holds a position of
preeminence.
- Between these extremes lies a wide variety of mixed
approaches, combining qualitative and quantitative perspectives in
different proportions.
56.14. The
Position of Impossible Probability in the Qualitative-Quantitative
Debate
The stance of Impossible
Probability within this debate is that the apparent contradiction between
qualitative and quantitative methods is ultimately irrelevant. Every
measurement of empirical reality must begin with the assessment of the singular
qualities of the particular subjects or objects under study, for which
measurement scales are established.
From this
perspective, the dialectical process shows that:
- The qualitative necessarily transforms into
the quantitative (since qualities must be measured).
- The quantitative necessarily transforms into
the qualitative (since numerical values must be interpreted in
meaningful terms).
Therefore, every
statistical study must integrate both: the interpretation of statistical
tables and the measurement of reality as objectively as possible.
However, in practice, such objectivity is extremely difficult—indeed impossible
in the case of humanity—because the root of error lies in the finitude of
human perception compared with the infinite singularities of empirical
reality. Human beings can never fully comprehend 100% of reality; there
will always be aspects that escape measurement, either due to their apparent
insignificance or the inadequacy of methods. This generates an inevitable
margin of error.
56.15. The
Role of Scientific Policy in Statistical Interpretation
In this sense,
the role of scientific policy is to determine which interpretative
paradigm should be used for statistical data and to make the necessary
decisions regarding their application. Scientific policy thus acts as the mediator
between the methodological potential of statistics and its practical deployment
in the pursuit of reliable knowledge.
56.16.
Descriptive and Inferential Statistics
Whether defined
as an analytical-mathematical discipline or as an epistemological method,
statistics is traditionally classified—depending on its ultimate
purpose—into two major branches:
- Descriptive statistics, which organizes and
summarizes empirical data from a defined sample.
- Inferential statistics, which goes further,
attempting to generalize from the sample to a wider population or
universe of subjects.
In both cases,
the starting point is a series of facts or phenomena, observed across a
space and time previously delimited.
56.17. The
Concept of the Sample in Impossible Probability
This series of
facts or phenomena within defined spatial and temporal boundaries is called a sample.
In Impossible Probability, two main types of samples are recognized:
- The sample of subjects or options,
symbolized as N.
- The sample of direct scores or frequencies,
symbolized as Σxi.
In any type of
universe—whether infinite universes of subjects or options (where
infinite qualities exist in empirical reality) or limited universes of
options (restricted to a finite set of possible alternatives)—both types of
samples are always obtained.
- The sample of subjects/options (N) provides
the set of elements on which measurements are taken.
- The sample of direct scores/frequencies (Σxi)
records the distribution of outcomes or occurrences.
Together, these
two samples constitute the foundation of statistical analysis in Impossible
Probability.
56.18.
Population Studies and Infinite Universes of Subjects
Population
studies fall within the category of universes of infinite subjects or
options. This is because any population observed at a given point in
history can only be understood as a sample of what is happening to that
population across its entire historical development. A population is never
static: it is part of a dynamic continuum, where individuals are born, die,
migrate, or transform their social and natural conditions. Thus, any study of a
population is necessarily limited, representing just one temporal slice
of an infinite and ongoing process.
56.19. From
Samples to Descriptive Statistics
Once a sample
has been defined, the work of descriptive statistics begins. This
involves describing what occurs within the sample through the calculation of
several key measures, such as:
- Empirical probability, derived directly from
observed frequencies.
- Theoretical probability, based on expected values under conditions
of equality of opportunities.
- Mean or typical deviation, as measures of
dispersion.
- Individual statistics, which reflect the
behavior of particular elements within the sample.
- Sample statistics, which aggregate and
generalize the results for the whole sample.
The purpose of
descriptive statistics is purely descriptive: to represent what occurs
in the sample without yet engaging in critical evaluation of hypotheses.
56.20. The
Purpose of Inferential Statistics
In contrast, inferential
statistics has a different purpose: to critically evaluate empirical
hypotheses. Its task is to determine whether such hypotheses are
sufficiently rational, based on a principle of critical reason.
If a hypothesis
is accepted, this acceptance is always provisional, never absolute. The
reason is that every statistical acceptance depends on a margin of error,
and within that margin there always exists the possibility of error—that is,
the hypothesis might ultimately prove false. Thus, inferential statistics
functions as the critical filter of empirical claims, placing them under
rational scrutiny before granting them temporary validity.
56.21.
Intra-meditional and Inter-meditional Statistics
In Introduction
to Impossible Probability: Statistics of Probability or Statistical Probability,
an additional classification of statistics is introduced: the distinction
between intra-meditional and inter-meditional statistics.
- Intra-meditional statistics (developed up to
section 15) deals with descriptive or inferential analysis based on a
single measurement. It studies data obtained from just one act of
measurement, focusing on the interpretation of that isolated result.
- Inter-meditional statistics (developed
between sections 16 and 20) extends the scope to multiple measurements.
Whether descriptive or inferential, it enables comparative analysis of
data across different measurements, allowing researchers to study the evolution
of subjects or options over time. This opens the possibility of
developing rational models of hypothesis testing concerning how a
given series behaves through successive measurements.
56.22.
Predictive Studies: Projection and Forecast
Within
inter-meditional statistics, one specific form of study is the predictive
analysis, elaborated in section 17 of Impossible Probability. Here,
a distinction is made between:
- Projection, which extrapolates from past
data to suggest possible future trends.
- Forecast, which makes stronger claims about
expected outcomes, attempting to anticipate empirical or theoretical
results.
This predictive
framework is crucial because it shows how, within the methodology of Impossible
Probability, the future can be studied not merely as speculation, but
through a structured process of empirical or theoretical prediction.
This augmented translation is based
on the post
published in https://probabilidadimposible.blogspot.com/,
On 7 April 2013,
Rubén García Pedraza
imposiblenever@gmail.com
