Statistics as the Logic of Science

\documentclass{article} \usepackage[utf8]{inputenc} % !BIB TS-program = biber \usepackage[backend=biber,style=numeric, citestyle=authoryear]{biblatex} \addbibresource{blog.bib} \usepackage{amssymb} \usepackage{dirtytalk} \usepackage{csquotes} \usepackage{amsmath} \usepackage{calc} \usepackage{textcomp} \usepackage{mathtools} \usepackage[english]{babel} \usepackage{fancyhdr} \usepackage{url} \def\UrlBreaks{\do\/\do-} \usepackage{breakurl} \usepackage[breaklinks]{hyperref} \usepackage{graphicx} \graphicspath{ {images/} } \usepackage{wrapfig} \usepackage{float} \usepackage[T1]{fontenc} \usepackage{outlines} \usepackage{enumitem} \setenumerate[1]{label=\null} \setenumerate[2]{label=\null} \setenumerate[3]{label=\roman*.} \setenumerate[4]{label=\alph*.} \newcommand{\midtilde}{\raise.17ex\hbox{$\scriptstyle\mathtt{\sim}$}} \usepackage{CJKutf8} \pagestyle{fancy} \fancyhf{} \title{Statistical Analysis by Communicative Functionals: \\ Lecture 2 – Statistics as The Logic of Science} \author{Justin Petrillo} \begin{document} \maketitle The question of $science \ itself$ has never been its particular object of inquiry but the existential nature, in its possibility and thereby the nature of its actuality. Science is power, and thus abstracts itself as the desired meta-good, although it is always itself about particularities as an ever-finer branching process. Although a philosophic question, the \textit{question of science} is inherently a political one, as it is the highest good desired by the society, its population, and its government. To make sense of science mathematically-numerically, as statistics claims, it is the scientific process itself that must be understood through probability theory as \say{The Logic of Science.} \footcite{QTSTATS} \section{Linguistic Analysis of the Invariants of Science: The Laws of Nature} The theory of science, as the proof of its validity in universality, must consider the practice of science, as the negating particularity. The symbolic language of science, within which its practice and results are embedded, necessarily negates its own particularity as well, as thus to represent a structure universally. Science, in the strict sense of having achieved already the goal of universality, is $de-linguistified$. While mathematics, in its extra-linguistic nature, often has the illusion of universal de-linguistification, such is only a semblance and often an illusion. The numbers of mathematics always can refer to things, and in the particular basis of their conceptual context always do. The non-numeric symbols of mathematics too represented words before short-hand gave them a distilled symbolic life. The de-linguistified nature of the extra-linguistic property of mathematics is that to count as mathematics, the symbols must themselves represent universal things. Thus, all true mathematical statements may represent scientific phenomena, but the context and work of this referencing is not trivial and sometimes the entirety of the scientific labor. The tense of science, as the time-space of the activity of its being, is the $tensor$, which is the extra-linguistic meta-grammar of null-time, and thus any and all times. \section{The Event Horizon of Discovery: The Dynamics between an Observer \& a Black Hole} The consciousness who writes or reads science, and thereby reports or performs the described tensor as an action of experimentation or validation, is the transcendental consciousness. Although science is real, it is only a horizon. The question is thus of its nature and existence at this horizon. What is knowable of science is thereby known as \say{the event horizon}, as that which has appeared already, beyond which is mere a \say{black hole} as what has not yet revealed itself – always there is a not-yet to temporality and so such a black hole can always be at least found as all that of science that has not and cannot be revealed since within the very notion of science is a negation of withdrawal (non-appearance) as the condition of its own universality (negating its particularity). Beginning here with the null-space of black-holes, the physical universe – at least the negative gravitational entities – have a natural extra-language – at least for the negative linguistic operation of signification whereby what is not known is the \say{object} of reference. In this cosmological interpretation of subjectivity within the objectivity of physical space-time, we thus come to the result of General Relativity that the existence of a black-hole is not independent of the observer, and in fact is only an element in the Null-Set, or negation, of the observer. To ‘observer’ a black-hole is to point to and outline something of which one does not know. If one ‘knew’ what it was positively then it would be not ‘black’ in the sense of not-emitting light \textit{within the reference frame (space-time curvature) of the observer}. That one $cannot$ see something, as receive photons reflecting space-time measurements, is not a property of the object but rather of the observer in his or her subjective activity of observation since to be at all must mean there is some perspective from which it can be seen. As the Negation of the objectivity of an observer, subjectivity is the $negative \ gravitational \ anti-substance$ of blackholes. Subjectivity, as what is not known by consciousness, observes the boundaries of an aspect (a negative element) of itself in the physical measurement of an ‘event horizon.’ These invariants of nature, as the conditions of its space-time, are the laws of dynamics in natural science. At the limit of observation we find the basis of the conditionality of the observation and thus its existence as an observer. From the perspective of absolute science, within the horizon of universality (i.e. the itself as not-itself of the black-hole or Pure Subjectivity), the space-time of the activity of observation (i.e. the labor of science) is a time-space as the hyperbolic negative geometry of conditioning (the itself of an unconditionality). What is a positive element of the bio-physical contextual condition of life, from which science takes place, for the observer is a negative aspect from the perspective of transcendental consciousness (i.e. science) as the limitation of the observation. Within Husserlian Phenomenology and Hilbertian Geometry of the early 20th century in Germany, from which Einstein’s theory arose, a Black-Hole is therefore a Transcendental Ego as the absolute measurement point. Our Solar System is conditioned in its space-time geometry by the MilkyWay galaxy it is within, which is conditioned by the blackhole Sagittarius A* ($SgrA*$). Therefore, the unconditionality of our solar space-time (hence the bio-kinetic features) is an unknown of space-time possibilities, enveloped in the event horizon of $SgrA*$. What is the inverse to our place (i.e. space-time) of observation will naturally only exist as a negativity, what cannot be seen. \section{Classical Origins of The Random Variable as The Unknown: Levels of Analysis} Strictly speaking, within Chinese Cosmological Algebra of 4-variables ($\mu$, X,Y,Z), this first variable of primary Unknowing, is represented by $X$, or Tiān (\begin{CJK*}{UTF8}{gbsn}天\end{CJK*}), for ‘sky’ as that which conditions the arc of the sky, i.e. “the heavens” or the space of our temporal dwelling ‘in the earth.’ We can say thus that $X=SgrA*$ is the largest and most relevant primary unknown for solarized galactic life. While of course X may represent anything, in the total cosmological nature of science, i.e. all that Humanity doesn’t know yet is conditioned by, it appears most relevantly and wholistically as $SgrA*$. It can be said thus that all unknowns ($x$) in our space-time of observation are within \say{\textit{the great unknown}} ($X$) of $SgrA*$, as thus $x \in X$ or $x \mathcal{A} X$ for the negative aspectual ($\mathcal{A}$) relationship \say{x is an aspect of X}. These are the relevant, and most general (i.e. universal) invariants to our existence of observation. They are the relative absolutes of, from, and for science. Within more practical scientific judgements from a cosmological perspective, the relevant aspects of variable unknowns are the planets within our solar system as conditioning the solar life of Earth. The Earthly unknowns are the second variable Y, or Di (\begin{CJK*}{UTF8}{gbsn}地\end{CJK*}) for “earth.” They are the unknowns that condition the Earth, or life, as determining the changes in climate through their cyclical dynamics. Finally, the last unknown of conditionals, Z, refers to people, Ren (\begin{CJK*}{UTF8}{gbsn}人\end{CJK*}) for ‘men,’ as what conditions their actions. X is the macro unknown (conditionality) of the gravity of ‘the heavens,’ Y the meso unknown of biological life in and on Earth, and Z the micro unknown of psychology as quantum phenomena. These unknowns are the subjective conditions of observation. Finally, the 4th variable is the “object”, or Wu (\begin{CJK*}{UTF8}{gbsn}物\end{CJK*}), $\mu$ of measurement. This last quality is the only $real$ value in the sense of an objective measurement of reality, while the others are imaginary in the sense that their real values aren’t known, and can’t be within the reference of observation since they are its own conditions of measurement within \say{the heavens, the earth, and the person}. \footcite[p.~82]{CHIN-MATH} In the quaternion tradition of Hamilton, ($\mu$, X,Y,Z) are the quaternions, ($\mu$, i,j,k). Since the real-values of X,Y,Z in the scientific sense can’t be known truly and thus must always be themselves unknowns, they are treated as imaginary numbers ($i=\sqrt{-1}$) with their ‘values’ merely coefficients to the quaternions $i,j,k$. These quaternions are derived as quotients of vectors, as thus the unit orientations of measurement’s subjectivity, themselves representing the space-time. We often approximate this with the Cartesian X,Y,Z of 3 independent directions as vectors, yet such is to assume Euclidean Geometry as independence. \printbibliography \end{document}

The Derivation of the Normal Distribution

\documentclass{article} \usepackage[utf8]{inputenc} % !BIB TS-program = biber \usepackage[backend=biber,style=numeric, citestyle=authoryear]{biblatex} \addbibresource{blog.bib} \usepackage{amssymb} \usepackage{dirtytalk} \usepackage{csquotes} \usepackage{amsmath} \usepackage{calc} \usepackage{textcomp} \usepackage{mathtools} \usepackage[english]{babel} \usepackage{fancyhdr} \usepackage{url} \def\UrlBreaks{\do\/\do-} \usepackage{breakurl} \usepackage[breaklinks]{hyperref} \usepackage{graphicx} \graphicspath{ {images/} } \usepackage{wrapfig} \usepackage{float} \usepackage[T1]{fontenc} \usepackage{outlines} \usepackage{enumitem} \setenumerate[1]{label=\null} \setenumerate[2]{label=\null} \setenumerate[3]{label=\roman*.} \setenumerate[4]{label=\alph*.} \newcommand{\midtilde}{\raise.17ex\hbox{$\scriptstyle\mathtt{\sim}$}} \pagestyle{fancy} \fancyhf{} \title{Statistical Analysis by Communicative Functionals: \\ Lecture 1 – A Geometric Derivation of the Normal Distribution} \author{Justin Petrillo} \begin{document} \maketitle \begin{abstract} The Internal Space-Time Geometry to Experiment as the Distribution of Measurement InterActions is set up by the Statistical Parameter. \end{abstract} \section{The Scientific Process} Statistics is the method of determining the validity of an empirical claim about nature. A claim that is not particularly valid will likely be true only some of the time or under certain specific conditions that are not too common. Ultimately, thus, within a domain of consideration, statistics answers the question of the universality of the claims made about nature through empirical methods of observation. It may be that two opposing claims are both true in the sense that they are each true half the time of random observation or within half the space of contextual conditionalities. The scientific process, as progress, relies on methods that over a linear time of repeated experimental cycles, increase the validity of the claims as the knowledge of nature approaches universality, itself always merely a horizon within the phenomenology of empiricism. This progressive scientific process is called ‘discovery,’ or merely $research$, although it is highly non-linear. The scientific process is a branching process as the truth of a claim is found to be dependent upon its conditions, and those conditions found dependent on further conditionals. This structure of rationality is as a tree. A single claim (C) has a relative validity (V) due to the truth of an underlying, or conditioning, claim, $C_i$, given as $V_{C_i}(C)=V(C,C_i)$. We may understand the validity of claims through probability theory, in that the relative validity of a claim based on a conditioning claim is the probability the claim is true conditioned on $C_i$, $V(C,C_i)=P(C|C_i)$. In general, we will refer to the object under investigation, of which C is a claim about, as the primary variable X, and the subject performing the investigation, of which $C_i$ is hypothesized (as a cognitive action), as the secondary variable Y. Thus, the orientation of observation, i.e. the time-arrow, is given as $\sigma: Y \rightarrow X$. An observer (Y) makes an observation from a particular position of an event (X) with its own place, forming a space-time of the action of measurement. An observation-as-information is a complex quantum-bit, which within a space of investigation is a complex variable, representing a tree of observation-conditioning rationality resulting from the branching process of hypothesis formation, with each node a conditional hypothesis and edge length the conditional probability. The gravitation of the system of measurement is the space-time tensor of its world-manifold, stable or chaotic of the time of interaction. We thus understand the positions of observers within a place of investigation, itself given at least in real-part component by the object of investigation. \section{Experimental Set-up} Nature is explained by a parameterized model. Each parameter, as a functional aggregation of measurement samples, has itself a corresponding distribution as it $occurs \ in \ nature$ along the infinite, universal horizon of measurement. \\ \\ Let $X^n$ be a random variable representing the n qualities that can be measured for the thing under investigation, $\Omega$, itself the collected gathering of all its possible appearances, $\omega \in \Omega$ such that $X^n:\omega \rightarrow {\mathbb{R}}^n$. Each sampled measurement of $X^n$ through an interaction with $\omega$ is given as an $\hat{X}^n(t_i)$, each one constituting a unit of indexable time in the catalogable measurement process. Thus, the set of sampled measurements, a $sample \ space$, is a partition of ‘internally orderable’ test times within the measurement action, $\{ \hat{X}^n(t): t \in \pi \}$. \\ \\ In this set up of statistical sampling, one will notice the step-wise process-timing of a single actor performing n sequential measurements can be represented the same as n indexed actors performing simultaneous measurements, at least with regard to internal time accounting. In order to infer the latter interpretational context, such as to preserve the common sense notion of time as distinct from social space, one would represent all n simultaneous measurements as n dimensions of X, although assumed to be generally the same in quality in such that all n actors sample the same object in the same way, yet are distinct in some orderable indexical quality. Thus, in each turn of the round time (i.e. one unit), all actors perform independent and similar measurements. It may be, as in progressive action processes, future actions are dependent on previous ones, and thus independence is only found within the sample space of a single time round. Alternatively, it may also be that the actors perform different actions, or are dependent upon each other in their interactions. Thus, the notion of actor(s) may be embedded in the space-time of the action of measurement. The embedding of a coordinated plurality of actors in the most mundane sense of ‘collective progress’ can be represented as the group action of all independent \& similar measurers completes itself in each round of time, with inter-temporalities in the process measurement process being similar but dependent on the previous one. The progressive interaction may be represented as the inducer $I^+:X(t_i) \rightarrow X(t_{i}+1)$, with the assumptions of similarity and independence as $\hat{x_i}(t) \sim \hat{x_j}(t) \ \& \ I(\hat{x_i}(t),\hat{x_j}(t))=0$. We take $\hat{X}(t)$ to be a group of measurement actors/actions $\{ \hat{x}_i(t): i \in \pi \}$ that acts on $\Omega$ together, or simultaneously, to produce a singular measurement of one round time. \section{Derivation of the Normal Distribution} The question with measurement is not, \say{what is the true distribution of the object in question in nature?}, but \say{what is the distribution of the parameter I am using to measure?}. The underlying metric of the quality under investigation, itself arising due to an interaction of measurement as the distance function within the investigatory space-time, is $\mu$. As the central limit states, averages of these measurements, each having an error, will converge to normality. We can describe analytically the space of our ‘atemporal’ averaged measurements in that the rate of change of the frequency $f$ of our sample measurements $x,x_0 \in X$ by the change in the space of measuring, is inversely proportional, by constant k, to the distance from the true measurement ($\mu$) and the frequency: \[\forall \epsilon > 0, \exists \delta(\epsilon)>0 \ s.t. \ \forall x, |x_0-x|<\delta \rightarrow \bigg| k(x_0-\mu)f(x_0) - \frac{f(x_0)-f(x)}{x_0-x}\bigg|<\epsilon \] or in the differential form \[\frac{df}{dx}=-k(x-\mu)f(x)\] \[f(t)=\int_{-\infty}^{+\infty}-k(x-\mu)f(x)dx\] the solution distribution is scaled by the constant of coefficient, C \[f(x)=Ce^{-\frac{k}{2}{(x-\mu)}^2}\] given the normalization of the total size of the universe of events as 1 \[ \int_{-\infty}^{\infty} f dx =1\] thus, \[C=\sqrt{\frac{k}{2\pi}}\] so the total distribution is, \[f(x)=\sqrt{\frac{k}{2\pi}}e^{-\frac{k}{2}{(x-\mu)}^2}\] \[E(X)=\int (x-\mu)f(x)dx=\mu\] \[\sigma^2=E(X^2)=\int {(x-\mu)}^2 f(x)dx=\frac{1}{\sqrt{k}}\] \[so, f(x)=N\bigg(\mu,\sigma=\frac{1}{\sqrt{k}}\bigg)\] \printbibliography \end{document} [/et_pb_text][/et_pb_column][/et_pb_row][/et_pb_section]