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800 | "Mathematical logic"
E. Shannon, A. N. Whitehead, H. Reichenbach, P. Fevrier), to biology ( J. H. Woodger, A. Tarski ), to psychology ( F. B. Fitch, C. G. Hempel ), to law and morals ( K. Menger, U. Klug, P. Oppenheim), to economics ( J. Neumann, O. Morgenstern ), to practical questions ( E. C. Berkeley, E. Stamm), and... |
801 | "Mathematical logic"
M. Bochenski, S. [Stanislaw] T. Schayer, D. Ingalls )." [45] "Applications have also been made to theology (F. Drewnowski, J. Salamucha, I. Thomas)." [45] The study of computability theory in computer science is closely related to the study of computability in mathematical logic. There is a differe... |
802 | "Mathematical logic"
Formal calculi such as the lambda calculus and combinatory logic are now studied as idealized programming languages. Computer science also contributes to mathematics by developing techniques for the automatic checking or even finding of proofs, such as automated theorem proving and logic programmin... |
803 | "Mathematical logic"
The use of infinitesimals, and the very definition of function, came into question in analysis, as pathological examples such as Weierstrass' nowhere- differentiable continuous function were discovered. Cantor's study of arbitrary infinite sets also drew criticism. Leopold Kronecker famously stated... |
804 | "Mathematical logic"
Mathematicians began to search for axiom systems that could be used to formalize large parts of mathematics. In addition to removing ambiguity from previously naive terms such as function, it was hoped that this axiomatization would allow for consistency proofs. In the 19th century, the main method... |
805 | "Mathematical logic"
With the development of formal logic, Hilbert asked whether it would be possible to prove that an axiom system is consistent by analyzing the structure of possible proofs in the system, and showing through this analysis that it is impossible to prove a contradiction. This idea led to the study of p... |
806 | "Mathematical logic"
Gentzen showed that it is possible to produce a proof of the consistency of arithmetic in a finitary system augmented with axioms of transfinite induction, and the techniques he developed to do so were seminal in proof theory. A second thread in the history of foundations of mathematics involves no... |
807 | "Mathematical logic"
A common idea is that a concrete means of computing the values of the function must be known before the function itself can be said to exist. In the early 20th century, Luitzen Egbertus Jan Brouwer founded intuitionism as a part of philosophy of mathematics. This philosophy, poorly understood at fi... |
808 | "Mathematical logic"
Brouwer's philosophy was influential, and the cause of bitter disputes among prominent mathematicians. Later, Kleene and Kreisel would study formalized versions of intuitionistic logic (Brouwer rejected formalization, and presented his work in unformalized natural language). With the advent of the ... |
809 | "Mathematical logic"
Dabei ist der Umfang des Buches angewachsen, so daß eine Teilung in zwei Bände angezeigt erschien." "Carrying out this plan [by Hilbert for an exposition on proof theory for mathematical logic] has experienced an essential delay because, at the stage at which the exposition was already near to its ... |
810 | "Mathematical logic"
A Precis of Mathematical Logic. Synthese Library, Vol. 1. Translated by Otto Bird. Dordrecht: Springer. doi: 10.1007/978-94-017-0592-9. ISBN 9789048183296. Cantor, Georg (1874). "Ueber eine Eigenschaft des Inbegriffes aller reellen algebraischen Zahlen" (PDF). Journal für die Reine und Angewandte M... |
811 | "Foundations of mathematics"
Foundations of mathematics is the study of the philosophical and logical [1] and/or algorithmic basis of mathematics, or, in a broader sense, the mathematical investigation of what underlies the philosophical theories concerning the nature of mathematics. [2] In this latter sense, the disti... |
812 | "Foundations of mathematics"
also called metamathematical concepts, with an eye to the philosophical aspects and the unity of mathematics. The search for foundations of mathematics is a central question of the philosophy of mathematics; the abstract nature of mathematical objects presents special philosophical challeng... |
813 | "Foundations of mathematics"
The development, emergence, and clarification of the foundations can come late in the history of a field, and might not be viewed by everyone as its most interesting part. Mathematics always played a special role in scientific thought, serving since ancient times as a model of truth and rig... |
814 | "Foundations of mathematics"
The systematic search for the foundations of mathematics started at the end of the 19th century and formed a new mathematical discipline called mathematical logic, which later had strong links to theoretical computer science. It went through a series of crises with paradoxical results, unti... |
815 | "Foundations of mathematics"
While the practice of mathematics had previously developed in other civilizations, special interest in its theoretical and foundational aspects was clearly evident in the work of the Ancient Greeks. Early Greek philosophers disputed as to which is more basic, arithmetic or geometry. Zeno of... |
816 | "Foundations of mathematics"
The discrepancy between rationals and reals was finally resolved by Eudoxus of Cnidus (408–355 BC), a student of Plato, who reduced the comparison of two irrational ratios to comparisons of multiples of the magnitudes involved. His method anticipated that of the Dedekind cut in the modern d... |
817 | "Foundations of mathematics"
Aristotle's syllogistic logic, together with the axiomatic method exemplified by Euclid's Elements, are recognized as scientific achievements of ancient Greece. Starting from the end of the 19th century, a Platonist view of mathematics became common among practicing mathematicians. [ citati... |
818 | "Foundations of mathematics"
Are they located in their representation, or in our minds, or somewhere else? How can we know them? The ancient Greek philosophers took such questions very seriously. Indeed, many of their general philosophical discussions were carried on with extensive reference to geometry and arithmetic.... |
819 | "Foundations of mathematics"
In the Meno Plato's teacher Socrates asserts that it is possible to come to know this truth by a process akin to memory retrieval. Above the gateway to Plato's academy appeared a famous inscription: "Let no one who is ignorant of geometry enter here". In this way Plato indicated his high op... |
820 | "Foundations of mathematics"
[4] In this view, the laws of nature and the laws of mathematics have a similar status, and the effectiveness ceases to be unreasonable. Not our axioms, but the very real world of mathematical objects forms the foundation. Aristotle dissected and rejected this view in his Metaphysics. These... |
821 | "Foundations of mathematics"
René Descartes published La Géométrie (1637), aimed at reducing geometry to algebra by means of coordinate systems, giving algebra a more foundational role (while the Greeks used lengths to define the numbers that are presently called real numbers ). Descartes' book became famous after 1649... |
822 | "Foundations of mathematics"
The strong implications of infinitesimal calculus on foundations of mathematics is illustrated by a pamphlet of the Protestant philosopher George Berkeley (1685–1753), who wrote "[Infinitesimals] are neither finite quantities, nor quantities infinitely small, nor yet nothing. May we not cal... |
823 | "Foundations of mathematics"
In his 1821 work Cours d'Analyse he defines infinitely small quantities in terms of decreasing sequences that converge to 0, which he then used to define continuity. But he did not formalize his notion of convergence. The modern (ε, δ)-definition of limit and continuous functions was first ... |
824 | "Foundations of mathematics"
Weierstrass began to advocate the arithmetization of analysis, to axiomatize analysis using properties of the natural numbers. In 1858, Dedekind proposed a definition of the real numbers as cuts of rational numbers. This reduction of real numbers and continuous functions in terms of rationa... |
825 | "Foundations of mathematics"
With these concepts, Pierre Wantzel (1837) proved that straightedge and compass alone cannot trisect an arbitrary angle nor double a cube. In 1882, Lindemann building on the work of Hermite showed that a straightedge and compass quadrature of the circle (construction of a square equal in ar... |
826 | "Foundations of mathematics"
Concepts of vector spaces emerged from the conception of barycentric coordinates by Möbius in 1827, to the modern definition of vector spaces and linear maps by Peano in 1888. Geometry was no more limited to three dimensions. These concepts did not generalize numbers but combined notions of... |
827 | "Foundations of mathematics"
Then the Russian mathematician Nikolai Lobachevsky (1792–1856) established in 1826 (and published in 1829) the coherence of this geometry (thus the independence of the parallel postulate ), in parallel with the Hungarian mathematician János Bolyai (1802–1860) in 1832, and with Gauss. Later ... |
828 | "Foundations of mathematics"
At that time, the main method for proving the consistency of a set of axioms was to provide a model for it. One of the traps in a deductive system is circular reasoning, a problem that seemed to befall projective geometry until it was resolved by Karl von Staudt. As explained by Russian his... |
829 | "Foundations of mathematics"
The purely geometric approach of von Staudt was based on the complete quadrilateral to express the relation of projective harmonic conjugates. Then he created a means of expressing the familiar numeric properties with his Algebra of Throws. English language versions of this process of deduc... |
830 | "Foundations of mathematics"
The algebra of throws is commonly seen as a feature of cross-ratios since students ordinarily rely upon numbers without worry about their basis. However, cross-ratio calculations use metric features of geometry, features not admitted by purists. For instance, in 1961 Coxeter wrote Introduct... |
831 | "Foundations of mathematics"
Additionally, De Morgan published his laws in 1847. Logic thus became a branch of mathematics. Boolean algebra is the starting point of mathematical logic and has important applications in computer science. Charles Sanders Peirce built upon the work of Boole to develop a logical system for ... |
832 | "Foundations of mathematics"
Frege's work was popularized by Bertrand Russell near the turn of the century. But Frege's two-dimensional notation had no success. Popular notations were (x) for universal and (∃x) for existential quantifiers, coming from Giuseppe Peano and William Ernest Johnson until the ∀ symbol was int... |
833 | "Foundations of mathematics"
The formalization of arithmetic (the theory of natural numbers ) as an axiomatic theory started with Peirce in 1881 and continued with Richard Dedekind and Giuseppe Peano in 1888. This was still a second-order axiomatization (expressing induction in terms of arbitrary subsets, thus with an ... |
834 | "Foundations of mathematics"
Several schools of the philosophy of mathematics ran into difficulties one after the other in the 20th century, as the assumption that mathematics had any foundation that could be consistently stated within mathematics itself was heavily challenged by the discovery of various paradoxes (suc... |
835 | "Foundations of mathematics"
Both meanings may apply if the formalized version of the argument forms the proof of a surprising truth. For instance, Russell's paradox may be expressed as "there is no set of all sets" (except in some marginal axiomatic set theories). Various schools of thought opposed each other. The lea... |
836 | "Foundations of mathematics"
E. J. Brouwer, which resolutely discarded formalism as a meaningless game with symbols. [7] The fight was acrimonious. In 1920 Hilbert succeeded in having Brouwer, whom he considered a threat to mathematics, removed from the editorial board of Mathematische Annalen, the leading mathematical... |
837 | "Foundations of mathematics"
Indeed, he used the words "formula game" in his 1927 response to L. E. J. Brouwer 's criticisms: And to what extent has the formula game thus made possible been successful? This formula game enables us to express the entire thought-content of the science of mathematics in a uniform manner a... |
838 | "Foundations of mathematics"
These rules form a closed system that can be discovered and definitively stated. [8] Thus Hilbert is insisting that mathematics is not an arbitrary game with arbitrary rules; rather it must agree with how our thinking, and then our speaking and writing, proceeds. [8] We are not speaking her... |
839 | "Foundations of mathematics"
Virtually all mathematical theorems today can be formulated as theorems of set theory. The truth of a mathematical statement, in this view, is represented by the fact that the statement can be derived from the axioms of set theory using the rules of formal logic. Merely the use of formalism... |
840 | "Foundations of mathematics"
It is closely connected with the further question: what impels us to take as a basis precisely the particular axiom system developed by Hilbert? Consistency is indeed a necessary but not a sufficient condition. For the time being we probably cannot answer this question... [10] In some cases... |
841 | "Foundations of mathematics"
Gödel's second incompleteness theorem establishes that logical systems of arithmetic can never contain a valid proof of their own consistency. What Hilbert wanted to do was prove a logical system S was consistent, based on principles P that only made up a small part of S. But Gödel proved t... |
842 | "Foundations of mathematics"
The foundational philosophy of intuitionism or constructivism, as exemplified in the extreme by Brouwer and Stephen Kleene, requires proofs to be "constructive" in nature – the existence of an object must be demonstrated rather than inferred from a demonstration of the impossibility of its ... |
843 | "Foundations of mathematics"
Others try to create a cognitive science of mathematics, focusing on human cognition as the origin of the reliability of mathematics when applied to the real world. These theories would propose to find foundations only in human thought, not in any objective outside construct. The matter rem... |
844 | "Foundations of mathematics"
Many researchers in axiomatic set theory have subscribed to what is known as set-theoretic Platonism, exemplified by Kurt Gödel. Several set theorists followed this approach and actively searched for axioms that may be considered as true for heuristic reasons and that would decide the conti... |
845 | "Foundations of mathematics"
This argument by Willard Quine and Hilary Putnam says (in Putnam's shorter words),... quantification over mathematical entities is indispensable for science... therefore we should accept such quantification; but this commits us to accepting the existence of the mathematical entities in ques... |
846 | "Foundations of mathematics"
Such a view has also been expressed by some well-known physicists. For example, the Physics Nobel Prize laureate Richard Feynman said People say to me, "Are you looking for the ultimate laws of physics?" No, I'm not... If it turns out there is a simple ultimate law which explains everything... |
847 | "Foundations of mathematics"
[11] And Steven Weinberg: [12] The insights of philosophers have occasionally benefited physicists, but generally in a negative fashion – by protecting them from the preconceptions of other philosophers.... without some guidance from our preconceptions one could do nothing at all. It is jus... |
848 | "Foundations of mathematics"
Precisely, for any consistent first-order theory it gives an "explicit construction" of a model described by the theory; this model will be countable if the language of the theory is countable. However this "explicit construction" is not algorithmic. It is based on an iterative process of c... |
849 | "Foundations of mathematics"
More precisely, it shows that the mere assumption of the existence of the set of natural numbers as a totality (an actual infinity) suffices to imply the existence of a model (a world of objects) of any consistent theory. However several difficulties remain: Another consequence of the compl... |
850 | "Foundations of mathematics"
It is abbreviated ZFC when it includes the axiom of choice and ZF when the axiom of choice is excluded. Starting in 1935, the Bourbaki group of French mathematicians started publishing a series of books to formalize many areas of mathematics on the new foundation of set theory. The intuitio... |
851 | "Foundations of mathematics"
His ambitions were expressed in a time when nothing was clear: it was not clear whether mathematics could have a rigorous foundation at all. There are many possible variants of set theory, which differ in consistency strength, where stronger versions (postulating higher types of infinities)... |
852 | "Foundations of mathematics"
In most of mathematics as it is practiced, the incompleteness and paradoxes of the underlying formal theories never played a role anyway, and in those branches in which they do or whose formalization attempts would run the risk of forming inconsistent theories (such as logic and category th... |
853 | "Information theory"
Information theory is the scientific study of the quantification, storage, and communication of digital information. [1] The field was fundamentally established by the works of Harry Nyquist and Ralph Hartley, in the 1920s, and Claude Shannon in the 1940s. [2]: vii The field is at the intersection ... |
854 | "Information theory"
Some other important measures in information theory are mutual information, channel capacity, error exponents, and relative entropy. Important sub-fields of information theory include source coding, algorithmic complexity theory, algorithmic information theory and information-theoretic security. Ap... |
855 | "Information theory"
Information theory studies the transmission, processing, extraction, and utilization of information. Abstractly, information can be thought of as the resolution of uncertainty. In the case of communication of information over a noisy channel, this abstract concept was formalized in 1948 by Claude S... |
856 | "Information theory"
[4] Coding theory is concerned with finding explicit methods, called codes, for increasing the efficiency and reducing the error rate of data communication over noisy channels to near the channel capacity. These codes can be roughly subdivided into data compression (source coding) and error-correct... |
857 | "Information theory"
The landmark event establishing the discipline of information theory and bringing it to immediate worldwide attention was the publication of Claude E. Shannon's classic paper "A Mathematical Theory of Communication" in the Bell System Technical Journal in July and October 1948. Prior to this paper,... |
858 | "Information theory"
Ralph Hartley 's 1928 paper, Transmission of Information, uses the word information as a measurable quantity, reflecting the receiver's ability to distinguish one sequence of symbols from any other, thus quantifying information as H = log S n = n log S, where S was the number of possible symbols, a... |
859 | "Information theory"
Much of the mathematics behind information theory with events of different probabilities were developed for the field of thermodynamics by Ludwig Boltzmann and J. Willard Gibbs. Connections between information-theoretic entropy and thermodynamic entropy, including the important contributions by Rol... |
860 | "Information theory"
Information theory often concerns itself with measures of information of the distributions associated with random variables. One of the most important measures is called entropy, which forms the building block of many other measures. Entropy allows quantification of measure of information in a sing... |
861 | "Information theory"
The latter is a property of the joint distribution of two random variables, and is the maximum rate of reliable communication across a noisy channel in the limit of long block lengths, when the channel statistics are determined by the joint distribution. The choice of logarithmic base in the follow... |
862 | "Information theory"
This is justified because lim p → 0 + p log p = 0 {\displaystyle \lim _{p\rightarrow 0+}p\log p=0} for any logarithmic base. Based on the probability mass function of each source symbol to be communicated, the Shannon entropy H, in units of bits (per symbol), is given by where p i is the probabil... |
863 | "Information theory"
Entropy is also commonly computed using the natural logarithm (base e, where e is Euler's number), which produces a measurement of entropy in nats per symbol and sometimes simplifies the analysis by avoiding the need to include extra constants in the formulas. Other bases are also possible, but les... |
864 | "Information theory"
The entropy of a source that emits a sequence of N symbols that are independent and identically distributed (iid) is N ⋅ H bits (per message of N symbols). If the source data symbols are identically distributed but not independent, the entropy of a message of length N will be less than N ⋅ H. If on... |
865 | "Information theory"
Between these two extremes, information can be quantified as follows. If X {\displaystyle \mathbb {X} } is the set of all messages { x 1,..., x n } that X could be, and p ( x ) is the probability of some x ∈ X {\displaystyle x\in \mathbb {X} }, then the entropy, H, of X is defined: [12] (Here, I ( ... |
866 | "Information theory"
The special case of information entropy for a random variable with two outcomes is the binary entropy function, usually taken to the logarithmic base 2, thus having the shannon (Sh) as unit: The joint entropy of two discrete random variables X and Y is merely the entropy of their pairing: ( X, Y ).... |
867 | "Information theory"
Despite similar notation, joint entropy should not be confused with cross entropy. The conditional entropy or conditional uncertainty of X given random variable Y (also called the equivocation of X about Y ) is the average conditional entropy over Y: [13] Because entropy can be conditioned on a ran... |
868 | "Information theory"
It is important in communication where it can be used to maximize the amount of information shared between sent and received signals. The mutual information of X relative to Y is given by: where SI ( S pecific mutual I nformation) is the pointwise mutual information. A basic property of the mutual ... |
869 | "Information theory"
This is often recalculated as the divergence from the product of the marginal distributions to the actual joint distribution: Mutual information is closely related to the log-likelihood ratio test in the context of contingency tables and the multinomial distribution and to Pearson's χ 2 test: mutua... |
870 | "Information theory"
If we compress data in a manner that assumes q ( X ) {\displaystyle q(X)} is the distribution underlying some data, when, in reality, p ( X ) {\displaystyle p(X)} is the correct distribution, the Kullback–Leibler divergence is the number of average additional bits per datum necessary for compressio... |
871 | "Information theory"
If Alice knows the true distribution p ( x ) {\displaystyle p(x)}, while Bob believes (has a prior ) that the distribution is q ( x ) {\displaystyle q(x)}, then Bob will be more surprised than Alice, on average, upon seeing the value of X. The KL divergence is the (objective) expected value of Bob'... |
872 | "Information theory"
Coding theory is one of the most important and direct applications of information theory. It can be subdivided into source coding theory and channel coding theory. Using a statistical description for data, information theory quantifies the number of bits needed to describe the data, which is the in... |
873 | "Information theory"
In scenarios with more than one transmitter (the multiple-access channel), more than one receiver (the broadcast channel ) or intermediary "helpers" (the relay channel ), or more general networks, compression followed by transmission may no longer be optimal. Any process that generates successive m... |
874 | "Information theory"
For memoryless sources, this is merely the entropy of each symbol, while, in the case of a stationary stochastic process, it is that is, the conditional entropy of a symbol given all the previous symbols generated. For the more general case of a process that is not necessarily stationary, the avera... |
875 | "Information theory"
The rate of a source of information is related to its redundancy and how well it can be compressed, the subject of source coding. Communications over a channel is the primary motivation of information theory. However, channels often fail to produce exact reconstruction of a signal; noise, periods o... |
876 | "Information theory"
We will consider p ( y | x ) to be an inherent fixed property of our communications channel (representing the nature of the noise of our channel). Then the joint distribution of X and Y is completely determined by our channel and by our choice of f ( x ), the marginal distribution of messages we ch... |
877 | "Information theory"
For any information rate R < C and coding error ε > 0, for large enough N, there exists a code of length N and rate ≥ R and a decoding algorithm, such that the maximal probability of block error is ≤ ε; that is, it is always possible to transmit with arbitrarily small block error. In addition, for ... |
878 | "Information theory"
In practice many channels have memory. Namely, at time i {\displaystyle i} the channel is given by the conditional probability P ( y i | x i, x i − 1, x i − 2,..., x 1, y i − 1, y i − 2,..., y 1 ). {\displaystyle P(y_{i}|x_{i},x_{i-1},x_{i-2},...,x_{1},y_{i-1},y_{i-2},...,y_{1}).}. It is often more... |
879 | "Information theory"
Information theoretic concepts apply to cryptography and cryptanalysis. Turing's information unit, the ban, was used in the Ultra project, breaking the German Enigma machine code and hastening the end of World War II in Europe. Shannon himself defined an important concept now called the unicity dis... |
880 | "Information theory"
The security of all such methods currently comes from the assumption that no known attack can break them in a practical amount of time. Information theoretic security refers to methods such as the one-time pad that are not vulnerable to such brute force attacks. In such cases, the positive conditio... |
881 | "Information theory"
However, as in any other cryptographic system, care must be used to correctly apply even information-theoretically secure methods; the Venona project was able to crack the one-time pads of the Soviet Union due to their improper reuse of key material. Pseudorandom number generators are widely availa... |
882 | "Information theory"
These can be obtained via extractors, if done carefully. The measure of sufficient randomness in extractors is min-entropy, a value related to Shannon entropy through Rényi entropy; Rényi entropy is also used in evaluating randomness in cryptographic systems. Although related, the distinctions amon... |
883 | "Information theory"
Information theory and digital signal processing offer a major improvement of resolution and image clarity over previous analog methods. [17] Semioticians Doede Nauta and Winfried Nöth both considered Charles Sanders Peirce as having created a theory of information in his works on semiotics. [18]: ... |
884 | "Information theory"
[20] Quantitative information theoretic methods have been applied in cognitive science to analyze the integrated process organization of neural information in the context of the binding problem in cognitive neuroscience. [21] In this context, either an information-theoretical measure, such as “func... |
885 | "Statistics"
Statistics (from German: Statistik, orig. "description of a state, a country") [1] [2] is the discipline that concerns the collection, organization, analysis, interpretation, and presentation of data. [3] [4] [5] In applying statistics to a scientific, industrial, or social problem, it is conventional to b... |
886 | "Statistics"
[6] When census data cannot be collected, statisticians collect data by developing specific experiment designs and survey samples. Representative sampling assures that inferences and conclusions can reasonably extend from the sample to the population as a whole. An experimental study involves taking measur... |
887 | "Statistics"
[7] Descriptive statistics are most often concerned with two sets of properties of a distribution (sample or population): central tendency (or location ) seeks to characterize the distribution's central or typical value, while dispersion (or variability ) characterizes the extent to which members of the di... |
888 | "Statistics"
A hypothesis is proposed for the statistical relationship between the two data sets, and this is compared as an alternative to an idealized null hypothesis of no relationship between two data sets. Rejecting or disproving the null hypothesis is done using statistical tests that quantify the sense in which ... |
889 | "Statistics"
[8] Multiple problems have come to be associated with this framework, ranging from obtaining a sufficient sample size to specifying an adequate null hypothesis. [7] Measurement processes that generate statistical data are also subject to error. Many of these errors are classified as random (noise) or syste... |
890 | "Statistics"
[10] Some consider statistics to be a distinct mathematical science rather than a branch of mathematics. While many scientific investigations make use of data, statistics is concerned with the use of data in the context of uncertainty and decision making in the face of uncertainty. [11] [12] In applying st... |
891 | "Statistics"
Descriptive statistics can be used to summarize the population data. Numerical descriptors include mean and standard deviation for continuous data (like income), while frequency and percentage are more useful in terms of describing categorical data (like education). When a census is not feasible, a chosen ... |
892 | "Statistics"
To draw meaningful conclusions about the entire population, inferential statistics is needed. It uses patterns in the sample data to draw inferences about the population represented while accounting for randomness. These inferences may take the form of answering yes/no questions about the data ( hypothesis... |
893 | "Statistics"
Mathematical statistics is the application of mathematics to statistics. Mathematical techniques used for this include mathematical analysis, linear algebra, stochastic analysis, differential equations, and measure-theoretic probability theory. [13] [14] The early writings on statistical inference date bac... |
894 | "Statistics"
Al-Kindi also made the earliest known use of statistical inference, while he and later Arab cryptographers developed the early statistical methods for decoding encrypted messages. Ibn Adlan (1187–1268) later made an important contribution, on the use of sample size in frequency analysis. [15] The earliest ... |
895 | "Statistics"
Today, statistics is widely employed in government, business, and natural and social sciences. The mathematical foundations of modern statistics were laid in the 17th century with the development of the probability theory by Gerolamo Cardano, Blaise Pascal and Pierre de Fermat. Mathematical probability the... |
896 | "Statistics"
[18] The first wave, at the turn of the century, was led by the work of Francis Galton and Karl Pearson, who transformed statistics into a rigorous mathematical discipline used for analysis, not just in science, but in industry and politics as well. Galton's contributions included introducing the concepts ... |
897 | "Statistics"
[21] Galton and Pearson founded Biometrika as the first journal of mathematical statistics and biostatistics (then called biometry), and the latter founded the world's first university statistics department at University College London. [22] Ronald Fisher coined the term null hypothesis during the Lady tas... |
898 | "Statistics"
Fisher's most important publications were his 1918 seminal paper The Correlation between Relatives on the Supposition of Mendelian Inheritance (which was the first to use the statistical term, variance ), his classic 1925 work Statistical Methods for Research Workers and his 1935 The Design of Experiments,... |
899 | "Statistics"
The final wave, which mainly saw the refinement and expansion of earlier developments, emerged from the collaborative work between Egon Pearson and Jerzy Neyman in the 1930s. They introduced the concepts of " Type II " error, power of a test and confidence intervals. Jerzy Neyman in 1934 showed that strati... |
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