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Home , History of algebra

A BRIEF HISTORICAL SURVEY OF AND ITS MAXIMIZED STATISTICAL IMPACT ON ALGEBRAIC STATISTICS

Professor habil. Gheorghe SĂVOIU, PhD University of Pitești Professor Sandra MATEI, MA Secondary School no. 198, Bucharest

Abstract This paper is written by a statistician and a mathematician, both enthusiasts of classical algebra and its historical evolution. Beyond the listing of some important moments, and the highlighting of a number of schools relating to the thought specifi c to algebra, the aim of the paper remains the impact of possible, and especially actual, joint development, exemplifying the coexistence of algebra and statistics, in the content of a new interdisciplinary solution, located interstitially, and naturally and called algebraic statistics. Key words: algebra, statistics, history of algebraic thought, algebraic statis- tics.

INTRODUCTION Contemporary algebra is a scientifi c concept placed between the status of an independent discipline and that of a component science within the broader range of . Algebra is defi ned both independently, as a theory of operations concerning real (both positive and negative) or complex numbers, solving , focused on substituting numerical values by letters and identifying the general formula of particular numerical calculus, and dependently or structurally, as a branch of mathematics, in which case algebra deals with the study of operations independent of their numerical values, or with the generalizations of arithmetic operations (DEX, 2012). Algebra is an essential, indispensable structural component of modern mathematics derived from arithmetic, as a generalization or extension of the latter, and its choice fi eld of study is comprised of the rules of mathematical operations and relations, the concepts derived from them, from to equations, from algebraic structures to numbers with special properties. In its millennial approach, algebra permanently had recourse to numbers, initially contextualizing absolutely elementary mathematics. All calculations and concepts of absolutely elementary mathematics were, and still are, governed by the sole act of counting with one, capable of generating the fi rst positive . Exploiting algebraic signs is a diffi cult and slow process, although fi gures certainly appeared long before the alphabet, as the hexagesimal (Babylonian) systems and the (Chinese) systems were the fi rst

66 Romanian Statistical Review - Supplement nr. 2 / 2016 essential steps of algebra, over four or fi ve millennia old. More than 1,200 years ago, the managed to achieve the unifi cation of fi gures as we known them today, and the famous fi gure zero or nought (0) would travel along the Silk Road for centuries on end, becoming zephir with the and defi nitively entering mathematical calculus in Europe as late as one millennium ago. The sign of equality (=) appeared 500 years ago, along with the plus sign (+), whereas negative numbers appear around the year 600, used clearly and widely nearly one thousand years afterwards; all of them paved the way for an analytical around 1650. It was not until the twelfth century that people understood that the birth of numbers by counting, or adding 1, actually meant creating things (Thierry de Chartres), and after four or fi ve centuries the phenomenon was unstoppable, with the huge leap taken from simple counting 0 → (0 + 1) → [(0 + 1) + 1] → … to the vital exponential identity xy × xz = (y + z) that produced the exponential and logarithmic explosion of the scientifi c revolution in the seventeenth century (Berlinski, 2013). Algebra is a collective work, spanning over millennia, as the work of traders and common people, the work of bankers and accountants, of both mathematicians and non-mathematicians (statisticians or logicians). Exaggerating the arithmetic source and the numerologic impact,conceptualizes algebra as a science of the numbers in the entire universe (Rees, 2008), the major implication is that both the cosmos and the microcosm can be defi ned using a algebraic synthesis, made up of numbers as well, with the sole specifi cation that only six signifi cant numbers are concerned: a) two of them, N and ε, referring to the fundamental forces of physics and chemistry of the universe (N = 1036, i.e. the intensity of the electrical forces that keep the atoms together, and ε = 0.007, which shows how strong the nuclei are interconnected, and how atoms formed on the Earth); b) another two, Ω and λ, determine the size and structure of the vital universe, indicating the continuity of its existence (Ω, or a decreasing number, which describes the process of the ratio of the actual density and the critical density, initially considered almost equal to the unity, and having a current value of only 0.04, which thus estimates the amount of matter in galaxies, rarefi ed gas, including “dark” matter, and λ or the value of antigravity, recorded in a small number approaching zero); c) the last two, Q and D, characterizing the properties of space itself (Q = 1/100000, or the ratio of two fundamental energies that can liminally lead to either an inert, structureless universe, or a violent universe, in which nothing could survive, and D = 3, or the number of spatial dimensions of our world). The ability to synthesize outer world through numbers, and the talent to think through numbers and substitute realities for numbers, are the main elements that defi ned the fi rst major contributions of algebra to the history of modern science. Together with geometry, mathematical analysis, combinatorics and , algebra was one of the main branches of pure mathematics. Algebra remains one of the multiple areas of modern mathematics together with topology, , (using computers) and discrete mathematics.

Revista Română de Statistică - Supliment nr. 2 / 2016 67 FORERUNNERS OF MODERN ALGEBRA, AND ITS GEOGRAPHIC, TIME OR PRINCIPLE LANDMARKS

The universe of space-time in algebra was generated by the needs and concerns of civilizations and ordinary people; the position of this science was by no means contra mundum for millennia on end. Simple problems found equally (and seemingly) simple solutions within the set of natural numbers or positive integers, in an amiable manner (Berlinski, 2013), while the fi rst negative numbers and placed mathematicians in open confl ict with the physical world. It was in this way that algebra began to live with reality, identifying operations with an uncompensated end, unacceptable in the practice of some money lenders left with loans partially paid, or rounded the baker’s solutions for slicinga loaf of bread in n parts. The main space- time areas that developed algebra and brought it to the level of a science that is part and parcel of modern mathematics are summarized in Box. 1

Forerunners, and geographic-temporal landmarks of modern algebra Box 1 Classical algebra has existed for over four millennia, being born two millennia BC; the Babylonians were the fi rst to use algebraic formulas to solve problems, while the Greeks and the Chinese used for the same purposes only geometric methods. However, the Babylonians cannot be called the inventors of algebra, though they were certainly important precursors, because they did not have suffi cient time to refi ne their procedures, or the ability to gener- alize them and transmit the legacy, although four could solve no less than four unknown variables. In the same period, Egyptians and Chaldeans were able to solve equations of fi rst and second degree through verbal operations, which are attested by documents that were preserved (Berlin papyrus from 1300 BC.). In parallel, ancient Greeks were familiar with the application of identities, expressed in a geometric forms, proving able to graphically solve some equations of the third, and even the fourth degrees, through conical intersections. It is , author of Arithmetic, the fi rst mathematical book that was assumed, a character who lived in ancient around 200 AD, than can be identifi ed as fi rst precursor of modern algebra (he was rightly called the father of algebra). Although he did not formulate a sole method of solving algebraic equations, Diophantus used for the fi rst time special letters for operations and numbers, being the initiator of symbolic (algebraic) notation, as well as the father of indeterminate equations (or Diophantine equations). In the person of , the same Greek culture and science introduces the fi rst woman mathematician of antiquity: she wrote a commentary on Diophantus’s work, and even extended it, thus becoming the mother of modern algebra. The contributions of the Indians to algebra are remarkable: the Treatise authored by (598-660) was a practical application of mathematics, in which its author developed several major algebraic formulas, including the linear , the equation of the second degree and undetermined equations, for the fi rst time generating negative numbers, introducing them into the world of real results. Strictly referring to the conceptual delimitation, Brahmagupta was confi rmedly as the major predecessor of defi ning algebra as a science, which was done by Muhammad Al-Khwarizimi (780-850), or Al-Hor- ezmi (whose name was Latinized as Algorithmi). An essential precursor of algebraic think- ing in mathematics, he managed, in two of his books that remaind memorable in the

68 Romanian Statistical Review - Supplement nr. 2 / 2016 history of algebra, the fi rst published in 825 and entitled The Book of Addition and Subtrac- tion by the Indian Methods (Kitab al-jam’wal tafriq bi hisab al-Hindi), and the second one, published in 830, Treatise of the Calculation by Completing and Compensation (Al-Kitāb al-muḫtaṣar fī ḥisāb al-jabr wa-l-muqābala), to become the father of algebraic algorithms, through the solutions of generalized computing provided (as his name, slightly deformed in pronouncing, apparently led to the emergence of the algorithm as a mathematical concept), as well as the fi rst mathematician who used algebra as a distinctive term. Algebra combines, through Al-Horezmi, the al-Jabr process of eliminating the negative units, roots and squares from the equation, by means of artifi cial addition of identical values on each side of the nodal equality (e.g. equation x2 = 40x − 4x2 is reduced to 5x2 = 40x), at the same time also using the al-muqabala through the transfer / bringing quantities of the same type on the same side of the equation (e.g. x2 + 14 = x + 5 is reduced to x2 + 9 = x). Dixit Algoritmi, as an assessment of Al-Horezmi’s rigorous and methodical calculation, was a major time landmark, Latiniz- ing and algebraically preserving for eternity the author’s name in the algorithm, as an art to operate with apparently Arab fi gures, which in fact has Hindu origins, spreading the Hindu- Arabic throughout the Middle East and Europe; it thus virtually covered a unifi ed system of solving generalized problems (unlike Diophantus’s customized approach). Al-Khwarizimi’s algebraic methods, nay even algebraic solutions, would be successfully used in modern algebra, as well; it is from them that the contributions of modern math- ematicians were clearly deduced, e.g. Gottfried Leibniz’s and George Boole’s, founders of . One should not leave out, from this brief historical survey of the emergence of algebra, the contribution of the Chinese decimal numerical system, the so-called rod nu- merals, which used distinct symbols for numbers between 1 and 10, and other symbols for powers of 10, which was already used a few centuries BC, and long before the develop- ment of the Indian numerical system. This prolifi c decimal system allowed representing large numbers, and calculations could be made through the Chinese counting system (suan pan). Therefore, from an imaginary fresco of the School of preclassical algebra Diophantus, Hypathia, Brahmagupta, and Al-Khwarizimi cannot be absent, all under the logo of suan pan. (Stokes-Brown, 2011). Preclassical algebra, and even classical algebra, fi nally brought together a compilation of rules, along with demonstrations, in order to identify solutions to some linear and quadratic equations, initially via intuitive geometric arguments, and fi nally with the help of abstract notation associated with the practical subject. Modern algebra brought into debate again a number of personalities of Arab mathematics at the beginning of the second millennium. Thus, Al-Karaji provided, in his treatise Al-Fakhri, the fi rst demonstration that used the principle of mathematical induction to prove the bino- mial theorem, Ibn al-Haytham developed number theory, emphasizing the importance of perfect numbers and initiating connections between algebra and geometry, became one of the actual founders of algebraic geometry, anticipating Descartes’s memo- rable contribution and amazing effort, Nasir al-Din Tusi developed the concept of algebraic , and to their centuries-old efforts were added those of Mahavira Bhaskara II, Indian mathematician, Al-Karaji, Persian mathematician, and , Chinese mathematician, all having important contributions to solving cubic, quartic, quintic equations by means of numerical methods. The Renaissance exponentially amplifi ed the development of mathemat- ics, used and generalized symbolic algebraic notation. Scipione del Ferro and Niccolò Fontana Tartaglia found the solutions to cubic equations, Gerolamo Cardano, in his book Ars Magna, published in 1543, made public the solutions to the equations of the fourth degree, discovered by his student Lodovico Ferrari. Modern algebra gradually stan-

Revista Română de Statistică - Supliment nr. 2 / 2016 69 dardized the outer quantitative horizon. François Viète (1540-1603) made use of literal for- mulas to describe, in what became famous formulas, the relations between roots and coeffi - cients, John Neper (1550-1617) invented the concept of logarithm, John Wallis (1616-1703) invented the concept of a string of rational numbers, Isaac Newton (1642-1727) extended the binomial formula for rational exponents and provided a method for calculating the ap- proximate values of irrational roots, G. von Leibniz (1646-1716) set out the convergence criterion of numerical alternating series, Michel Rolle (1652-1719) provided the solution for the separation of the roots of algebraic equations, James Stirling (1696-1770) conceptualized the factorial (n!), Gabriel Cramer (1704-1752) invented the , Leonhard Euler (1707-1783) expanded them and introduced the orthogonal , Étienne Bézout (1730-1783) eliminated the unknown between two equations, Edward Waring (1734-1798) identifi ed the interpolation, Alexandre-Théophile Vandermonde (1735-1796) es- tablished the properties of determinants, Joseph-Louis Lagrange (1736-1813) summarized the theory of algebraic equations, Pierre-Simon Laplace (1749-1827) created the rule of developing determinants in keeping with minors of various orders, Carl Friedrich Gauss (1777-1855) formulated the fundamental theorem of algebra and provided the representation of complex numbers in the , William Rowan Hamilton (1805-1865) developed non- and the abstract theory of operations, De Morgan (1806-1871) gener- ated the formal logic of operations, Kummer (1810-1893) created the ideal numbers, and Clifford (1845-1879) the dual numbers, Cauchy (1789-1857) and van der Waerden (1903- 1996) described the rules of calculation, etc. (MathWiki, Algebra, 2015). The year 1830 is the landmark of the complete maturation of algebra: Évariste Galois’s personality (1811-1832) and his theory marked the clear delineation of necessary and suffi cient condi- tions for an equation to be solved by resorting to radical (Năstăsescu, Niță, 1979). Modern algebra after Galois seems seemed to be fully delineated, as communication, hierarchies and geographical and time landmarks were, at fi rst sight, relatively well concluded (confl icts of precedence like the Kowa Seki – Leibniz on the paternity of the determinants in solving systems of linear equations lose their reason in the algebraic world heritage)… Sources: MathWiki, Algebra, (2015) available at: http://ro.math.wikia.com/wiki/Algebră (Stokes-Brown, 2011) and (Năstăsescu, Niță, 1979).

The main landmarks of modern algebra are almost entirely due to René Descartes (1596-1650), who abstracted and generalized algebraic calculation into its modern signifi cance, offering algebra and its devotees the satisfaction of limiting the number of positive roots for an equation. In 1637 René Descartes introduced modern algebraic notation in his published paper titled La Géométrie, which represented the advent of . The principles of algebra were captured in the most elegant manner by Descartes in his famous Discours de la méthode, as early as 1637. The four principle axioms, which are apparently purely personal, as they were drafted by Descartes, seem to meet the requirements of both logic and geometry, but especially of algebra, once we place them in the profound mathematician’s contra mundum orientation, which: 1. will accept as true only that which cannot be questioned, i.e. algebraic proof or demonstration, validated algebraically, is prioritized over the potentially contradictory demonstrability.

70 Romanian Statistical Review - Supplement nr. 2 / 2016 2. will divide every issue in as many parts as will be needed to resolve it correctly, i.e. algebraic decomposition is done according to necessity dictated by the accuracy of the fi nal aggregate solutions. 3. will present its ideas from the simplest to the most complicated ones, i.e. the algebraic solving approach follows an ascending order, from simple to complex, from the singular to the general, from the particular case to the rule (formula, theorem, theory). 4. will list all the concepts, so that nothing pertinent is omitted, i.e. the completeness of conceptual assumptions ensures the generality of the approach of algebraic demonstration (Descartes, 1637). Morality, addressed as compliance to laws (algebra theory) and cultivation of reason and rationality in application of algebra, along with thought and meditation as an existential approach, not only philosophical, but also algebraic (cogito, ergo sum) are the major conceptual of Cartesian principle-based orientation. The history of algebra in Romania is much more recent, more didactic, and especially subject to adverse temporality and impermanence, as well as repressive, nay even unprincipled territoriality. The fi rst evidence of the use of algebra, and the concept itself, occurred in the 1729 volume Satire authored by Antioh Cantemir, the son of Prince Dimitrie Cantemir: Antioh Cantemir allegedly translated a book on Newton into Russian, and even wrote an algebra book, again in Russian, which was left in manuscript form (Marcelle, 1938):

Satire I (excerpt) “We don’t have any need for ’s science, To divide the land into portions and lots, And can go without algebra to know how many farthings go in a leu. Sylvan loves only that knowledge which Multiplies the income, as th’expense decreases; If the money purse doesn’t get any bigger, then there is no use in all that, But only foolishness to society…”

Most of the opinions published in this country claim that the fi rst algebra textbook would have been authored by Gheorghe Asachi, who wrote, edited and pu- blished several didactic books in the mathematical register (Babără, 2005): Elements of Mathematics, Part I, Arithmetic (1836), Part II, Algebra (1837), Part III, Basic Geometry (1838), Elements of Mathematics (1843), Elements of Arithmetic (1848). In 1870 the translation of Cursu elementaru de algebră (A Coursebook in ) appears in Romania, authored by Franz Mocnik, and in 1900 B. Ionescu published Tratatul de algebră elementară pentru școalele secundare după noua programă (An elementary Algebra Treatise for Secondary Schools in keeping with the New Curriculum), and Al. Manicatide published his Manualul de algebră elementară (A Handbook of Elementary Algebra), as an alternative manual. In both its substance and its topics, Al. Manicatide’s textbook did not signifi cantly differ from that of Franz Mocnik, and it kept being reprinted until 1941, when N. Abramescu’s

Revista Română de Statistică - Supliment nr. 2 / 2016 71 version appeared, to be replaced, in 1950, by another algebra textbook, translated from Russian and unsigned, where the single major point of novelty was the intro- duction of combinatorial analysis. The symptoms of the paradigm of algebra in the methodology of teaching mathematics in the Soviet style were mainly that algebra lost its connection with the essence of classical and modern algebra by: its a-historical character (removing algebra from its historical context), discipline-wise confi nement, excessive abstraction and formalization, mechanization of the process of solving al- gebra problems, according to the criterion More and faster, virtually identifi ed with better in algebra (Diaconu, 2006). The great Romanian mathematicians and acade- mics were signifi cantly involved in supporting teaching mathematics in secondary and high-school education, though many of them, due to their inherent specialization, only published books for students or textbooks. Thus, a particular example is that of Traian Lalescu (1882-1929), who pu- blished valuable didactic books, among which, in 1924, Calculul algebric (Algebraic Calculation), or Grigore Moisil (1906-1973), in his 1954 book Introducere în algebră (Introduction to Algebra), etc. Gheorghe Ţiţeica (1873-1939), another exemplary ma- thematician, founded, together with Dimitrie Pompeiu (1873-1954), the Mathematica journal, and participated, with Ion Ionescu, Andrei Ioachimescu and Vasile Cristescu, to the founding (in 1895) and development of the prestigious Romanian journal Gazeta matematică (Mathematical Gazette). The fi rst more realistic textbook in ana- lysis, and therefore in modern algebra, in the postwar period, was published as late as 1968, disclosing enough algebra under the title Elemente de analiză matematică si mecanică (Elements of Mathematical Analysis and Mechanics), by academician Caius Iacob (1912-1992), renamed Elemente de analiză matematică – pentru anul IV de liceu (Elements of Mathematical Analysis, for the Fourth Year of High School) in 1974, a book that alaso focused on algebra and, for the fi rst time, provided a relative synchronisation with the pedagogical approach of European mathematics, and confi r- med a split between mathematics and physical sciences in Romania. The structural, impactive approach attempted by Gheorghe Asachi’s algebra can still be detected, however, after a century and a half, in 1989, when they taught algebra together with arithmetic in the same manual for the sixth grade of the middle school. Contemporary algebra has become increasingly extensive, and brings toge- ther: i) elementary / classic algebra, where the presentation of the properties of real numbers resorts to symbols, identifying differently constants and vari- ables, while emphasizing the study of mathematical expressions and equa- tions; ii) abstract / modern algebra, which studies algebraic structures (groups, rings, bodies); iii) , devoted to vector spaces and matrixes; iv) , analyzing properties that are common to all algebraic structures); v) algebraic number theory, which studies the properties of numbers and applies results from number theory;

72 Romanian Statistical Review - Supplement nr. 2 / 2016 vi) algebraic geometry or applications of algebra in geometry. vii) combinatorial algebra, bringing together the methods of abstract alge- bra, applied to matters of combinatorial analysis, etc.; viii) algebra of logic, or linguistic algebra, where the three basic operations are and, or and non, which was practically invented by George Boole (1815-1864), etc. The new technologies, computers and specialized computer software packages have amplifi ed, in modern algebra, too, its involvement in research, and especially simpli- fi ed its methods and reduced its stages. As a natural consequence, new areas of mathema- tics and algebra were constantly developed; some relatively handy examples are the theory of calculability of Alan Turing, the complexity theory, the information theory, initiated by Claude Shannon, the theory of signal processing, mining and optimization, etc. The incredibly high speed of data processing and the performance resulting from time predictions by computers have enabled a new approach to problems of algebra otherwise restricted by their huge amount of data, which used to be impossible to solve promptly, and yet are today but natural in point of real-time calculation, and thus new algebraic areas of study were generated, such as numerical analysis and symbolic computation…

CONTEMPORARY COEXISTENCE AND ASSOCIATION OF ALGEBRA AND STATISTICS Based on the process of acquiring information on the principle of reducing information that are useful with regard to theri share, in the context of today’s infor- mation explosion, and also on the need for inter- and transdisciplinary knowledge, the authors have identifi ed at least ten arguments for a new cohabitation or coexistence of statistics and algebra, more complex and adapted to the needs of modern reality (be they economic, or social, educational, cultural, etc.): I. The specifi c , simultaneously algebraic and statistical (discrete, continuous, etc.), including variables and characteristic variation; II. Similar sets, groups and associations; III. Coincidence of some of the conditions posed by Yule and Kendall for average indicators and their simultaneously algebraic and statistical selection in descriptive and algebraic analyses (algorithm of quick and easy calculation, existence of a formula of algebraic calculus and the condition for it to lend itself to algebraic calculations, and an increasingly clear and accurate algebraic defi nition); IV. The expanding universe of Cartesian graphs and algebraic and statistical charts; V. Dispersion and the rule of adding dispersions as a starting point in dispersion or variance and in purely algebraic analyses of determination; VI. The quick and easy transfer of classical variables into binary variables, and resorting to the mean value and the (Boolean) dispersion of binary variables; VII. Regression, correlation and association of variables (which is

Revista Română de Statistică - Supliment nr. 2 / 2016 73 simultaneous algebraic and statistical); VIII. Extrapolation in the time series and the typology of classical algebraic progressions; IX. Non-random surveys (e.g. quotas, layered, etc.) and random surveys, compared to algebraic calculation algorithms; X. Ranking methods and algebraic sets of associated numbers, etc.

Some recent trends in the curriculum of US universities identify an accelerating trend of preference for algebra and statistics at the same time. The tendency itself is called “Algebra for all” & “Statistics for all” (Zalsman Usiskin, 2015) and get closer, between them and to the center of the curriculum, statistics and algebra, in accordance with the choice of the students and the pressures of reality caused by the increased volume of data required to make decisions, to understand, diagnose, teach, test, validate, evaluate, compete, create, innovate, build, substitute, etc. The relationships between statistics, mathematics (underlying algebra) and a lot of other subjects in the academic curriculum can be synthetically described, or at least suggested, through a graph, assembled or aggregated, of the main fi elds of academic education (Figure 1)

Association and and focus on statistics and algebraic calculus, as demanded by students in American universities Fig. 1

Sources: (Zalsman Usiskin, 1995; 2015 A growing number of US students chose and passed the examination of statistics

74 Romanian Statistical Review - Supplement nr. 2 / 2016 and algebraic calculus in a growing range of universities all over the academic education in the US, which requires the presence of the discipline of statistics in K12 – curriculum (the major cause that can be deduced is the Big Data phenomenon). The statistics (Zalsman Usiskin, 2015) show that between 1999 and 2013 the number of students who have applied for statistics and algebraic calculation in their personal curriculum, expressed in thousands, rose, in the US, from 25 to 170 (statistics), and from 30 to 104 (algebra or algebraic calculus). The premisses that could be identifi ed for both subject areas remain the same: a) modern application in almost all fi elds of human activity dominated by the Big Data phenomenon; b) the fundamental ideas of statistics, including variability, disorder, the laws of probability, etc., are available to almost all students and graduates; c) the need for every high-school student, undergraduate and college graduate to be able to use sound statistical reasoning focused on rigorous algebraic calculations in order to intelligently cope with the requirements to ensure a healthy, satisfi ed and productive life. Hence the implicite coexistence of statistics and algebra, defi ned by reciprocity or, in a concrete manner, by inter-, trans- and cross-disciplinarity, and practically, through the multidisciplinary approach. The consequences are logical and immediate, and they have already been put into practice. Research in statistics has an impact on algebra and mathematical analysis, whi- ch is as notable as the innovations due to the probability theory, while algebra and analy- sis conduce to improved statistical methods (Pons, 2012). The unifying multidisciplina- rity of algebra and statistics lead to the emergence of a new science: algebraic statistics. It already exists, and consists in using algebra for advanced statistics. Algebra has been, and remains useful for experimental design, parameter estimation and hypothesis testing. Traditionally, algebraic statistics was associated with experimental design and multivariate analysis (especially that of time series or chronological series). In recent years, the sense of the term algebraic statistics was sometimes limited, and at other times expanded, being also used to defi ne the use of algebraic geometry and commutative algebra in statistics. Moreover, there had been some traditions in using algebraic statistics, going back a fairly logn way in time, some nearly century-old. In the past, statisticians used algebra to amke advanced statistics in research, e.g. Sir R.A. Fisher, who used Abelian groups for experimental design, and Karl Pearson, who used polynomial algebra to study Gaussian models… Two main directions defi ne the major stimuli or motivations in algebraic statistics, which emerged and developed in the last two decades. The fi rst one was described in a book by Pistone and Wynn (1996), using ideas for databases specifi c to the range of issues of experimental design. This book was only the impulse that generated a whole line of research, which eventually led to the publication of a book on algebraic statistics authored by Pistone, Riccomagno and Wynn (2001), addressing all the issues of experimental design. The other direction was traced by the book of Diaconis and Sturmfels (1998) on algebraic methods of distribution of discrete expo- nential families. This work infl uenced statistical work and led to the volume edited by Pachter and Sturmfels (2005), which focused on the relationships between algebraic statistics and computational biology… Consequently, algebraic statistics outlined a distinct science, or rather a new discipline in the last 15-20 years.

Revista Română de Statistică - Supliment nr. 2 / 2016 75 Recent advances in algebraic statistics have extended its fi eld beyond the fi eld of the ‘traditional Algebraic Statistics’, which focused on experimental design, in fi ve areas:

The fi ve ‘traditional’ areas of the Algebraic Statistics Table no. 1 1. The study of Gaussian models has become an important part of algebraic statistics. 2. Its aspects of computing (i.e. computational aspects) are based on numerical algebraic geo- metry and put forth statistical questions crucial to the validity of statistical inference. 3. In the theory of learning algebraic statistics it is permitted to study asymptotic statistics for models with hidden variables. 4. Establishing identifi able statistical models becomes an integral part of algebraic statistics. 5. There is a strong emphasis on applications of these methods to phylogenetics, learning and teaching machines, biochemical reaction networks, social sciences, economics and ecological inference. Source: The Journal of Algebraic Statistics, the fi rst publication of the new science, already appeared in the last fi ve years, available at: http://library.iit.edu/jalgstat/

The two classical disciplines having seen one of the longest histories in the his- tory of modern science come together, in an apparently unexpected manner, to form one of the newest multidisciplinary disciplines in the recent taxonomy of the 21st century.

INSTEAD OF CONCLUSIONS, OR A PARAPHRASE FOR DEFINING THE BEHAVIOUR OF ALGEBRAIC STATISTICS Algebra was long known only by devotees, or people dedicated to science (i.e. scholars), and thus it became a by-word: when talking about something diffi cult and unknown to anyone, one would use the idiom this is algebra to him. Analogously, the scientifi c discipline of statistics was nuanced as imminently and necessarily moral, i.e. it was identifi ed with a method or a combination of methods that show what must be seen and what should be concealed from a moral, social, economic, political, etc. point of view. It now becomes clear that algebraic statistics has already become a new morality of scientists faced to the phenomenon of information explosion, called simply big data.

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