In this paper we present the relations between mathematics and mathematics education in Italy between the 12th and the 16th century. Since the subject is extremely wide, we will focus on two case-studies to point out some relevant aspects of this phenomenon: 1) Fibonacci’s studies (12th-13th century); 2) Abacus schools. More particularly, Fibonacci, probably the greatest European mathematician of the Middle Ages, made the calculations with Hindu-Arabic digits widely spread in Europe; Abacus schools were also based on the (...) teaching of the calculation with Hindu-Arabic digits. These case-studies are significant for understanding the connections between science, science education and the development of science within Western civilization. We think that the knowledge of such significant relations can be useful for the scholars who are nowadays engaged in mathematics education and in the research field of science-society relations. Finally, we attempt to outline the interaction between mathematics education and advanced mathematics in that period, focusing on the figure of Leonardo Pisano (c. 1170-c.1250), called Fibonacci, who played an influential role both in mathematics education and in advanced mathematics. (shrink)

A syntactically correct number-specification may fail to specify any number due to underspecification. For similar reasons, although each sentence in the Yablo sequence is syntactically perfect, none yields a statement with any truth-value. As is true of all members of the Liar family, the sentences in the Yablo sequence are so constructed that the specification of their truth-conditions is vacuous; the Yablo sentences fail to yield statements. The ‘revenge’ problem is easily defused. The solution to the semantical paradoxes offered here (...) revives the mediaeval cassatio approach, one that largely disappeared due to its incomprehending rejection by influential contemporary writers such as William Shyreswood and Thomas Bradwardine. The diagnosis readily extends to the set-theoretic paradoxes. (shrink)

This study looks at the relation between mathematical discovery and semiosis, focusing on the famous Fibonacci sequence. The serendipitous discovery of this sequence as the answer to a puzzle designed by Italian mathematician Leonardo Fibonacci to illustrate the efficiency of the decimal number system is one of those episodes in human history which show how serendipity, semiosis, and discovery are intertwined. As such, the sequence has significant implications for the study of creative semiosis, since it suggests that symbols (...) are hardly arbitrary products of human reason, but rather unconscious probes of reality. (shrink)

This study looks at the relation between mathematical discovery and semiosis, focusing on the famous Fibonacci sequence. The serendipitous discovery of this sequence as the answer to a puzzle designed by Italian mathematician Leonardo Fibonacci to illustrate the efficiency of the decimal number system is one of those episodes in human history which show how serendipity, semiosis, and discovery are intertwined. As such, the sequence has significant implications for the study of creative semiosis, since it suggests that symbols (...) are hardly arbitrary products of human reason, but rather unconscious probes of reality. (shrink)

ABSTRACT At the beginning of the 13th century Fibonacci described the rules for making mixtures of all kinds, using the Hindu-Arabic system of arithmetic. His work was repeated in the early printed books of arithmetic, many of which contained chapters on ‘alligation', as the subject became known. But the rules were expressed in words, so the subject often appeared difficult, and occasionally mysterious. Some clarity began to appear when Thomas Harriot introduced a modern form of algebraic notation around 1600, (...) and he was almost certainly the first to express the basic rule of alligation in algebraic terms. Thus a link was forged with the work on Diophantine problems that occupied mathematicians like John Pell and John Kersey in the 17th century. Joseph Fourier's work on mechanics led him to suggest a procedure for handling linear inequalities based on a combination of logic and algebra; he also introduced the idea of describing the set of feasible solutions geometrically. In 1898, inspired by Fourier’s work, Gyula Farkas proved a fundamental theorem about systems of linear inequalities. This topic eventually found many applications, and it became known as Linear Programming. The theorem of Farkas also plays a significant role in Game Theory. (shrink)

Fibonacci’s treatment of so-called recreational problems in his Liber Abbaci has been interpreted as an early episode both in the history of systems of linear equations, and in the history of negative numbers. However, these problems are also interesting in their own right. We discuss some of the conventions which seem to have governed these problems. By considering certain pairs of problems, where one problem is unsolvable and its partner is solvable, we show that Fibonacci went to a (...) significant effort to conform to these conventions. We also examine the methods which he could have used to construct his problems. (shrink)

Abstract«La divine proportion ne peut cependant pas être exprimée en nombres de façn exacte; néanmoins elle peut être exprimée de telle façon que, à travers un processus infini, nous pouvons en rapprocher de plus en plus et en délimitant le carré nous ne sommes jamais à plus d'une unité.» [Kepler, 1608].

In this article, we explore the extraction of recursive nested structure in the processing of binary sequences. Our aim was to determine whether humans learn the higher-order regularities of a highly simplified input where only sequential-order information marks the hierarchical structure. To this end, we implemented a sequence generated by the Fibonacci grammar in a serial reaction time task. This deterministic grammar generates aperiodic but self-similar sequences. The combination of these two properties allowed us to evaluate hierarchical learning while (...) controlling for the use of low-level strategies like detecting recurring patterns. The deterministic aspect of the grammar allowed us to predict precisely which points in the sequence should be subject to anticipation. Results showed that participants’ pattern of anticipation could not be accounted for by “flat” statistical learning processes and was consistent with them anticipating upcoming points based on hierarchical assumptions. We also found that participants were sensitive to the structure constituency, suggesting that they organized the signal into embedded constituents. We hypothesized that the participants built this structure by merging recursively deterministic transitions. (shrink)

The presented study introduces a new theoretical model of collapse for social, cultural, or political systems. Based on the current form of quantum anthropology conceptualized by Heidi Ann Russell, further development of this field is provided. The new theoretical model is called the spiral model of collapses, and is suggested to provide an analytical framework for collapses in social, cultural, and political systems. The main conclusions of this study are: 1) The individual crises in the period before a collapse of (...) social, cultural, and political systems form the trajectory of a conical helix similar to a vortex. 2) The occurrences of crises in the period before a collapse have the shape of the trajectory on the surface of the circular cone with a convex wall narrowing up to its peak. The shape of this cone is based on the Fibonacci sequence coiled into the three-dimensional space. 3) The constant circular movement along the trajectory of crises can occur in exceptional situations in the development of social, cultural, and political systems; however, such a state is always temporary. In such cases, the trajectory of the crisis does not follow the Fibonacci sequence, but the shape of a regular helix. Remaining on the trajectory of a regular helix in the long-term is highly improbable for social, cultural, and political systems. 4) The creation of new potentialities after the final collapse of a system is explained by the conception of topological inversion, when the heretofore embodied part of the energy-information field returns to the global, wave-particle energy-information potential. 5) The global, wave-particle energy-information potential is a source of energy-information for future embodiments in the sense of the future collapses of wave functions. (shrink)

Who were the five strangest mathematicians in history? What are the ten most interesting numbers? Jam-packed with thought-provoking mathematical mysteries, puzzles, and games, Wonders of Numbers will enchant even the most left-brained of readers. Hosted by the quirky Dr. Googol--who resides on a remote island and occasionally collaborates with Clifford Pickover--Wonders of Numbers focuses on creativity and the delight of discovery. Here is a potpourri of common and unusual number theory problems of varying difficulty--each presented in brief chapters that convey (...) to readers the essence of the problem rather than its extraneous history. Peppered throughout with illustrations that clarify the problems, Wonders of Numbers also includes fascinating "math gossip." How would we use numbers to communicate with aliens? Check out Chapter 30. Did you know that there is a Numerical Obsessive-Compulsive Disorder? You'll find it in Chapter 45. From the beautiful formula of India's most famous mathematician to the Leviathan number so big it makes a trillion look small, Dr. Googol's witty and straightforward approach to numbers will entice students, educators, and scientists alike to pick up a pencil and work a problem. (shrink)

We investigate interpolation properties of many-valued propositional logics related to continuous t-norms. In case of failure of interpolation, we characterize the minimal interpolating extensions of the languages. For finite-valued logics, we count the number of interpolating extensions by Fibonacci sequences.

Positional notation and related numerical manipulation techniques of Indian origin were introduced to Europe during the twelfth century through Arabic mediation and vividly described by Fibonacci as modus Indorum, the method of the Indians. This article aims to juxtapose Sanskrit and Latin texts to highlight the connections and differences between matrix and reflection in a complex cultural process of diffusion and assimilation. With reference to positional notation, this contribution examines a conceptual distinction between the graphical notion of position and (...) a logical-classificatory principle hitherto perhaps not fully recognized, although present in the texts of both traditions. The thesis of a consequent diagrammatic conception of positional notation in its origins, as a system of rules relating graphical position and order of magnitude, is also advanced. (shrink)

The twelfth century was a period of transmission and absorption of Arabic learning though it filtered outside of the Arabic world as early as the second half of the tenth century. In general, the lure of Spain began to act only in the twelfth century, and the active impulse toward the spread of Arabic mathematics came from beyond the Pyrenees and from men of diverse origins. The chief names are Adelard of Bath, Robert of Chester, Hermann of Carinthia and Gerard (...) of Cremona. In this time the Latin world became acquainted with the Hindu numerals, the Arabic Algebra and Euclid'sElements. However, not only Spain, but also the Norman kingdom of southern Italy and Sicily occupies a position of peculiar importance, though the works of the translators did not become very influential. There were made direct translations from Greek into Latin. One had to wait a century more to obtain a translation from Greek into Latin of the chief Archimedean scientific and mathematical treatises by William of Moerbeke. In the thirteenth century Fibonacci and Jordanus Nemorarius stand at the threshold of European mathematics. Not only was Fibonacci the first to explain Arabic arithmetic, but his works, especially his later ones, contain many original ideas. Jordanus continued the Greco-Roman tradition rather than the Greco-Arabic one, but he did so with much independence. To Nicole Oresme (fourteenth century) was due a broadened view of proportionality, a geometric proof to determine the summation of convergent infinite series and the proof, evidently the first in the history of mathematics, that the harmonic series is divergent. The Configuration Doctrine was treated by Merton College authors and by Oresme. In the fifteenth century theDe triangulis omnimodis of Regiomontan, a systematic account of the methods for solving triangles, marked the rebirth of trigonometry. (shrink)

With In Focus Sacred Geometry, learn the fascinating history behind this ancient tradition as well as how to decipher the geometrical symbols, formulas, and patterns based on mathematical patterns. People have searched for the meaning behind mathematical patterns for thousands of years. At its core, sacred geometry seeks to find the universal patterns that are found and applied to the objects surrounding us, such as the designs found in temples, churches, mosques, monuments, art, architecture, and nature. Learn the fundamental principles (...) behind: Interpreting the sacred symbolism and principles behind shapes Calculating numbers used in sacred geometry Applying the golden ratio and Fibonacci’s Sequence to objects Translating symbolic and geometrical letters and alphabets This accessible and beautifully designed guide to sacred geometry includes a frameable poster of the main sacred geometrical shapes and their unique, innate arrangements. The In Focus series applies a modern approach to teaching the classic body, mind, and spirit subjects. Authored by experts in their respective fields, these beginner's guides feature smartly designed visual material that clearly illustrates key topics within each subject. As a bonus, each book includes reference cards or a poster, held in an envelope inside the back cover, that give you a quick, go-to guide containing the most important information on the subject. (shrink)

This book introduces readers to the art of doing mathematical proofs. Proofs are the glue that holds mathematics together. They make connections between math concepts and show why things work the way they do. This book teaches the art of proofs using familiar high school concepts, such as numbers, polynomials, functions, and trigonometry. It retells math as a story, where the next chapter follows from the previous one. Readers will see how various mathematical concepts are tied, will see mathematics is (...) not a pile of formulas and facts, but has an orderly and beautiful edifice. The author begins with basic rules of logic, and then progresses through the topics already familiar to the students: numbers, inequalities, functions, polynomials, exponents, and trigonometric functions. There are also beautiful proofs for conic sections, sequences, and Fibonacci numbers. Each chapter has exercises for the reader. (shrink)

_A concise history of “just price,” from Aristotle to the present day_ The question of what constitutes a fair price has been at the center of market interactions since the time of Aristotle. Should a seller sell to the highest bidder, or is there some other standard, such as a morally defined price, to be applied? Charles R. Geisst traces the ways that philosophers, religious leaders, and economists have sought to answer that question, from antiquity through the modern era. Aristotle’s (...) thinking on usury influenced the idea of pricing well into the Renaissance. In his view, money was barren and should not be used to beget more money. As trade became more extensive, the strictures placed on pricing by Aristotelian thinking began to fall away, replaced by Roman and common-law conceptions of value and interest. Geisst’s book follows the evolution of that thought—influenced along the way by figures such as Copernicus, Fibonacci, Adam Smith, Marx, Cassel, and Keynes—and charts parallel developments in European and Islamic notions of fair pricing. Today, pricing is seen as an economic inevitability, dictated by the laws of supply and demand. But this has not always been the case. As Geisst argues, the idea of a just price was once a moral concept, long before it was an economic one. (shrink)

This paper critically assesses the claim by Gavin Menzies that Regiomontanus knew about the Chinese Remainder Theorem (CRT) through the Shù shū Jiǔ zhāng (SSJZ) written in 1247. Menzies uses this among many others arguments for his controversial theory that a large fleet of Chinese vessels visited Italy in the first half of the 15th century. We first refute that Regiomontanus used the method from the SSJZ. CRT problems appear in earlier European arithmetic and can be solved by the method (...) of the Sun Zi, as did Fibonacci. Secondly, we pro-vide evidence that remainder problems were treated within the European abbaco tradition independently of the CRT method. Finally, we discuss the role of recreational mathematics for the oral dissemination of sub-scientific knowledge. (shrink)

This seminal, multidisciplinary book shows how mathematics can be used to study the first principles of DNA. Most importantly, it enriches the so-called "Chargaff's grammar of biology" by providing the conceptual theoretical framework necessary to generalize Chargaff's rules. Starting with a simple example of DNA mathematical modeling where human nucleotide frequencies are associated to the Fibonacci sequence and the Golden Ratio through an optimization problem, its breakthrough is showing that the reverse, complement and reverse-complement operators defined over oligonucleotides induce (...) a natural set partition of DNA words of fixed-size. These equivalence classes, when organized into a matrix form, reveal hidden patterns within the DNA sequence of every living organism. Intended for undergraduate and graduate students both in mathematics and in life sciences, it is also a valuable resource for researchers interested in studying invariant genomic properties. (shrink)

This paper presents an experimental study investigating artificial grammar learning (AGL) in monolingual and bilingual children, with and without dyslexia, using an original methodology. We administered a serial reaction time (SRT) task, in the form of a modified Simon task, in which the sequence of the stimuli was manipulated according to the rules of a simple Lindenmayer grammar (more specifically, a Fibonacci grammar). By ensuring that the subjects focused on the correct response execution at the motor stage in presence (...) of congruent or incongruent visual stimuli, we could meet the two fundamental criteria for implicit learning: the absence of an intention to learn and the lack of awareness at the level of resulting knowledge. The participants of our studies were four groups of 10-year-old children: 30 Italian monolingual typically developing children, 30 bilingual typically developing children with Italian L2, 24 Italian monolingual dyslexic children and 24 bilingual dyslexic children with Italian L2. Participants were administered the modified Simon task developed according to the rules of the Fibonacci grammar and tested with respect to the implicit learning of three regularities: (i) a red is followed by a blue; (ii) a sequence of two blues is followed by a red and (iii) a blue can be followed either by a red or by a blue. Results clearly support the hypothesis that implicit learning took place, since participants of all groups became increasingly sensitive to the structure of the input, implicitly learning the sequence of the trials and thus appropriately predicting the occurrence of the relevant items, as manifested by faster reaction times in predictable trials. Moreover, group differences were found, with bilinguals being overall faster than monolinguals and dyslexics less accurate than controls. Finally, an advantage of bilingualism in dyslexia was found, with bilingual dyslexics performing consistently better than monolingual dyslexics and, in some conditions, at the level of the two control groups. These results are taken to suggest that bilingualism should be supported also among linguistically impaired individuals. (shrink)

This unusual book takes the form of a dialogue between a linguist and another scientist. This unusual book takes the form of a dialogue between a linguist and another scientist. The dialogue takes place over six days, with each day devoted to a particular topic--and the ensuing digressions. The role of the linguist is to present the fundamentals of the minimalist program of contemporary generative grammar. Although the linguist serves essentially as a voice for Noam Chomsky's ideas, he is not (...) intended to be a portrait of Chomsky himself. The other scientist functions as a kind of devil's advocate, making the arguments that linguists tend to face from those in the "harder" sciences. The author does far more than simply present the minimalist program. He conducts a running argument over the status of theoretical linguistics as a natural science. He raises the general issues of how we conceive words, phrases, and transformations, and what these processes tell us about the human mind. He also attempts to reconcile generative grammar with the punctuated equilibrium version of evolutionary theory. In his foreword, Massimo Piatelli-Palmarini says, "The vast number of readers who have been enthralled by Goedel, Escher, Bach may well like also this syntactic companion, a sort of 'Chomsky, Fibonacci, Bach.'". (shrink)

Introduction -- The fundamentals and their role in design -- Underneath it all -- Tiling the plane without gap or overlap -- Symmetry, patterns and fractals -- The stepping stone of Fibonacci and the harmony of a line divided -- Polyhedra, spheres and domes -- Structures and form in three dimensions -- Variations on a theme: modularity, closest packing and partitioning -- Structural analysis in the decorative arts, design and architecture -- A designer's framework.

For a given numeration system U, a set X of integers is said to be U-star-free if the language of the normalized U-representations of the elements in X is star-free. Adapting a result of McNaughton and Papert, we give a first-order logical characterization of these sets for various numeration systems including integer base systems and the Fibonacci system. For k-ary systems, the problem of the base dependence of this property is also studied. Finally, the case of k-adic systems is (...) developed. (shrink)

In lieu of an abstract, here is a brief excerpt of the content:Form and Discontent*Rosmarie Waldrop (bio)1. Composition as ExplanationIn the beginning there is Gertrude Stein, who says in “Composition as Explanation”: “Everything is the same except composition and as the composition is different and always going to be different everything is not the same” [520].I could also say, in the beginning is Aristotle: “the fable is simply this, the combination of the incidents” [1460].2. A Look AroundThe forms that have (...) been striking me as “wild” have tended to be Steinian in this sense of stressing the horizontal, the axis of composition. I’m most struck by 3 tendencies: one is literally spatial composition, e.g. Susan Howe’s work; another, an emphasis on discontinuity, leaps on the level of syntax, of logic, of grammar. Breaks that do not immediately fold back into a smooth unity and may occur in mid-sentence:I was on my way from Carthage, it was night. It is not wax I am scorching was dead about her with knots.—Norma Cole, MarsI hide behind a category by misbehaving.—Carla Harryman, Under the BridgeLoop conceived in a line, the spine with its regions, reasons. On another, sweaters hang by the wrist to dry. The list is sweet. You lie. The eye is met by the season.—Ron Silliman, ParadiseYou took my temperature which I had meant to save for a more difficult day.—Rosmarie Waldrop, The Reproduction of ProfilesWent out so I’d take the car and a whole system of banking and money is based on a hierarchy.—Leslie Scalapino, Considering How Exaggerated the Music IsROAST POTATOESRoast Potatoes for.—Gertrude Stein, “Tender Buttons”Or, a subtler example, Mei-mei Berssenbrugge’s almost unnoticeable shifts from one grammatical structure into another: [End Page 54]Never mind if he calls, the places you get through inwardness take time, and to drift down to the shore of the island, you know by the sand moving, even the coarse sand here It’s hard to say if you can even stand up, there....[13]The third tendency is mathematical. There is a lot of counting. Not stresses, not even syllables—I’m not considering that most unimaginative bunch, the new formalists—but words per line, sentences per paragraph. You all know about Ron Silliman’s love for the Fibonacci number series, or Lyn Hejinian’s My Life, which began with 37 sections of 37 sentences each. When she published a second edition at age 45, she added 8 sentences to each section plus 8 new sections.Jacques Roubaud: “No ideas but in numbers” [291].Then, in a somewhat more remote way, a number of people use ideas of permutation. For example, Jackson Mac Low’s 4 Trains, of which I will quote the first two stanzas:Train rule atom intricate nitrogen. Rule unusual leotard entropy. Atom train ostrich might. Intricate nitrogen train rule intricate casual atom train entropy. Nitrogen intricate train rule ostrich genealogy entropy nitrogen.Rule unusual leotard entropy. Unusual nitrogen unusual social unusual atom leotard. Leotard entropy ostrich train atom rule dies. Entropy nitrogen train rule ostrich pope yearly.[7]As Zukofsky puts it: “[Thus] poetry may be defined as an order of words that as movement and tone (rhythm and pitch) approaches in varying degrees the wordless art of music as a kind of mathematical limit” [27].All these examples draw attention to arrangement, composition. They all listen to language. I once wrote:I don’t even have thoughts, I say, I have methods that make language think, take over and me by the hand. Into sense or offense, syntax stretched across rules, relations of force, fluid the dip of the plumb line, the pull of eyes.... No beginnings. All unrepentant middle.[A Form / of Taking / It All 74]And none of these forms are “organic form.” None relies primarily on metaphor, though from the Romantics on poetry has been more or less identified with it.3. Frameworka. HistoricalLet me step back for a moment to the beginnings of the concept of organic form, what it reacted against.When Goethe started to make a splash writing free verse, around 1770, the dominant aesthetic... (shrink)

Summary In terms of control of composition, the fabrication of money was arguably the most demanding of all pre-Industrial Revolution metallurgical practices. The calculations involved in such control needed arithmetical computations involving repeated multiplications and divisions, not only of integers but also of mixed numbers. Such computations were possible using Roman numerals, but with some difficulties. The advantages gained by employing arithmetic using Indo-arabic numerals for alloying calculations would have been the same as for other types of commercial calculations. A (...) method of calculating the proportions of two coinage alloys of different compositions that were needed to produce a third alloy of specified intermediate composition, eventually known as the rule of alternate alligation, was however a specific requirement in the minting of money. This requirement made a particular demand on medieval mathematics, as a solution was not possible using arithmetic alone. Furthermore, the problem became indeterminate when more than two components were available for mixing. All the necessary methods of calculation were described by Fibonacci in his Liber abbaci. These methods are to be found in a sequence of practical arithmetics that stretch through the trattati d'abbaco of the Italian city states during the fourteenth and fifteenth centuries, to the first printed books of Continental authors such as Pacioli. Of the earliest sixteenth-century printed English books on practical arithmetic, The Grounde of Artes by Robert Recorde contained the first advance in the methodology of alternate alligation since Fibonacci. In the seventeenth century, formal proof of the rule of alternate alligation was given by Kersey in his practical arithmetics and in his treatise on algebra. The topic was also addressed by several authors of academic algebras. It was in this century that the problem of indeterminacy in the solution of alternate alligation involving the mixture of more than two components, was finally resolved. English mint accounts provide little direct information on the alloying calculation carried out, and nothing on the methods used. However, the continued description of such calculations in other mint documents, merchants notebooks, and in goldsmiths' handbooks spanning four centuries imply that the calculations described in practical arithmetics were used in practice. Monetary matters had high economic and political profiles during the whole of the time span covered in this study. The influence of such considerations pervades the history of the technology of minting and the mathematics on which it called, and is discussed in context. (shrink)

In terms of control of composition, the fabrication of money was arguably the most demanding of all pre-Industrial Revolution metallurgical practices. The calculations involved in such control needed arithmetical computations involving repeated multiplications and divisions, not only of integers but also of mixed numbers. Such computations were possible using Roman numerals, but with some difficulties. The advantages gained by employing arithmetic using Indo-arabic numerals for alloying calculations would have been the same as for other types of commercial calculations. A method (...) of calculating the proportions of two coinage alloys of different compositions that were needed to produce a third alloy of specified intermediate composition, eventually known as the rule of alternate alligation, was however, a specific requirement in the minting of money. This requirement made a particular demand on medieval mathematics, as a solution was not possible using arithmetic alone. Furthermore, the problem became indeterminate when more than two components were available for mixing. All the necessary methods of calculation were described by Fibonacci in his Liber abbaci. These methods are to be found in a sequence of practical arithmetics that stretch through the trattati d'abbaco of the Italian city states during the fourteenth and fifteenth centuries, to the first printed books of Continental authors such as Pacioli. Of the earliest sixteenth-century printed English books on practical arithmetic, The Grounde of Artes by Robert Recorde contained the first advance in the methodology of alternate alligation since Fibonacci. In the seventeenth century, formal proof of the rule of alternate alligation was given by Kersey in his practical arithmetics and his treatise on algebra. The topic was also addressed by several authors of academic algebras. It was in this century that the problem of indeterminacy in the solution of alternate alligation involving the mixture of more than two components, was finally resolved. English mint accounts provide little direct information on the alloying calculations carried out, and nothing on the methods used. However, the continued description of such calculations in other mint documents, merchants notebooks, and in goldsmiths' handbooks spanning four centuries imply that the calculations described in practical arithmetics were used in practice. Monetary matters had high economic and political profiles during the whole of the time span covered in this study. The influence of such considerations pervades the history of the technology of minting and the mathematics on which it called, and is discussed in context. (shrink)

This wide ranging discourse covers many disciplines of science and the human condition in an attempt to fully understand the manifestation of time. Time's Paradigm is, at its inception, a philosophical debate between the theories of 'Presentism' and 'The Block Model', beginning with a pronounced psychological analysis of 'free will' in an environment where the past and the future already exist. It lays the foundation for the argument that time is a cyclical, contained progression, rather than a meandering voyage into (...) infinity, bringing into question the validity of a commensurate 'Big Bang'. Following, the proposal widens to encompass physics. It tackles clock rates and time dilation, acausality and the nuisance of a universal clock, and demonstrates that conscious consideration creates the present moment - time's flow - separating the solid state past and future whose reality is devoid of space. Arguments relating to Quantum Physics theory, including the Uncertainty Principle and a Superposition of States, lend credibility to key areas involving cognitive awareness. It is posited that defined points in time and space prohibit progress in linear models for progression. Thus motile paradoxes can be resolved with the absence of infinities; temporal perception, it is concluded, being the result of uncertainty. Time's Paradigm takes the bold step of asking us to consider a tangible dimension of time, representing an intimate extension of our three, known spatial dimensions. Chaos theory is briefly introduced leading to the configuration of a fractal fourth dimension of time whose assumption demands only one direction of flow. Further, it asks whether our universe is expanding or contracting. It considers the simple physics of bodies contracting in a fourth dimension of time (UC), and how that marries comfortably with standard scientific models such as Special Relativity. The rate at which matter is contracting in the universe is illustrated in a reduction factor of 1.618... coinciding with Fibonacci's Ratio and countering Time Dilation. Lastly the more complex aspects of relativistic velocities are tackled together with the conundrum of Zero Velocity and The Speed of Light being attributes of the same event in Cyclical Space-Time, and ultimately, the prospect of superluminal velocities by interaction with parallel time-zones in a multi-layered block universe. (shrink)

Eine Reihe von mittelalterlichen Schriften zur Landmessung (vom 9. islamischen Jahrhundert bis zu Fibonacci und Pacioli) enthält eine besondere Art von „algebraischen” Aufgaben. Darin werden z.B. die Summe der Fläche und einer oder alle vier Seiten eines Quadrates beschrieben und nach der Seite gefragt. Es zeigt sich erstens, daß dieser Aufgabentyp mindestens seit dem frühesten 2. vorchristlichen Jahrtausend von geometrischen Praktikern tradiert wurde, und zweitens, daß er die Entwicklung einer „Algebra” in der altbabylonischen Schreiberschule inspirierte. Der Aufsatz untersucht, in (...) welcher Weise die Überführung der „sub-wissenschaftlichen” Praktikertradition in einen systematischen Schulunterricht den mathematischen Inhalt und den Denkstil künftiger Praktiker präge. Im letzten Kapitel wird diskutiert, inwieweit es sinnvoll ist, innerhalb dieser Schulalgebra von „Schulen” zu sprechen. (shrink)

The “unknown heritage” is the name usually given to a problem type in whose archetype a father leaves to his first son 1 monetary unit and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\frac{1}{n}}$$\end{document} (n usually being 7 or 10) of what remains, to the second 2 units and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\frac{1}{n}}$$\end{document} of what remains, and so on. In the end, all sons get the same, and nothing remains. The earliest known (...) occurrence is in Fibonacci’s Liber abbaci, which also contains a number of much more sophisticated versions, together with a partial algebraic solution for one of these and rules for all which do not follow from his algebraic calculation. The next time the problem turns up is in Planudes’s late thirteenth century Calculus according to the Indians, Called the Great. After that the simple problem type turns up regularly in Provençal, Italian and Byzantine sources. It seems never to appear in Arabic or Indian writings, although two Arabic texts (one from c. 1190) contain more regular problems where the number of shares is given; they are clearly derived from the type known from European and Byzantine works, not its source. The sophisticated versions turn up again in Barthélemy de Romans’ Compendy de la praticque des nombres (c. 1467) and, apparently inspired from there, in the appendix to Nicolas Chuquet’s Triparty (1484). Apart from a single trace in Cardano’s Practica arithmetice et mensurandi singularis, the sophisticated versions never surface again, but the simple version spreads for a while to German practical arithmetic and, more persistently, to French polite recreational mathematics. Close examination of the texts shows that Barthélemy cannot have drawn his familiarity with the sophisticated rules from Fibonacci. It also suggests that the simple version is originally either a classical, strictly Greek or Hellenistic, or a medieval Byzantine invention; and that the sophisticated versions must have been developed before Fibonacci within an environment (located in Byzantium, Provence, or possibly in Sicily?) of which all direct traces has been lost, but whose mathematical level must have been quite advanced. (shrink)

While for the majority of physicists the problem of the deciphering of the brain code, the intelligence code, is a matter for future generations, the author boldly and forcefully disagrees. Breaking with the dogma of classical logic he develops in the form of the conversion postulate a concrete working hypothesis for the actual thought mechanism. The reader is invited on a fascinating mathematical journey to the very edges of modern scientific knowledge. From lepton and quark to mind, from cognition to (...) a logic analogue of the Schrodinger equation, from Fibonacci numbers to logic quantum numbers, from imaginary logic to a quantum computer, from coding theory to atomic physics - the breadth and scope of this work is overwhelming. Combining quantum physics, fundamental logic and coding theory this unique work sets the stage for future physics and is bound to titillate and challenge the imagination of physicists, biophysicists and computer designers. Growing from the author's matrix operator formalization of logic, this work pursues a synthesis of physics and logic methods, leading to the development of the concept of infophysics. The experimental verification of the proposed quantum hypothesis of the brain is presently in preparation in cooperation with the Cavendish Laboratory, Cambridge, UK, and, if proved positive, would have major theoretical implications. Even more significant should be the practical applications in such fields as molecular electronics and computer science, biophysics and neuroscience, medicine and education. The new possiblities that could be opened up by quantum level computing could be truly revolutionary. The book aims at researchers and engineers in technical sciences as well as in biophysics and biosciences in general. It should have great appeal for physicists, mathematicians, logicians and for philosophers with a mathematical bent. (shrink)