Automatic Processing of Historical Japanese Mathematics (Wasan) Documents

“Wasan” is the collective name given to a set of mathematical texts written in Japan in the Edo period (1603–1867). These documents represent a unique type of mathematics and amalgamate the mathematical knowledge of a time and place where major advances where reached. Due to these facts, Wasan documents are considered to be of great historical and cultural significance. This paper presents a fully automatic algorithmic process to first detect the kanji characters in Wasan documents and subsequently classify them using deep learning networks. We pay special attention to the results concerning one particular kanji character, the "ima" kanji, as it is of special importance for the interpretation of Wasan documents. As our database is made up of manual scans of real historical documents, it presents scanning artifacts in the form of image noise and page misalignment. First, we use two preprocessing steps to ameliorate these artifacts. Then we use three different blob detector algorithms to determine what parts of each image belong to kanji Characters. Finally, we use five deep learning networks to classify the detected kanji. All the steps of the pipeline are thoroughly evaluated, and several options are compared for the kanji detection and classification steps. As ancient kanji database are rare and often include relatively few images, we explore the possibility of using modern kanji databases for kanji classification.Experiments are run on a dataset containing 100 Wasan book pages. We compare the performance of three blob detector algorithms for kanji detection obtaining 79.60% success rate with 7.88% false positive detections. Furthermore, we study the performance of five well-known deep learning networks and obtain 99.75% classification accuracy for modern kanji and 90.4% for classical kanji. Finally, our full pipeline obtains 95% correct detection and classification of the "ima" kanji with 3% False positives.

Diagnostics of the ability to understand mathematical texts by school graduates

The ability to understand the text plays a leading role among the basic educational competencies of students. In this paper the technology for diagnosing the ability to understand mathematical texts by school graduates is presented. The discussed technology is built in accordance with the requirements of ontological diagnostics, which makes the diagnostic result unambiguously interpreted, that is, the tools used in this diagnosis meet the validity requirement. Ontological diagnostics is based on the method of “dialectical deduction” (Hegel’s method), which assumes the choice of an initial predicate for its sequential morphologization with a specific content. In the work, the concept of ontological diagnostics is being clarified as a methodological basis for constructing a conceptual and technological scheme for diagnosing the ability to understand mathematical texts. The construction of a system of postulates that provide the formation of a conceptual and technological scheme of the considered diagnostics is discussed. On the basis of this study, a list of abilities was obtained that provide an adequate understanding of mathematical texts by respondents. As a result, the proposed diagnostics makes it possible to reveal the presence or absence of the certain abilities to understand mathematical texts from this list: the ability to choose a basic predicate corresponding to the proposed subject of thought, i. e. ego; the ability to construct complex predicates by the addition and refinement method based on the selected basic predicates; ability to construct the content of the predicate; the ability to read the content of the predicate and the ability to compare the content of the predicate with ego for their identity. A brief description of the conceptual-technological scheme of the regarded diagnostics is given, and a specific example of the constructed technology application is considered for diagnosing the ability to understand mathematical texts by school graduates.

Reading Mathematical Texts as a Problem-Solving Activity: The Case of the Principle of Mathematical Induction

Reading mathematical texts is closely related to the effort of the reader to understand its content; therefore, it is reasonable to consider such reading as a problem-solving activity. In this paper, the Principle of Mathematical Induction was given to secondary education students, and their effort to comprehend the text was examined in order to identify whether significant elements of problem solving are involved. The findings give evidence that while negotiating the content of the text, the students went through Polya’s four phases of problem solving. Moreover, this approach of reading the Principle of Mathematical Induction in the sense of a problem that must be solved seems a promising idea for the conceptual understanding of the notion of mathematical induction.

On the correctness of problem solving in ancient mathematical procedure texts

It has been argued in relation to Old Babylonian mathematical procedure texts that their validity or correctness is self-evident. One “sees” that the procedure is correct without it having, or being accompanied by, any explicit arguments for the correctness of the procedure. Even when agreeing with this view, one might still ask about how is the correctness of a procedure articulated? In this work, we present an articulation of the correctness of ancient Egyptian and Old Babylonian mathematical procedure texts – mathematical texts presenting the solution of problems. We endeavor to make explicit and explain how and why the procedures are reliable over and above the fact that their correctness is intuitive.

The Isabelle/Naproche Natural Language Proof Assistant

Abstract"Image missing" is an emerging natural proof assistant that accepts input in the controlled natural language ForTheL. "Image missing" is included in the current version of the Isabelle/PIDE which allows comfortable editing and asynchronous proof-checking of ForTheL texts. The dialect of ForTheL can be typeset by "Image missing" into documents that approximate the language and appearance of ordinary mathematical texts.

Kemampuan pemahaman membaca teks dan komunikasi matematik sebagai landasan dalam pemecahan masalah

This article is a review of international research articles that specifically emphasize the discussion of the ability to read mathematical texts and communicate mathematics as a basis for problem solving. The detailed descriptions include the criteria for understanding in reading a mathematical text; the effectiveness of language-based programs in school mathematics on student understanding; the use of learning approaches and media in building conceptual understanding and communication in solving mathematical problems; and strategies to build mathematical communication.

Printing the Bespoke Book

In 1482, Erhard Ratdolt, a prominent German printer in Venice, issued the editio princeps of Euclid’s Elements. Ratdolt experimented with the new technology of printing to overcome the difficulty in arranging geometric diagrams alongside the text. This article examines the materials and techniques that Ratdolt used in his edition of Elements including his use of vellum, gold printing, and illumination for special copies as well as his use of woodcuts, movable type, and metal-cast diagrams. Significantly, the legacy of Ratdolt’s innovations continued almost one hundred years later in subsequent editions of Elements. In 1572, Camillo Francischini printed Federico Commandino’s Latin translation and commentary, and today, there are at least two surviving copies of this edition printed on blue paper. Both printers, Ratdolt and Francischini, used the printing press to produce unique and bespoke books using material and visual cues from luxury objects like illuminated manuscripts. These case studies of Euclid’s Elements brings together the fields of art history, history of the book, and the history of geometry, and analyzes the myriad ways that printers employed the printing press in the early modern period to elevate and modernize ancient, mathematical texts.

Dedekind on Continuity

In this chapter we present Richard Dedekind’s conception of continuity and his various approaches to continuous domains in a historical context. In addition to his seminal work on foundations of irrational numbers (Stetigkeit und irrationale Zahlen, 1872), we also include a discussion of more mathematical texts (both published and unpublished) in which Dedekind also treats other continuous domains, such as Riemann surfaces, spaces, and multiply extended continuous domains. Dedekind’s reflections on these matters illustrate the wide range and general coherence of his thoughts. In particular, while Dedekind’s approach to mathematics can be characterized as being axiomatic, mapping-based, structuralist, and increasingly abstract, we argue that there is also a more general outlook underlying his methodology, which can be described as being, broadly understood, arithmetical.

Mathematical texts

Approximately a dozen mathematical papyri have survived from ancient Egypt. Based on their script (but also their stage of the Egyptian language) they fall into two groups—hieratic and demotic texts. These papyri constitute our primary source material to learn about ancient Egyptian mathematics. Because of the procedural style that they were written in, it is assumed that they were used in teaching junior scribes the mathematical techniques they would need for their job; however, the procedural format may also have constituted the way of collecting mathematical knowledge at the time. It is only if this format is taken into account in the (modern) analysis of Egyptian mathematical texts that their sophistication becomes visible, and a deeper understanding of Egyptian mathematics beyond rudimentary similarities to modern equivalents can therefore be achieved.