Paradigm: The Evolution of Scientific Thought and Numbers
“God made the integers, everything else is man’s work”
Introduction
Although there is more than one definition of science, these definitions may vary according to the countries and the history of science (Şen, 2016:3–6). As a matter of fact, science is not only an activity where we can obtain information. It also has an aspect that requires action (Şenel, 2012:13). In this context, science can be seen as a way of seeking the truth (Şen, 2016:4). On the other hand, knowledge is defined as “The fact of knowing or recognizing something, person, etc.; acquaintance; familiarity gained through experience.” (oed.com). In addition, knowledge is comprehensive and reduces uncertainty (plato.stanford.edu). Nevertheless, we can say that this definition may vary depending on some factors, just like the definition in science. Scientific knowledge, on the other hand, is considered independent of everyday knowledge, but we can say that it is a structure that differs according to the degree of abstraction it contains, the methods and techniques used, and the generalization of the results of scientific knowledge (Ramsperger, 1939). In addition to obtaining scientific knowledge, one of the important aspects of scientific knowledge is that it helps to use other scientific knowledge. Discovery is one of the important factors for scientific knowledge. Scientific discovery is the result of a successful scientific research (plato.stanford.edu). In other words, an effort is necessary for a scientific discovery and it is important that this effort is successful. Although there is an accumulation of scientific knowledge that emerges as a result of such actions of people, it can be concluded that scientific knowledge is a constantly developing and progressing structure rather than a static structure. The progress made can be considered as scientific progress, and this is constantly cumulative. The historian of science George Sarton emphasizes that the acquisition and progress of this knowledge is the only activity of man. We can say that scientific progress is accompanied by development. Although there is a belief that these developments should be named in a scientific world, the name “scientific revolution” has come to the fore. For example, in one of Copernicus’ works, it is referred to as the “Copernican Revolution”, in fact, what is mentioned is “a term originally applied to rotating wheels and including the revolution of celestial bodies” (plato.stanford.edu), but later different definitions specific to different fields were made. However, scientific revolutions also tend to persist.
Thomas Kuhn, one of the most influential philosophers of science of the twentieth century, comes to mind in relation to scientific revolutions and the structure of these revolutions because he wrote a very important work in the history of science on the structure of these revolutions, “The Structure of Scientific Revolutions” (plato.stanford.edu). This book describes the development of science differently from other books published before (Masterman, 1965, 67). In fact, the advisory committee of this book includes important figures in the history of science such as Niels Bohr and John Dewey (Kuhn, 1970). The reason why this important work has been so influential is that while initially there was a belief that the progress of science was cumulative, Kuhn (1970) argues that this situation has changed and that historians of science now believe that the paths of development may no longer be cumulative and that they can build a different structure on the existing science (Kuhn, 1970:6). Therefore, Kuhn, who deeply examined the structure of Scientific Revolutions, brought an important work in this field to the forefront. He emphasized that a scientific revolution is a very famous part of a scientific work (Kuhn, 1970:7). Although it is mostly based on examples of physical sciences, there is an interpretation that covers all sciences. While the most important features of scientific revolutions are that they should be in a standardized form and that they should be dependent and continuous from past knowledge, it can be said that this is not the case when we look at the revolutions that have been made, it can be said that it has opened a completely different field ( Kuhn, 1970:7). In other words, we can infer from this that there is more than one way of the progress of science. Some of these are that it has a revolutionary character. Therefore, when there is a revolutionary structure in a normally progressing scientific process, it is characterized as a scientific revolution. This can be clearly perceived as two headings. Kuhn divided this situation into two headings as normal science and revolutionary science (plato.stanford.edu). But of course, there is no easy transition between normal science and revolutionary science; sometimes it can take years, sometimes centuries. Therefore, this process of how a revolutionary science emerges was tried to be explained by Kuhn (1970) and he tried to define this process as Paradigm. In other words, there is actually a situation about how a revolutionary science is modeled and this can be described as a paradigm. On the other hand, the world of science is constantly exposed to such paradigms. Because science is in a constantly expanding and changing structure. Therefore, the way to express this process in a clear way (dictionary.cambridge.org) is through paradigm.
Paradigm Concept and Thomas Kuhn’s Paradigm Theory
The term paradigm is a Greek word meaning “model” or “example” (dictionary.cambridge.org). In the world of science, paradigms are the cornerstone of a particular field and guide scientists working in that field. A paradigm represents a collection of basic scientific theories, methodologies and beliefs accepted in a given period. In sociological terms, a paradigm refers to a set of scientific habits (Masterman, 1965:66).
The paradigm gained popularity especially in the mid-20th century with the work of the American philosopher of science Thomas Kuhn. Thomas Kuhn first introduced the paradigm in detail in his book “The Structure of Scientific Revolutions” published in 1962. Importantly, Kuhn argued that the paradigm existed before the theory was formed (Masterman, 1965: 66). This implies that there are certain conditions and appropriateness for a paradigm to emerge. Kuhn (1970) emphasizes that there is no need to have a set of rules for a paradigm. Although this situation brings some problems, it is thought that this situation can be overcome through definitions and nomenclature (Kuhn, 1970). In addition, a relationship can be established through modeling thanks to social support, which proves that paradigm is used as a word for paradigm. On the other hand, since these models are made by scientists, there is a flexibility in terms of the progress of science. The emergence of a paradigm depends on certain conditions and suitability and may not need any set of rules. In this case, replacing rules with paradigms in science can significantly improve the understanding and development of science (Kuhn, 1970).
The concept of paradigm and its stages, as outlined in Kuhn’s “The Structure of Scientific Revolutions” (Kuhn, 1996), has significantly influenced the philosophy of science. He defines a paradigm as a set of practices within a discipline that encompasses the questions, objects, methods and explanations commonly practiced by a particular scientific community (Walz and Adrian, 2008). He distinguishes between two meanings of paradigm: disciplinary matrices and instances (Shan, 2018). The concept of paradigm is crucial in understanding the progression of scientific knowledge and the dynamics of scientific revolutions. Kuhn’s model of scientific development consists of several stages. It starts with a pre-paradigm stage, characterized by a lack of consensus on the basic theories and methods within a scientific discipline (Chibeni & Moreira- Almeida, 2007). This phase is followed by the establishment of a dominant paradigm leading to the period of normal science, where scientists work within the established framework to solve puzzles and anomalies (Sterman, 1985). But as anomalies accumulate, a crisis arises that challenges the existing paradigm and leads to a revolutionary shift towards a new paradigm (Chibeni and Moreira-Almeida, 2007). The new paradigm then enters its period of normal science and the cycle continues. Kuhn’s work has been influential in various fields beyond the natural sciences. For example, it has influenced the study of paradigm shifts in religious education (Gearon, 2017), the development of paradigms in information science (Budd & Hill, 2013), and the understanding of paradigm shifts in strategic management (Solesvik, 2018). Kuhn’s paradigm theory has also been used to investigate the development of competing paradigms in different disciplines (Chen et al., 2002). In sum, the concept of paradigm and its stages as articulated in Kuhn’s “Structure of Scientific Revolutions” has had a profound impact on the philosophy of science and has been widely applied in various academic fields to understand the evolution of scientific scholarship.
According to Kuhn, the five basic stages of paradigm are as follows:
- Pre-Science Phase
- Normal Science
- 3- Crisis- Revolution
- New Normal Science
- New Depression
Kuhn’s definition of paradigm is a crucial tool for capturing the essence of change in science. This theory provides a framework for understanding how scientific development and scientific revolutions take place (Kuhn, 1970). On the other hand, the paradigm is in a continuous cycle. This cycle has a certain progression. You can see this cycle in table 1 below.
There can be more than one example of a scientific development, and Kuhn (1970) in his book mentions the scientific revolutions and processes of scientists such as Newton and Kepler. However, to these paradigm phases, we can consider numbers, which are the basic building blocks of Mathematics. Therefore, we can see how the development of these numbers takes shape according to the paradigm phases.
The history of numbers in mathematics can be understood through the lens of Thomas Kuhn’s paradigm phases, which consist of five basic stages: pre-scientific stage, normal science, crisis-revolution, new normal science and new crisis. Each of these stages can be associated with specific groups of numbers that reflect the evolution of mathematical understanding and representation. The pre-scientific stage can represent the early development of numerical systems and counting methods, with a focus on primitive number systems and basic counting methods. This stage is associated with the group of natural numbers, which forms the basic foundation of numerical representation. The next stage, normal science, can correspond to the period when a scientific community operates within an established paradigm. In the context of numbers, this stage is aligned with the inclusion of integers and rational numbers, satisfying the need to represent quantities. Natural numbers are excluded. This phase can reflect the fundamental understanding of arithmetic principles and the extension of numerical systems to adapt to new mathematical challenges and complexities. The crisis-evolution stage may represent a period of critical evaluation and transformative change in mathematical understanding. This phase may be associated with the introduction of irrational numbers and real numbers, signaling a fundamental shift in numerical representation to include quantities that cannot be fully expressed as fractions or whole numbers. In addition, the development of complex numbers and other advanced mathematical structures represents a revolutionary paradigm shift that reshapes the understanding of numbers and their applications in mathematics. The new normal science phase is characterized by advanced may represent the establishment of a new paradigm in mathematical understanding, reflecting the incorporation of mathematical structures and the refinement of numerical systems. This phase can be associated with the continued exploration and use of advanced number groups, such as transcendental numbers and hyperreal numbers, reflecting the ongoing development and complexity of numerical representation. Finally, the new crisis phase refers to a period of reassessment and innovation in mathematical understanding, reflecting the ongoing search for new mathematical challenges and the need to improve and extend numerical systems to address emerging complexities. In summary, Thomas Kuhn’s paradigm stages provide a comprehensive framework for understanding the historical development of numbers in mathematics and can reflect the progression from basic arithmetical principles to the incorporation of advanced mathematical structures at different stages of scientific progress.
Pre-Science Phase
This stage represents the early development of numerical systems and counting methods, with a focus on primitive number systems and basic counting methods. This stage is associated with the group of natural numbers (1, 2, 3, …), which form the basic foundation of numerical notation.
Mathematics is an important structure that has maintained its existence throughout human history and carries its continuity to the present day. However, mathematics dating back to the Paleolithic period was expressed in a very different way than today (Pejlare & Brating, 2019). In addition to people’s mental development, they may have also made progress in their language development. Thus, it can be said that they can express their thoughts in their minds even if they do not have a developed language structure (Mazur, 2017:5). As language developed, the act of experiencing life and sustaining it accordingly also developed. Since transferring different aspects of life to other people requires a skill, the language used may have been insufficient in this regard. Therefore, as mental development developed and people experienced more of the experiences of life, they may have felt the need to transfer this to the objects around them. In this sense, by making certain symbols in caves or tree hollows, they seemed to have completed the function that language could not fulfill. These symbols are thought to be about calculations and mathematics as well as providing information about daily life (Pejlare & Brating, 2019). In other words, people symbolized the mathematics they used and the purpose of these symbols was to be easily understandable (Mazur, 2017: 5). Therefore, the first expressions in mathematics were symbols that enabled understanding of each other.
The first known symbol in the history of mathematics is the Lebombo bone, estimated to be about 37,000 years old. Located on the border of South Africa, this bone is a small piece of the fibula of a baboon. It has 29 distinct notches on it. This bone can be considered a clue to the emergence of calculation in human history (Pejlare and Brating, 2019).
Given that the Lebombo bone has 29 notches, humans may have tried to predict when a month would be full. Thus, while the month-to-month calculation is the simplest possible early calculation, it seems likely that the first mathematicians were women, since women needed a lunar calendar for a menstrual cycle (Pejlare and Brating, 2019). Another of the first signs of the use of prehistoric mathematics is the Ishango bone found in the Democratic Republic of Congo.
This 20,000-year-old bone has three rows of asymmetrical notches. This bone structure is much more complex than the Lebombo bone, but there are different opinions.
There is evidence of another artifact used in early mathematics in Europe. This evidence is thought to be a tally stick, which is a wolf bone (Pejlare & Brating, 2019). Therefore, the first appearance of mathematics is the symbols used, and although the main purpose of its use varies, it plays an important role as the beginning of mathematics.
After the use of symbols, people developed a desire to write in order to meet their communication needs more effectively (Mazur, 2017: 9). The emergence of mathematical writing took place in the West and the East almost long before literature (Mazur, 2017: 12). When dealing with mathematics, people were able to develop various methods and approaches on how to use it. On the other hand, after the development of writing, mathematical number systems began to emerge. As the images below show, different civilizations used different symbols and systems.
The number systems we are talking about are systems created using the symbols of the regions in which they are located. The bases of these systems also differ. For example, the Babylonians used a 60-based counting system. It should be noted that not all numbers correspond to a symbol in this system (Beveridge, 2016: 19). The Egyptians, on the other hand, expressed numbers with independent symbols they encountered in daily life (Beveridge, 2016: 21).
Many civilizations such as the Hebrew, Greek, Roman, Chinese and Indo-Arabic developed their own unique number symbols and systems. Even in the recent past, some communities living in some polar regions continue to use their own numbers and systems (Artifexian, 2021). However we define numbers and systems, there must be a harmonious order between the meaning and what we perceive (Mazur, 2017: 45).
Although today’s numerals are derived from Hindu-Arabic numerals, their spread and educational use in Europe became evident in Johannes de Sacrobosco’s Algorismus of 1523 (Mazur, 2017: 89). In the image above (see Figure 4), the Algorismus book is illustrated with contemporary numerals. This image is a copy from The Tomash Library on the History of Computing. As a result, the meanings attributed to symbols formed numerals, and specific number systems were developed based on these numerals. The grouping of these systems gave rise to certain number groups.
The grouping of these systems revealed certain number groups. Natural numbers are at the beginning of these number groups. People knew and used numbers like 1, 2, 3, 4, … and then zero was added to this group. The main reasons for the addition of zero are that it is a number to be used instead of absence and to emphasize the difference between 27 and 207 so that people do not confuse the numbers. The first person to add zero to the number system was the Indian Mathematician Brahmagupta (Crilly, 2017: 23–24). Later, in communities such as Arab, Spanish and Italian, an order such as 0, 1, 2, 3, 4, … was formed (Stewart, 2017: 55). In this way, the set of natural numbers was formed and people continued their daily lives and scientific studies in this direction (encyclopedia.com).
Today, although the smallest natural number 0 is not included in this group, there are various opinions about whether the largest natural number exists (Nesin, 2019: 14). However, as accepted by societies, natural numbers are considered as 0,1,2,3, As a result, no matter what symbols and systems are used, civilizations have adopted natural numbers such as 0, 1, 2, 3, 4, to perform arithmetic calculations for years. However, natural numbers may have been insufficient to meet the needs of societies. This shortcoming could be due to the lack of numbers or symbols that could express the situation that merchants owed (Crilly, 2017: 40). In addition, the inability to subtract natural numbers supports this deficiency. In other words, if we want to subtract the number 9 from the number 6, we cannot find the result, or even if we express it as -3, the result we find is not a natural number (Nesin, 2017: 89).
Normal Science
This phase c o r r e s p o n d s to the period when a scientific community operates within an established paradigm. In the context of numbers, this phase is aligned with the inclusion of integers (…, -3, -2, -1, 0, 1, 1, 2, 3, …) and rational numbers, meeting the need to represent quantities outside the scope of natural numbers. This stage reflects the basic understanding of arithmetic principles and the extension of numerical systems to adapt to new mathematical challenges and complexities.
The shortcomings of natural numbers have led people to different searches. In fact, another name for natural numbers is positive integers (Stewart, 2019: 56). That is, positive integers can also include negative integers. It is possible to attribute the beginning of these searches to the Chinese. The Chinese thought of positive numbers of the same magnitude in response to the negative results found in the number sticks they used for operations (Stewart, 2019: 57).
As another example, people have used negative numbers to emphasize how cold it is where they are (Crilly, 2017: 41). In addition, Indian Mathematician Bhaskara found more than one answer to a problem and stated that negative numbers were included in these answers. However, over time, the idea that negative number results should not be taken into account has changed and negative numbers have been included in the group of integers (Stewart, 2019: 59, Crilly, 2017: 41).
Although integers include negative numbers, a problem arose about how to express the parts of a whole and how to use this knowledge (Crilly, 2017: 41). However, this situation has been fully integrated into the group of numbers over time. With the addition of negative numbers to whole numbers, although it is perceived that there are no more problems to meet the needs of people in daily life, new problems may have arisen. This is because people realized the inadequacies of the negative numbers they were using mostly when making measurements (encyclopedia.com). In this sense, the number obtained as a result of the ratio of two whole numbers to each other did not correspond to a whole number. Therefore, it could cause confusion in the world of mathematics and in daily life. For example, a merchant did not have much information about what would happen when he divided a 27-unit fabric in half ( Crilly, 2017:43). Therefore, a new group of numbers seemed to be necessary to solve this problem. And the set of numbers called rational numbers was formed. People were now able to show and use the ratio of any multiplicity to any other multiplicity. In fact, what we are talking about is a ratio, but it is a structure that was frequently used in Ancient Greece. Moreover, Euclid’s Elements contains expressions about ratios (Euclid, 2018:8). However, a problem started to emerge here, because in land uses, an expression that could not be calculated or whose result could not be made sense of started to emerge. Not only land uses but also people dealing with mathematics realized that there was a problem or a missing situation (Richeson, 2021:79).
Crisis- Revolution
This phase refers to a period of critical evaluation and transformative change in mathematical understanding. This phase may be associated with the introduction of irrational numbers and real numbers, which marks a fundamental shift in numerical representation to encompass quantities that cannot be fully expressed as fractions or whole numbers. In addition, the development of complex numbers and other advanced mathematical structures represents a revolutionary paradigm shift that reshapes the understanding of numbers and their applications in mathematics.
The Greeks had a slightly different understanding of numbers and concepts. They did not even think of number groups such as natural numbers and integers as numbers. In fact, according to Euclid, even the number 1 was not a number, but only a value. Therefore, they did not consider rational numbers as numbers either, but only as ‘the ratio of two numbers to each other’. In the figure below, the ratio of circles to pentagons or the ratio of stars to triangles should be a rational number, but again, they did not even consider the result of the ratios as numbers. To them, it was just a relationship between two numbers (Richeson, 2021: 81). So what we actually understood here was the idea that a number was indivisible. It was not thought of as a certain ratio, i.e. the division of three into four, such as 3:4, but as 3 out of 4 parts. Thus, they would definitely find a ratio in every two multiplicities they compared and they would operate accordingly (Richeson, 2021: 83).
Pythagoras and his students seemed to take this for granted. What was not taken into account, however, was the idea that a rational number had in fact disappeared. It is said that Hippasus, a student at the Pythagorean school, had derived a non-rational, i.e. disproportionate, number from the case where a rational number is equal to 2 (Beveridge, 2016: 24). While most of what is known about Pythagoras is doubtful information, the same is true for Hippasus (Richeson, 2021: 84). Therefore, it is very difficult to reach precise and clear information. However, let’s try to convey it as accepted by the scientific community. It is said that Hippasus found and used numbers that are disproportionate, that is, numbers that are not a certain ratio. The disproportion here causes us to call them non-rational numbers (Richeson, 2021: 86).
According to another view, based on the square root of the number 2, the square root 2 cannot be written as a proportional number considering that it is equal to a ratio such as a²/b² and that the result is an even number a and b. Because the divisors of two even numbers must have the number 2 (Beveridge, 2016: 24). As a result, with these statements, disproportional, that is, irrational numbers are obtained. Starting with this process, other mathematicians have studied irrational numbers and expanded the examples of irrational numbers. The contribution of this situation played a major role in volumes 10 and 13 of Euclid’s Elements (Richeson, 2021: 88). As a result of this expansion, the group of real numbers came to the fore.
The concept of real numbers evolved over time as mathematicians discovered new types of numbers and tried to define them rigorously. The earliest concept of real numbers was probably based on the idea of magnitude, which refers to the size or extent of a geometric object (mathshistory.st-andrews.ac.uk). The ancient Greeks, such as Pythagoras and Euclid, developed the concept of ratio, which compares two magnitudes of the same kind. The development of proportional numbers laid the foundation for real numbers. However, the modern understanding of real numbers emerged in the 17th century. During this period, mathematicians’ interest in geometric representations of numbers and the concept of limits led to a better understanding of the role of real numbers in mathematics. Scottish mathematician John Wallis developed an analytic expression of real numbers by studying decimal fractions and infinite series (Stillwell, 2010).
A turning point in the history of mathematics was the work of Tartaglia and Viète in converting fractions into decimals, which extended fractions and decimals to real numbers. This conversion made decimals easier to work with and advanced mathematical thinking (britannica.com).
In the 15th and 16th centuries, the invention of the decimal system made it easier to represent and manipulate real numbers. The decimal system uses a decimal notation where each digit represents a power of ten, separating the integer part of the decimal number from the fractional part. The decimal system also allows for infinite decimal expansions that can approximate irrational numbers to any desired accuracy. In the 17th and 18th centuries, the discovery of new types of numbers, such as complex numbers and transcendental numbers, challenged the concept of real numbers, and the need for a precise definition of real numbers arose in the 19th century when mathematicians developed the field of calculus, which studies limits, continuity and derivatives (mathshistory.st-andrews.ac.uk).
With the development of science, people’s imagination became more and more boundary-pushing. This meant that not only did numbers and unknown numbers appear as equations in the world of mathematics, but also that the numbers needed to solve these equations were already sufficient. Later, it was no longer enough for mathematics to develop in a meaningful way. This was pioneered by Newton and Leibniz, who are considered to be one of the turning points of mathematics, and who discovered a strong side of mathematics with their development in calculating the rates of change (Crilly, 2017: 51).
As new conditions emerged, irrational numbers became unable to meet the expectations of mathematics and scientists. In this situation, which challenges the human imagination, numbers called ‘Imaginary Numbers’ emerged (Crilly, 2017:51). Imaginary numbers were first mentioned in 50 BC by Heron of Alexandria, who tried to find the impossible cross-section of an imaginary pyramid for some unknown reason, but gave up because he had problems with some square root expressions. Nevertheless, he was the first to mention it (Beveridge, 2016:62). Later, it was Descartes who mentioned imaginary numbers (Descartes, 1997:14). On the other hand, the names that advanced equations, such as Ferro, Tartaglia and Cardano, inadvertently or without realizing it, actually talked about imaginary numbers. While the starting point of the idea of imaginary numbers appeared as a number with a negative square root, it developed further in the idea of the square of a number being equal to a negative value (Crilly, 2017:52).
If we try to explain imaginary numbers in a simple way: ‘Let’s say we have 5 apples in the fruit bowl, if we take 4 of them, we have 1 left. But if we take all 5, we have 0 left. But we cannot take 6 apples, because there are only 5 apples.’ As seen in this case, we are faced with a situation that does not actually exist. In other words, there is a situation that does not actually exist but that we calculate (Cirlly, 2017:52). The need for these numbers are important steps on equations; they are likely to have been discovered during the existence and solution of second and third order equations. Imaginary numbers are also called complex numbers. This group of numbers is defined as numbers that contain an imaginary ‘i’ (Crilly, 2017:53). The person who tried to explain these numbers simply on a virtual plane is John Wallis (Stewart, 2017:164). The image below shows Wallis’s plane.
The existence of complex numbers may have contributed significantly to mathematics and may have enabled the discovery of other different number groups. In fact, the group of numbers we call algebraic numbers is one of them. Algebraic numbers are complex numbers that are the roots of non-zero polynomial equations with rational coefficients. They are a fundamental concept in number theory and mathematical analysis, arising from the study of polynomial equations and their roots. The historical and theoretical foundations of algebraic numbers have been the subject of extensive research in the mathematics literature. The development of algebraic numbers, number theory and algebraic are deeply connected to the historical evolution of the structures. Foundational works such as “Algebraic Numbers and Algebraic Functions” can provide insights into the historical and theoretical foundations of algebraic numbers, showing a comprehensive view of their origins and importance in mathematics (Cohn, 2018). This work not only explored the basic principles and properties of algebraic numbers, but also paved the way for their formalization and application in mathematical theory. The study of algebraic numbers is also closely linked to various mathematical topics such as polynomial equations, field theory, and algebraic geometry. For example, research on bounding the equivalent Betti numbers of symmetric semi-algebraic sets has implications for the representation and characterization of algebraic numbers in mathematical structures and geometric spaces (Basu and Riener, 2017). Moreover, the study on the distribution of Salem numbers sheds light on the algebraic nature of some mathematical structures and their distributional properties (Götze and Gusakova, 2020). On the other hand, algebraic numbers have practical implications in various fields such as cryptography, computational geometry and machine learning. Work on formalizing class groups of Dedekind fields and spherical fields highlights the practical applications of algebraic numbers in formal reasoning and encryption protocols, highlighting their role in modern cryptosystems (Baanen et al., 2022). Similarly, research on complete block algebras of classical type Hecke algebras underlines the importance of algebraic numbers in the context of modular representation theory and mathematical physics (Ariki, 2018). In conclusion, the origins of algebraic numbers can be traced back to the historical development of number theory and algebraic structures; fundamental studies shed light on their theoretical foundations and practical applications. The interdisciplinary nature of algebraic numbers underlines their importance in various mathematical, scientific and cryptographic fields. The formation of algebraic numbers may have opened the door to advanced mathematical structures.
New Normal Science
This phase represents the establishment of a new paradigm in mathematical understanding, reflecting the incorporation of advanced mathematical structures and the refinement of numerical systems. This phase can be associated with the continued exploration and use of advanced number groups, such as hyperreal numbers and transcendental numbers, reflecting the ongoing development and complexity of numerical representation.
The concept of hyperreal numbers originates from the development by Abraham Robinson of non-standard analysis, a mathematical framework introduced by Massas (2022) in the 1960s. Non-standard analysis extends the real number system to include infinitely small and infinitely large numbers, leading to the creation of hyperreal numbers. These numbers can be useful in providing a solid foundation for mathematics and mathematical analysis, as they allow for a more intuitive treatment of infinitely small and infinitely large quantities (Dawson, 2021). The construction of hyperreal numbers involves the use of ultrafilters and the implementation of non-standard models of the real number system (Borovik et al., 2012). This approach allows the representation of numbers that are larger or smaller than any standard real number, thus extending the mathematical framework to cover a wider range of quantities and phenomena (Koper et al., 2018). Hyperreal numbers have found applications in various fields, including mathematics, mathematical physics and probability theory (Zemanian, 2003). In the context of mathematical analysis, hyperreal numbers provide a powerful tool for dealing with divergent series and discontinuous functions, offering a more comprehensive and tractable framework for handling these mathematical concepts (Jin, 2019). Moreover, the arithmetic and levels of hyperreal numbers contribute to the definition of limits and approximations in mathematical analysis, further emphasizing their importance in this field (Fleuriot & Paulson, 2000). The development of hyperreal numbers has also led to advances in mathematical formalization and axiomatization, as researchers have contributed to the theoretical foundations of hyperreal numbers by trying to establish finite arithmetic axiomatizations for hyperrational non-standard analysis (Lovyagin & Lovyagin, 2021). As a result, hyperreal numbers have emerged as a fundamental extension of the real number system, providing a rich mathematical framework for dealing with infinitesimals, infinite quantities and divergent series. Their applications span a variety of fields, including mathematics, mathematical physics and probability theory, making them a valuable and versatile tool in mathematical analysis and beyond. Besides the Hyperreal numbers, another group of numbers that developed around the same time were the Transcendental numbers.
Transcendental numbers have a rich broad and comprehensive background of distinctions that distinguish algebraic and supernumerary numbers. An algebraic number is the root of a non-zero polynomial equation with coefficients, while a transcendental number we satisfy such an equation. In other words, we can call any non-algebraic complex number a redundant number. The concept of transcendental numbers has been extensively researched and developed in the field of number theory and comprehensive analysis. A detailed explanation of the indefinitely comprehensive theory of algebraic equations with rational coefficients was first presented in the classic book “Transcendental Number Theory” in 1975 (Baker, 1975). This foundational work may have provided a comprehensive coverage of redundancies and their comprehensiveness in mathematics. The study of the love profile is also linked in detail to various mathematical topics such as the diophantine approximation, continued fractions, and exponents of irrationality. For example, Diophantine’s approximation of exponents and Sturmian’s work on continued fractions are shaped by extreme profiles and what is one of them (Bugeaud and Laurent, 2005). In addition, extensive work on Hankel determinants, Padé treatments, and irrationality exponents has expanded the scope of studies and their relationship to approximation theory ( Bugeaud et al., 2015 ). He also had a variety of interests, including a wide range of physics, astrophysics and computer science. For example, work on the algebraic structure of functional matrices of special form has shed light beyond detailed extended structures (Zudilin, 1996). Additionally, the design of a redundant function accelerator is available in practical computations in its abundance in hardware accelerators ( Song et al., 2023 ). The length and dimensions of love are different. Harris’s book “How to Explain Number Theory at Dinner” examines the context of extreme love, providing insight into its origins and significance (Harris, 2017). As a result, it has been the subject of numerous, diverse, general and comprehensive comprehensive studies and reviews. Fundamental studies, interdisciplinary studies, and more are everywhere to be seen collectively adding to the general scope of a vast number of them and their roles in mathematics and beyond.
New Depression
This phase refers to a period of reassessment and innovation in mathematical understanding, reflecting the ongoing search for new mathematical challenges and the need to improve and extend numerical systems to address emerging complexities.
The development of numbers over the last 50 years has seen significant advances in fields as diverse as mathematics, computer science and physics. However, there are still unresolved issues and dead ends in number systems that continue to pose problems. One of the biggest unsolved problems in mathematics is the Riemann Hypothesis, which remains an open question in number theory (“Prime obsession: Bernhard Riemann and the Great Unsolved Problem in Math”, 2003). In addition, deadlock in computer systems has been a persistent problem, especially in distributed systems where deadlock prevention is complex (Malhotra, 2016). Furthermore, the generation of random numbers has been a topic of interest with research focusing on quantum random number generation and its applications in cryptography and data transmission. Moreover, the study of grand unified theory (GUT) has contributed to the understanding of unsolved problems in mathematics, such as Fermat’s last theorem, and its consequences for the broader field of mathematics and physics (Escultura, 2013; Escultura, 2008). These cases can be indicators that can help us to improve and change numerical systems.
In the world of mathematics, the process of solving the big problems that will occupy mathematicians in the coming years seems to be based on slowly emerging solutions and proofs. With the impact of the computer age, mathematicians have gained computational capacity that was unthinkable in previous eras, which has enabled mathematical experiments and research to reach new heights. However, as a result of these developments, specialization has increased and mathematicians have focused on deep knowledge in their own fields. This is a natural consequence of the diffusion of knowledge, technology and mathematical applications. In the future, it is envisaged that mathematicians will be able to assess their current skills with more sophisticated computer evidence. In this process, it has been challenging for mathematicians to communicate and understand each other with colleagues in different fields of expertise (Crilly, 2017:211- 212–213).
Conclusion
Paradigm is the cornerstone of scientific thought and development. In the world of science, the limits of old paradigms are recognized and new paradigms are developed. Thomas Kuhn’s paradigm theory provides a framework that helps us understand this process (Kuhn, 1970).
The evolution of number systems is a mathematical example of a paradigm shift. Symbols have been used to express and make sense of mathematics in line with the needs that have emerged since the dawn of humanity. Although these symbols have differed according to the needs and discoveries of civilizations, they have continued their lives with mathematics as a common aspect. In line with the relationship and commonality of these symbols with mathematics, an accepted number system or numbers began to form. Later, zero, which is called “nothing”, was included in the group of numbers in line with the needs. With the inclusion of zero, people’s needs were met and they continued using natural numbers.
However, after a certain progress, the inadequacy of natural numbers was realized and people started to search for this direction. This inadequacy may lie in the idea of how the numbers or symbols formed by merchants, especially in debt-credit situations, will be and how they will be used. Therefore, mathematicians of some civilizations may have come to the fore and contributed to the discovery and symbolization of negative numbers.
The group of numbers we call whole numbers was formed. However, a new problem arose when whole numbers were proportioned to each other or which numbers we should use to represent a certain part of a whole. This problem was solved with the emergence of rational numbers. Rational numbers are thought of as proportional numbers. In other words, the ratio of multiplicities to each other and will always appear as another proportional number, but as a result of a strange event that took place in an indeterminate part of history, a mathematician named Hippasus tried to find a disproportionate number and as a result of the studies carried out after that, disproportionate, that is, irrational numbers emerged.
The irrational number group was enough to answer the problems in people’s daily lives by causing further deepening and development of mathematics. Of course, since the imagination of mankind is limitless, this number group was not enough either, could there be an imaginary number group? Of course, mathematicians overcame this and found the group of numbers we call complex numbers and used in advanced engineering. In fact, in case complex numbers were insufficient or did not meet the current needs, number groups such as algebraic numbers, hyperreal numbers and transcendental numbers were formed.
As a result, we use natural numbers in areas such as basic needs, measurement and trade. We use whole numbers to indicate a coldness or to emphasize debt in any commercial situation. We express and use rational numbers as a part of a whole or as a ratio of multiplicities to each other. We use irrational numbers in the same way, albeit indirectly, in engineering and in situations related to the basic needs of human beings. Complex numbers are used in physics and engineering, in dynamic systems, and in the control of flying objects (Stewart, 2017:172). Algebraic numbers have important implications in various fields such as cryptography, computational geometry and machine learning, hyperreal numbers have applications in various fields including mathematics, mathematical physics and probability theory, and physics, astrophysics and computer science. As can be seen, each type of number has its own fascinating and complex structure.
It is seen that science and mathematics are important for the progress and development of humanity and their emergence is due to the basic needs of people. Although the paradigm has an important place in this context for humanity to continue this development, some scientific studies in history, or ‘Scientific Revolutions’ as Kuhn (1970) calls them, are an important turning point for human life.
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