Abstract
As an academic discipline, mathematics is more than a set of computing mechanisms; it is argued by scholars that the quality of mathematics education is in decline today in human society. Historically, mathematics and philosophy had a close interrelationship enabling the understanding of various natural phenomena. According to Platonism of pure mathematics, the dynamics of natural phenomena and human societies follow invariant and absolute mathematical principles. This article presents a socio-philosophical argument that the concept of society within the natural space can be classified into natural society and synthetic society based on the concept of mathematical purifications, and mathematics education has a role in it. The existence of logical inversions of various forms in the synthetic societies are analysed and corrective roles of mathematics education are explained. The Pythagorean philosophy of mathematics in building an improved as well as just society is an appropriate solution that calls for a relook into mathematics education in order to reduce utilitarian distortion in mathematics education today and to promote intellectualism as well as harmony while reforming mathematics education.
Contribution: This article presents detailed analysis of the existing critical issues related to mathematics education today in human society and it compares various social forms in nature, including human, in light of the concepts of socio-philosophy. It is illustrated that these two aspects are interrelated in view of developments in education and evolution of human society. The historical perspectives of mathematics education forming a free and harmonious society following the Pythagorean philosophy of mathematics and society are presented, which can be a solution to the multidimensional problems in human society and in mathematics education today.
Keywords: Pythagoras; Platonism; ethnomathematics; mathematics education; philosophy of mathematics; society.
Introduction
In human histories of civilisations, mathematics plays a significant and crucial role for the survival of the species within the natural universe due to applications of mathematical principles either by discovering the associated mathematical principles or by empiricism without knowing the underlying inherent mathematical principles. This is to emphasise that the very existence of every species, including the human species, within the natural universe purely depends on the laws or principles of mathematics. For example, the numbers and the numerical counting systems (i.e., algebraic relations and orders) are metaphysical entities with profound applications in human society including science and technology (Peacocke, 2019). There is an agreement between the philosophical views of Plato and Aristotle that ‘things are somehow made of numbers’ (Taylor, 1926). The reason is that the natural universe is maintained and manifested on the basis of ever-consistent mathematical principles and structures, which are often termed the Platonic (mathematical) universe, by following the metaphysical principles of philosophy (Bagchi, 2021; Côté, 2013; Gregory, 2021). Note that the concepts of metaphysical forms of numbers include the elements of philosophy of Carnap (Peacocke, 2019). Interestingly, access to pure mathematical principles and the associated Platonic mathematical universe is not insularly limited to human species. Almost every species (other than human species) also employs mathematics intrinsically for survival in the challenging natural universe (Howard et al., 2019; Rodríguez et al., 2015; Rugani et al., 2016). For example, honeybees can count and construct geometrically hexagonal structures, which is one of the most stable geometric shapes, defying gravity during natural disturbances such as vibration and oscillations due to earthquakes and storms. Even small creatures such as spiders survive by using mathematical principles to capture prey. Note that spiders spin webs to form geometric shapes in various mathematical symmetries invisible to insects, which fall into the web to be captured as food for the spiders. There are numerous other examples of similar situations where species other than human employ mathematics for survival in nature (Kalmus, 1964).
On the other hand, the socio-cultural histories of the human species and its evolution over time partly alienated it from the natural universe, although humans (as the elements of nature) reside within the similarly challenging natural universe subjected to the similar proportions of natural forces. In the natural societies, various species coexist and coevolve, which can be viewed as the execution of natural cause-effect relations of mathematical logic. However, synthetic as well as unstable structural relations are formed among the species if the natural cause-effect logic is violated or synthetically altered using tools. An example is the introduction of biological species from one geographic location to another location by incidental or accidental transportation by humans, disturbing the ecological harmony at the destination location. The concept of balanced materialism (i.e., finite materialism founded upon natural cause-effect relations) promotes coexistence in the natural universe. In contrast, the unjustifiable as well as synthetic (infinite) materialism of the human species has resulted in increasing alienation, which has caused mutual harm between nature and the human species, challenging the very foundation of their survival within the natural universe. Fundamentally, the alienation has given birth to the synthetic society (based on synthetically altered cause-effect relations employing scientific and technological innovations and tools) of the human species within the natural universe, which has invited a series of inversions within the synthetic societies, causing distortions, conflicts, alternating hegemony and disparities in multiple dimensions. This is to emphasise that this article does not intend to view the positive impacts of technological innovations in human society negatively; however, it can be argued that the illogical, irrational, unmindful and unjustified technological utilitarianism and materialism would bring associated negative effects in nature as well as in human society. It is not surprising that synthetic materialism (i.e., infinite and illogical materialism depleting finite natural resources) within a synthetic society of human species and the fully utilitarian education of mathematics in the synthetic society are interrelated. This article analyses these interrelated aspects in detail. This article emphasises the reformation of mathematics education through the Pythagorean philosophy of mathematics and society to improve the quality of mathematics education by instilling creativity and to form a just society of humans through appropriate mathematics education by promoting intellectualism. First, we present the motivation for and the contributions made by this article.
Discussion
Motivation and contributions
In ancient times, the intellectual inquiries in various natural science disciplines were drawn from the domains of mathematics and philosophy to understand the forces of nature, formation as well as dynamics of human society and to get answers to even more, deeper questions. Note that today the positive and fruitful interrelationships and interactions between the disciplines of mathematics and philosophy are well known (Blanchette, 1998; Hacking, 2014). In ancient times, the domains of science and mathematics were viewed as natural philosophy. The respective domains of education and schools of thought were not distinctly separated, which provided fertile ground for radical discoveries and inventions. However, in later times the traditional scholastic developments in various educational schools of thought have walled out each other, eliminating cross-talk between various disciplines and causing various modes of confinement of knowledge, resulting in distortions. An existing example is the persistent issues related to the deterioration of education systems in human society caused by utilitarian education models (Gamson, 1966; Haslam et al., 2021). Repeated utilitarian education-theoretic attempts to revitalise mathematics education at different levels have failed to provide positive results (Van Zanten & Van den Heuvel-Panhuizen, 2021). Note that the philosophy of education needs to seek a symbiotic relational balance between the two aspects, such as the usefulness of learning and the intrinsic value of gaining new knowledge. The same principle is applicable to mathematics education; however, it often results in conflicts between the two approaches in the classroom (Huckstep, 2000). On the other hand, the number of female students participating in mathematics education is in decline and it is proposed that the social role has a key responsibility (Martínez et al., 2023). In other words, the notion of inclusive and gender un-biasness in mathematics education deserves serious attention in order to revitalise mathematics education in human society to instil creativity and to bring in associated improvements by forming a just society through mathematics education.
On the other hand, the evolution of human society has invited a set of irrationalities or logical irregularities within the structures of social sciences, which are repeatedly manifested in terms of socio-ethnic conflicts, econo-political disparities and related distortions. In this article we will analyse such phenomena in the views of Pythagorean philosophy of mathematics education and the improper emphasis on (only) utilitarianism in mathematics education in present day. We detail the concepts of synthetic society, natural society, inversions in synthetic society and the need for mathematical purification through appropriate mathematics education, reducing utilitarian distortions and promoting intellectualism and harmony. The intellectualism indicates a logically thoughtful, revisionary and analytical understanding and action about: (1) understanding self-being; (2) discovery of knowledge and preservation of natural spaces; (3) mutual interactions preserving harmony; (4) good for all; and (5) maintaining stability. This article proposes that we need to understand mathematics and mathematics education in the views of Pythagorean philosophy of nature, human society and mathematics as interconnected streams. This article argues that mathematics education today needs to be revitalised, reducing utilitarian distortion to form a just, harmonious, liberal and free human society based on mathematical logicism and intellectualism preserving natural habitat on Earth (see Figure 1).
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FIGURE 1: Role of revitalised mathematics education in society. |
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In this article we encourage revisitation of the elements of philosophy of mathematics of Pythagoras, which would provide directions to improve mathematics education systems and to establish a more just as well as harmonious society today through the holistic education of mathematics. The contributions made in this article can be summarised as follows:
- It is explained in detail in this article by following the historical contexts that the Pythagorean philosophy of mathematics and society helps to discover new knowledge about complex mathematical reflective abstract forms and it prepares the logical mind of citizens by exciting rational thinking through appropriate mathematics education, which would result in formation of a just as well as liberal (harmonious) society.
- We argue that such abstract knowledge of mathematics can be translated into application through empiricism to solve real-world problems in society (such translation of knowledge from pure mathematics to applied mathematics has occurred many times in the history of human civilisations).
- We propose that, in this way, the Pythagorean philosophy of mathematics can revitalise mathematics education and it can help to reshape human society through appropriate mathematics education, which is not merely dependent on ‘problem-solution-practice’.
The article is organised as follows. First, we present the importance of mathematics in science as well as in society. Next, we explain the philosophical notions of natural society and synthetic society. A brief historical description about Pythagoras and Pythagorean society along with the contributions made in mathematics in ancient times are given next followed by an illustration of the issues related to mathematics education in present time. After corrective suggestions are made, the article concludes.
Mathematics and nature
It is well known that laws of mathematics are foundational to explain the natural universe and, as a result, mathematics has profound applications in almost all branches of natural sciences, including physics, biology, chemistry, Earth sciences and social sciences. In 1960, physicist Eugene Wigner convincingly stated that there are numerous cases where mathematical theories were developed independent of physics and later, in 20th century physics, applied those mathematical theories to explain natural phenomena (Wigner, 1960). Interestingly, astronomer and mathematician Galileo expressed a similar opinion: that the grand book of nature is written in mathematics (Galilei, 1623). The same observation can be extended to the dynamics of social sciences (Kim et al., 1992).
In summary, scholars have convincingly proposed that the laws of physics explaining the sensual and physical world are understood through mathematical representations and, moreover, the properties of mathematical structures are independent of human inventions, meaning that mathematics is essentially a discovery of existing knowledge in nature (Aguirre et al., 2016; Wigner, 1960). One of the reasons for the effectiveness and importance of mathematics is that mathematics helps to identify regularities in the natural world (Aguirre et al., 2016). In other words, logic and mathematics help us to detect the violation of regularities in synthetic forms including synthetic human creations such as technology and social structures. Moreover, the manifestations of nature can be understood through the language of mathematics, which predicts the outcome or phenomena in natural as well as human (synthetic) societies. From the viewpoint of philosophy of mathematics, four mathematical principles are fundamental to explain nature: (1) axiom of commutative addition; (2) existence of symmetries in nature; (3) numerical quantification of natural elements and algebraic operations on them; and (4) algebraic ordering of elements (Aguirre et al., 2016). Recall that in ancient times Thales philosophised that manifestations of nature have two distinct realms: ‘Being’ and ‘Becoming’. Abstract mathematics essentially encapsulates the understanding of the ‘Being’ realm of nature, which is a perfect, permanent world (i.e., time invariant) and reachable by the rational mind. In contrast, the physical world is the ‘Becoming’ realm of nature and is imperfect, transient and sensual.
Nominalist contradictions
The traditional philosophy of mathematics in ancient times was based on the concept of Platonic realism to discover knowledge about the natural world, which is elaborated in the dialogue of Plato in Menon; Socrates made an effort to establish it firmly in the philosophy of mathematics through experimental empiricism. The philosophical schools of thought of Aristotle, Descartes and Kant also emphasised Platonic realism in mathematics education and philosophy (Winslow, 2000). It is important to note that during the period of ‘scientific revolution’ covering the 20th century, discussions about the foundations of mathematical natural philosophy were deeply engaged with the conceptual forms of mathematics and considering the Platonic philosophy of mathematics. However, historically, a different philosophical school of thought emerged in mediaeval times, called nominalism, which departed from Platonic conceptualism and abstractionism of mathematics (Sepkoski, 2005). Some notable nominalists are Gassendi and Hobbes. The nominalists eventually gave birth to another school of thought, called constructivism, in the philosophy of mathematics. It is argued that the constructivist philosophy of mathematics tried to bridge the domains of discrete events of experiences in the physical world and the underlying conceptualism of mathematics as principles (Sepkoski, 2005). This can be viewed as the epistemological approach. Several researchers have explained that constructivism essentially tries to deny the objectivist approach of science and is actually rooted in the social context of understanding knowledge (Sfard, 1994; Winslow, 2000). It is noted that the formation of the body of knowledge is based on two diverging lines: empirical abstractionism and reflective abstractionism (Piatelli-Palmarini, 1979). The constructivism philosophy of mathematics relies on the sensory worldview based on empiricism, whereas reflective abstractionism is in a higher position in the hierarchy of rising complexities of knowledge (Winslow, 2000). Interestingly, the nominalist philosophy of mathematics potentially invites scepticism in explaining the natural universe. This is because, on the one hand, nominalism in the philosophy of mathematics advocates understanding of the real world through the lens of Aristotelian formalism and, on the other hand, asserts that formal (mathematical) description of the real world is a fiction of the human mind, which apparently has diverging interpretations. Moreover, in this view it is difficult to explain why applications of mathematics as a fiction of the mind are so effective in technological innovations in the real (physical) world. This is important to note as it is argued by researchers that the constructivism philosophy of mathematics embodies dilemma and can be reduced to a form of Platonic realism of mathematics (Kahn, 2021).
Natural society and synthetic society
Let us consider the entire universe where Earth is a habitable planet for numerous species. Note that geometric shapes, visible objects (such as matter), invisible objects (such as dark matter) and all dynamics in such a universe are fully governed by purely mathematical reasoning (Aguirre et al., 2016; Galilei, 1623; Wigner, 1960). The habitable space on planet Earth is a natural subspace where natural societies of several biological species (other than human) originated, evolved and coexisted for millennia. The harmony of such natural societies was kept for a comparatively long time by pure mathematical reasoning, which can be viewed through the lenses of Aristotelian logicism and Darwinism (Aguirre et al., 2016; Linde, 1994; Smolin, 1997). The natural societies of biological species (other than human) successfully avoided inversions because such interactive and interdependent societies are structured and maintained by pure mathematical reasoning (i.e., a form of mathematical purification). For example, the theory of natural selection and evolution of Darwin and Lamarck is a manifestation of mathematical purification of natural societies residing with in the natural space. The concept and the elements of mathematical purification of the natural societies within the natural space are illustrated in Figure 2.
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FIGURE 2: The concept and elements of mathematical purification. |
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It is important to note that by inversion we indicate the inherent socio-structural defects and the resulting degenerative effects. For example, in the natural space the ratios of numbers of predators and prey are kept in balance by the mathematical rules ensuring survival of every species; however, the inversions in synthetic society of human species have resulted in global overpopulation, global gender inequalities and the erosion of the natural environment. In the views of natural (mathematical) philosophy, the natural societies are purified by omnipresent pure mathematical reasoning or mathematical laws, which is another version of pure logicism. It is important to note that pure mathematical reasoning, logicism and the existence of the Platonic (mathematical) universe are independent of any species, including humans. For example, the geometric statement of Pythagoras about a right-angled triangle (i.e., Pythagorean theorem), group algebraic theory of Galois (i.e., Galois group algebra) and Ramanujan functions do not depend on time, differences of human ethnicities and different socio-cultural contexts. In other words, the purely mathematical reasons and truths are absolute and invariant constituents of the Platonic mathematical universe. The pure mathematical principles and associated logicism intrinsically preserved the balance between finite natural resources and the evolution of biological species within natural society. However, the history of social developments of the human species within natural society caused disruption in the balance due to the fact that humans created a set of synthetic societies within the subspace of natural society. The socio-philosophical concept is illustrated in Figure 3.
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FIGURE 3: The socio-philosophical concept of natural society and synthetic society. |
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The synthetic societies of humans are structured and evolved based on self-perceptions, belief systems and the constraining forces. The self-perceptions and belief systems (set A) can be presented in algebraic terms as follows (Equation 1):
In Equation 1, RI stands for Religious Identities, CI stands for Cultural Identities, NI stands for National Identities and HI denotes Hierarchical Identities encompassing socio-economic and ethnic varieties as well as political ideologies. The subspace of synthetic societies is guarded by the formation of an alienation boundary separating them from natural society to ensure self-management and self-propelled survival, and avoiding purely mathematical reasoning and logicism. The other historical aim of such creation, as well as alienation from natural society, is to construct a set of synthetic (artificial) social self-hierarchies within the same species to ensure formation of synthetically controlled structure and flow or diversions of resource consumption as and when desired. Once more, the omnipresence as well as everlasting effects of pure mathematical reasoning and logicism are ignored within synthetic societies. In the view of natural philosophy, it is an avoidance of mathematical purification of synthetic societies of humans. Moreover, the required natural resources were transported within the subspaces of synthetic societies from the mathematically purified natural space to sustain synthetic societies of different forms. Due to the absence of pure mathematical reasoning and logicism (i.e., mathematical purification), synthetic societies started to adversely interfere with the natural societies present within the natural space. Moreover, inherently present structural defects and self-contradictions (for example, the depletion of oxygen in polluted air, due to over-industrialisation; racism; and female foeticide, to name a few) within the synthetic societies of humans resulted in the formation of a series of inversions within the synthetic societies themselves. The inversions can be presented as set B as follows (Equation 2):
In Equation 2, GI denotes gender inequalities in synthetic societies under inversions, SC stands for Social Conflicts (internal), EC stands for Ethnic Conflicts (external), ED stands for Economic Disparities (internal as well as external) and ND denotes Natural Depletion. This last, ND, has further resulted in the instability and erosion of natural space, inviting the possible extinction of the species both in natural society and in the subspace of synthetic societies. This is because every species resides within the respective natural spaces and the erosion or depletion of such natural spaces would lead to survival crisis; examples are the disappearances of the dodo and Tasmanian tigers, and currently global warming inviting the possibility of mass extinction in the future. The symbolic representations of inversions present within synthetic societies are illustrated in Figure 4 and Figure 5.
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FIGURE 4: Type-I inversion in synthetic society. |
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FIGURE 5: Type-II inversion in synthetic society. |
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It can be noted that plausibly a trade-off was made while forming the synthetic societies by enhancing the survivability of humans in artificial confinement within the alienation boundary, separated from the natural societies at the expense of natural freedom based on free will and free choice. However, the synthetic societies reside within a subspace of natural space in the universe and a resulting struggle ensues due to the continuing effort of natural space to enforce necessary mathematical purification of synthetic societies to eliminate inversions as well as structural irregularities, whereas the synthetic societies try to resist such mathematical purification admitting inherent inversions within. From the viewpoint of natural philosophy, the resistance of synthetic societies of humans to conform to the demands from the omnipresent as well as invariant purely mathematical principles and reasons in the universe for the restoration of natural balance is the root cause of natural degeneration and possible extinction of species. In this context, the Pythagorean society of ancient times and the Pythagorean philosophy of mathematics play important roles towards solving the problems of synthetic societies of present time.
Ancient mathematics and Pythagoras
In this section, we discuss Pythagoras as a mathematician and Pythagorean society along with their contributions to mathematics education in ancient times. The early history of Pythagoras, recorded by Diogenes Laertius and Porphyry, indicates that Pythagoras was a compassionate person and a mathematician who believed in natural ways of harmonious living. Some of the literature today debates verifiable records about the life of Pythagoras; however, there are interpretable historical chronologies and we cannot simply ignore the history of contributions made by Pythagoras and Pythagoreans to mathematics and its applications to understand the natural world in ancient times (Levin, 1980; Maor, 2007; O’Meara, 1990; Zhmud, 2012). The elements of Pythagorean society and its philosophy of mathematics are important to understand the relevance of the development of mathematics education in ancient times and the persisting issues of mathematics education today (Shulte, 1964). It is rightly stated by scholars that ‘the Pythagorean theorem constitutes one of the first great ideas of mathematics’ (Danesi, 2020). Interestingly, Pythagoras followed the methods of numerical experimentations in mathematics in order to discover the laws of strings (Shulte, 1964). In general, the contributions of Pythagoras in mathematics are attributed mainly to number theory today. However, the mathematics curriculum in algebra and geometry in schools today are also indebted to the contributions made to mathematics by Pythagoras and the Pythagoreans in ancient times (Shulte, 1964).
It is noted earlier that the metaphysical concept that the natural world can be understood by employing numbers is rooted in the Pythagorean philosophy of mathematics and it received momentum in history through the involvement of Iamblichus (O’Meara, 1990). Iamblichus actively revived the Pythagorean philosophy of mathematics. Later, the Neoplatonist school of philosophers and mathematicians accepted the Pythagorean philosophy of mathematics in describing the natural world, which resulted in early attempts to explain physics using mathematics (i.e., mathematising physics) and greatly influenced the scientific discoveries made in mediaeval as well as Renaissance periods (O’Meara, 1990). Even today, the Pythagorean theorem is a fundamental theorem in mathematics taught in schools around the world to measure distances in Euclidean spaces. Moreover, the understanding of various conic sections (see Figure 6) needs the knowledge of Pythagorean theorem and its applications in geometry.
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FIGURE 6: Pythagorean theorem and conic sections. |
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The Pythagorean society
According to Empedocles, visionary Pythagoras had an extraordinary mind (Dutsch, 2020). Historical records suggest that Pythagoras visited Egypt and Babylon to study further and to enrich the knowledge of mathematics. Interestingly, a contemporary of Pythagoras named Ion claimed that the philosophy of Pythagoras had some sophist elements (Dutsch, 2020). The school of mathematics of Pythagoras admitted several women scholars, who became Pythagorean themselves (Dutsch, 2020; Pomeroy, 2013). The women mathematicians of Pythagorean school such as Timycha, Phintys or Philtys, Occelo, Eccelo, Perictione, Myia, Theano, Eurydice and Panthea are some of the notable scholars; however, there are several other Pythagorean women mathematicians. Over time, the Pythagorean society was formed where common citizens joined to learn mathematics. The Pythagorean school of mathematics had two circles: inner circle and outer circle, based on the variations of degree of scholastic participation of students in mathematics. The members of the inner circle were dedicated mathematics researchers. On the other hand, the outer circle was open to any common citizens who wanted to learn mathematics. Historical records suggest that Pythagorean mathematicians were radical thinkers, who practised a liberal and natural way of living. Moreover, Pythagorean society was a secretive society, which was necessary to protect it from the constraints as well as socio-cultural-political restrictions of ancient Greek socio-political environments (Dutsch, 2020; Rowett, 2014). The Pythagorean society was based on the principles of harmony and symmetry of the natural world and it was derived from the Pythagorean philosophy of mathematics. It is generally understood from the available history about Pythagoras that the Pythagorean society was a very inclusive society, where everything, including judgement and music, was based on mathematical logic and geometric proportions to establish a just society (Dutsch, 2020; Pomeroy, 2013; Rowett, 2014). This shows that Pythagorean society was a liberal as well as progressive intellectual club of mathematicians and followers incorporating gender equality. The Pythagoreans in later times made efforts to reform human society based on logic to establish a just society. It is interesting to note that in ancient Greece an intellectual system of logical rhetoric was developed, where the logical arguments in the society were kept in symmetrically diagonal (opposite) directions to evaluate multiple thoughts of different proportions. The purpose of such logical rhetoric was to review the established societal structures and norms of human society, figuring out the existing faults in it (see Figure 7).
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FIGURE 7: The relations between antilogies and social refinements. |
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It is now called antilogies and Plato, along with others, did participate in such intellectual as well as logical rhetoric (Rossetti, 2023). It could be understood today as an intellectual effort at that time towards the formation of a just society through the logical refinement of common minds in a human society, where a common individual might not have education in sophisticated mathematical arguments based on pure logic. Moreover, the historical records suggest that an agreement between the opinions of Pythagoras and Aristotle (at a later time) can be found, which emphasised that doing good for the commons is essential in any society (Dutsch, 2020).
Utilitarian distortion of mathematics education
The study of mathematics is equivalent to the study of abstract objects in pure forms, which are often within the time-invariant (Platonic) universe. The contributions of mathematics education (for example, the Science-Technology-Engineering-Mathematics [STEM] system) in maintaining the socio-economic progress in a synthetic society is well recognised. Moreover, mathematics education has a fine role of refining any synthetic society by educating citizens and intellects imparting logical and rational thinking. In ancient times, mathematics education in societies or schools of Pythagoras, Euclid and Plato maintained the distinctions by preserving the similar foundations in contrast to the role of mathematics education today. However, later, mathematics education transformed into a real-life problem-solving mechanism in society. For example, the main emphasis today in mathematics education is to present mathematics as the practice of problem-solving tools to solve practical problems in societies and to produce more materials promoting consumption (Star, 2016). It is argued that mathematics education today needs the infusion of creativity and the recognition of mathematical values among learners, which is currently downplayed in the curriculum as well as in the teaching methods (Kim et al., 2019). Evidently, the over-emphasis on practising mathematical problem-solving techniques using technological tools does not aim to nurture creativity among students or learners. The use of technological tools to learn mathematics will enhance only problem-solving techniques, which can be viewed as skill development to prepare a skilled workforce. Let us consider two interesting examples illustrating the limitations of emphasised computation-oriented and practice-based learning of mathematics.
Example 1
Let us consider that r > 0 is the radius of a circle in Euclidean 2-space. The circumference of the circle is computed as 2πr. We ask the following computational question: How do we deterministically compute the circumference of the circle in the physical perceivable world when it involves an irrational (never-ending decimal) number (π)? In other words, how do we apply this algebraic formula in the physical world to deterministically form a circle of a predefined shape accurately? Moreover, the question is: Can we ever manufacture a circle in the physical world in exactly measured form? Evidently, if we thoughtfully try to address these questions then it leads to incoherent understanding of mathematics and its applications in problem-solving in the physical world due to (only) computation-oriented learning.
Recall that in the mathematical (as well as philosophical) domain of logic, it is consistent that a TRUE proposition cannot lead to a FALSE proposition. We illustrate in Example 2 a fallacious contra-example, which arises due to the computational-and-practice-oriented learning of mathematics.
Example 2
Let us consider the TRUE proposition: eiπ + 1 = 0. Further computation leads to the following conclusion (Equation 3):
We ask the following question: How does the multiplication of two non-zero numbers lead to zero? This indicates that procedural-practice-based mathematics education would lead to the possibility of inviting FALSE proposition even though we started from a TRUE proposition.
The reason for the appearance of such mathematically misleading interpretations is that we blindly followed a few practising steps in Example 1 and Example 2 without understanding the underlying mathematical principles. These two examples illustrate the fundamental limitations of mathematical education by employing (only) computation-oriented and practice-based learning without paying any attention to the underlying mathematical principles.
Apparently, this distortion of mathematics education over time is a gradual decline and is mostly pronounced during industrialisation as well as in post-industrialised societies, where continuously increasing materialism appeared. The distortion and decline of mathematics education in present time is well recognised and often the responsibility of such distortion is shouldered upon the educators and mathematics curriculum (Star, 2016). However, it is a systematic distortion of mathematics education within the framework of education at all levels and the skewed perception about mathematics is percolated within synthetic societies. The main reason for such distortion and decline in mathematics education today is over-emphasis on the utilitarianism of mathematics education as a field of study. Recently, there are efforts to regenerate the curriculum and teaching of mathematics education starting from the K12 system; however, it appears to be ineffective or difficult to achieve (Schmidt & Houang, 2012; Star, 2016; Tyack & Cuban, 1995). Unfortunately, the proposed changes to improve mathematics education in the synthetic societies emphasise the strategies of teaching and extensive practices only (Star, 2016). This is not an appropriate direction. The utilitarian system of education aims to produce an elite class to maintain properties of synthetic societies and such education emphasises: (1) rigidly structured instruction-based learning; (2) examination-oriented education; and (3) creation of a sterile learning environment separating educators and learners into two classes to reduce free interactions (Gamson, 1966; Haslam et al., 2021). In other words, a utilitarian education system does not aim to promote open exchange of knowledge or free and open engagement between educators and learners. We need to relook at the definition of education suggested by Jean-Jacques Rousseau. According to Rousseau, education should include three critical elements: (1) naturalism (i.e., natural education); (2) curiosity; and (3) freedom of learning, which brings happiness (Burch, 2017; Jimack, 1983). The ideal of natural education is to prepare a synthetic society driven by autonomous and responsible humans (Burch, 2017). Evidently, the vision of Rousseau about natural education overlaps with the ancient wisdom of Pythagoras about forming a natural (free and just) society through mathematics (i.e., Pythagorean mathematical society). The distortion, decline and failing efforts to rejuvenate mathematics education in synthetic societies and the following social inversions can be possibly mitigated by revisiting the natural philosophy of Pythagorean society and the learning of mathematics in a holistic way.
Pythagorean philosophy of mathematics and education
It is known that mathematics is a science of abstract mathematical structures which are coherently and consistently ordered or interrelated in the Platonic (mathematical) universe (Aguirre et al., 2016). In the history of science of explaining natural world, mathematics always played a crucial and essential role such that without mathematics proper understanding of the laws of the physical world and applications of such laws in human inventions could not be made. A set of interrelated and dynamic natural observations would appear chaotic without appropriate mathematical understanding of it. For example, the Darwinism in evolutionary biology is a manifestation of a mathematical principle, where an ordered structure emerges from the apparent chaos in natural space (Aguirre et al., 2016; Linde, 1994; Smolin, 1997). The Pythagorean philosophy of mathematics and the natural world validated such observations in the ancient history of science (Aguirre et al., 2016). According to Pythagorean philosophy of mathematics, the elements of the universe are quantifiable and harmony in the universe is preserved by mathematical laws following geometric proportions or ratios (Aguirre et al., 2016). Some historians and scholars have pointed out that Aristotle classified the ‘mathematical Pythagoreans’ as the mathematicians who employed geometry to reason about the natural universe (i.e., ‘reason why’ by using axiological hierarchy) and the mathematical Pythagorean Eurytus applied the method of numerical quantifications of natural objects in ancient times (Horky, 2013). These are some of the characteristics of Pythagorean philosophy of mathematics, where efforts were made to understand nature through the lens of mathematics in ancient times. Later, physicists and mathematicians such as Galileo and Dirac asserted similar opinions, saying that the natural universe can be properly understood by employing the principles of mathematics (Aguirre et al., 2016). In brief, the Pythagorean philosophy of mathematics attempts to uncover the truth and harmonious beauty of the natural world by using number theory, geometry and the algebraic laws of order, proportion and symmetry (Hemenway, 2005). It indicates that education in mathematics needs to prepare intellectual minds to support a harmonious, progressive, liberal and inclusive society with proportional equalities based on mathematical logicism and mathematical rationalism, which invite natural balance within natural space. Moreover, education in mathematics has a role to shape a just society based on pure logicism. For example, the Pythagorean philosophy of mathematics plays a vivid role in the formation and shaping of a just (human) society based on the concepts of geometry, which was later adopted and refined by Aristotle (Ambrosi, 2013). Even today, a similar proposition is applicable that appropriate mathematics education brings good to human society and helps in establishing a harmonious (just) society (Fischer, 1992).
Influences of Pythagorean philosophy of mathematics education
The influences of Pythagorean philosophy of education can be found in later schools of learning mathematics in ancient times. The school of learning established by Plato distinctly followed the Pythagorean philosophy of education. The detailed accounts of refinements and interpretations of Pythagorean philosophy of mathematics education in the school of Plato, as explained by Xenocrates, are discussed by Dillon (2019). According to Xenocrates, who was the head of a Platonic school of learning mathematics, the need for interpretations of Pythagorean philosophy of mathematics education and formalisation of philosophy of mathematics of Plato was due to the establishment of a school of learning by Aristotle in Lyceum. The conflicting intellectual relationship between the Aristotelian school and Platonic school of learning had given birth of a series of anomalous interpretations of philosophy of mathematics education by Plato as well as Pythagoras. According to Xenocrates and Speusippus, the philosophical interpretations of the generation and existence of the cosmos were attributed to three mathematical principles: (1) multiplicity of elements; (2) infiniteness of cosmos; and (3) indefinite Dyad (Dillon, 2019). Xenocrates emphasised that Monadism and Dyadism philosophical principles can be put together to form numbers in mathematics (not soul). Later the Hellenistic code of ethics was influenced by such philosophy when Cicero wrote in his Book IV (De Finibus) about the first principle of nature saying that ‘every natural organism aims at being its own preserver’. Interestingly, the doxographic tradition of interpretations of philosophy and mathematics indicates that the schools of Xenocratean formalisation are founded upon the Pythagorean philosophy of mathematics education (Dillon, 2019).
It is important to note that the philosophy of mathematics of Pythagoras of Samos and Aryabhata I of India had similarities: both indicated the metaphysics (transcendental property) of numbers, including irrational numbers such as π, and the conservation of the universe (Puttaswamy & Aryabhata, 2012). It is of note that the utilitarian model of mathematics education today based on the ‘elementarising’ methodology of mathematics learning to implement the ‘practice of teaching mathematics and to solve teaching problems’ in a class obstructs the appropriate learning of mathematics and the growth of the discipline (Biehler, 1994). We need to revisit the philosophy of Kant about the historical processes of discovering laws in natural science disciplines and the role of metaphysics. According to Kant’s philosophy of natural science, apart from pure reasons the theory of objects plays a vital role in understanding the principles of natural science using the lenses of metaphysics and transcendental analytics (Meer, 2018). As an example, it is argued that Newton did not need to physically see the trajectories of planets in the cosmos to formulate the mathematical laws of gravity governing them (Meer, 2018).
Future directions
According to Ernest (2018), in the domain of philosophy of mathematics education, the role of metaphysics is closely related to ontology, which aims to explain the nature and the existence of elements. Moreover, Ernest and others suggest that the goals of mathematics education can be broadly classified as follows:
- Preparing manpower for industries: numerical application-oriented, training-based learning in an authoritarian environment.
- Technology-centred education to prepare skilled manpower: application-oriented education of mathematics, emphasising skill certification.
- Traditional knowledge-centric education: aiming to impart a body of knowledge in mathematics to learners without specifically paying any attention to application.
- Creativity-promoting education: aims to infuse creativity in mathematics education and promotes growth of body of knowledge in mathematics.
- Preparing socially aware citizen-centric education: aims to prepare intellectual minds through mathematics education to form a liberal and just society.
It is easy to conclude that the combination of the latter three in the list above closely resembles the Pythagorean philosophy of mathematics education and society in a broader sense, which is in line with the proposal made in this article to revive mathematics education today. The combination of these three elements would emphasise the values of mathematics-centric learning of knowledge, promotion of creativity and the preparation of minds to form a just society. Note that the interdisciplinary approach to learn a complex science subject, such as mathematics, would add epistemic values to education (North, 2024; Robinson et al., 2016). Moreover, mathematics education empowers people in any society bringing in multidimensional improvements in the lives of its citizens (North, 2024). It is important to understand that mathematics is not merely a computing mechanism or process nor merely a practising tool for application to sustain synthetic materialism. Rather, education in mathematics should include the broader and holistic approach to uncover the invariant truth as well as beauty of nature, which would prepare an improved and sustainable society in the natural space. We can view it as a mathematically purified synthetic society in the present time of human history.
Conclusion
The dynamics of all natural phenomena including the societies of various species within the natural space follow a set of mathematical principles, which are time invariant. There is a fine role of mathematics education in reshaping human society. The concept of mathematical purification distinguishes the categories of societies depending on the presence of inversions violating the principles of mathematical logicism. The importance of appropriate mathematics education in building a liberal and just society is recognisable by following the Pythagorean philosophy of mathematics. The reasons for failure to rejuvenate mathematics education are rooted in the failure to reduce the over-emphasised utilitarianism and distortion of mathematics education today. The history of liberal, open and inclusive Pythagorean society indicates that mathematics education today needs to reduce the over-emphasis on utilitarianism and promote intellectualism and creativity to purify the synthetic societies of the human species. The Pythagorean philosophy of mathematics and nature may provide a solution to the synthetic societies today by improving mathematics education.
Acknowledgements
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S.B. is the sole author of this research article.
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