A Descriptive Study of Conceptual Structures and Image Schemas Corresponding to Algebra Based on Cognitive Semantics

According to the hypothesis of embodied mathematics, the basis and origin of mathematics is the human-being embodiment, and therefore mathematical Platonism and similar hypotheses, which assume the existence of mathematics independent of man, can only have a place of faith and not science. Based on this hypothesis, and since mathematics and its branches such as algebra are also the output of the system of thought and human cognition, they must be describable on the basis of basic conceptual structures such as the image schemas, the conceptual metaphors, the conceptual metonymy, and the conceptual blending. therefore, in this research, based on the descriptive-analytical method and the hypothesis of embodied mathematics, the fundamental metonymy of algebra and its role in the formation of mathematical concepts are discussed, and the cognitive origin of various branches of algebra such as identities, algebraic equations and inequalities, locus, the idea of analytic geometry, vector algebra and algebra of sets are analyzed based on a cognitive perspective. The mathematical generalization of the conceptual structures based on the algebraic foundation is the generator of effective branches in other experimental and analytical sciences. Since, according to the embodied mathematics hypothesis, mathematics, like language, is derived from the human thought system and cognitive faculties, it is possible to describe basic algebraic concepts based on conceptual structures such as metonymy, metaphor and conceptual blending, and this result can be used to optimize mathematics education.
Introduction
The word algebra in Arabic means unification and linking of broken parts, and is also used in the sense of necessity and as the opposite of option (Dehkhoda Dictionary). Similarly, the word jabbar means both bone-setter and tyrant. This word, following the great work of Abu Ja'far Muhammad ibn Musa al-Khwarizmi in the third century AH called Jabr va Moghabeleh (full name: Kitab al-Mukhtasar fi Ḥisab al-jabr wa al-mughabaleh), caused a branch of mathematics in Europe to later be called Algebra, and the name of al-Khwarizmi himself remaind in the mathematics in the form of the word algorithm. Al-Khwarizmi used the words Jabr and Moghabeleh to describe the transposition of positive and negative terms on both sides of an equation and the elimination of equations from both sides. In simpler terms, algebra means transferring a negative expression from one side of an equation to the other and transforming it into a positive expression (algebra), and Moghabeleh means eliminating similar expressions from both sides of the equation. Al-Khwarizmi used linguistic descriptions instead of symbols to describe algebraic expressions. Today's mathematical symbols and notation have been formed through a long and evolutionary process. Today, algebra encompasses a much larger scope of mathematical knowledge and has numerous branches.
The hypothesis of embodied mathematics, presented by Lakoff and Núñez, attempts to identify the origin of mathematics and its place in human cognitive faculties. Since, according to this hypothesis, mathematics as we know it is not a transcendent and external thing, but an embodied thing, traces of conceptual structures as discussed in cognitive sciences and cognitive linguistics can be observed in it. Therefore, the role of structures that affect the conceptualization system, including image schemas, metonymy, metaphors, and conceptual blending, is also prominent in embodied mathematics. This research describes some of the basic concepts of algebra as a branch of mathematics from the perspective of cognitive semantics and the hypothesis of embodied mathematics. From the perspective of Lakoff and Núñez, algebra has a kind of metonymy (whole for part), which they called the fundamental metonymy of algebra. This research describes algebraic concepts based on the fundamental metonymy of algebra and conceptual structures derived from human cognitive faculties, based on the perspective of embodied mathematics, which itself stems from the second generation of cognitive sciences and within the framework of cognitive semantics.
 
Method
The method of this researched is descriptive-analytical method based on language intuition and introspection-based viewpoint. The Algebraic concepts are used as research data.
Results
Algebra as a whole is a whole for part metonymy for other branches of mathematics like arithmetic and geometry. This metonymy is called fundamental metonymy of algebra. Based on this metonymy, basic algebraic concepts can described and analyzed in the cognitive semantics framework. In this viewpoint, analytical geometry in a product of the metonymy generalization which extends the algebra from arithmetic as source domain of a metaphor toward the geometry as target domain of the same metaphor. The concept of locus is a metaphoric blend which integrates arithmetic and geometry in a two-conditional statement and algebra plays the role of a whole for part metonymy in this integration. Identities, Equations and inequalities can be described based on the container image schema and the relation between part and whole in the metonymy. Mathematical relations are kind of blending that integrates two or more sets and mathematical functions are conceptual metaphors which project a domain on a target. Vectors are product of motion perception based on the source-path-goal image schema, and sets, also the sets algebraic operations, are mathematical embodiment of container image schema and its different aspects.
Conclusion
The description of Algebra in the cognitive semantics framework and based on embodied mathematics hypothesis leads us to conclude some generalizations which extends algebra and algebraic concepts to other branches of science and these generalizations are foundations of new concepts and fields as mechanics, trigonometry, topology, vector space and etc. This viewpoint can be used in the mathematical educations and the increasing effect and efficiency of using cognitive semantics in the educational methods of algebra-learning can be tested.
Ethical Considerations
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Funding
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Conflict of interest
The authors declare no conflict of interest