The Role of Mathematical Methods in the Formation of Scientific Conceptions: Elementarity and Mathematics

Abstract

This article presents a systematic philosophical and methodological analysis of the role of mathematical methods in the formation of scientific knowledge and theoretical frameworks. Particular attention is given to the concept of elementarity and its function as a fundamental methodological principle in mathematical reasoning, scientific modeling, and the construction of theoretical structures. The study argues that mathematics should not be regarded merely as a technical instrument for describing empirical phenomena, but rather as a constitutive and normative foundation of scientific thought. In addition to classical philosophical perspectives—those of Plato, Aristotle, and Kant—the article examines major developments in contemporary philosophy of science, including Hilbert’s axiomatic program, Gödel’s incompleteness theorems, and Lakatos’s theory of research programmes. Through a comparative analysis, it is demonstrated that mathematical methods provide the conditions for universality, precision, and logical coherence in the formation of scientific conceptions. It is concluded that the principle of elementarity, together with the systematic application of mathematical methods, constitutes one of the essential foundations for the rational development and methodological integrity of science.

Keywords

• mathematical methods
• scientific conceptions
• elementarity
• philosophy of science
• methodology
• mathematical abstraction

Bilimsel Kavramsallaştırmaların Oluşumunda Matematiksel Yöntemlerin Rolü: Elemanterlik ve Matematik

Öz

Bu çalışma, matematiksel yöntemlerin bilimsel bilginin ve kuramsal yapıların oluşumundaki rolünü felsefi ve metodolojik bir çerçevede sistematik olarak incelemektedir. Çalışmada özellikle elemanterlik kavramına odaklanılarak, bu kavramın matematiksel düşünme, bilimsel modelleme ve kuramsal yapıların kurulmasında temel bir yöntemsel ilke olarak nasıl işlev gördüğü ortaya konulmaktadır. Bu bağlamda matematiğin yalnızca ampirik olguların betimlenmesinde kullanılan teknik bir araç olmadığı, aksine bilimsel düşüncenin kurucu ve normatif temellerinden biri olduğu gösterilmektedir. Platon, Aristoteles ve Kant’ın yaklaşımlarının yanı sıra, Hilbert’in aksiyomatik programı, Gödel’in eksiklik teoremleri ve Lakatos’un araştırma programları kuramı gibi çağdaş bilim felsefesi yaklaşımları karşılaştırmalı olarak ele alınmaktadır. Yapılan çözümlemeler, matematiksel yöntemlerin bilimsel kavramsallaştırmaların oluşumunda evrensellik, kesinlik ve mantıksal tutarlılık sağlayan temel koşulları oluşturduğunu göstermektedir. Sonuç olarak, elemanterlik ilkesi ile matematiksel yöntemlerin sistematik kullanımı, bilimin rasyonel gelişimi ve metodolojik bütünlüğünün sağlanmasında temel belirleyiciler olarak ortaya konulmaktadır.

Anahtar Kelimeler

• matematiksel yöntemler
• bilimsel kavramsallaştırmalar
• elemanterlik
• bilim felsefesi
• metodoloji
• matematiksel soyutlama

Kaynakça

1. Abasova, L., & Mahammadli, D. (2025). The establishment and activities of the Mahammad Fuzuli Institute of Manuscripts under the Azerbaijan National Academy of Sciences (up to 1987). Akademik Tarih ve Düşünce Dergisi, 12(5), 1015–1030.
2. Aliyeva, G., İsmayılov, N., & Mahammadli, D. (2025a). The publication of Jack London’s works (1876–1916): Issues of history, literary form, and technical design. Akademik Tarih ve Düşünce Dergisi, 12(4), 960–974.
3. Aliyeva, G., İsmayılov, N., & Mahammadli, D. (2025b). The methodology of forming the print run policy in Western Europe. Akademik Tarih ve Düşünce Dergisi, 12(4), 975–987.
4. Alizadeh, H., & Mahammadli, D. (2025). Problems of promoting multiculturalism ideas in libraries. Akademik Tarih ve Düşünce Dergisi, 12(3), 808–823.
5. Aristotle. (1998). Metaphysics. Oxford University Press.
6. Askerova, S. V., & Mammadov, E. E. (2025). Children in Azerbaijan at the beginning of the third millennium: Social analysis based on demographic and cultural indicators. Pakistan Journal of Life and Social Sciences, 23(1), 8304–8313.
7. Balayeva, E., & Mahammadli, D. (2025). Contemporary trends in the publication of scientific literature: The experiences of the United States and Canada. Akademik Tarih ve Düşünce Dergisi, 12(4), 853–869.
8. Balginova, K. M., Maydangalieva, Z. A., Satygalieva, G. B., & Mahammadli, D. (2018). The digital Kazakhstan. Vestnik NAN RK, 6, 82–94.

9. Bayramov, A., & Mahammadli, D. (2025). Systematic approach methodology for studying the library collection. Akademik Tarih ve Düşünce Dergisi, 12(2), 844–851.
10. Bayramov, A., İsmayılov, N., & Mahammadli, D. (2025). Book publishing in South America: Historical development and contemporary issues. Akademik Tarih ve Düşünce Dergisi, 12(5), 1055–1077.
11. Bayramzadeh, S. Z. O., & Kazimi, P. F. O. (2020). Ethnogenesis of minor peoples or the search for eternal history. American Scientific Journal, 42(2), 4–10.
12. Gödel, K. (1986). Collected works. Oxford University Press.
13. Heydar, M. D. (2023). Library and information infrastructure of the regions of Azerbaijan Republic. Technium Social Sciences Journal, 39, 782.
14. Hilbert, D. (1964). Foundations of geometry. Open Court.
15. Ismayilov, K., Ismayilov, N., & Mammadova, V. (2019). Library information services in academic libraries of Azerbaijan: A comparative study. Library Management, 40(6/7), 461–477.
16. Ismayilov, N. (2015). Development of Azerbaijani bibliography in the late 19th and early 20th centuries. Bibliotekovedenie, 4, 88–91.
17. Ismayilov, N. (2022). Library resources: Allocation and usage problems. Grani, 26(1), 38–43.
18. Ismayilov, N., & Aliyeva, G. (2023). Creation and formation of document flow in library bibliography. Grani, 26(3), 165–170.
19. Ismayilov, N., & Bayramova, I. (2022b). Methodology of studying document flow in tourism. Grani, 25(3), 70–76.
20. Ismayilov, N., & Khalafova, S. (2022). Library sites that provide information to users. Grani, 25(2), 22–28. Ismayilov, N., & Mahammadli, D. (2024). Qərbi Azərbaycan tarixinə dair sənəd axınının təşəkkülü və inkişafı. Akademik Tarih ve Düşünce Dergisi, 11(6), 4483–4496.
21. Ismayilov, N., & Sadigova, S. (2022). Synergetic foundations of modern bibliography methodology. Grani Journal, 26(5), 45–51.
22. Ismayilov, N. I., Mahammadli, D., & Khudiyeva, V. (2022). Methods and means of information search in the digital environment. Grani, 25(5), 31–34.
23. Ismayilov, N., Mahammadli, D., & Gasimli, H. (2023a). Organization and development of document flow in tourism regions. Grani, 26(2), 122–127.
24. Ismayilov, N., Mahammadli, D., & Gasimli, H. (2023b). Characteristics of library information services. Akademik Tarih ve Düşünce Dergisi, 10(3), 878–890.
25. Ismayilov, N., Bayramova, I., & Mahammadli, D. (2025). Historical and contemporary issues of book publishing in Southern Europe. Akademik Tarih ve Düşünce Dergisi, 12(5), 905–928.
26. Kant, I. (1998). Critique of pure reason. Cambridge University Press.
27. Karabalina, A. A., Maydangalieva, Z. A., Satygalieva, G. B., Ahmetalina, G. A., & Mahammadli, D. (2018). Family pattern and academic performance. Bulletin NAS RK, 6, 58–66.
28. Kazimi, P. (2021). The problem of reliability of information in the global network. Journal of Practical Studies in Education, 2(5), 32–34.
29. Kazimi, P. (2025). Informasiya təminatında tələbatçı məmnuniyyəti. Akademik Tarih ve Düşünce Dergisi, 12(4), 681–689.
30. Kazimi, P. F. (2018). Information engineering. Baku.
31. Kazimi, P. F. O., & Aliyeva, A. (2019). Evaluation criteria of censorship. Historical Sciences, 17, 65–72.
32. Kazimi, P. F. O., Oqlu, I. I. A., & Qizi, Y. G. Y. (2022). Philosophical view on information theory (The path from the divine to the digital world). Technium Social Sciences Journal, 32, 724–731.
33. Kazimi, P., & Balayeva, E. (2024). Ethnocultural description of the Persians (Socio-historical analysis). Akademik Tarih ve Düşünce Dergisi, 11(1), 193–199.
34. Kazimi, P., & Kunanets, N. (2014). Library education center in Azerbaijan. Bibliotechnyi Visnyk, 3, 26–31.
35. Kazimi, P., & Mahammadli, D. (2021). Digital use in library services. Technium Social Sciences Journal, 25, 819.
36. Kenzhebayeva, D. K., Urmurzina, B. G., & Mahammadli, D. (2018). The modern youth values in Kazakhstan. News of NAS RK, 6(322), 51–66.
37. Khalafova, S., & Ismayilov, N. (2024a). National bibliography in cryptographic information systems. Journal of Theoretical Studies, 27(4), 32–36.
38. Khalafova, S., & Ismayilov, N. (2024b). Information provision of tourism and economy. GeoJournal of Tourism and Geosites, 56(4), 1809–1815.
39. Khalafova, S., & Mahammadli, D. (2025). Book publishing in the Baltic States. Akademik Tarih ve Düşünce Dergisi, 12(5), 139–160.
40. Khalafova, S., Mahammadli, D., & Manafova, U. (2025). Cataloguing literature in Azerbaijan. Akademik Tarih ve Düşünce Dergisi, 12(5), 871–891.
41. Kushzhanov, N. V., & Dashgin, M. (2019a). Digital agenda of EAEU. Bulletin NAS RK, 2-S, 55–61.
42. Kushzhanov, N. V., & Dashqin, M. (2019c). Digital agenda of EAEU. Vestnik NAN RK, 2, 55–61.
43. Kushzhanov, N. V., & Mahammadli, D. (2019b). The digital transformation of the oil and gas sector in Kazakhstan: Priorities and problems. News of the National Academy of Sciences of the Republic of Kazakhstan, 3(435), 203–212.
44. Lakatos, I. (1976). Proofs and refutations. Cambridge University Press.
45. Mahammadli, D. (2024). Functions of library websites. Akademik Tarih ve Düşünce Dergisi, 11(6), 4708–4717.
46. Mahammadli, D. H. (2024). Library infrastructure informatization issues. Akademik Tarih ve Düşünce Dergisi, 11(1), 611–619.
47. Mammadov, A., & Ismayilov, N. (2023). National libraries: Typology, main functions and modern directions of development. Scientific and Theoretical Almanac Grani, 26(1), 15–20.
48. Mammadov, E. E. (2013). Role of the periodical fund of the National Library. IJISM, 11(1), 95–104.
49. Mammadov, E. E. (2022a). The national library of Azerbaijan: Functioning during COVID. International Leads, 37(2), 10–17.
50. Mammadov, E. E. (2022b). History of books and libraries in Azerbaijan. Baku.
51. Mammadov, E., & Mahammadli, D. (2025). Modern problems of library resources management. Akademik Tarih ve Düşünce Dergisi, 12(3), 996–1007.
52. Mammadov, E., Mahammadli, D., & Karimov, E. (2025). Book culture in Southeast Asia. Akademik Tarih ve Düşünce Dergisi, 12(5), 13–36.
53. Mehraliyeva, N., & Mahammadli, D. (2025). Museum activities and the book. Akademik Tarih ve Düşünce Dergisi, 12(5), 226–239.
54. Mustafayev, A., & Kazimi, P. (2023). Motivation and nature of choosing teaching profession. Akademik Tarih ve Düşünce Dergisi, 10(5), 1823–1830.
55. Mustafayev, A., Yusifova, G., & Kazimi, P. (2023). Post-traumatic stress disorders. Grani, 26(2), 51–54.
56. Nadir, I., & Sevda, K. (2022). Local lore documental network resources. Technium Social Sciences Journal, 33, 628.
57. Oqlu, K. P. F., Nadir, I., & Tofiq, R. Z. (2023). Origins of Azerbaijani scientific library activity. Grani, 26(4), 125–129.
58. Plato. (2008). The Republic. Oxford University Press.
59. Popper, K. (2002). The logic of scientific discovery. Routledge.
60. Qasimli, H., & Mahammadli, D. (2024a). Information demand in tourism. Akademik Tarih ve Düşünce Dergisi, 11(6), 4588–4595.
61. Qasimli, H., & Mahammadli, D. (2024b). Bibliographic service in tourism. Akademik Tarih ve Düşünce Dergisi, 11(6), 4718–4726.
62. Rzayeva, Z., & Mahammadli, D. (2025). Some aspects of book publishing in Eastern European countries: Historical and methodological specifics. Akademik Tarih ve Düşünce Dergisi, 12(5), 95–110.
63. Takhirov, K. M., & Ismayilov, N. I. (2011). Development of national bibliographic resources. Bibliografiya, 2, 50–55.
64. Tofiq, R. Z., Oqlu, M. M. A., & Kazimi, P. F. O. (2022). Library management application issues. Technium Social Sciences Journal, 37, 774.
65. Wigner, E. (1960). The unreasonable effectiveness of mathematics. Communications in Pure and Applied Mathematics, 13(1), 1–14.