CAN SYMBOLIC COMPUTATION AND FORMALIST SYSTEMS ENHANCE MATH EDUCATION WITH ARTIFICIAL INTELLIGENCE?

Year 2024, Volume: 26 Issue: 2, 487 - 504, 13.12.2024

Selçuk Yazar

https://doi.org/10.26468/trakyasobed.1435490

Abstract

In recent years, a solution developed using deep learning methods has been used to solve difficult problems in a field. The capability of deep learning models is that they require large and heavily sampled data sets. Computer Algebra Systems developed over time have made significant progress, especially in the field of symbolic mathematics solutions solved by machine learning. It is a persistent problem how appropriate it is to use such formal systems in some aspects of algorithmic decision-making. In this paper, we discussed the suitability of artificial intelligence applications to formal propositions by evaluating a deep learning study conducted especially in the field of symbolic mathematics and Math education. Symbolic computation systems have a strong potential for enhancing math education. Furthermore, within the framework of the Incompleteness Theorem, to show why the construction of a mathematical grammar is not a complete solution for Mathematics education systems.

Keywords

Symbolic Mathematics, Formal Systems, Computer Algebra Systems, Artificial Intelligence, Mathematics Learning.

SEMBOLİK HESAPLAMA VE FORMALİST SİSTEMLER MATEMATİK EĞİTİMİNİ YAPAY ZEKA İLE GELİŞTİREBİLİR Mİ?

Abstract

Son yıllarda, derin öğrenme yöntemleri kullanılarak geliştirilen bir çözüm, bir alandaki zor problemleri çözmek için kullanılmaktadır. Derin öğrenme modellerinin özelliği, büyük ve yoğun örneklenmiş veri setlerine ihtiyaç duymalarıdır. Zaman içerisinde geliştirilen Bilgisayar Cebiri Sistemleri, özellikle makine öğrenmesi ile çözülen sembolik matematik çözümleri alanında önemli ilerlemeler kaydetmiştir. Bu tür biçimsel sistemlerin algoritmik karar vermenin bazı yönlerinde kullanılmasının ne kadar uygun olduğu süregelen bir sorundur. Bu bildiride özellikle sembolik matematik ve Matematik eğitimi alanında yapılan bir derin öğrenme çalışması değerlendirilerek yapay zeka uygulamalarının formal önermelere uygunluğu tartışılmıştır. Sembolik hesaplama sistemleri matematik eğitimini geliştirmek için güçlü bir potansiyele sahiptir. Ayrıca, Eksiklik teoremi çerçevesinde, matematiksel bir gramer yapısı oluşturmanın Matematik eğitim sistemleri için neden tam bir çözüm olamayacağının gösterilmesi amaçlanmıştır.

Keywords

Sembolik Matematik, Biçimsel Sistemler, Bilgisayar Cebir Sistemleri, Yapay Zeka, Matematik Öğrenimi.

References

• Ardon, L. (2022). Reinforcement Learning to Solve NP-hard Problems: an Application to the CVRP. arXiv preprint arXiv:2201.05393.
• Bansal, K., Loos, S., Rabe, M., Szegedy, C., & Wilcox, S. (2019, May). Holist: An environment for machine learning of higher order logic theorem proving. In International Conference on Machine Learning (pp. 454-463). PMLR.
• Brown, T., Mann, B., Ryder, N., Subbiah, M., Kaplan, J. D., Dhariwal, P., ... & Amodei, D. (2020). Language models are few-shot learners. Advances in neural information processing systems, 33, 1877-1901.
• Chomsky, N., & Schützenberger, M. P. (1959). The algebraic theory of context-free languages. In Studies in Logic and the Foundations of Mathematics (Vol. 26, pp. 118-161). Elsevier.
• England, M. (2018). Machine learning for mathematical software. In Mathematical Software–ICMS 2018: 6th International Conference, South Bend, IN, USA, July 24-27, 2018, Proceedings 6 (pp. 165-174). Springer International Publishing. doi: 10.1007/978-3-319-96418-8_20
• Flavio, P., Alberto, T., Alessandro, S. (2023). A Hybrid System for Systematic Generalization in Simple Arithmetic Problems. 289-301. doi: 10.48550/arXiv.2306.17249
• Gödel, K. (1931). Über formal unentscheidbare Sätze der Principia Mathematica und verwandter Systeme I. Monatshefte für mathematik und physik, 38, 173-198.
• Heid, M. K., Thomas, M. O., & Zbiek, R. M. (2012). How might computer algebra systems change the role of algebra in the school curriculum?. In Third international handbook of mathematics education (pp. 597-641). New York, NY: Springer New York. doi: 10.1007/978-1-4614-4684-2_20
• Hilbert, D., & Ackerman, W. (1928). Theoretische logik. Julius Springer, Berlin.
• Irving, G., Szegedy, C., Alemi, A. A., Eén, N., Chollet, F., & Urban, J. (2016). Deepmath-deep sequence models for premise selection. Advances in neural information processing systems, 29.
• Kaliszyk, C., Urban, J., Michalewski, H., & Olšák, M. (2018). Reinforcement learning of theorem proving. Advances in Neural Information Processing Systems, 31.
• Kaneko, M., Maeda, Y., Hamaguchi, N., Nozawa, T., & Takato, S. (2013). A Scheme for Demonstrating and Improving the Effect of CAS Use in Mathematics Education. 2013 13th International Conference on Computational Science and Its Applications, 62-71. doi: 10.1109/ICCSA.2013.19
• Lample, G., & Charton, F. (2019, September). Deep Learning For Symbolic Mathematics. In International Conference on Learning Representations.
• Long, Z., Lu, Y., & Dong, B. (2019). PDE-Net 2.0: Learning PDEs from data with a numeric-symbolic hybrid deep network. Journal of Computational Physics, 399, 108925. doi: https://doi.org/10.1016/j.jcp.2019.108925.
• Lucas, J. (1996). Minds, machines and Gödel: A retrospect. Artificial intelligence: critical concepts, 3, 359-76. Magma Computational Algebra System. http://magma.maths.usyd.edu.au/magma/
• Makishita, H. (2014). Practice with computer algebra systems in mathematics education and teacher training courses. In Mathematical Software–ICMS 2014: 4th International Congress, Seoul, South Korea, August 5-9, 2014. Proceedings 4 (pp. 594-600). Springer Berlin Heidelberg. doi: 10.1007/978-3-662-44199-2_89
• Müller, D., Gauthier, T., Kaliszyk, C., Kohlhase, M., & Rabe, F. (2017, June). Classification of alignments between concepts of formal mathematical systems. In International Conference on Intelligent Computer Mathematics (pp. 83-98). Cham: Springer International Publishing.
• Peitgen, H. O., & Richter, P. H. (1986). The beauty of fractals: images of complex dynamical systems. Springer Science & Business Media.
• Pickering, L., del Río Almajano, T., England, M., & Cohen, K. (2024). Explainable AI Insights for Symbolic Computation: A case study on selecting the variable ordering for cylindrical algebraic decomposition. Journal of Symbolic Computation, 123, 102276.
• Pochart, T., Jacquot, P., & Mikael, J. (2022, March). On the challenges of using D-Wave computers to sample Boltzmann Random Variables. In 2022 IEEE 19th International Conference on Software Architecture Companion (ICSA-C) (pp. 137-140). IEEE.
• Rav, Y. (2007). A critique of a formalist-mechanist version of the justification of arguments in mathematicians' proof practices. Philosophia Mathematica, 15(3), 291-320.
• Searle, J. R. (1980). Minds, brains, and programs. Behavioral and Brain Sciences, 3(3), 417–424. https://doi.org/10.1017/S0140525X00005756.
• Seidametova, Z. (2020). Combining Programming and Mathematics through Computer Simulation Problems. In ICTERI Workshops (Vol. 2732, pp. 869-880).
• Turing, A. M. (1936). On computable numbers, with an application to the Entscheidungsproblem. J. of Math, 58(345-363), 5.
• Q. Wang, C. Kaliszyk, and J. Urban, "First experiments with neural translation of informal to formal mathematics," 2018: Springer, pp. 255-270.
• Wolfram Mathematica. https://www.wolfram.com/mathematica/ Zhao, C., Yang, L. (2022). Symbolic Deep Learning for Structural System Identification. Journal of Structural Engineering-asce, 148(9) doi: 10.1061/(asce)st.1943-541x.0003405