Research ArticleOpen AccessGoogle Scholar indexed
Algebraic Structures of Mathematical Foundations
University of Duesseldorf, Duesseldorf, Germany
- 1 University of Duesseldorf, Duesseldorf, Germany
Open Journal of Philosophy·Volume 10 (2019)·Pages 137–142·Published 13 December 2019·DOI10.4236/ojpp.2020.101010
Copy link · social · email
Abstract
In this paper, we continue to examine the role of algebra underlying geome- try. The mathematics that we examine are abstract and are quite far removed from applications. Those who may wish to examine the possible uses of ab- stract geometry in physics might well begin by reading (Schlichenmaier, 2007; Bub, 1997) . The method of research that we apply is to use the logic of H. S. Leonard to extract the underlying algebra of mathematical proofs and then use the results for the purpose of finding the algebra that underlies mathe- matical results.
KeywordsRingsFactorizationIdealsPolynomials
- Apéry, F. (1987). Models of the Real Projective Plane. Braunschweig: Vieweg. https://doi.org/10.1007/978-3-322-89569-1
- Bézout, é. (2006). General Theory of Algebraic Equations. Princeton, NJ: Princeton University Press. (Translated by Eric Feron from Theorie général des équations algébrique, Paris, 1770)
- Bub, J. (1997). Interpreting the Quantum World. Cambridge: Cambridge University Press.
- Gibson, C. (1998). Elementary Geometry of Algebraic Curves. Cambridge: Cambridge University Press. https://doi.org/10.1017/CBO9781139173285
- Hartshorne, R. (1997). Geometry: Euclid and Beyond. New York: Springer Verlag.
- Jones, R. M. (2014). Model Theories of Set Theories and Type Theory. Open Journal of Philosophy, 4, 54-58. https://doi.org/10.4236/ojpp.2014.41008
- Kendig, K. (2015). Elementary Algebraic Geometry (2nd ed.). Chicago, IL: Dover.
- Knudson, K. P. (2015). Morse Theory, Smooth and Discrete. Hackensack, NJ: World Scientific. https://doi.org/10.1142/9360
- Leonard, H. S. (1969). The Logic of Existence. Philosophical Studies, 7, 49-64. https://doi.org/10.1007/BF02221764
- Moore, G. H. (2002). Hilbert on the Infinite, the Role of Set Theory in the Evolution of Hilbert’s Thought. Historia Mathematica, 29, 40-64. https://doi.org/10.1006/hmat.2001.2332
- Murphy, G. (1990). C*-Algebras and Operator Theory. San Diego, CA: Academic Press.
- Pelletier, F. J. (1999). A Brief History of Natural Deduction. History and Philosophy of Logic, 20, 1-31. https://doi.org/10.1080/014453499298165
- Sharpe, D. (1987). Rings and Factorization. Cambridge: Cambridge University Press. https://doi.org/10.1017/CBO9780511565960
- Sylvester, J. (1973). The Collected Mathematical Papers of James Joseph Sylvester (Volume I (1837-1853), pp. 247, 263, 249). New York: Chelsea Publishing Company.
- Schlichenmaier, M. (2007). An Introduction to Riemann Surfaces, Algebraic Curves and Moduli Spaces. Berlin: Springer.