**Mathematics Has Become a Part In Our Modern Civilizations**

**"The World of Mathematics and The Outside World"**

In the process for the spread on development of mathematical topics that we have been seen in our Introduction 01 and Introduction 02, the discoverer and inventor in Mathematics have received enough information from their analysis throughout an environment surround them, in which this model at this moment called Modelling and Formulating (Schwartz, 1996). For the example, from surrounding us that we have seen about, then the ancient mathematicians have:

· the concept of a straight line, which based on the height of the trees that can be seen.

· the concept of circle, which based on the natural circular object like a sun.

· the concept of a sphere, which based on the natural object like a fruit.

· the concept of a cylinder which based on natural objects like the falling tree.

· the concept of angle in geometry from the natural object like an angle that has been formed by hand or foot in several kinds of position, that exist in our nature.

The discoverer and the inventor of theory mathematics have designed this natural object into a recognizable named object in the study of Mathematics we called geometry with their connection or meaning among them, in which at this moment called as Transforming and manipulating (Schwartz, 1996). In the very early age of Mathematics, those inventors and those discoverers in Mathematics testing the idea of this mathematics into several kinds of a simple practical project, like the process of building the canal, or dividing a section on the land into several sections in order for taxing, in which this method in this 21^{st} century of math being called Inferring (Schwartz, 1996). Then, they starting to build the connection between several kinds of geometrical objects for satisfying their intellectual curiosity. In the beginning, they just collecting a pure conceptual of mathematics. While the passionate of mathematics developing those concepts, most of the time some of them will be looks like going to the other world. They are not worried about using the equations that related to 4 dimensions or 10 dimensions, or any other multi-dimensions. German Mathematician, Georg Friedrich Bernhard Riemann design a geometric foundation for complex analysis through one dimensional manifold with an atlas of charts to the open unit disk, such that provides a way of comparing two charts of an atlas, by comparing the composition of one chart with the inverse of the other under restriction to the intersection of their domains of the definition are a function that has a subset of the complex numbers as a domain and the complex numbers as a codomain, that is generally supposed to have a domain that contains a nonempty open subset of the complex plane that is differentiable at every point of an open subset of the complex plane (Murray Ralph Spiegel, 1964). Another geometry design by Russian Mathematician, Nikolai Ivanovich Lobachevsky, and Hungarian Mathematician Janos Bolyai started with the assumption for any given line *R* and point *P* not on *R*, in the plane containing both line *R* and point *P* there are at least two distinct lines through *P* that do not intersect *R*, in which at that time called as Bolyai-Lobachevskian Geometry and in the modern version called as Euclid’s parallel postulate. Another, Germany Mathematician named Georg Cantor define the theory of “set” in the simplest way, and the importance of one-to-one correspondence between the members of two sets, defined infinite and well-ordered sets, with also proven that the real numbers are more numerous than the natural number (Joseph Warren Dauben, 1993), and in that time, seems like not so useful on the perception of application, but now his theory becomes a fundamental theory in mathematics learning.

Furthermore, mathematics has been informed about the attention to the pure abstract idea has a danger. As is explained by American-Jewish Mathematician Morris Kline in “*Mathematics and the Physical World (1960)*”, the most mathematician may reach the abstract of the universe, but they have and surely will have to back to Earth for recharging and/or will be out of breath in the mental.

Moreover, a form of the speculation of abstract in mathematics, those speculations may be applicable in some years later. At the end of this moment, we know now about geometries by G. F. B. Riemann has been proven how useful it for Albert Einstein in his time in developing the theory of relativity at that time and for us to learn and develop it at this moment. Also, the Theory of “Set” by Georg Cantor has been applied in several kinds of applications, including in algebra and statistics. Regarding the concept of multi-dimension, those speculations have been tested, especially in the test of view on the screen either as tube television, flat, curve television, etc.

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**Reference**:

Ahlfors, L. *Complex Analysis, 3 ^{rd} ed*. Published by: McGraw-Hill, 1979

Dauben, Joseph W. (1977). *Georg Cantor and Pope Leo XIII: Mathematics, Theology, and the Infinite*. Published by Journal of the History of Ideas. DOI: 10.2307/2708842. JSTOR: 2708842

Dauben, Joseph W. (1979). *Georg Cantor: His Mathematics and Philosophy of the Infinite*. Published by: Boston: Harvard University Press. ISBN: 978-0-691-02447-9

Dauben, Joseph W. (2004). *Georg Cantor and the Battle for Transfinite Set Theory*. Proceedings of the 9^{th} ACMS Conference (Westmont College, Santa Barbara, California). Pp. 1-22. Published in Journal of the ACMS 2004

Dénes, Tamás (2011). *Real Face of János Bolyai*. Published by Notices of the American Mathematical Society. P.42-52. Retrieved 2011-06-18

du Sautoy, Marcus (2003). *The Music of the Primes: Searching to Solve the Greatest Mystery in Mathematics*. Published by: Harper Collins. ISBN: 978-0-06-621070-4

Hallett, Michael (1986). * Cantorian Set Theory and Limitation of Size*. Published by: New York: Oxford University Press. ISBN: 978-0-19-853283-5

Hawking, Stephen (2005). *God Created the Integers*. Published by: Running Press, pp. 697-703. ISBN: 978-0-7624-1922-7

Hawking, Stephen (2005). *God Created the Integers*. Published by: Running Press, pp. 814-815. ISBN: 978-0-7624-1922-7

Ji, L., Papadopoulos, A., and Yamada, S., (2017). *From Riemann to Differential Geometry and Relativity*. Published by: Springer, 2017. ISBN: 978-3-319-60039-0

Kline, Morris (1959). *Mathematics and the Physical World*. Published by John Murray in London, 1960.

Rudin, W. *Real and Complex Analysis, 3 ^{rd} ed*. Published by: McGraw-Hill, 1986

Spiegel, Murray R. *Theory and Problems of Complex Variables – with an introduction to Conformal Mapping and Its Application*. Published by: McGraw-Hill, 1964

Lobachevskii, N. I. (1856). *Geometrical Researches on The Theory of Parallels, … tr. from the original by George Bruce Halsted*. Published by: Ann Arbor, Michigan: University of Michigan Library, 2005. Retrieved from: https://quod.lib.umich.edu/cgi/t/text/text-idx?c=umhistmath;idno=AAN2339 [Accessed on Feb 16, 2021]

Weyl, Hermann (2009). *The Concept of a Riemann Surface, 3 ^{rd} ed*. Published by: Ney York Dover Publications, ISBN: 978-0-486-47004-7, MR 0069903

Victor J. Katz (2009). *A History of Mathematics: Introduction*. Published by: Addison-Wesley, 2009