I'm sorry to ask the following question because it's seems that is one of bad myths in the opinion of mathematicians and students , But really i'm very interesting to imagine mathematics without Geometry how it w'd be , Then My question here is :Can we build mathematics without Geometry ?
This is an important question.
The short answer: Yes, via abstract algebra and provided we do not lose sight of the benefits of algebra vis-a-vis the many links between algebra and geometry. For a good look at topics in algebra that are independent of geometry, see
https://marinazahara22.files.wordpress.com/2013/10/i-n-herstein-topics-in-algebra-2nd-edition-1975-wiley-international-editions-john-wiley-and-sons-wie-1975.pdf
Aside: It is inadvisable to carry out the proposed program of building mathematics without geometry. I say this because, after all, we do live on a planet richly endowed with geometry and gain out intuitions about mathematical structures and relations from our experiences in the physical world.
Dear Rafik, you have posed an interesting question. We definitely cannot build mathematics in the way we know it today. However, there are many mathematical theories whose systems of axioms are free of geometrical statements (or intuition). For example, we can build number theory from the axioms of Peano, which make no reference to distances or angles. On the other hand, set theory can be consistently developed from the Zermelo's axiomatic system, which are also free from geometry.
In addition many concepts, initially based on geometry, have been conveniently redefined without their related geometric intuition. For instance, we can think of the n-th roots of unity as the vertices of the unit n-gon seated in the coordinate plane centered at the origin and with a vertex at (1,0). On the other hand, we can think of the n-th roots of unity as the zeros of the polynomial p(x) = x^n -1. The last concept, being more algebraic, allows us to generalize the definition of "n-th roots of unity" to any field (for example, finite fields) that is not necessarily embedded in Q, R, or C, even though the geometrical intuition is not that clear a priori in this context.
Having said that it is not only possible, but sometimes useful to do mathematics without geometry, I will agree with the aside note of James Peter. Without geometry, the richness of mathematics would be considerably reduced and, in particular, a significant number of applications will be lost as most of our mathematical intuition comes from the geometry of the world we live in.
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