Is geometry obligatory in higher mathematics?

Question

higher mathematics

In the future I want to start higher mathematics, I understand algebra perfectly, but I don’t understand geometry, I can’t solve problems, and when I go to the GDZ, it turns out that everything is elementary there, so, is it possible to do without geometry in higher mathematics?

Answer 1

It is possible
When studying higher mathematics, it is quite possible to do without knowledge of school geometry. Usually in technical specialties in higher educational institutions they study analytic geometry, which is almost entirely based on algebra, because. all constructions can be carried out in one or another coordinate system using analytical relations.
To fully study mathematics, I advise you to start with the foundations - to master mathematical logic, set theory, then the basics of algebra and mathematical analysis. All mathematical theorems can be proved using only axioms and mathematical logic. The construction of mathematics in this case turns out to be abstract, without appeal to visibility; using formulas, but without drawings.
Geometry can also be developed on an abstract-axiomatic basis, using the axioms of Euclid and his followers - here I can recommend D. Hilbert's book "Fundamentals of Geometry". But the resulting theory is quite complex, and most studies not directly related to the foundations of mathematics now use analytic geometry and its branches, and not the geometry that is studied in school, based on the axioms of Euclid. In two or three dimensions, both points of view of geometry give equivalent results, and in the study of spaces with four or more dimensions, analytic theory is used almost exclusively. Studies that would attempt to generalize Euclid's axioms to more than three dimensions belong more to mathematical exoticism than to the "mainstream".