Is it not logically wrong to make proofs in one field of mathematics, by using tools belonging to another one, since they start from different axioms?

<p>It is as long as one does not prove the existence of an isomorphism (which between algebra and geometry is provable).</p> <p>If one postulates the continuity of real numbers and the continuity of points on a line, it can be shown that there is a one-to-one correspondence between numbers and points and that this correspondence preserves distances and proportions.</p> <p>If one postulates parallelism between the lines of two systems, one can construct a flat grid. And it can be shown that there is a similar correspondence between the points in the plane and those in the grid. That, again, preserves vector distances and geometric similarities.</p> <p><a href="https://frisellagiuseppe.medium.com/is-it-not-logically-wrong-to-make-proofs-in-one-field-of-mathematics-by-using-tools-belonging-to-4a8657ccf20c"><strong>Read More</strong></a></p>