The paper summarizes the contributions of the three philosophies of mathematics—logicism, intuitionism-constructivism (constructivism for short) and formalism and their rectification—which constitute the new foundations of mathematics. The critique of the traditional foundations of mathematics reveals a number of errors including inconsistency (contradiction or paradox) and undefined and vacuous concepts which fall under ambiguity. Critique of the real and complex number systems reveals similar defects all of which are responsible not only for the unsolved long standing problems of foundations but also of traditional mathematics such as the 379-year-old Fermat’s last theorem (FLT) and 274-year-old Goldbach’s conjecture. These two problems require rectification of these defects before they can be resolved. One of the major defects is the inconsistency of the field axioms of the real number system with the construction of a counterexample to the trichotomy axiom that proved it and the real number system false and at the same time not linearly ordered. Indeed, the rectification yields the new foundations of mathematics, constructivist real number system and complex vector plane the last mathematical space being the rectification of the complex real number system. FLT is resolved by a counterexample that proves it false and the Goldbach’s conjecture has been proved both in the constructivist real number system and the new real number system. The latter gives to two mathematical structures or tools—generalized integral and generalized physical fractal. The rectification of foundations yields the resolution of problem 1 and the solution of problem 6 of Hilbert’s 23 problems.
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