Neutrosophic mathematics is a branch of mathematics that deals with the representation and analysis of indeterminate, inconsistent, and uncertain information. It provides a mathematical framework to handle ambiguity and contradiction in various domains. This paper presents a comprehensive study of the evolution of Neutrosophic math- ematics, covering its foundations, applications, and future prospects. The foundations of Neutrosophic mathematics are explored, in- cluding axiomatic systems, mathematical structures, and logical rea- soning. Axiomatic systems establish the rules and principles govern- ing neutrosophic sets and logic, while mathematical structures provide frameworks for representing and manipulating neutrosophic informa- tion. Logical reasoning enables the systematic handling of uncertainty and contradiction in mathematical reasoning.