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When discussing monumental achievements in the world of mathematics, the name Andrew Wiles stands out prominently. His journey is not just of personal triumph but also a milestone in the history of mathematics. In the early 1990s, Wiles, an English mathematician, set out to prove a conjecture that had puzzled mathematicians for over 300 years: Fermat's Last Theorem.

The theorem, simply stated, posits that there are no whole number solutions to the equation \( x^n + y^n = z^n \) for any integer value of n greater than two. What Wiles accomplished was to link this problem with another conjecture, the Taniyama-Shimura-Weil conjecture, which involves the properties of elliptic curves. Through years of dedicated work, often in solitude, Wiles was able to provide a proof that not only resolved Fermat's Last Theorem but also helped advance other areas of mathematics, such as number theory and algebraic geometry.

Andrew Wiles's approach to solving this problem exemplifies the essence of complex mathematical problem-solving. The characteristics of perseverance, creativity, and the relentless pursuit of truth, despite the difficulties or challenges faced, are as essential to understanding his success as the mathematical techniques he employed. His story is not only about solving an old mathematical riddle, but it's also an inspiration for anyone who faces seemingly insurmountable challenges.

One of the essential tools in the mathematician's toolkit is the principle of mathematical induction, an approach that's particularly significant in proving statements related to an infinite number of cases. Mathematical induction is a form of proof that is used to establish the truth of an infinite sequence of propositions.

The process of induction involves two main steps. First, prove that the base case, often the statement for the integer 1, is true. Second, prove that if any one statement in an infinite sequence of statements is true, then so is the next one. This is often referred to as the inductive step. If both parts are proven, it's as if one has an unbroken chain of dominoes falling one after another, and thus the statement is proven for all natural numbers.

It's important to note that while mathematical induction was not directly used to prove Fermat's Last Theorem by Andrew Wiles, understanding this technique is crucial when delving into higher mathematics. It illustrates one of the fundamental strategies used for proving a wide array of theoretical concepts. Inductive reasoning underlines the beauty of mathematics; from one truth, an infinite number of others can be born.

Complex mathematical problem solving is an intriguing and challenging task that necessitates a deep and comprehensive understanding of various mathematical concepts and an ability to connect these concepts in novel ways. Solving intricate problems is much like a detective solving a mystery; it requires keen insight, persistence, and sometimes, moments of sheer genius.

The pursuit of solutions to complex problems often includes dealing with moments of frustration, facing dead ends, and experiencing eureka moments. One essential characteristic of complex problem solving is the ability to see the problem from different perspectives and to use a combination of analytical approaches. It is these multi-faceted and creative strategies that are the hallmark of great mathematical minds like Andrew Wiles.

Furthermore, a supportive community, rigorous testing of hypotheses, and the accumulation and synthesis of knowledge across the discipline all play a crucial role. Consequently, problem-solving in mathematics goes beyond mere computation; it encompasses the formulation of theories, testing them with logic and proof, and often, invoking them in broad, uncharted intellectual territories. It's a testament to human intellect and its ceaseless journey to understand and explain the universe through the universal language of mathematics.