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We hope that a student, interested in mathematics as a major or because of its role in their career path, will ponder these questions guided by the brief re ections given here. Mathematics has been called the "Queen of the Sciences". History reveals to us the reasons for this and reveals the dual nature of mathematics providing guiding light toward both the above questions. On one hand, mathemat- ics is a content driven discipline constructed via pure mathematical research following the deductive paradigm of axioms, theorems and proofs. This be- gan primarily in the Greek Era over 2000 years ago with the development of Euclidean geometry and overtime has owered into deep areas of mathemat- ical thought: algebra, analysis, combinatorics, geometry, and number theory to name a few. On the other hand, mathematics is a powerful language to model and understand the physical world. Indeed, mathematics provides a tool in physics providing foundational models in mechanics, thermodynam- ics, electricity and magnetism, quantum mechanics, and relativity. In the 19th and 20th centuries, the breadth of mathematical applications grew im- mensely: the industrial age spawned applications of mathematics thoughout the enginneering world; applications in modeling population dynamics, cel- lular function and dynamics were discovered leading to a diversity of mathe- matical thought in the biological sciences and medicine; the computer age has led to new results in discrete mathematics and in applying number theory to encryption algorithms; and most recently mathematics has found signi cant application in the analysis of nancial markets. The meaning of the word "dual" should be clear: often, the results of mathematics provide the tool in the applied world and just as often, the desire to construct a model leads to new mathematical ideas on which to apply the deductive paradigm. Download free Undergraduate Handbook for Mathematics & Statistics.pdf here