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This book collects and streamlines materials I have developed for my course on learning mathematics as a foreign language. These ideas, originally conceived as teaching aids, serve as the foundation for this publication. The book includes approximately 190 exercises of varying difficulty, with detailed answers to all of them provided in the Appendix, supporting independent study and self-assessment. Let me highlight several features that distinguish this book from typical mathematics textbooks. First, the goal is not to teach mathematics or mathematical logic per se, although the text abounds with examples from these disciplines. Rather, the aim is to reveal how transparent and comprehensible the lingua franca of the modern exact sciences can be once one undergoes a period of adaptation—a journey for which this book is intended to be your guide. This endeavor is particularly relevant today, given the rapid advancement of artificial intelligence—entities for which mathematical language is native. The algorithms controlling robots, neural networks analyzing data, and large language models generating text are all built upon the strict laws of logic and algebra that constitute the grammar of mathematics. While mastering this language does not grant a complete understanding of the ”mind” of a machine, it provides essential insight into the formalized structures that govern auto- mated decision-making. Second, this work is designed primarily as supplementary reading rather than a traditional textbook. The language of mathematics differs profoundly from any spoken language, and mastery is defined by criteria distinct from standard proficiency scales. Therefore, this book is best used to bridge the gap between intuitive understanding and rigorous formalism, complementing standard courses in logic, set theory, or algebra. Third, the book does not claim to provide a scientifically precise linguistic description of mathematical phenomena. Passages venturing beyond mathematics and logic should be viewed as illustrative descriptions rather than rigorous scientific linguistic analysis. While this work explores the internal structure of mathematical language from a logico-mathematical perspective, formal linguistic analysis lies beyond its scope and remains an exciting direction for future research. Fourth, particular attention is paid to the anatomy of proofs. The text offers a level of detail often omitted in standard textbooks, where technical nuances are frequently left to the reader. For instance, the proof of ordinal exponentiation via functions with finite support is presented with exhaustive thorough- ness. Furthermore, we explicitly analyze the connection between different definitions of ordinal operations—recursive ones versus those based on ordered sums, products, and powers. This duality of approaches is rarely presented so fully in educational literature, yet it is essential for a deep understanding of how mathematical structures are built. Just as fluency in a foreign language requires mastering the subtleties of grammar, fluency in mathematics requires seeing the microscopic details of its logical foundations. A note on the depth of exposition is also necessary. While we strive for rigor, the primary objective of this book is to demonstrate the architecture of mathematical thought. Consequently, for some monumental results—such as Gödel’s Incompleteness Theorems, Morley’s Categoricity Theorem, or the independence results in Set Theory—we limit ourselves to a conceptual discussion of their significance and consequences rather than formal derivations. Conversely, we pay scrupulous attention to the “microscopic” foundations of- ten glossed over in standard texts: for instance, the rigorous derivation of elementary arithmetic properties from the Peano axioms is presented in full detail. Additionally, we emphasize a model-theoretic approach to proofs in predicate logic, highlighting the semantic link between language and structure. The book’s structure mimics the stages of language acquisition, as reflected in the title. Its content supplements standard mathematical literature by focusing on the descriptive aspect of mathematics—expressed through formalisms and structures.