This is a short chapter finalizing the previously mentioned duality idea: mathematics is well connected and at the same time chaotic and well organized.

This is “wrap-up” chapter of the first part of the book. It is designed to put all the ideas presented so far together. First two chapters described the mystery and the last two chapters offer a plausible solution: The Rare axioms hypothesis. This is the third deep idea of the book. In short: There are very few complex and interesting games that are also consistent. Our universe is one such a game. Mathematicians study all possible consistent games. Thus, it should not be that surprising that some of our games (math theories) are in agreement with our universe.

The main purpose for the first two chapters is to convince the reader that we indeed have a great and interesting mystery to solve! This is a short chapter and is designed to “drill-in” this message. It also offers an independent tale on Alien Board Game, in which we introduce the first deep idea in this book – our universe is nothing but a game. The scientists practice “reverse engineering”; they study the final product, the nature, and try to decipher the rules of this game (laws of nature). Mathematicians start from the opposite end. They make their own rules (axioms) and from them they build their own game (mathematical theories).

Building on the idea introduced in Intermezzo, we now offer three essays on games, computer games in particular. One is about Chess, the other Conway’s Game of Life, and the last one is about Bouncing Balls on computer screen. Each chapter could be read independently, but put together they present a curious thought, the second deep idea of this book. The tradeoff: Consistent games are typically boring and interesting games are typically inconsistent.

In a tradition of Socrates, we offer a self-criticism of Rare Axiom hypothesis and we convince the reader that yes, this hypothesis has some merit and no, it cannot be the full answer. There are so many missing pieces. We also introduce a tale Down the Rabbit Hole, which can be read independently, but more importantly, it offers a glimpse of the fourth deep idea of the book: Mathematics is incredibly interconnected! For whatever reasons, almost any two mathematical ideas, regardless if they come from unrelated fields, can be linked to each other. This is a well-known and accepted fact among professional mathematicians. Yet, nobody has an explanation for it.

This is an important chapter. It offers three essays, which are (for most parts) independent but together they tell an amazing tale. They offer several examples on this superbly interconnected mathematics, but this time we also see the design. The interconnections are surprising but not chaotic. We show the glimpse of an order, a hierarchy. Which is the fifth deep idea of the book: Mathematics is incredibly interconnected and yes, it is haphazardly connected. Nevertheless, within this chaos there exists a very strong structure, a blueprint.

Now that the reader is persuaded, about this strange mathematical connections, we go into some details. All examples thus far we brief, and here we take our time and explain, in some detail, four strange cases where an ancient mathematics resurfaces and shows its face in modern world. Each of these four essays stands alone and can be read independently. They cover: Earths trajectory, music and trigonometry, logic and logic circuits, and erratic path of atoms and stock market. At the end, we summarize our findings and declare that we seem to be none-the-wise. True, we are now even more convinced about this spooky action of mathematics, but have no idea why.

According to G. H. Hardy, the “real” mathematics of the greats, such as Fermat and Euler, is “useless,” and thus the work of mathematicians should not be judged on its applicability to real-world problems. Yet, mysteriously, much of mathematics used in modern science and technology was derived from this “useless” mathematics. Cell phone technology is based on trig functions, which were invented centuries ago. Newton observed that the Earth’s orbit is an ellipse, a curve discovered by ancient Greeks in their futile attempt to double the cube. It is as if some magic hand had guided the ancient mathematicians, so their formulas were perfectly fitted for the sophisticated technology of today. Using anecdotes and witty storytelling, this book explores that mystery. Through a series of fascinating stories of mathematical effectiveness, including Planck’s discovery of quanta, mathematically curious readers will get a sense of how mathematicians develop their concepts.

The concluding chapter combines all the five deep ideas of the book and offers a plausible solution to the mystery introduced within the first two chapters of this book. As expected, the proposed solution opens another door, and we seem as confused as ever, but this time on higher level and about more important things.

The previous chapter raised the issue of consistency, which has its own place within the world of mathematics, in particular mathematical logic. The next four chapters talk about the ways mathematicians struggled with issues very similar to the ones computer game developers do: Are their games (theories) consistent or maybe there is an inherent contradiction (bug) lurking behind the scene? We tackle the problems of The Fifth Axiom, The Axiom of Choice, Measure Theory, and Russell’s Paradox. As before, these chapters can be read independently, but put together, they offer a plausible answer to our mystery introduced in the first two chapters.

This is a more technically and mathematically demanding chapter. Here we expand on the idea of orderly structured mathematics. The chapter has two purposes: first, to convince the reader that there exists a tree-like structure within mathematics, and second, to encourage an original research. Maybe a talented and eager reader could take up the challenge and actually construct such a “tree-of-life” within mathematics.

Here, we familiarize the reader with the main theme. How is it possible that the formulas of ancient Greek mathematicians are perfectly fitted for our technology that was just recently invented? We show of several concrete and well-known examples, where an obscure, ancient, mathematical idea reappears in modern science and technology: ellipses, imaginary numbers, trig formulas, and such. At the end of the chapter, the reader is “hooked” and he/she believes that there is a mystery to be solved.