In publishing this book
introduction
Starting the Seminar: What is Mathematics?
Let's start with a simple math activity | Calculating Shapes | Do Proofs Really Matter in Math? | Is It Math or Physics?

Part 1 | Foundations of Mathematics
Lecture 1: The Crisis in the Number System
How did the discovery of numbers transform human thought? Height, intelligence, address, latitude and longitude, temperature and humidity… Time and space, and everything that expresses our identity are all numbers.
Looking at how we are more familiar with mathematics using numbers than with geometry, isn't it possible that our thinking is becoming increasingly computerized?
Mathematicians who created the tradition of mathematics│Pythagoras and the discovery of numbers│The crisis of numbers│The origin of integrals│A modern version of Zeno's paradox│Back to geometry

Lecture 2: Long Thoughts on the Essence
Defining something like 'what is X' is always difficult.
In a world of uncertainty, 19th-century mathematicians, who wanted mathematics to be certain, tried to establish an unbreakable foundation by defining every single entity in mathematics.

Is Mathematics Clear Thinking? │ An Extreme Theory of Numbers │ An Obsession with Certainty

Lesson 3: Building a Machine That Finds Answers
It is said that the advancement of civilization occurs by increasing the number of tasks that can be done automatically without thinking.
The ability to calculate mechanically is extremely important in mathematics.
So, is it possible to create an algorithm that mechanically finds the answer to every equation in the world?
The ability to calculate mechanically│The mathematical puzzle that shook the world│An algorithm that makes all calculations possible│There is no such algorithm│Questions to find questions

Lesson 4: Logical and Mathematical Thinking
"If this sentence is true, then Kim Min-hyung is a billionaire." Is this sentence true or false? What is the basis for determining whether something is true or false, and what constitutes logically correct reasoning? The ability to draw accurate inferences even when the truth or falsity of a proposition is unknown is crucial to mathematical thinking.
Math Through Conversation│If This Sentence Is True, Kim Min-hyung Is a Billionaire│What Is Logic│Conversation in Wonderland

Lecture 5: Functions That Make Up the World
Let's review some basic concepts about functions.
Mathematics is the product of thousands of years of accumulated systematic language and conceptual tools for describing natural phenomena.
When we use such language effectively, we can get one step closer to mathematical thinking.
What is a function? What are coordinates? Mastering sines and cosines.


Part 2 | Adventures in Math
Lesson 6: Calculating without Numbers
Can we find the sum of A and B without numbers? Ancient Greek mathematicians used geometry and ratios instead of numbers to perform calculations.
If they had perfected their geometric number system, mathematics might have developed much earlier.
Calculating in the Ancient Greek Way│Calculating on the Plane│Proof, and a Better Proof│Mathematical Theory of Different Viewpoints│Relationships Between Viewpoints

Lesson 7: Information of a Different Dimension
By finding as many correlations as possible between seemingly infinite pieces of information, we can effectively reduce the 'dimension' of the information.
In the era of information science driven by big data and AI, understanding the "multidimensionality" underlying visible information will become a crucial sensibility.
Imagining Abstract Space│Dimensions of Information│Infinite Dimensions!│The 'Information' of Sound│Fundamental Frequency and Elementary Particles

Lesson 8: Equations for Finding the Shape of the Universe
Einstein's equations revolutionized science by providing crucial insights into the deepest phenomena of the universe. Statements like "time is relative" and "spacetime is curved" resonated deeply with artists of the time, even if they didn't understand the specific mathematics.
Roger Penrose's Macroscopic Mind│The Shape of the Universe│Music, Mathematics, and Modernism│Linear Functions│The Linearity of Time│Laws and Equations

Lesson 9: Seeing the World Through Mathematics
To 'see' means to perceive shape and substance.
And this means discovering interactions with light, ultrasound, gravity, and so on.
Mathematical civilization, too, is a journey of constantly discovering the algebra behind geometry, the geometry behind that, and the algebra behind that, in order to see the reality of the world.
Back to the axiom│Can we see the shape of the universe│What happens in the human brain│What it means to ‘see’ the world│Geometry followed by algebra, geometry followed by algebra…

Concluding the seminar
Special Lecture: Foundation of Mistakes
People who attended the seminar
Recommendation

Professor Kim Min-hyung explores the world of mathematics with seven readers from across generations and professions.
: “How do mathematical questions move thought and the world?”

As AI and big data become deeply embedded in various industries and the daily lives of individuals, the consensus that understanding data and statistics is a vital survival skill is growing, leading to ongoing efforts to understand the world's problems and social issues through mathematical thinking.
Amidst this, the book "When Mathematics is Needed," which was adapted from the renowned lectures of world-renowned mathematician Professor Minhyung Kim, became a hot topic, receiving acclaim from 80,000 readers.
Professor Minhyung Kim, who has established himself as a power writer by being selected as one of the seven powerful people in science in 2019 by the Dong-A Ilbo and one of the ten authors of the year in 2018 by the Kyunghyang Shinmun, has returned in 2020 with “Again, the Moment We Need Mathematics” to present a deeper and more profound world of mathematics to readers who have overcome their fear of mathematics and are beginning to feel curious about it.
《Again, the Moment When Math is Needed》 is a book that was translated from a seminar held on a summer night in 2019, where seven readers, some of whom had taken their first steps into the world of mathematics through 《The Moment When Math is Needed》 and others with different understandings of mathematics, experienced the beautiful world of mathematics, which is difficult but essential and touches our lives.
The following is a list of those who participated in the seminar called 'Summer Math School'.
Journalists who need math to read physics books, developers who constantly have to create formulas for programming, middle and high school students who believe that math is a tangible subject, artists who are curious about the math behind art, math teachers who don't want to produce many dropouts, job seekers who are traumatized by rigid math classes, etc.
Professor Kim Min-hyung's seminars, which are tutorial-style, aim to deepen understanding through everyday conversations rather than one-way lectures, attempt to approach mathematical concepts from basic numerical concepts to nature, the universe, and future common sense.
What kind of world of mathematics did they encounter in this seminar, which heated up a midsummer night with persistent questions?

■ How has human thought evolved from the Greek era to modern mathematics?
: “The conversation we have had together feels like a long journey of mathematical civilization.”

“Everything in the world is a number.” Just like the Pythagorean maxim, everything that expresses our identity, such as height, intelligence, address, temperature, humidity, time, and space, is a number.
However, according to legend, Pythagoras killed his student who discovered that the diagonal of a square with side length 1 is √2.
For him, who believed that only rational numbers were numbers, the existence of irrational numbers was a crisis in the world itself (Chapter 1: The Crisis that Became the Number System).
But now, thousands of years later, we take the concept of √2, as well as more precise and much larger numbers, for granted, and we can easily understand the meaning of a graph showing the trend of infectious disease infections.
In "Again, the Moment We Need Mathematics," Professor Kim Min-hyung begins his journey to discover "what is mathematics" with an anecdote from the Greek era.
“Human thinking is evolving into mathematics.”

This book explores the formation of mathematical thinking that has accumulated along with human civilization through wide-ranging conversations with mathematical masters.
Part 1, “Foundations of Mathematics,” covers the historical context in which we became familiar with “numbers” from Greece to Newton, as well as the origins of 19th-century mathematical theories that became the foundation of modern science, including information science and quantum mechanics.
During the turbulent 19th century, various attempts were made to establish the conceptual foundation of mathematics, including numbers and calculations, under the belief that mathematics must be certain.
The remarkable stories of contemporary mathematicians who struggled to present new frameworks of thought, such as Hilbert (Chapter 2), who believed in the absoluteness of the number system, Matyasevich (Chapter 3), who defined algorithms and discovered the impossibility of mechanical calculation, and philosophers who identified mathematical thinking with logic (Chapter 4), vividly demonstrate how much mathematics contributed to the leap forward of human thought.
Mathematics, a product of thousands of years of civilization that has accumulated systematic language and conceptual tools to clearly and accurately think about nature and the world, has now permeated our lives in every way.
This book naturally guides readers into the beauty of mathematics, an academic discipline that has continued to ask questions throughout its long history.


■ A head-on challenge to formulas and concepts, showcasing the convergent thinking necessary for the hyper-connected era.
: “Experience the joy of mathematical thinking and reaching the deepest recesses of thought!”

“Mathematical thinking is the process of formulating precise answers to all the questions that arise in everyday life.
The idea is that if you systematize your understanding of things in a more detailed way, mathematics will naturally follow.
However, going through this process is not easy.
“Isn’t this the core of why we have difficulty with math?”
-From the text

Math is difficult.
Even Einstein found mathematics difficult enough that he had to seek advice from mathematicians for his research.
As the value of science and technology increases, science communication is emerging as an important topic, but mathematics is always left out of this change.
The problem of education that tries to avoid difficult mathematical language and difficult mathematics cannot be overlooked.
However, without understanding mathematics, the invisible hand at the center of change in the world, the fear of being swept away by the changes in the world will only grow.
Professor Kim Min-hyung, who has been giving various public lectures and mentoring activities related to mathematics education, feels the need for a model for popularizing mathematics to overcome these limitations.
This book, which began with such a critical awareness, emphasizes mathematics as a universal language for exploring the world, boldly crossing the boundaries of everyday logical conversations, mathematics, physics, the humanities, and the arts.

Unlike the previous work, "When Math is Needed," which focused on interesting topics while excluding formulas as much as possible, this book directly challenges concepts that have made "math dropouts" hit a wall, from basic formulas like the Pythagorean theorem to vectors, geometry, trigonometry, and statistics.
Vectors, introduced in Part 2: Mathematical Adventures, are tools for expressing AI learning, while matrices enable vector spatial transformation and learning calculations. Furthermore, understanding the dimensions of information underlying visible information is essential for understanding the correlations in big data. (Chapter 7: Dimensions of Information) For example, how do you analyze 1 million cells when 20,000 genes are expressed per cell? This book poses questions like these, meticulously explaining the complex and challenging mathematical concepts that will drive the future.
Professor Kim Min-hyung has dedicated his life to algebraic geometry, a research topic that is currently at the core of AI technology, and he explains it in a specific yet intuitively understandable language, allowing you to experience the essence of modern mathematics.


■ Tracing the long journey of questions that have sought the essence of human thought, the world, and nature.
: A master's beautiful and persistent exploration of the relationships and order of an evolving world.

What does it mean to see the world through mathematics? This book poses profound questions such as, "Can a blind person understand geometry?" and defines what humans see and hear, and further, what we see, hear, and understand about the reality of the universe, as "understanding shapes and reality," that is, reading the interactions between substances (Chapter 9, "Seeing the World through Mathematics"). The author, who presents the process by which humans approach the reality of the world by understanding the interactions between light, gravity, and ultrasound, explains that mathematical civilization is a journey of constantly discovering the algebra behind geometry, the geometry behind that, and the algebra behind that, in order to see the reality of the world.

While explaining Einstein's theory of relativity, which reveals the vast structure of the universe in concrete equations, it also moves between 20th-century artists and works influenced by relativity, such as Penrose's triangle, Escher's prints, and the structure of the music of contemporary musician Xenakis (Chapter 8: Equations for Finding the Shape of the Universe). These topics are not easy to read in one sitting.
However, as if embarking on a journey into an unknown world, as you gradually delve into the unfamiliar language of mathematics, made up of numbers and geometry, you will discover the joy of breaking free from familiar thought patterns and exploring questions together, the intellectual joy of reaching the deepest depths.