Introduction 1

Where on earth did the numbers come from?

Chapter 1: The Two Pillars of Philosophy 13

Plato vs. Aristotle

Chapter 2: The Universals Debate 27

Does the number '2' exist in this world?

Chapter 3: Modern Rationalism and Empiricism 39

Descartes vs. Locke

Chapter 4: What Kind of Knowledge Is Mathematics? 55

Hume vs. Kant

Chapter 5: The Structure of Mathematical Revolutions 71

Thomas Kuhn vs. Bachelard

Chapter 6: On the Nature of Mathematical Research 81

Is it a discovery or an invention?

Chapter 7: Reasoning in Worldly Affairs 93

Induction vs. Deduction

Chapter 8: Probabilistic Judgment 103

Probability and Expected Value

Chapter 9: The Fundamental Laws of Logic 115

The law of contradiction and the law of the excluded middle

Chapter 10: The Utility of Symbolic Logic 127

Conjunctions and logical operations

Chapter 11: Proof in Mathematics 143

Mathematicians' Secret Weapon Revealed

Chapter 12: The Mystery of Infinite Sets 157

Infinite size comparison

Chapter 13: Paradox 173

A Crack in Logic and a New Beginning

Chapter 14: Is Mathematics a Game of Symbols? 185

Formalism in the Philosophy of Mathematics

Chapter 15: Is Mathematics Completely Truthful? 199

Gödel and Turing

Chapter 16: Theo-Intuitionism 211

The excluded middle problem, the concept of mathematical truth

Outgoing Post: No Answer, But the Questions Continue 227

Philosophy of Mathematics in the AI ​​Age

Mathematics is one of the oldest and most widely taught subjects in schools.
However, not many people have actually asked fundamental questions like, "What kind of knowledge is mathematics?", "Where did numbers come from?", and "Is mathematics an invention or a discovery?"

Moreover, the fact that one of the important thoughts that runs through the entire history of Western philosophy has been the discussion of the nature and status of mathematics, a unique type of knowledge, is a topic that is often forgotten or overlooked even among philosophy majors.
From Plato and Aristotle in ancient times, to Descartes and Locke in modern times, and even to modern mathematical philosophers, mathematics has been at the heart of philosophy, going beyond simple calculations and asking questions like, "What is truth?" and "How is knowledge possible?"

It is unfortunate, however, that most people who study mathematics, and even those who major in philosophy, rarely have the opportunity to engage with this fascinating and essential discussion.

This book started from such a problem awareness.

Initially, I intended this book to be an introductory book to the philosophy of mathematics for young people, with the goal of writing it in an easily understandable way so that middle and high school students could look at mathematics through a new lens of philosophy.
However, as I continued writing, I became convinced that this topic could be a meaningful and engaging intellectual journey for the general educated public as well. Accordingly, I carefully refined the sentences and structure to ensure an accessible read while maintaining the depth of thought.
This book avoids difficult symbols or technical mathematical theories, and instead focuses on questions such as, "Where do numbers come from?", "Is mathematics a human invention, or is it a real truth that exists somewhere in the universe?", and "Does the concept of infinity actually exist?"
By following the historical debates of philosophers, we focus on helping readers think for themselves and develop their own philosophical perspectives.

Above all, this book is an attempt to revisit the intuitions and questions I had as a pure mathematician before I began studying philosophy, and to translate them into the language of philosophy.
In that respect, I would like to express my deep gratitude to Professor Lim Il-hwan of Hankuk University of Foreign Studies, who opened the door to philosophy to me, a subject I had no idea was even a part of, and led me into the fascinating world of mathematical philosophy.
Without that teaching, this book would never have been started.

I hope this book will provide a new perspective for students who love mathematics and ignite the intellectual curiosity of readers who have felt distant from mathematics.
Let's embark on a journey together to see mathematics through the lens of philosophy, mathematics above mathematics.