Hold on a second. Mathematics. Logic. When we combine those two in the same sentence, let alone the same phrase, I can almost feel some of your brain cells saying, “Alright, I’m heading out,” and going on vacation. “Mathematical Logic” sounds like a topic whispered about only by ultra-intelligent people living in ivory towers, a subject whose name alone requires three doctoral theses, right?
I used to think so, too. That was until I stumbled upon this small, unassuming, but surprisingly packed book on one of the universe’s dusty shelves: “What is Mathematical Logic?”. This book was written by a group of authors (C. J. Ash, J. N. Crossley, et al.) with names as intimidating as the subject itself. But when you open the cover, you’re not greeted by a professor’s boring lecture notes, but by a conversation that feels like a smart, funny friend saying, “Hey, come over here and let me explain this thing my way.”
The book doesn’t just dive in with, “Okay, now divide formula A by B.” Instead, we travel back in time. On one side, there’s a path starting with the formal deductions of Aristotle and Euclid, like, “If all men are mortal, and Socrates is a man, then Socrates is probably not going to turn into a potato.” On the other side, there’s a more “number-crunching” trail followed by guys like Archimedes, who used mathematics to understand the universe.
For centuries, these two paths went their own ways, unaware of each other. One was busy with the “art of correct thinking,” while the other explored the mysterious world of numbers and shapes. Then, in the 17th century, Newton and Leibniz entered the scene and, with a magic wand called “calculus,” merged these two paths. That was the beginning of the romantic comedy where math and logic asked, “Hey, maybe we work better together?”
The book narrates this historical journey so fluently that one moment you find yourself in Ancient Greece, wearing a toga and debating philosophy, and the next, you’re having coffee with a periwigged mathematician.
Now for the real question: do you really need to be a math genius to understand this book? Honest answer: No, but a little background definitely helps. The author doesn’t just throw you into the deepest part of the ocean. They let you roll up your pants, step into the water, and get used to the temperature first.
And that’s the beauty of this book. Instead of proving everything from the ground up, it explains the “why” behind the big ideas. Take, for example, the famous Gödel’s Incompleteness Theorem. In essence, this theorem states that within any mathematical system, there will always be statements that are “true but unprovable.”
This is a mind-bending idea. It’s as if the architect of the universe winks at you and says, “You thought you could solve all the secrets? Sweet dreams.” The book breaks down this colossal idea into digestible pieces using tools like Turing machines (the imaginary ancestors of modern computers) and recursive functions. It doesn’t teach you how to build a Turing machine (thankfully), but it explains what a Turing machine does and what that has to do with logic, often through a metaphor.
Think of it this way: You have the world’s most advanced cookbook. You can make any dish. But could that book contain a chapter titled “A Recipe for a Dish That Cannot Be Made with the Recipes in This Book“? No, because if it did, it would have been made! Gödel’s theorem is a bit like that. Mathematics defines its own limits from within.
If you absolutely hate math and numbers give you an allergic reaction, this book will probably just collect dust on your nightstand. But:
- If you’re curious about how mathematics is not just about numbers, but actually a “way of thinking,”
- If you’re a coder who wonders, “Where do the roots of this ‘logic’ thing really come from?”
- Or if you’re just looking for an intellectual adventure that will stretch your mind without overwhelming you,
…then “What is Mathematical Logic?” is for you.
This book isn’t a textbook; it’s more of a guide, a tour guide. It stops you in front of the great monuments of mathematical logic (the Gödel-Henkin Completeness Theorem, Model Theory, Set Theory, etc.), tells you how impressive they are, and then says, “If you want, you can check out more detailed books to explore inside, but for now, just enjoy the view.”
Ultimately, What is Mathematical Logic? takes a frightening beast like mathematical logic, tames it, and lets you play with it. You might not be able to fully train it, but at least you won’t have to be afraid of it anymore. And that, in itself, is a major victory.
