mertilhan.ecevit A Crazy Math Student’s Journey into Mathematics – Arash M. Rezaeinazhad / Kadir Has University, Istanbul, Turkey
Paving the Road to Mathematics: Logic!
Logic? Really? Aren’t we supposed to do some crazy math and get sick of memorizing integrals? Actually no! Enough is enough. You are at the gates of advanced calculus, this means no more nonsense. You may have heard how mathematics is useful in real-life problems, but do you also remember the answer for the question that goes like this: “Where are we going to use these integrals and formulas?” (If you have ever got a satisfying answer, please write an email to me.) But mathematics is, in fact, useful! The key to this wonderland is to be able to ask the right questions (This I have learned from Madam Mathematics).
Modern mathematics aims to give individuals the independence of thought and the power to face different problems and to make the individual know “abstraction” as soon as possible. What does abstraction mean? It means to cut off nonsense and see the essence. This will be clearer as we proceed, but to make it clear, let me give you an example. Every system of mathematics is a game like chess that has certain pieces (for instance, in algebra our pieces are x, a, A, ⊕, and so on) and certain rules called axioms that bind the manipulation of pieces to themselves. If we follow the algebra example, one of these rules in a field of mathematics called group theory states that in a group there is a neutral element called 0 which, when added up to every other element, does not change it (let’s call this ‘I’) and every element has an inverse that when you add it to the element itself you get the neutral element (let’s call this ‘II’). I know, I know! It got boring. But did you read the quote? Whenever you tried to solve an equation like x − 1 = 0 when you were saying take the −1 to the right hand side and it changes sign, then ‘eureka!’ you were actually playing with axioms. Here’s how it is: By ‘II’ add the inverse of −1 to both sides, then on the left hand side you get x + 0, which by ‘I’ is x, and on the right hand side, you get 0 + 1, and again by ‘I’ it is equal to 1. Hence, our black cat in the dark room, which we were blindly following, was 1!
My aim by giving such an example at the beginning (which I must confess is not a really rigorous one, but hey, we are at the beginning, just don’t be tough on me for now) is to explain how modern mathematics works. It is not about solving and memorizing integrals that you are never going to use. It is about being able to move forward being bound to certain rules; it is about seeing that if a goal can be achieved using the pieces you have by obeying rules (You can go the illegal way here as much as you want, the result will be only that your achievement, being a theorem or a solution to an equation, will be wrong. It is strange that the more illegal activities you do in mathematics, you get better and better doing it with rules, so don’t be afraid of making mistakes!)
This is how modern mathematics works, and if you want to learn this heavenly gift, you need to know the pieces and know about the rules. The better you know the rules, the better you can work with them, and the way to be able to enjoy mathematics is to learn about these rules, hence Logic! Yes, really! You may possibly know how important the terms like definition, if and only if, reductio ad absurdum, and so on are, and without being able to know them, not only will you not be able to enjoy mathematics, but you cannot even prove a single theorem of yours. If a person is not able to distinguish between a definition, reasoning, fallacy, and likewise terms, then after defining the natural power of an element by means of products, he will start “proving” that a^0 = 1.
This is your last chance. After this, there is no turning back. You take the blue pill—the story ends; you wake up in your bed and believe whatever you want to believe. You take the red pill—you stay in Wonderland, and I show you how deep the rabbit hole goes. Remember: all I’m offering is the truth. Nothing more.
So get ready! If you take the red pill, then in this introduction to logic, I am aiming to brainwash you. To show a little bit of logic that is very useful in studying mathematics (call it observer’s logic). If I can only be able to tell you that it is highly inappropriate to write sentences like ‘a number > 0’, or to be able to stop you from the madly use of ⇒ between your lines of reasoning, and to be able to show you what really the universal quantifier is and an ϵ is not a very small number (if you are surprised by this, you are welcome), and “The whole is greater than the sum of its parts.” is a wrong statement, then my work is done. In mathematics, the ability to solve integrals or equations is of secondary nature; the primary ones are definitions and reasoning. You take the blue pill, and I quote the following:
“It will be very important as we proceed to keep in mind this distinction between the logic we are studying (the object logic) and our use of logic in studying it (the observer’s logic). To any student who is not ready to do so, we suggest that he close the book now and pick some other subject instead, such as acrostics or beekeeping.”
Let’s get going!
Kleene, S. (1967). Mathematical Logic (p. 3). Dover Publications.