A student may ask the same question to himself/herself or his/her teachers, whenever s/he gets stuck on a specific mathematical proof problem.
We agree that proofs are hard and especially, in higher classes, when the difficulty level reaches its peak. But why are proofs so important in mathematics? Why are certain mathematicians crazy about proofs? Let’s see.
What is a proof?
A proof is said to be a form of logical argument that can establish, beyond doubt, that a certain thing is true. There should not be any contradictions to the fact and no involvement of double meanings.
The fact that something is clear should be established beyond doubt. This is an integral part of mathematics. Q.E.D or “quod erat demonstrandum” is a Latin word that means “which is what had to be proven.”
We generally use 2 forms of reasoning in our daily life to come to conclusions. These 2 forms are:
- Inductive reasoning,
- Deductive reasoning.
Inductive reasoning: In this form of reasoning, a general conclusion is drawn from what’s been presented in front of the reasoner. A simple example can clear this thing up. Our protagonist in this example will be a sheep.
For example, you have the past experience of seeing all sheep white. So now, you probably have come to a conclusion that all sheep are white on the basis of your past experience, without verifying the fact whether it’s true or not.
That’s the general conclusion drawn from observations. But is that true? Partially, it is but the statement isn’t correct because there are other coloured sheep on our planet as well. Just because you haven’t seen it, you have drawn that conclusion. That’s an example of inductive reasoning. Now we should also add some of our views to this issue.
Scientists generally come to a hypothesis on the basis of inductive reasoning. That’s not necessarily a bad thing. They actually believe what they see. If the concept’s not based on general principles and there are some exceptions, the conclusion would change accordingly. But for that, the scientists need to see those features first before they change their hypothesis but there’s one thing that we should point out here is the fact that they are open to changes. The same thing is applicable to all inductive reasoning. The reasoner should be open to changes in conclusions on the basis of appropriate circumstances.
Deductive reasoning: This form of reasoning is a little different than inductive reasoning. Here, the reasoners start their deductions from a general statement.
You know one thing for sure to the best of your knowledge that the specific statement is true. Now you’ll try to draw various conclusions from the same statement that you know for sure. Let’s begin our discussion with an example. That statement would be- All sheep loves to eat grass. Now you encounter an animal and you know for certain, the animal standing in front of you is actually a sheep. So now you conclude that the animal in front of you loves to eat sheep. Now that conclusion is completely true; there’ll be no doubt about it. That can only go wrong if you misjudge the animal standing in front of you if the animal in front of you isn’t a sheep and you reckon it to be so out of misjudgment. If the 2 statements are at par, your conclusion is right.
Proofs and Mathematics go hand in hand
Mathematics is based completely on proofs. You’ll have to prove certain statements of math in a manner that are universally true everywhere.
For example, Pythagoras theorem; that proof has to be true everywhere and for an infinite period of time. Hence, we can safely say that mathematics is truly based on deductive reasoning. There’s no place for inductive reasoning in mathematics. Trial and error would not work here. Mathematical proof is basically an argument which deduces a statement that’s derived from another proven statement which, to the best of your knowledge is true. For example, you know that the sum of three angles in a triangle is 180°. From this statement, you’ll be able to find out various angles of the triangle. If you know 2 angles of a triangle, you can easily find out the 3rd one by subtracting the sum of those 2 angles forms 180°.
Importance of deductive reasoning in mathematics is known from earlier times. These were called axioms and you can actually see some of these Euclid’s axioms on this page. All these were based on deductive reasoning.
But there are certain flaws in deductive reasoning as well. We can show you one such example- a paradox proof of “1=2”. You are probably thinking that this is absurd and is definitely wrong but after you go through the entire procedure, I am sure that you’ll start wondering about the correct ways through which that seemingly faulty proof’s done. Let’s begin.
The proof will show you that an absurd math statement is valid. That statement is 2=1. After you have gone through the entire proof, you can decide whether we have made a mistake or not. All we’ll say that we have followed the math operations in the exact manner and have not improvised in any aspect.
Let, a = b.
Multiplying “a” on both sides,
or, a^2= ab
Adding a^2 on both sides,
or, a^2+a^2 = ab + a^2
or, 2a^2 = a^2 + ab
Subtracting with 2ab on both sides,
or, 2a^2- 2ab = a^2 + ab – 2ab
or, 2a^2- 2ab = a^2 – ab
or, 2 (a^2 –ab) = 1 (a^2 – ab)
Dividing both sides with a^2 – ab, we get
or, 2 = 1.
We have come to our result which is definitely faulty. But the process is right, there’s no denying that fact but is the result feasible? Certainly not! Therefore, this proof is a mistake.
It is seen that mathematicians are particularly crazy about proofs. Let’s compare the same with our daily life, for example, “law and order”. Proofs are specifically very important in court judgments.
Say, there’s a convict for a certain criminal case and the court requires proofs to pass the judgment. Proofs are provided; the court’s satisfied and declares the convict as a criminal. Now, what will you have to say in this matter?
You might agree with the judgment or you might not. You will probably ask a hundred people and will not get the satisfactory answers, meaning there’ll be at least some people who should definitely believe the opposite.
Therefore, it can be said safely that proofs in such cases are not concrete and the judgment is mainly passed on the decision of the majority. There’ll still be at least one people who will have an entirely different view. But in the field of mathematics, there’s no such chance.
Mathematical proofs are concrete and have to be accepted by every individual on our planet. At least that’s how such proofs are developed. There’s an absolute certainty in math proofs which makes mathematicians crazy about them. Hence, these proofs are an important part of mathematics.
Another point to remember is that proofs are the only way to evade mistakes in formulae. Albert Einstein’s special theory of relativity has been formulated from hyperbolic geometry. But the general theory of relativity has actually been obtained from the special theory itself. Without this general theory of relativity, modern satellite devices, as well as GPS, would not have worked. So it’s seen that many proofs are actually derived from other proofs. Therefore, there’s no denying the fact that proofs are mighty important in the field of mathematics.
Proofs and People
This has been a problem in modern generations with the advancements of technology. A computer or a robot is capable enough of deriving certain equations to prove the fact that proof is valid. Previously, all those long equations are proved by human beings. But now machines are capable enough of doing the same job. It’s good enough but there’s one thing that should be kept in mind. There should be the involvement of a human being because s/he is the only one who can check whether the proof is correct or not.
Limitations of Proofs
Mathematics is generally seen upon as a subject that claims to have universally true statements in each and every topic covered present in its realms. But how true is that statement? Isn’t there a limitation involved? Surely, there is one. We have already provided you with an example up above. There are a few advanced examples as well. You can check some of them our here but do remember that these are for advanced grades.
Proofs are hard and especially, in higher classes, the difficulty level reaches its peak. But why are proofs so important in mathematics? Why are certain mathematicians crazy about proofs? Let’s see.