Why do we require proofs in Mathematics?

A student has the tendency to question the same to himself/herself or the teachers whenever he/she gets stuck on a specific mathematical proof problem. We agree that proofs are hard and specially, in higher classes the difficulty level reaches its peak. The importance of such problems increases in mathematics and also in the subject- physics where students face repeated problems on derivations. If you ask a student a question which would say “what’s the best possible thing s/he wants to see on his/her exam answer sheet?”, s/he would probably come up with a prompt reply of “Q.E.D (quod erat demonstrandum)” or “Proved” beside a successfully completed math proof problem. Students breathe a sigh of relief after completing that hurdle in their math paper. But why is proof so important in mathematics? Why are certain mathematicians crazy about proofs? We will try to clarify the issue to the best of our abilities in this article.

What is a Q.E.D or a proof?

Proof is said to be a form of logical argument that can establish, beyond doubt, that a specific 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.”

Reasoning

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 to be 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 colored 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 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 for 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 form 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,
 a^2= ab
Adding a^2 on both sides,
 a^2+a^2 = ab + a^2
 2a^2 = a^2 + ab
Subtracting with 2ab on both sides,
 2a^2- 2ab = a^2 + ab – 2ab
 2a^2- 2ab = a^2 – ab
 2 (a^2 –ab)= 1 (a^2 – ab)
Dividing both sides with a^2 – ab, we get
 2 = 1.

We have actually come to our result which is definitely a faulty one. But the process is right, there’s no denying that fact but is the result feasible? Certainly not! Therefore this proof is a faulty proof.

Why proofs?

It’s seen that mathematicians are particularly crazy about proofs. Let’s compare the same with our daily life, more specifically with “law and order”. Proofs are specifically very important in court judgments. Say for example there’s a convict for a certain criminal case and the court requires certain 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 proofs 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 of your views. Hence, it can be said safely that proofs in such cases are not concrete and the judgment’s mainly passed on the decision of 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 the 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 a 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 examples are for advanced grades.

Mathematics, a seemingly abstract subject is pretty fascinating indeed. After all it’s the bed rock of science and is of utmost importance in most fields of science. It also finds significant application in your daily life and various activities. Hence, it’s necessary to develop a strong base in this subject to benefit in the long run. After-school online math tutorials can help you out in that aspect. We’ll sign off for now. Hope you had a good read.

 

Image Credit:242community