Why proofs

In everyday life, when we're not just being completely irrational, we generally use two forms of reasoning. One of them, called inductive reasoning , involves drawing a general conclusion from what we see around us. For example, if all the sheep you have ever seen were white, you might conclude that all sheep are white. This form of reasoning is very useful — scientists form their theories based on the observations they make in a similar way — but it's not water tight.

Since you can't be sure that you have seen every single sheep in the Universe, you can never be sure that there isn't a black one hiding somewhere, so you can't be sure your conclusion is really true. If you use inductive reasoning, you have to be open to revising your conclusion when new evidence comes to light, and that's what scientists generally do.

The other form of reasoning, called deductive reasoning , goes the other way around. You start from a general statement you know for sure is true and draw conclusions about a specific case. For example, if you know for a fact that all sheep like to eat grass, and you also know that the creature standing in front of you is a sheep, then you know with certainty that it likes grass.

This form of reasoning is water tight. It can only go wrong if your premise is false, that is if you're wrong about all sheep liking grass, or if your observation is wrong, that is, the creature you're looking at is not actually a sheep. But if those two things are correct, then your conclusion follows necessarily from your premise: it is true everywhere and for eternity.

Dividing both sides by gives. Mathematics is all about proving that certain statements, such as Pythagoras' theorem, are true everywhere and for eternity. This is why maths is based on deductive reasoning. A mathematical proof is an argument that deduces the statement that is meant to be proven from other statements that you know for sure are true. For example, if you are given two of the angles in a triangle, you can deduce the value of the third angle from the fact that the angles in all triangles drawn in a plane always add up to degrees.

The importance of deductive reasoning in maths has been known since the ancient Greeks. Euclid of Alexandria , known as the father of geometry, came up with a collection of axioms , statements he thought were clearly true and needed no further justification click here to see them. These included in a slightly different form the statement that the internal angles of a triangle add up to degrees.

Any other statement about geometry, for example Pythagoras' theorem, should be deduced from these axioms by deductive reasoning.

Euclid's famous maths book The Elements was based on this approach. It's one of the most successful books in history — some say that it has gone through more editions than the bible. But of course, you still need to be very careful with deductive reasoning as mistakes can easily slip in.

To be certain your conclusion is right, you need to be certain that your general assumptions are correct and that you've used them correctly. Can you spot the flaw? Why do mathematicians insist on proving everything? In normal life, we're not as pedantic.

If all the evidence in a murder case points to a particular suspect, we're happy to convict them and say their guilt has been proved "beyond reasonable doubt". But then, we can never be really sure. As any innocent convict will tell you, there's always a chance they didn't do it.

Mathematics is perhaps the only field in which absolute certainty is possible, which is why mathematicians hold proofs so dearly. Also, if we don't insist on proofs, mistakes can creep in that aren't easily spotted otherwise. A famous example comes from the above-mentioned triangles. One of Euclid's axioms is equivalent to saying that the sum of the internal angles of all triangles is degrees — he thought this was so obvious, we should just accept it.

Mathematicians that came after him, however, thought they could do better. They tried to derive this fact from Euclid's other axioms. That way, we don't just have to believe it, but can consider it as proven assuming the other axioms are correct.

Mathematicians were struggling with this proof for hundreds of years. But what makes you think that what you have written is correct? Testing can certainly help, but testing cannot be the whole answer, for two reasons. While testing can demonstrate the presence of errors, it cannot guarantee their absence. If testing were all that were required, software with bugs in it would never be released. If testing were the only way to understand whether what you have written is correct, you would have no recourse but to try random programs until after a very long time you stumbled across a correct program.

Likewise with proofs--one could say it is unimportant to know how to "prepare" the "food" of a theorem via proof because there is the "McDonalds" of the math book nearby. But, after years of just relying on memorizing theorems, a person will never be able to come up with a sound theorem of their own.

Being able to prove something makes it much more solidified in one's mind, and gives you a tool that is applicable to many circumstances, not just a single instance.

This is where proof is much more powerful than memorization. One approach would be to give an example of a theorem that sounds so absurd that noone in his right mind would accept it without proof.

For example, mention the Banach-Tarski result to the effect that the unit ball can be decomposed into 5 parts which can then be reassembled by rigid motions into Of course, this one he won't believe even after you give him a proof Suppose we're interested in solving some particular real-life problem. We're going to make some "let" statements. Why are we allowed to do this? We're actually leveraging a theorem:.

This legitimizes the "Let" statement. So theorems tell us the rules of the game; and therefore, the range of problems the human race is capable of solving is hard-limited by the theorems we know.

So theorems are important. Does that mean you personally ought to know the proof of a theorem? That we're not allowed to use it 'til we know the proof? Of course not. That being said, if you want the ability to prove your own theorems and thereby move the human race forward, expanding the range of problems that we're capable of solving, well you'd better start going beyond the question: "What can I do with this theorem?

By the way, most theorems in mathematics aren't proved because of their direct applications to real world problems. This begs that we ask: "Why should we care about theorems that don't have direct applications to real-life problems?

I think the dislike for proofs, and the belief that they must not be important, comes from two misconceptions:. The first is blatantly false. Most proofs are paragraph proofs, and don't have to show every tiny little step. No one will be displeased with you if you commute a few variables here and there without explicitly saying so.

Often, proofs are taught in high school with a very specific format, and straying from it is penalized. This makes it seem pretty arbitrary and dumb. The second is a little more subtle. Proofs do, in fact, prove that things are true. One benefit of that is that we can be absolutely certain we're working "on solid ground", so to speak.

But if you aren't sure if something is true, you don't go trying to prove it first. The more important aspect of a proof is that it is a justification as to why something is true. Proving the "why" of something gives you. But if you look at the proof, you might notice that it doesn't actually involve computing the totient function at all, and it just uses the fact that it's multiplicative.

Well, more likely it'd take a bit of needling to give up that fact, but it's still possible to pry it out This gives you a far more general theorem, one that would be much harder to find if you only knew the special cases, without proof.

For the second point, anorton's answer has a very good example involving double and triple angles. Ultimately, proofs are for verifying results that you're pretty sure are true.