What is a proof?

I am just a high school student, and I haven't seen much in mathematics (calculus and abstract algebra).

Mathematics is a system of axioms which you choose yourself for a set of undefined entities, such that those entities satisfy certain basic rules you laid down in the first place on your own.

Now using these laid-down rules and a set of other rules for a subject called logic which was established similarly, you define certain quantities and name them using the undefined entities and then go on to prove certain statements called theorems.

Now what is a proof exactly? Suppose in an exam, I am asked to prove Pythagoras' theorem. Then I prove it using only one certain system of axioms and logic. It isn't proved in all the axiom-systems in which it could possibly hold true, and what stops me from making another set of axioms that have Pythagoras' theorem as an axiom, and then just state in my system/exam "this is an axiom, hence can't be proven".

EDIT : How is the term "wrong" defined in mathematics then ? You can say that proving fermat's last theorem using the number-theory axioms was a difficult task but then it can be taken as an axiom in another set of axioms .

Is mathematics as rigorous and as thought-through as it is believed and expected to be ? It seems to me that there many loopholes in problems as well as the subject in-itself, but there is a false backbone of rigour that seems true until you start questioning the very fundamentals .

There are really two very different kinds of proofs:

• Informal proofs are what mathematicians write on a daily basis to convince themselves and other mathematicians that particular statements are correct. These proofs are usually written in prose, although there are also geometrical constructions and "proofs without words".

• Formal proofs are mathematical objects that model informal proofs. Formal proofs contain absolutely every logical step, with the result that even simple propositions have amazingly long formal proofs. Because of that, formal proofs are used mostly for theoretical purposes and for computer verification. Only a small percentage of mathematicians would be able to write down any formal proof whatsoever off the top of their head.

With a little humor, I should say there is a third kind of proof:

• High-school proofs are arguments that teachers force their students to reproduce in high school mathematics classes. These have to be written according to very specific rules described by the teacher, which are seemingly arbitrary and not shared by actual informal or formal proofs outside high-school mathematics. High-school proofs include the "two-column proofs" where the "steps" are listed on one side of a vertical line and the "reasons" on the other. The key thing to remember about high-school proofs is that they are only an imitation of "real" mathematical proofs.

Most mathematicians learn about mathematical proofs by reading and writing them in classes. Students develop proof skills over the course of many years in the same way that children learn to speak - without learning the rules first. So, as with natural languages, there is no firm definition of "what is an informal proof", although there are certainly common patterns.

If you want to learn about proofs, the best way is to read some real mathematics written at a level you find comfortable. There are many good sources, so I will point out only two: Mathematics Magazine and Math Horizons both have well-written articles on many areas of mathematics.

Starting from the end, if you take Pythagoras' Theorem as an axiom, then proving it is very easy. A proof just consists of a single line, stating the axiom itself. The modern way of looking at axioms is not as things that can't be proven, but rather as those things that we explicitly state as things that hold.

Now, exactly what a proof is depends on what you choose as the rules of inference in your logic. It is important to understand that a proof is a typographical entity. It is a list of symbols. There are certain rules of how to combine certain lists of symbols to extend an existing proof by one more line. These rules are called inference rules.

Now, remembering that all of this happens just on a piece of paper - the proof consist just of marks on paper, where what you accept as valid proof is anything that is obtained from the axioms by following the inference rules - we would somehow like to relate this to properties of actual mathematical objects. To understand that, another technicality is required. If we are to write a proof as symbols on a piece of paper we had better have something telling us which symbols are we allowed to use, and how to combine them to obtain what are called terms. This is provided by the formal concept of a language. Now, to relate symbols on a piece of paper to mathematical objects we turn to semantics. First the language needs to be interpreted (another technical thing). Once the language is interpreted each statement (a statement is a bunch of terms put together in a certain way that is trying to convey a property of the objects we are interested in) becomes either true or false.

This is important: Before an interpretation was made, we could still prove things. A statement was either provable or not. Now, with an interpretation at hand, each statement is also either true or false (in that particular interpretation). So, now comes the question whether or not the rules of inference are sound. That is to say, whether those things that are provable from the axioms are actually true in each and every interpretation where these axioms hold. Of course we absolutely must choose the inference rules so that they are sound.

Another question is whether we have completeness. That is, if a statement is true under each and every interpretation where the axioms hold, does it follow that a proof exists? This is a very subtle question since it relates semantics (a concept that is quite illusive) to provability (a concept that is very trivial and completely mechanical). Typically, proving that a logical system is complete is quite hard.

I hope this satisfies your curiosity, and thumbs up for your interest in these issues!

Since you're a high-school student, here's an answer that's less sophisticated and much less rigorous:

I suppose you could make up any set of axioms you want, and start using them to prove theorems. So, as you say, you could make Pythagoras' theorem an axiom in your world, and then you wouldn't need to "prove" it.

But, if you're going to start making up your own system of axioms, and doing mathematics in this private world, there are a few things you need to worry about:

(1) If no-one else uses the same axioms as you, then no-one will be very interested in your "theorems", since they are only true in your private world. Your private world might be a bit lonely. So, better to use the same axioms as everyone else.

(2) It's useful (though not absolutely necessary) to have a system of axioms that bears some relationship to reality. That way, the theorems you prove will sometimes give you information that has value in the "real" world -- in fields like economics and engineering, for example. Your private world might be quite different from physical reality, if you don't choose the axioms carefully. So, your results could be misleading or even dangerous, even though they are provably "true" in your world.

(3) If you're not careful, the system of axioms you invent might lead to contradictions, or it might have other fundamental logical flaws. The axioms can't be completely arbitrary (as far as I know).

There are some areas of mathematics where part of the game is making up modified systems of axioms and seeing what happens. But most of us play by a fairly well established set of rules, for the reasons outlined above (and for other reasons, too, I expect).

Additions

Regarding your added comment that "there is a false backbone of rigour that seems true until you start questioning the very fundamentals". It seems to me that the rigour is in the reasoning that's used to derive theorems from the chosen set of axioms. I don't think this rigour is "false".

What's bothering you, I suppose, is that there is some freedom when choosing the set of axioms, and, depending on what choices you make, you get a different set of theorems -- a different version of the truth, and different statements of what is "right" and "wrong". I understand your concern -- I can see how it might be disturbing to find out that the axioms of mathematics are not universally agreed. One example of a debatable axiom is the "Axiom of Choice" (read more here). Most mathematicians assume that this axiom is true, but some don't, and, of course, the two groups get a different set of theorems. Not entirely different, but different.

But, on the other hand, the choice of axioms is not completely arbitrary, and there is a very large overlap in the sets of axioms that are in common use. So, in practice, things typically work just fine, despite the fact that the foundations are not entirely cast in stone.

Questioning the fundamentals, as you are doing, is a valid thing to do, and mathematicians have been doing it for a long time. If you want to know more about this, from sources that are at least somewhat "credible and reliable", then this Wikipedia page might be a good place to start.

I'm not sure, but to me your specific question doesn't seem to have been given the simple answer to why assuming Pythagoras as an axiom is wrong in that situation.

The reason is: because you're actually being asked "Given the set of axioms you've been taught, derive Pythagoras." The question implicitly assumes some particular axiom system.

In general a proof could be considered formally a set of symbols obeying some rules (of logic) which begins with a set of axioms and assumptions and ends with the statement you want to prove.

A proof is a completely convincing argument. Thus, a proof of the Pythagorian theorem would be a completely convincing argument that the Pythagorian relation is correct as stated. The notion of "proof" is arguably more fundamental than this or that axiom system or system of formal logic. This point of view is that of Errett Bishop. It is the underlying theme of his book 1967 "Foundations of Constructive Analysis" (for a review see http://www.ams.org/journals/bull/1970-76-02/S0002-9904-1970-12455-7/home.html as well as http://www.jstor.org/stable/2314383?origin=crossref).