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:
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).
There is not complete agreement among mathematicians about which axioms or rules to use in every case. There are, however, no “loopholes” in a consistent set of axioms and rules. If they are inconsistent, then you can prove absolutely anything!
Quite often, however, mathematicians use axioms and rules that have been found to be very useful, but it is not known with certainty whether they are consistent or not, e.g. the ZFC axioms for set theory. After over a century of intensive study by the experts, no inconsistencies have been found in ZFC. As a foundation for mathematics, it just seems to work.
As for exam questions, the “usual axioms and rules” of logic and mathematics (those most widely used) can be assumed to be available to you unless otherwise stated on the exam paper or in the course materials. Yes, that is a bit vague, but you exploit such ambiguity at your peril. Most examiners have no sense of humour in this regard.
A rigorous mathematical argument which unequivocally demonstrates the truth of a given proposition. A mathematical statement that has been proven is called a theorem.
In mathematics, a proof is a demonstration that if some fundamental statements (axioms) are assumed to be true, then some mathematical statement is necessarily true. Proofs are obtained from deductive reasoning, rather than from inductive or empirical arguments; a proof must demonstrate that a statement is always true (occasionally by listing all possible cases and showing that it holds in each), rather than enumerate many confirmatory cases. An unproven proposition that is believed to be true is known as a conjecture.
Proofs employ logic but usually include some amount of natural language which usually admits some ambiguity.
In fact, the vast majority of proofs in written mathematics can be considered as applications of rigorous informal logic. Purely formal proofs, written in symbolic language instead of natural language, are considered in proof theory. The distinction between formal and informal proofs has led to much examination of current and historical mathematical practice, quasi-empiricism in mathematics, and so-called folk mathematics (in both senses of that term).
http://en.wikipedia.org/wiki/Mathematical_proof and go to methods of proof
You might be interested to learn that we didn’t have a very good definition of what a proof is until fairly recently. Before the birth of modern logic and set theory, mathematicians had managed to use accepted proof methods to derive contradictions (a famous example is Russell’s paradox). A great deal of work was put into defining proof systems that didn’t allow for such inconsistencies. Among other things, this involved creating much stricter rules for how you can construct sets.
Most people are content with our current proof methods, even though we rarely (if ever) use completely rigorous proofs. Unfortunately, it has been proven that any sufficiently advanced proof system can’t prove its own consistency. In fact, if someone could prove that our current proof methods are consistent (i.e. can’t generate contradictions), then that would imply that our proof systems is inconsistent. So in that sense, yes, the “false backbone of rigour” does fall appart a bit when you start digging. If our proof system is consistent, we will never know…
There are lots of strange and interesting phenomena related to proofs and logic. One thing you might have heard of is that certain statements can’t be proven or disproven. They are, in a sense, unknowable. Unless I’m mistaken, the continuum hypothesis and the axiom of choice are two such statements. You can choose to assume they’re true or false and write perfectly valid proofs in either case.
The same can’t be said for provable statements, though. If we take your example and add Fermat’s last theorem to our axioms we could face a problem. What if Fermat’s last theorem wasn’t true? (it is, but for the sake of argument suppose it isn’t). You could use the other axioms to prove it false. You now have a system of logic that is inconsistent; you can use it to prove paradoxes. And if you can do that, you can literally prove anything. Not good…
(I’m essentially parroting what my logic prof said during our first lecture. I’ve fact-checked this to the best of my abilities, but this is still mostly from memory. If something is wrong, please let me know.)
Excellent Question. The concept of a proof carries authority; the conclusion of a proof is demonstrated from the premises. Proofs of theorems still assume a proof theory, which, as you’ve noted, depend on accepting a logic and some axioms. One can think of a proof as a convincing demonstration (informal sense) or as a mathematical object (formal sense). I take it that you are more interested in the idea of a convincing demonstration. All demonstrations must make certain assumptions. An opponent can’t be put in checkmate unless they accept the rules of chess. Similarly, one can’t offer a convincing demonstration unless one accepts the premises from which the demonstration proceeds. The more general question of whether anything is ever convincingly demonstrated is a philosophical question, but first-order logic and the Peano axioms provide many natural cases of convincing demonstrations. The Stanford Encylopedia of Philosophy has a nice article on the development of proof theory here.
Well, of course you’re free to adopt the Pythagorean as an axiom in your system. But that doesn’t remove the burden of proof; it merely transforms it, as now you have to prove that a) your system as stated is consistent; and, almost as important, b) that your new axiom is not redundant i.e. could not itself be derived from other axioms in that system. That is, you’ll have to prove that you can neither prove nor disprove the Pythagorean using the other axioms of your system. If I was your teacher I’d definitely take you up on that. But if I was you, I might not want to go there. Unless you really are game.
Basically, axioms aka postulates aren’t universal truths; they’re Lego bricks. At the end of the day it doesn’t matter what bricks you have or decide to use, so much as what you can build with them, how that relates to other builds, and occasionally whether it’s useful for anything.
I could adopt as my only axiom that I’m God or Chuck Norris, and hence everything I say is true just because I say so – or; it is true that I said it, since “things Jesper said” is the only universe of which this system is valid. And even then fairly useless. You can’t build things with it. Even I can’t build things with it. You may note, say, that Jesper said A, and then he said not-A. But you can’t even take that and derive a proper contradiction. Systems with that property are called dialethian – and they may not all be as useless as this.
Or I might adopt as axiomatic that only those propositions that have been proved or disproved are allowed to have a truth value, so that there will be propostions that have none; and consequently the classical dictum, known as the law of the excluded middle, that a proposition is either true or false, is itself false – in this kind of system, called intuitionistic or constructivist. And that has some very interesting consequences, or builds, or meta-builds known as category theory and topos theory.
All I’m saying is, you’re on the right track. Challenge everything up to and including your ability to challenge everything. But derive the consequences. Cuz that’s where the fun is. You’re absolutely right that you can take anything at all to be your axioms in your system. The challenge is still to do something interesting with it, and hopefully tell us something of how your system relates to everything else.
But right now you’re at school. And school tells you what system to use. Use it, then. It doesn’t have to become part of your identity.
You can prove the Pythagorean theorem given Euclid’s postulates. Once you’ve done that, you’ll probably just go right on using it wherever those postulates hold, without even bothering to remember or recount your proof. And hence it is, in a sense, axiomatic for your new build.
You can also quite easily disprove the Pythagorean: just do your geometry on a sphere or a saddle (hyperboloid). In non-Eucledian geometry, while still incredibly important, the Pythagorean is only true in the limit of infinitesimal triangles. And may that be a lesson to us all: today’s universal truth is likely tomorrows special case. Maybe.
Very interesting question. You certainly seem to have at least one of the major qualities a well to do mathematician would possess, namely, the drive for rigor.
To answer your question, and to paraphrase what you have said, you have to get to the very fundamentals. Let’s expand a little more.
Although I want to avoid a technical and philosophical discussion of what Mathematics is (is not or can be), let’s just say any system with a set of strictly, i.e. rigorously, predefined rules and some kind of fundamental object defines a mathematical system. So, for example, just rules would make up a logical system; we’re treading on thin ice here too, but let’s move on.
An example of a mathematical system is topology. This is one of the prime examples of what most people don’t think mathematics to be, since most of the time you’ll be dealing with sets and operations on them, which aren’t operations you’re used to seeing up till now. However, we still may use ideas and operations from another system, say the system of numbers (I want to avoid the phrase “number theory” for fear of a potential technical slight), on objects here as long as we can verify these such objects satisfy all the requirements those ideas and operations state for them to be put in use. So for example, we can count how many sets we’re working with, and forgoing any concerns regarding the applicability of operations on numbers in this context, we can even make statements such as, “There are $2n$ sets.” And we need not prove here that $2n = n + n$, since we already have confirmed that the system of numbers can, although perhaps not entirely, be used in the system of topology. In other words, what’s proven once, is proven totally. For more on this you may wish to see this question, and in particular, I also talk about this in my answer there.
How can we be completely sure that something is true? Never. Indeed, we agree to set up axiomatic systems, as you have mentioned, and then proceed to determine certain truths in these such systems. And we can only be sure that the truths we have obtained are true given that our axiomatic system is valid and that our process of reason holds. In the absolute sense, there is no way to know for sure if those two things are actually true. But there are a few things I want to mention here. First, Mathematics was never to concern itself with such matters. Mathematics simply stands on the premise, let’s say this and that is true; given these assumptions, what else can we say is true? And the various other answers to the question I linked to provide and excellent discussion of this. If you still are concerned about such things, however, you should know that now you’re entering the domain of philosohpy; namely the philosophy of relatity (Ontology) and knowledge (Epistemology); interestingly enough there is also Philosophy of Mathematics. Furthermore, despite all this, there are still certain proposition in Mathematics which can not shown to be either true or false; this is known as Godel’s Theorem.
Lastly, I want to address what might be an underlying concern here. I ask you, where is the concern in accepting the validity of such a statement as,
$$2 + 2 = 4.$$
Is it in determining what ideas, exactly, are referred to by the symbols “$2$” or “$4$”? Or maybe the “$+$” and “$=$”? We have already agreed on the definitions of the ideas referred to by those symbols, and we can agree in our application of those definitions and ideas. Indeed, some may disagree with this application, but that’s ok. That is not necessarily to say our conclusion is incorrect, but just that not everyone’s on the same page. And disagreements like this can happen in higher mathematics; in fact, that is precisely why we have proofs of conjectures peer reviewed.
Proving something is no more than just saying why something is true mathematically. There are many ways to prove something. Here are the thre most used:
Logic: Using established rules and thinking to get to a coherent answer.
Contradiction: Trying to prove that what we are proving is false. If this fails, what we are trying to prove is true.
Alternate thinking: Putting the problem in a different way to clarify previously unseen things.
Advanced Mathematical proofs generally come in abstract and obscure ways of rationalization. But a proof can come as simple as some simple logic.
I don’t know if the following comes as an adequate classification of wrong statements in mathematics and at least some parts of logic.
That said, plenty of wrong statements have counterexamples. For instance, if someone were to claim that “all (Euclidean… all triangles are assumed Euclidean in this answer) triangles are isoceles” or “all triangles are equilateral”, these statements are wrong, because there exist triangles which are not isoceles and triangles which are not equilateral. To convince someone that a mathematical statement is wrong thus requires either constructing a counterexample or indicating how in principle a counterexample can get constructed, or indicating how a counterexample can exist within the theory.
You can use already proven theorems when constructing a counterexample or indicating how one could get constructed for the theory you work with. With the triangle example, you could use Thales Theorem, which in effect gives you a method to construct several triangles, and then confirm that at least one of such triangles is not isoceles or not equilateral.
A proof is a chain of statements
$p_1\implies\cdots\implies p_n$ which can be braked down so that each implication essentially correspond to a conclusion of type modus ponens or a substitution – due to Kurt Gödel.
From a formal point of view, $p_1\iff a\wedge c$ is a conjunction of all axioms and some conditions which is formulated in the theorem to be proved ($c\implies q_n$ is the theorem), and the chain have links like
$(p_k\wedge(p_k\implies q_k))\implies p_k\wedge q_k$.
Example. Goldbachs conjecture can be formulated:
If $m>2$ is an even natural number (the condition $c$), then it exists two prime numbers $p,q$ such that $m=p+q\;$ (the conclusion $q_n$).
Here the axioms are the axioms of Peano ($a$), and by finding each modus ponens and each substitution needed in the chain, the conclusion $q_n$ could be made.
From a more informal point of view the axioms and a lot of theorems are supposed to be known by the reader and don’t have to be pointed out.