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.