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Behind the conjectures and definitions

Mathematics comes from personal intuition. Before (almost) any theorem is proved, there is some intuition guiding the prover as to why it would be true. Most of Mathematics along history was done based only on intuition, not proofs. In all ancient cultures other than Greece, later through Arabia and into Europe until around the 19th century, there were hardly any proofs. So what is the point of proofs? Without proofs, you might understand the intuition behind a theorem, or you might not. In some cases, you would have to fall back to trusting the prover as to the correctness of the theorem (which is sometimes called "proof by authority"). And who knows? The prover might be wrong (although historically the great Mathematicians were almost never wrong).

Nowadays, all theorems are proved, and the system of proof has evolved into Mathematical rigor, which allows even computers to check whether proofs are correct or not. All the older theorems that weren't proved or were only loosely proved are now proven with modern rigor. In theory, anyone may read these proofs, go step by step, and understand the theorem. Actually, this isn't really true, because truly understanding a theorem is not just "knowing" its proof, but rather understanding its proof. A person who truly understands a theorem can recreate the proof out of his mind, and not out of his memory. But this is another issue -- In order to change this, Mathematics must be based on human interaction and iterative questions and answers. A person should not feel that he understands a theorem until he feels like he could've proved it. I hope the School of Mathematics will address this problem in the current system of studying Mathematics.

But, still, proofs have changed the way we do Mathematics and has brought "anyone" closer to understanding theorems. But this is not the end. A modern Mathematics book or article basically contains three elements - Definitions, Conjectures and Proofs. I say conjecture instead of theorem, because before a theorem is proved it is a conjecture. So, I would say that when a book has theorem and then proof it is actually conjecture, then proof, which turns the conjecture into a theorem. We have a system of axioms and logical deduction that allows us to ask "why" on each step of the proof and we should be able to get an answer. But what about the definitions and conjectures? There is no explanation as to why we define a certain concept and why we conjecture something in the first place (before we knew it was true). When reading Mathematics, we are left to trust the author as to the importance of a definition or the reasons for conjecturing something. It is just like it was with proofs back in the day.

Yet, we don't really think that any conjecture is just as important, and the same goes for definition: If I ask which prime numbers can be represented as the sum of a square, a cube, and another number to the power of 7223, we would all feel that this is a strange question and probably not be too interested at its truthness. And if I define a "wonderful number" as a number whose decimal expansion, when written backwards and multiplied digit by digit by its original decimal expansion turns out to be an arithmetic progression, then this definition would not seem important to most of us. So, just like we don't think any theorem is true, we also don't think any definition or conjecture is significant.

There is the typical example of Fermat, who didn't prove any of the conjectures he suggested, but (almost) all of them turned out to be true, after many years of attempts of later Mathematicians to prove them. Now we have basically completed the lack of proofs in Fermat's Mathematics. But what about the questions themselves and the definitions. Why did he look at primes of the form x^2 + y^2? Why did he think it would be important to solve Pell's Equation? Why did he look at Fermat's Last Theorem? Fermat had some amazingly strong intuition, and if he were alive today, we might ask him about these things. But even so, he might not have answers. We don't know if Fermat had exact proofs of his results or not, but even if he didn't his intuition worked well for proving conjectures. It also worked well for creating conjectures. There is no reason to assume he had answers as to the motivation behind his many conjectures. But we cannot ignore the vast amounts of significant Mathematics that had been developed as part of the attempt to understand Fermat's ideas. Later Mathematicians proved Fermat's results, sometimes developing complex fields of Mathematics that were definitely now known to Fermat. Our task now it to properly find what is hidden behind the conjectures of Fermat, even if Fermat himself didn't understand it.

We have developed the system of proofs in order to understand what a theorem truly is. We now need to develop two new system in order to understand what the significance of conjectures and definitions truly are. Some of this has begun in Motivation.

Some people have pointed out to me that different people have different intuitions and a different understanding of what a 'natural' step is. That is also true for proofs, and it is wonderful to have many proofs of the same theorem, each for a different type of viewpoint. The same goes for motivating definitions and conjectures - we will end up with many different "answers" for the "why?" question -- each just as important.