Infinite number of mistakes in a proof

Writing my Bachelors Thesis has opened my eyes to what seems to be a horrible paradox. I am turning my thesis in this Friday, and have been proof reading for weeks now. Every time I print my thesis, read it through with critical eyes or have someone else do it for me, I will find a number of mistakes. After fixing these mistakes, explaining things in more details and carefully rethinking the proof I print the thesis to redo the process. But the next time, about the same number of mistakes will be present.

This process does not seem to converge, and after weeks, I still find small and large mistakes in the proofs. The reason seems to be, that when I find a mistake, I try to explain more carefully what happens, introducing more details in the proof. And with these new details, I make room for more mistakes. A mistake is not necessarily something which makes the proof wrong, it is just a slightly wrong argument for a correct statement, but nevertheless, they have to go.

Maybe this is a general problem, since opening up for more details also opens up for more things to be explained, and to explain those to an extent, will open up for even more details.

Do you experience this when writing proofs?

Yes, yes, we all know what you are talking about.

Although many young mathematicians starting out expect that once they prove a theorem, then it is an easy matter to just write it up, I always caution my graduate students that writing up a piece of mathematics is at least as much work as coming up with the original proof or proof idea. It is during this writing-up process, if taken honestly and carefully, that one checks through all the details of an argument, and very often subtle mistakes or misunderstandings are uncovered. Perhaps at the initial idea stage, one had implicitly assumed that a certain property would be true, but it turns out in the end to be a nontrivial thing to ensure, or perhaps a certain hypothesis turns out not be the case in the situation where you need it, and so complicated workarounds are introduced. Usually one learns an enormous amount of mathematics during this process, and one's knowledge of the fundamental issues surrounding the topic is strengthened.

But meanwhile, as your paper is revised, if the mathematics is sound then the process does converge. Eventually, you will find yourself taking care of smaller and smaller issues, and towards the end you may find perhaps that you are tweaking font sizes and spacing issues. (I recall in my own dissertation eventually spending a lot of time to make sure that the relative parenthesis sizes were systematically correct in nested expressions.) At this point, you have finally reached the limit point towards which you had been converging all along.

So I encourage you to stick with it, and eventually you will get there. But be sure not to stop too soon!

It's like the length of a coastline: it grows without limit as you measure it with shorter and shorter rulers. But take heart: eventually you will reach the atomic scale, and then you can invoke Heisenberg's Uncertainty Principle to explain your errors.