Can we discuss mathematics?
Well, most would tend to say no, we can’t. Either something is wrong or right. 1+1=2 is right, and 1+1=0 is wrong. But wait, even in this clear cut example, it makes also sense to assume the latter is true. OK, you say, but once the assumptions about the numbers we are working with are fixed, we can no longer discuss. Theorems hold or do not hold. Again, wait! Isn’t there the result by this Goedel saying that there are theorems which are neither wrong nor right, theorems that cannot be proved, nor denied by counterexamples, yet even theorems that cannot be proved, but are true in some other sense? You will argue, that at least a proof is either correct or not correct. That beats me almost. But years of experience tells me that there is nothing more disputable than a mathematical proof, at least, if it got some substance.
First of all, arguments are not always as dense as we wish. Once we try to fill in all those annoying details, the proof gets too long to be checked easily. The history is full of examples, and errors were discovered in long established proofs. A simple proof of a complicated theorem is a rare thing, but of course desirable. At least, in theory we can check everything. So, OK, a proof is either wrong or right.
In some respect, mathematical theories can be wrong, however. We should always keep in mind that the assumptions are arbitrary. So there is ample room for discussion right at the start. Next, we could ask ourselves if the time spent with a theory is well spent. I could name numerous theories in mathematics which I would not classify as worth thinking about, and others which are completely neglected. But I’d better not.
Finally, if we turn to apply mathematics to reality, we open a new can of worms. Suddenly, the correct hypothesis is far from clear, and consequently the truth of our results. Do they really apply for the situation they are made for? And if it is the result of a numerical computation, things get even worse. For then we have to deal with rounding errors, wrong programming etc. In short, we are in no way better off than any other empirical scientist.
But I am not writing this blog post to ruin the golden tower of mathematics. My main intention goes towards the teaching of our subject. Students tend to follow the books. The books contain the truth, and they also contain the important mathematical content. Things not in the book are not important. And they are not part of tests. So the idea of a canonical mathematical wisdom builds in the student.
I wish we could break this idea. In my lessons, I am not so bold, and I do stick to the canonical path, because I feel I have to. But sometimes, I’d like to break apart and discuss with the students the problems mentioned above. I often like to present applications, where the correct theory to apply is not obvious, and weather the result has really some meaning. E.g., in statistics such discussion could lead to fruetful ideas. Or I l’d like to ask why a specified subject is in the lesson. I’d also like to discuss proofs. Is the proof convincing? Are the assumptions necessary? Is the result trivial? Students do not like this, however. Humans do prefer the straight road as in the book. Only, that there is really no such thing in mathematics.
i recently read another article that i think you might find worth reading (my very limited judgment), maybe i can find it again:
„I’d like to close by saying a couple more things about the Russell-Whitehead system. We all know that symbolic logic is a closed abstract system that does not say anything directly about logical thinking. The question, though, is how useful is the system. I’d be inclined to say it is not very useful. The system has had a couple of failures. They tried and failed to extend the system to provide a basis for mathematics. G�del’s Theorem is related to this failed attempt. They also tried to extend the system to include inductive as well as deductive logic, and all they got here were some amusing paradoxes. There is really no way to codify inductive logic. The only way to study it is to play games of Eleusis.“
Nice link. I like that student.
The simple truth is that we cannot understand mathematics completely, not even simple arithmetic. We might be unwilling to accept that, but it the essence of Gödel’s work.
In real life, that is meaningless anyhow since there is so much we can understand, but failed to understand to date.
„And how many geometries can there be anyhow. So far, no answers to any of these questions.“ This question has been answered by Hilbert to some extend, by the way.