Improving my mathematical rigour

I have reached a point in my mathematical journey where I feel the need to learn how to make sound arguments for the validity of a mathematical claim. Or in other words, I want to learn more on how to prove things.

The path I took through the Danish educational system has never dealt much with mathematical proofs, but rather on how to apply the mathematics we have learned. I have developed an intuition for mathematics in some areas. But I lack mathematical rigour, so I often time have to resolve to hand waving instead.

The usual approach to learning proving techniques is through a taught topic where you are presented with some proofs. Through that you will expand your toolbox and learn how to do proofs. However, I would through a series of blog posts dabble into how to prove mathematical things and study different techniques.

Ben Tilly pointed me through his blog – random observations – to a document he wrote on how to do proofs. It has a flow chart which you can also see to the below, which I think is a very thorough way to ensure that you get through the proof. It doesn’t say anything about how to actually make your arguments, but it helps you break down the problem.

Let me spend the rest of this blog post to go through the flow chart and interpret it.

Explaining the flowchart

Flowchart on how to do proofs

Click on the image for a full size view

Read it: As Ben Tilly notes it is the obvious first step to read the problem that we want to prove.

Do I understand it?: It is important to be honest when we ask if we understand the problem? If not we need to work it out.

Work it out: This is a place where we won’t have to spend a whole lot of time during the first blog posts, since the problems will be pretty straight forward and easy to understand. However, in general it is the place where we make sure we understand the definitions and you might want to run through some examples. I love to do a couple of examples (if I can come up with them) to get the feel for the claim and see what it does.

How many parts: Once we understand the problem we can ask how many parts the claim exists of. There may be several claims in problem which we need to prove individually.

One example is the fundamental theorem of arithmetic, which claims that any integer larger than one can be written as a unique product of prime numbers. In order to prove this we would need to prove both the existence of a prime factorisation and the uniqueness which means two parts.

Another example is the the subtle “if and only if”, which means that we need to prove that something is both Necessary (the “if” part) and sufficient. (The “only if” part).

For the first part of the series of blog posts I think that many of the problems will only contain one part.

Select a part: Well not much to say here. However it is important to realise that we should only work on one part at a time, otherwise we will end up killing our self. Which one to choose first is most likely a matter of personal taste. I tend to start with the parts I know how to approach as they are then removed as problems.

Do known techniques apply?: This is where we open the toolbox which and see if we can use some of the already known techniques we have for proving things. This is where we will dabble for the next while, describing some of the tools and techniques which may be applied to the problem. Some tools could be direct proof, induction proofs or proof by contradictions, but don’t worry this is what we will work with for the next while.

Any Idea on why it is true?: If we have an idea of why the result makes sense, but can’t form it into something we can apply the methods to. Try to describe why it makes sense and we may end up with a game plan for proving the claim. Once again working with examples can sometimes help us understand the problem and get the feel for what is going on.

Ben Tilly notes one pit fall of this step, where we might end up making a heuristic proof, something which supports that the result is true through a somewhat plausible line of reasoning. However, the arguments are lacking the rigour for the claim to be a real proof. This is where I often end up and what we want to avoid.

Try to find a new technique: Sometimes none of the tools we have in our toolbox will solve the problems. That means one of two things. Either the problem is unsolvable or we need a new tool in order to be able to prove it. The first is likely not the truth, at least not for now, but at some points in life something might just not be solvable.

Being side tracked a bit here, but just need to state something I have learned in my academic life so far, is that sometimes a negative result is a good result as well. At least if we can show why something we thought was possible is not. And yes, I know there is a difference between not being able to prove something and proving that something is false.

What do I need to do?: Now that we have found a tool we think is appropriate, so let’s dig out the recipe for using that tool.

Try it: Until we actually apply the technique we have decided to try, we won’t know if it works out. Trying it has two possible outcomes, either it works and we are happy, or it doesn’t work and we are sad.

Are you frustrated?: I would love to say that I have never been at this stage, but that would be a huge lie. The hard part is recognizing and accepting that we might be frustrated so we can do something about it.

As Margaret J. Wheatley once noted “In virtually every organization, regardless of mission and function, people are frustrated by problems that seem unsolvable.” Doing proofs is no different from that.

Do something else: Either we can try another part of the proof which may shed some light on the problem where we are stuck, or maybe it is time to do something else. At least for me things need time to sink in, so walking the dog, cooking dinner or taking a shower oftentimes let my brain get a rest and suddenly a new idea for an approach pops out.

Am I done?: This is pretty self explanatory, if the this we have proven all parts then we are done. Otherwise we still have some parts to prove.

Write it all up: This step is made mainly for students who are supposed to hand in their work, but it is actually a pretty important step. Writing this up in one place and checking that everything is there and that everything is correct is important to improve the rigour

And finally we are done with proving something. It is a rather long process to explain, but I think it is a good explanation of how it can be done, and I do believe that it will give me just the structure to work within when proving things.

I hope that we together can go on a mathematical journey and learn something from each other.

Posted by Kristian

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