Does writing a bachelor thesis make sense?

Yes. It's true that the undergraduate knows almost nothing and would certainly be not qualified to write an original scientific paper. But there are some advantages:

• You do not know enough to write an original essay, but if you write it, you'll get used to writing academic papers: You get used to $\LaTeX$, you get used to the schedule of an academic paper, etc.

• At some universities, these papers made by undergraduate students do not need to be something original, suppose you make a small calculus book that has a nice (and easy to write) feature, you're helping others to learn. The work might not be original, but it's useful anyway.

You'll have to first look carefully at what are the requirements of a bachelor's thesis if you choose to embark on writing one. Doing original mathematics research that you can write up in a form that is basically suitable for publication is a complex process that usually cannot be undertaken solely by an undergraduate. You would need a good adviser to help you in your project. Such an adviser should ideally be able to (1) come up with one or more problems that are suitable for undergraduates (this often involves knowing the literature in one field of mathematics fairly well and translating a more general problem in the literature into a special case that is suitable for an undergraduate's current background level); (2) be able to offer hints and suggestions when your first attempts to solve the problem inevitably fail (i.e. you're supposed to struggle with a problem if it is one that is in need of an original solution); and (3) guide you through the process of turning your discovered results into a well-written publication.

At many universities, the requirement that an undergraduate contribute original research in this strict sense of solving an unsolved problem is typically relaxed. Often, it is acceptable for an undergraduate to write an expository paper on an advanced topic in mathematics. Thus, issues (1) and (2) above are replaced with an adviser being able to (1') select an appropriate level of advancement for the type of project you are interested (i.e. not too simple and not too difficult) and (2') suggest specific references to consult and specific examples to work out to help you understand the topic well enough to be able to explain to others.

The latter task is much more feasible as a first exposure to research. It involves many of the skills that you will use in future research if you decide to pursue a post-graduate degree in mathematics. But it relieves you of the pressure of coming up with new results. You would still learn a lot about mathematical writing and it is certainly not trivial to learn about an advanced topic well enough to be able to write a good expository paper on the topic.

If you decide to pursue a project along the former lines, then you should be sure that you pick an adviser who has some experience with guiding students in this type of project. (Because your university does not require such a bachelor's thesis, this may be a non-trivial task.) You should have a long period of time to work on the project and should get started as soon as possible. Without any experience in mathematical writing, you should aim to complete your research first and obtain results suitable to write up several months before the bachelor's thesis is due. You will need that additional time to learn how to write up your work using LaTeX and will likely have to create many drafts before you produce a suitable document. Trying to do original research at the same time as you are trying to learn how to write a substantial mathematical document for the first time is a recipe for likely disaster.

I think it's a fine idea to embark on a thesis involving research.

• Showing you can "learn math" and have learned a solid core of undergraduate math is certainly important.

• But it is also important to show that you have the independence, ambition, creativity, and determination to "do math".

The first you can accomplish through taking lectures/classes that cover some of the core concepts expected of a math undergraduate.

The second you can do by deciding upon a thesis topic, doing some preliminary work, as in a literature review, presenting the proposal to a potential advisor, securing an advisor, and most importantly, doing the research and "doing the math" required to write an appropriate thesis. Even better, you might even want to submit it to an appropriate journal that publishes undergraduate and/or graduate research in mathematics.

So if the idea of writing an undergraduate thesis appeal to you, and you are willing to commit to the work involved, then GO FOR IT!