u/Salt-Rutabaga-8870

Hi there, I have recently posted here about my journey, as an amateur mathematician, which led to a scientific collaboration and a paper in Bulletin of London Mathematical Society. The post received a lot of positive attention, for which I am grateful, however, a lot of people in the comments, understandably assumed that I was a crank at first. It fact, most times an amateur mathematician claims that they have made a contribution to the math body of knowledge, their work is trivial, wrong, or not even wrong. So, as an amateur mathematician who "managed to do it", I wanted to write a list of advices for others, if they want to be taken seriously and not viewed as maths cranks. This has been done before, here is one example. But most of those advices are written by established mathematicians, and while they are correct, and I agree most of the points they make, I feel like me being an amateur who somewhat succeeded gives me an interesting perspective on the issue. I also have interacted with several people who can be considered "math crackpots" personally, so I can reflect on that experience as well.

This is actually not my first time doing this. I have written an answer to this question on Academia Stack Exchange, I could not post it since the question was protected, so the author of the question opened a chat room, where I have written my answer, but, unfortunately that was taken down. You can read the comments under the question, they give a bit of a story, but in any case that text was lost, so here is my second attempt. Hopefully it won't be taken down.

As we are here in the domain of mathematics, let us start with definitions. I define an amateur mathematician as a person who:

1. Does not have any formal mathematics education beyond high school. Not a Bachelors, Masters, or any other equivalent or higher qualification. That would include people like myself, who might have a degree in chemistry, or engineering, or some other areas which are related to maths but not maths.
2. Engages in doing maths in their own free time, out of personal interest and amusement. That can be anywhere on a spectrum between solving sudokus to learning about sphere packing in 8 dimensions and trying to crack unsolved mathematical problems. Naturally my advice is targeted towards people on the latter half of this spectrum.

Disclaimer: everything I will write here is based on my personal experience as an amateur mathematician who "did it". While I do think I can provide a valuable perspective, it is still based on personal story and I do not imply that what I write is an absolute truth.

With that, let's begin. My list of advices to amateur mathematicians who think they have found something interesting and new in their pursuit of maths:

1) Make sure you understand the basics.

This is crucial. In my case, while I do not have a maths degree, I did get A* in A-level maths and A in Further maths in school. I also have worked as a maths and further maths teacher and/or tutor for more than half a decade, and my style of teaching involves deriving vast majority of mathematical statements I teach to my pupils, rather than just throwing formulae and rules at them. The necessity to explain maths as part of my job boosted my understanding of foundational concepts over the years, and when I got to work on what later became Fractional Residue Theorem, while I did make mistakes and assumptions, I was not steering too much into the crazy territory, and when I did, I was aware of that. My guesses and methods might not have been rigorous, in a strict mathematical way, but they were sensible.

I think having this baseline knowledge is essential. I have seen people who claim that they have made discoveries, while being unable to wrap their head around idea of irrational numbers. Thinking they are "infinite" or "incomputable" or "contain every truth on earth". There is often a lot of "magic thinking" around mathematical concepts which are actually very trivial. You also need to know stuff like that 0.(9) =1. Not very close, not almost, not infinitely close, 0.(9) identically equal to 1, and why that is the case at least at some level of rigour.

This advice from a real life example I have encountered, where a person had no understanding of irrational numbers, and tried proving pi and e were rational. The same is true about basics of calculus, algebra, and axioms. I do not think you need very deep knowledge, for example I have only very basic understanding of ZF(C) set theory, but you need to understand why you need axioms and what axioms are.

So, advice number 1: Make sure you understand the basics very well. At least at the level of A-level maths and some topics in further maths. When I say well, I don't mean well enough to get a C on the exam. I mean well enough to explain to a very demanding student, why those concepts and ideas are the way they are, while being able to answer non trivial questions and connect different topics to each other.

2) Invest in a tutor. (No, AI will not do)

While I do think AI can be useful for an amateur mathematician, and I will mention it in the latter advice, I think if you want to pursue advanced maths, having a human being with an expertise is essential. They do not need to be a PhD, my tutor, who taught me the basics of complex analysis necessary for me to come up with the foundations of FRT, has a masters degree. They were able to clearly explain to me the ideas behind CR equations, contours, poles, Laurent series, etc while answering my questions which arose on the spot, in a way I don't think AI or watching videos on YouTube would be able to do. My tutor was also able to provide guidance on my initial work, for which he is thanked in the acknowledgments in the paper. He spotted obvious mistakes and weak arguments, and was giving me sanity checks. He also suggested a few venues on how I can explore the thing I found, even though he was unable to help me with the fractional calculus part of my work, as he did not know about it.

A separate note regarding using AI. I have known at least two people personally, on top of many internet stories, where AI convinced people that they are Euler level geniuses. AI is a tool, but it is also a product, and companies are invested in you continuing to use it. So AI can be very sycophantic and tell you that your ideas are interesting and correct even if they are not, to keep you engaged. Also, AI can hallucinate. Even if it does that 1 time out of 50, you cannot count on it in serious mathematical research. Aside from maybe asking to provide references or useful literature, but then you need to go and read those works yourself.

So, advice number 2: Invest in a tutor who will be willing to help you learn, and be polite but firm when you are making mistakes. They can also be the first person to ask about interesting things you find on your math journey.

3) Assume you are wrong.

Seriously. Assume what you found is either wrong or trivial. Because it probably is. Before I got my interesting results which laid the foundation for FRT and the collaboration with Arran Fernandez, I had some other ideas which turned out to be incorrect or already known. One time I had an idea about exploring a non zero number which squares to give 0, similar to how i squares to give -1. And then I learned about Dual numbers :D.

The thing is, maths is a vast, vast, vast body of knowledge which accumulated over thousands of years by minds like Euler (the GOAT), Gauss, Riemann, Galois, Cantor, Noether, Ramanujan, Euclid, Perelman, Wiles, Viazovska and many many others who had much more training and expertise than you, and the chances that you have spotted something those giants have missed is miniscule. These odds are stacked so much against you, that it is better to assume your thing is wrong or trivial. But the pursuit of finding out while it is wrong or trivial can be an interesting journey in of itself.

My exploration of interplay between complex analysis and fractional calculus resulted in a scientific paper, but even if it did not, it was an interesting journey by itself. I got some answers on MSE, learned about a book by Prof Samko, got in touch with several maths professors. There is value to exploring your idea, even if it ends up being not new, or completely incorrect. But you need to keep an open mind and accept that you are wrong if it turns out to be like that. So I think it is better to assume you are wrong from the start.

So, advice number 3: Assume that what you found is wrong or trivial. Value the journey of exploration by itself, and who knows, maybe you will be pleasantly surprised.

4) Avoid "popular" problems and areas.

I have already mentioned how odds are stacked against you, when you are an amateur, regarding making a contribution to maths. This is even more true when we talk about exploring stuff like Collatz Conjecture, Riemann Hypothesis, Twin prime conjecture and other massive unsolved problems in massive and well studied areas like number theory, complex analysis, topology etc. All of those areas and problems have hundreds or thousands of brightest and most trained mathematicians working on them every day of the week, probably every hour, that you, with your amateur level of knowledge cannot even fathom 1% of their work, let alone make a contribution.

Here you might object and point out that my own work was on the interplay between complex analysis and fractional calculus. And that is true, but the devil is in the details. Fractional calculus is not a massively studied area of maths like number theory or real analysis or topology. There are not so many researchers who study fractional calculus, and most of them do it from real point of view. So you can explore interplay between two areas of mathematics, one popular and one less so, but I encourage you to explore those less popular areas. The odds are still against you, but much less so when we are talking about area like fractional calculus, as opposed to area like linear algebra.

I truly believe that the only reason I was able to make a contribution is because I went to explore a rather obscure area of maths, following the footsteps, without knowing it, of the people who laid the foundations of fractional calculus hundreds of years ago, like Lacroix and trying to apply it in a creative way to a topic I have learned with my tutor, namely the residue theorem. I did not try to solve a massive unsolved problem, I was playing around with some half forgotten concepts. And even then, a lot of the theoretical knowledge that leads to FRT was already known, I encourage you to watch the seminar led by my co-author for details. But basically most of the knowledge, like fractional Cauchy formula was already discovered. I just took it one small step further, trying to apply fractional calculus to complex analysis in the same way normal calculus does.

So, advice number 4. Explore less popular, more obscure areas of maths. This increases your chances of actually finding something new. But also it is cool even without making a contribution! Learning about fractional calculus is interesting and now I can impress my maths tutor and some other maths experts in my life with math knowledge they don't have, since fractional calculus is usually not a part of the standard undergrad or even postgrad course. Lesser known areas of maths are interesting!

5) Don't do maths for the sake of discovering something.

If you want to be a maths researcher, go and get a degree, and then a PhD, and then postdoc, etc. This is hard but it is the only proper way. If you are an amateur, do maths for the fun of it. You are not constrained by grants, or deadlines, or university administration, or by peer pressure. You can do maths for fun. And maths for fun is the best kind of maths.

If you only engage in mathematical activity, as an amateur, in hopes of discovering something, you will probably be disappointed (See advice number 3). Yes, I managed to do it. But just like it would be irresponsible for person who won a jackpot in the casino to start giving financial advice to people to go and play slots in Vegas, it would be irresponsible for me to say that you should do maths at an amateur level with the aim of discovering something.

The difference between doing maths for fun and wasting your money on casino slots is that there is value in the former even if you don't get lucky like I did. Doing maths is fun! You learn a lot! It trains your brain! You can impress people with it (or sometimes be called a nerd, but who cares!). It is a nice skill to have, and a nice knowledge pool dive in. There is value to it. And I encourage you to explore maths at amateur level for that, not for the slim chance of discovering something new. If you want to do that, go and get a degree. Let the discovery be a welcome surprise, if it happens at all, rather than the almost unattainable goal.

So, advice number 5. Don't do maths in order to discover something. Do it for fun, and then maybe you will get lucky. But don't focus on getting lucky. Focus on the fun.

6) If you think you found something interesting, and your tutor cannot explain it, go to MSE before bothering an expert.

The fact that someone with moderate expertise in maths cannot explain your findings does not mean they are actually new. The last true polymath died at the dawn of 20th century, and today nobody knows every single bit of contemporary mathematics. However, the thing you have found may be extremely trivial to an expert, and rather than wasting their time, go to a forum like Mathematics Stack Exchange and ask a question about the interesting thing you found. If what you found is already known, the collective expertise of forum users will be enough it most cases to point it out to you, as well as provide useful links and literature references, should you explore the topic further.

Also posting a question on MSE in the way which is acceptable will train you for the next bit, if you you won't get an explanation from MSE users regarding the interesting thing you found and decide to go further and get in touch with an expert.

Posting a question on MSE before contacting an expert will also show them that you are serious and that you took you time for due diligence and won't bother them with some trivial or crazy stuff.

So, advice number 6: Before getting in touch with an expert, post a question on MSE. Ask if this has been done before. What are the limits of your finding. Is there any literature, etc. But always remember advice number 3 :D.

7) If you think you found something interesting and you want to share it with an expert, make sure you present it in a proper way.

If you think you discovered something interesting, as an amateur, you will eventually need to get in touch with an expert in the field. Even more so, it would have to be an expert in the specific area of mathematics you worked on, not just a random PhD or even professor. You have to search for those people, and there maybe only a few dozen of them in the world, who have expertise in the area you are interested in.

That means no word documents for maths proofs. Use LaTeX and other established conventions. If you want someone with the expertise to take their time and look at your notes, make sure you take your time to make them presentable.

Here is where AI can actually be useful. It saved me a lot of time and effort to just ask AI to "Write this integral in LaTeX..." pasting it to Overleaf and then editing the errors, rather than learning all the intricacies of LaTeX myself. But just remember that you are the one responsible for what is written in the document, not ChatGPT.

I cannot attach the file here, so here is the link to a LinkedIn page where I uploaded my original notes, which I have sent to Prof. Fernandez in October 2025, when we got in touch and based on which he offered me to co-author a paper. Now, this represents significant point in my journey, as this is the final stage of it where I got by myself (not counting help from my tutor). This document was written by me (Using AI for LaTeX, as mentioned), and I think this is an absolute minimum level of presentation you should aim for. My document is not even that perfect, there are typos and even a single math mistake (I wonder if you can find it, it is pretty trivial, and being able to do so is actually a good indicator of having basic understanding of maths, see advice 1), but my document offers a clear format for an expert to look at my ideas.

So, advice number 7. If you are sharing the findings with an expert, make sure you present your work in an acceptable way.

8) If you get in touch with an expert, be polite and cordial. Ask questions instead of making claims that you discovered something interesting.

For reference, here is the text of my first email I have sent to Prof Fernandez, again, with all the original typos:

"Dear Dr Fernandez

I am a high school maths teacher and an amateur mathematician, who recently encountered an odd phoenomenon when trying to apply fractional calculus to solve some problems involving residue theorem. While the mathematical rationale for doing it the way I did was shaky at best, i managed to get correct answers for various classes of problems. And I do not understand why what I did works. I first tried asking my maths tutor who has a masters degree in mathematics, and he could not explain why this works. I tried posting on mathstackexchange, a forum for mathematical problems, and while I got some answers, people who wrote them said that they were not exactly sure why this works. One of the answers referenced a book written by Prof Stefan Samko, I tried reading ithe chapters referenced, but I don't think it explains why I got the results that I got, although the book is highly technical and I might have missed something. So I tried getting in touch with the author himself, but I have learned from his university, that he had retired, so I got contacts of his two former students, one of which gave me your contacts, as one of the leading experts on fractional calculus, which brings me here.

I ask you if you could take some time out of your busy day and have a look at my notes. I have compiled them in pdf format via overleaf, so they are clear and easy to read. I really want to make sence of this set of problems I encountered and your insight, as one of the experts in this subfield of mathematics can give me the clarification I have been searching for. Just to give you an idea of how much time it woukd take, my notes are about 13-15 pages of pdf. + some references.

Please let me know if you are happy to look at my notes and I will send them to you as well as be forever grateful for your time and expertise. 

I look forward hearing from you.

Kind regards

Egor"

Few things to point out:

First of all, be better than me regarding typos, I can really improve on that front. I am very grateful to Prof. Fernandez for looking past that. But some others may not be so forgiving. And it is totally reasonable for them to not be forgiving.

Secondly, I do not claim I discovered something. I ask polite questions. Remember advice number 3. It still holds even at this stage. I don't even send my notes at this stage. I sent them in my second email after receiving his reply, confirming that he is happy to see the notes.

Thirdly, I am polite. I am asking a person with an expertise less than a hundred people in the world have to take their time and look at my amateur notes. I give an outline of my journey to show that I am not crazy, as this can be assumed. Remember, they do not owe you their time, and you better make sure you present yourself as someone who will not waste their time.

Also, If the expert tells you that you are wrong, accept it. Don't try to argue or insult them. They may say something like "this does not work and the reason this does not work requires 10 years of studying topic X." If that is the case, the accept that and move on. If you are really curious you may try to find another expert in the same field, but again, be polite, and if they give you the same answer as the first one, just stop and move on with your life. There is more fun maths stuff to do.

So, advice number 8. Make sure you are polite with an expert you are getting in touch with. Do not make claims that you discovered something, show interest in the area of maths you are exploring, and they presumably have expertise in, and don't waste their time. And accept that you are wrong if they tell you.

9) Have some results to show.

This really depends on the exact area of maths you are exploring. But in my case, I found 3 different types of integrals where my amateur approach worked (actually I found 4, but the 4th one was trivial so I did not include it in the original notes sent to Prof Fernandez). But the fact that I manage to get correct answers, I think, positively contributed to the conversation.

Instead of having vague hypothesis and claims, show that you can actually apply the interesting thing you exploring to solve a problem. Not a massive unsolved problem but a simple maths problem, like an integral in my case. That means you are acting in good faith, and you can actually achieve provable correct results.

So, advice number 9. If you get to the stage of getting in touch with an expert, and perhaps even earlier, make sure you can show some mathematical results which stem from applying your work.

I was thinking about the 10th advice, but I don't want to chase the round number, just for the sake of it. If I think of anything else, I will add it. I also encourage you to write your suggestions and disagreements with my list in the comments. I will do my best to read them :)

All the best.

Egor