You may have thought about it in some point of your life, especially if you are a math person. "How to come up with an original math idea?" The question is not a simple one. You may or may not need to have some level of understanding in mathematics in order to think of something unique, as both the experts and newcomers in a particular field could possibly make a meaningful contribution, with of course the experts having an upper hand.

There are at least two sides to creating original ideas in mathematics. The sides are expert and newcomer levels.

The first one is at the expert level: you understand some field in math at so deep level that you can come up with ideas that advance the field in question. In other words, you can count on your reliable skills and knowledge so much that you have the power to think of something new.

The second is the newcomer level, and actually this is the one I am most familiar with. You have no knowledge in the particular math field, and may not come up with new ideas so easily as the expert would. But you have one kind of an advantage the expert may not have: you know nothing, and thus you are not bound by any field specific mathematical laws and rules that would naturally restrict your thinking processes if you knew of them in the first place.

The experts have the knowledge, and may not think as freely as you do. I, for example, started thinking of trees as sums of numbers the moment I was introduced to the generation problem of series-reduced free trees. As I had no previous knowledge in the theory of trees and especially in the ways one would represent them, I was not bound by some rule that trees are represented by matrices, for example.

Coming up with an idea in math is not so hard, but coming up with an original one is. Many have come up with an idea only to later realize it is not new, and thus maybe regret coming up with the idea in the first place. No sane person would throw away days or even years of their life. Being an original thinker is a gamble, and in the first days of discovering something, always check if the idea already exists. Ask a mathematician for clarification, for example. I myself did not, and had maybe worked on the idea for about half a year until I finally started to seek the answer to the question: "Is it original?" The luck seems to be on my side, though, as to this day I haven't seen any similar idea that is publicly available.

There is also the one aspect in an original idea that you may not be aware of: The process of making others agree on it. This may be even harder than the actual process of coming up with the original idea. I have one tip: use general, widely accepted notation that the mathematical community uses. I was blinded by my 'superior' notation only to realize no one dared to check out my ideas as they were too cryptic to them. I have, therefore become more humble in the notation practices, but still honored the identity of my ideas by not discarding every new notation I have come up with. Mainly as there previously has not been the tool I have created.

If you have actually an original idea and mainly have generally accepted notation, you have at least four options:

1) Publish your findings in your website and send your ideas to yourself in a well known email provider

2) Make a book out of your ideas and publish it with an isbn code

3) Show your results to a mathematician and proceed to publish with his/her help

4) Make a scientific paper and publish it in a well known mathematics journal

I myself can say that I have done the options 1) and 2) and tried the options 3) and 4). The main point is you should have some kind of proof that you have come up with the idea on a particular date that if someone else discovers the same idea you at least have the priority in case a dispute happens.

In my opinion the options are in a difficulty order with the first option being the easiest and the last one being the hardest. You should always first at least do the first option's part in which you mail the ideas to yourself, at least then you have the date of the ideas in the email provider. Also make sure that the website is yours and not someone else's as you would not possibly have the control over your ideas if the person in question would turn against you.

If you make a book make sure it has an isbn code: this is mandatory for public books, and I really recommend that you make it public. Include the year the book was created and a copyright notice.

You should try the option 3) with caution, as you need to be prepared to reason your arguments and possibly even show a proof of your results. Make sure that you have at least done the email part of the option 1) before you try this one.

The option 4) is the hardest, especially if you are an amateur. You need to have done proofs, formatting, general notation and citing in a correct way, and make sure you try to publish your findings in a WELL KNOWN journal and not some shady one, as there is a possibility of your ideas being stolen.

So there you have it. I hope this does not discourage you but rather that you give things a try should you be lucky enough to come up with an original idea of your own. Start with creating mathematical problems for yourself that you can actually solve. This worked for me, as creating problems one can solve is actually harder than solving them in the first place, increasing your skills and creativity in math and also possibly giving you some new insights and ideas. Try to create problems that would have been a challenge for you. Or, if you feel being tricky, maybe even create a math problem book with your own challenges! That could be fun and original at the same time...

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