# Practical Tips: Mathematical research and discoveries

How to behave when you have the feeling of working on something innovative? What to do if there is a chance (even the 1%) that your work is leading you to something original?

For example what if I don't know mathematicians I trust to ask? Is a good idea to talk about your results (even if are not real results) to someone?

Question:

What do you do if you have this feeling?

I'm looking for a list (or links to related SE questions/guides/books or others) of practical tips to use while studying a certain subject you have the feeling of having found an innovative approach that can provide a new solution.

I am interested too in books about the mathematicians' way to research. (I think is related)

I already asked this question on Math.SE but was closed two times. Here the link to MathematicsSE

• What do you mean by trust? Is this question related/duplicate? May 10, 2013 at 12:10
• @DanielE.Shub yes this qeustion is a bit related thanks for the link . Anyways that was only an example, my qeustion was more general. May 10, 2013 at 15:55
• There is an art to balancing "diverging" and "converging." By diverging, I mean using your intuition to figure out what "should" be true - what you think you can prove, if you try. Converging means actually proving it. You don't want to get too far out there into the unknown without proving milestone lemmas to anchor you down, or you may find that much of your research is predicated on a false assumption. On the other hand, forcing yourself to prove every little thing while suppressing your imagination simply won't lead to interesting research. May 19, 2013 at 3:36
• @AlexanderGruber yea, that is one of my problems too, if I try to formalize all I my ideas, the work becomes long and more i continue and more i lose the meaning of what I'm doing, but I'f travel with fantasy I continue to find a great number of ideas and connections and I come in a huge landscape of conjectures... Do you know some reading about this? there is maybe some point that I must remember while working in order to find a balance between "diverging" and "converging."? May 19, 2013 at 8:55
• @MphLee Actually, the terminology "diverging and converging" comes from my previous career in design. I don't know a good place to read about this, but it is a relatively common topic of conversation in art school. The rules on when to diverge and converge aren't set in stone and are a little different for most people; however, my rule of thumb is not to get more than about "two steps ahead" of what I've already proven before I try to converge. (After all, sometimes going back and proving the conjectures helps you think of new ones, too.) May 20, 2013 at 0:20

`0.` To give original contribution to the understanding of mathematical objects is what mathematician do, and talk about all the time. There is a priori no need to fear for ideas being stolen; if you want to play secure, you can publish a preprint to ensure priority.

Given the wording of your question, I assume that you are an amateur mathematician, and that you feel that you may have found something about an important, well-known question. Sorry if this is not the case.

As most mathematicians, I receive demand for advice of this kind from time to time, so here is my usual answer.

1. Be prepared to have made a mistake,
2. be prepared to have found something known for a long time,
3. be prepared to have found something that will not attract interest.

This may sound very negative, but these are the worries that are much more likely to be relevant than seeing your contribution stolen. I have seen recently an amateur publishing on viXra after seeking advice from me, that was afraid of having her ideas stolen. It turned out that her contribution was a few hundred years behind current knowledge.

Researchers in mathematics only succeed in advancing knowledge because they spend much time learning their specialty and keeping up with what is being proven, and we do sometimes reinvent the wheel (I got scooped by 130 years once, realizing that Camille Jordan already solved a cute problem I was interested in), or make mistake, or do things that do not interest our colleagues. It is tremendously difficult to avoid this pitfalls when you don't have access to the literature, don't have colleagues to speak with about your research, don't have a regular seminar to listen to, don't have had a PhD advisor to guide you through your first problems.

So, for a positive piece of advice:

`4.` learn the field you are interested in (e.g. read books, from the point you are in your knowledge to the field you are interested in). Be prepared for this step to take much time.

• 1,2,3 are all very true. And sometimes, re-inventing the wheel can result in a highly cited paper. E.g. "Preferential Attachment" (en.wikipedia.org/wiki/Preferential_attachment) was re-invented several times (1925,1955,1976 etc) but still managed to get 16,787 citations as a Nature Paper in 1999. May 10, 2013 at 12:34
• @Benoît Kloeckner In order to be more concrete, I talk about me: 1st I don't know if my question is know, but I know that I never saw something similar (and I searched keywords), I feel to have found a way to look in this direction like a new theory, but more I work on this idea and more I undesrtand that I must study totally (at the first look) different fields because are all linked. So In thit moment I'm seriously asking myself if is good to ask someone or not (because I do not have many theorems): May 10, 2013 at 15:22
• if I don't ask probably I'll stay forever on a fake problem building useless and wrongs constructions (or already known with other names) and anyways very slowly, If I do and there is a good idea, an expert mathematician can do 100 times faster what I was going do and "cut me out" forever. May 10, 2013 at 15:23
• @MphLee: if you feel that someone else could easily appropriate your ideas, there is even less probability that they avoid 1.2.3. : innovative mathematics are often (though not always) hard to understand for everyone, possibly (but not necessarily) except for its author. May 10, 2013 at 21:24
• Another point: you are reluctant to discuss your ideas, but it is often more difficult to get people interested in them than likely that they steal them from you. May 10, 2013 at 21:26