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Recently I've noticed a pattern in all of my "researches" (if you can call them that), and that is I will not allow myself to use known identities if I can't prove them (or at least understand a given proof of them). For example, I was recently looking into integrals of Bessel Functions, and I read about the Sonine-Formula, that is, $$ \int_0^\infty{J_z(at)J_z(bt)J_z(ct)t^{1-z}dt} = 2^{z-1}\Delta(a, b, c)^{2z-1}\left(\sqrt{\pi}\Gamma\left(z+\frac{1}{2}\right)(abc)^z\right)^{-1} $$ where $\Delta(a, b, c)$ is the area of a triangle with side lengths $a, b$, and $c$ (or 0 if it doesn't exist).

This might be a bit of an extreme example, as I don't think I'd ever come across this at the level I'm working at, but if I did, I wouldn't go ahead and use it. Instead I'd be stopped dead in my tracks. I wouldn't allow myself to proceed until I knew exactly what was going on this formula.

I do this simply because I'm scared of what might happen if I don't fully understand the tools that I'm using.

But I've also noticed that a lot of higher level mathematics makes reference to identities not found or entirely understood by the original mathematician.

So should I be scared of not knowing everything about a problem? Should I readily use other peoples' work regardless of whether or not I understand it?

EDIT: It might be useful to note that I'm still only in high school, so I'm nowhere near the level of experience most of the people have on this website, and this fear I'm feeling might just be a side-effect of being young and inexperienced.

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    $\begingroup$ I can think of two things worth noting. First, when you are experimenting and you encounter an identity you don't know or can't prove, it's often good to ask the follow-up question: if I manage to prove this, will it yield anything immediately useful? The second is that I find this feeling generalises not only to identities, but various known theorems, theories and tools. In this case I find it increasingly common that I use something I don't fully understand, because there is simply not enough time to understand everything. On the other hand, when doing math, Don't. Trust. Nobody. $\endgroup$
    – snar
    Oct 9, 2014 at 1:32
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    $\begingroup$ Hi; there does not seem to be any way to avoid that. We drive and ride in vehicles we do not understand all the time. We use programs and the internet and most users no little about either. I think it was Doron Zeilberger that said math is now a religion. We are forced to accept other results as true and move on. Sometimes, we can check them and sometimesd not. $\endgroup$
    – bobbym
    Oct 9, 2014 at 1:45
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    $\begingroup$ I've come across that very identity in my own research and I have to say.. I looked at that proof and just accepted that it was right. It's not an easy result at all and was the basis for a single, somewhat long paper in a mathematical physics journal I believe. Some things you just have to take on faith because the details are very painful to wade through. $\endgroup$ Oct 9, 2014 at 1:52
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    $\begingroup$ Echoing what the others have said here, while your thirst for knowledge is commendable, please note that this is a common historical theme in mathematics. Calculus was in wide use for quite possibly a thousand years (Gk method of exhaustion) with no one providing a (rigorous) proof for its use. That's because the tools to prove calculus hadn't been invented yet. Proof of intuitive results is often a retrospective process. $\endgroup$
    – avgvstvs
    Oct 9, 2014 at 12:18

2 Answers 2

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You'll never be able to know the proof of every mathematical truth. If you find a peer-reviewed result, you should feel comfortable using it. Some proofs span hundreds, even thousands, of very terse pages.

That said, if a particular identity is central to what you're researching you should try to learn a lot about it.

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  • $\begingroup$ Yes. One mathematician I recall actually gained notoriety by understanding the proofs of every theorem he used. In general, though, professional mathematicians don't bother understanding every single theorem's justification--even merely within their field. $\endgroup$
    – geometrian
    Oct 9, 2014 at 3:52
  • $\begingroup$ It is however important to also know the limits of an identity. $\endgroup$ Oct 9, 2014 at 9:14
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While I basically concur with David Peterson's answer one important thing to keep in mind is that such identities might pose restrictions, which may prove slightly too strict for your requirements. In that case, you should not blindly trust in the identity still being valid despite leaving its (currently established) boundaries.

As a simple example take the identity

$$ (-1)^{2n} = 1\ \forall n\in\mathbb Z$$

(which reads "even powers of minus one are plus one"). Now at some point you notice in your case $n$ might also take real values, in which case this "identity" actually becomes a falsehood and must be corrected to

$$ (-1)^{2r} = e^{2\pi i r},\quad |(-1)^{2r}| = 1\quad \forall r\in\mathbb R$$

(which reads "for real exponents, powers of minus one become phase factors, i.e. complex numbers of absolute value one"). And that also becomes invalid once you allow for complex valued exponents...

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