Are Ph.D. dissertations always required to include new discoveries? I was wondering if a dissertation could, instead, consist of a reorganization or reformulation of known material. Inventing new things is not my strength. But I think I'm pretty good at organizing information and making it clear and easy to understand. In some cases, I think that this activity might even be more valuable than discovering new things, and I find it to be just as satisfying. I can see that it might lead to a nice book, but I don't know if it could ever get me a doctorate degree. The field is mathematics, if that matters.
Are Ph.D. dissertations always required to include new discoveries?
What distinguishes a PhD from all other academic degrees is the requirement of a novel, sufficiently substantial intellectual contribution. In practice, what "novel" and "sufficiently substantial" mean are handled on a case-by-case basis.
The word "discovery" does not appear in the above description and would not be appropriate in certain academic fields, but in mathematics a PhD thesis is required to contain new theorems or new proofs. It is often contrasted (e.g. in various programmatic descriptions and graduate handbooks) with a master's thesis, which may contain only novelties of exposition.
So if you go around mathematics departments asking "Can I write a PhD thesis in which the novelties are all expository?" I would expect the answer to be a quite consistent no.
1) "Inventing new things is not my strength." I too was once extremely daunted by the prospect of inventing/discovering significant new mathematics...so much so that I spent five years as a PhD student learning how to do it. (I got it now.) My point being: if you are otherwise talented and interested in mathematics, there is no good reason to think that you will not become good at this.
2) If you are otherwise talented and interested in mathematics, you will very likely become good enough at discovering new mathematics to write and defend a PhD thesis. The requirements of a PhD thesis are partly aspirational: in reality, we settle for a bit less much of the time. I have seen some PhD theses where I was confident that there was nothing really new in them. To be a bit specific, I know a thesis in which a student attained the main result through five pages of calculation, whereas a more conceptual framework would yield it in about a paragraph. The result was of the form "For all n, there is a [mathematical object of type X] for which the [parameter value associated to any mathematical object of type X] is n." It had not been stated before in the literature, but when I searched further I found that the theorem could have been proven simply by citing extant examples in the literature. And by the way, this was not the worst PhD thesis I've seen -- on the contrary, the theorem was natural and interesting, and I rather liked it. It resulted in a paper published in a reputable journal.
3) You write
In some cases, I think that this activity might even be more valuable than discovering new things,
Yes, I agree most heartily. I suspect that the number of theorems proved in 2016 lies in the interval [10^5,10^6]. Without smart people taking time to reorganize, simplify and expose these results, we're all going to get carried away in the flood.
and I find it be just as satisfying.
I do a lot of expository work, and I find it more satisfying in some ways and less satisfying in others than discovery-oriented research. It is an unfortunate professional reality that its perceived value by the overall mathematical community is lower.
However, really good exposition is rare enough so that people who are good enough at it can make academic careers out of it. Michael Spivak is a legendary figure for his lifetime of mathematical exposition. Keith Conrad, a mathematician of my generation (and a friendly colleague of mine) is well on his way to the same achievement. Both of these people can do discovery-oriented research and have the papers to show for it. However it seems to be the case that they are more interested in exposition and have spent more of their time on that. To become the Spivak or Conrad of your generation would be a worthy goal.
4) There are doctoral degrees that are not so heavily focused on discovery-oriented research. In comparison to the PhD they are much fewer and they are not very widely known, but they exist. Let me quote from my UGA colleague Michael Klipper's website:
Before being hired by UGA, I received a Doctorate of Arts (D.A.) degree from Carnegie Mellon University in 2011. Instead of doing research, I wrote a textbook draft for an analysis course.
I know nothing about this degree program other than what I just said, but I know Dr. Klipper and the value he brings to my department. Based on that alone I recommend that you give it serious consideration and search for similar programs if you feel that this is where your interests lie.
Generally: yes, you have got to do something "new" in mathematics.
It is open to debate what constitutes "new", however. It doesn't need to be the proof of the Riemann hypothesis.
Keep in mind that new discoveries and inventions don't appear out of nothing, but usually build upon existing work. That might mean that the basic ideas are already floating in the air at the beginning of the PhD. You just fill in the details of the possibilities that other people have conjectured.
Summarizing and organizing, however, can be a valuable contribution to your research. You will typically find holes in the theory as you organize it, and so you can fill those in. If you find among those data a good abstract principle that has been overlooked previously, all the better.
The thesis is supposed to be publishable ... So if there are, in your field, published papers that do not include new discoveries, then (theoretically) there could also be Ph.D. theses that do not include new discoveries.
But in general what is acceptable for a thesis is something you decide with your advisor, not with us here.