Chances of academically publishing a book without professional degree
As someone who has served as referee for math textbook proposals for reputed academic publishers, I can tell you that if your proposal were sent to me, I could not care less about whether you have a degree or what your title is. I would care about the content of your proposal and whether it convinced me that you can write a high quality, correct, and interesting book that serves a need for students and researchers of mathematics.
However. You should have no illusions that convincing me of such a thing is an easy standard to meet. It’s not. It wouldn’t be easy even for a professional mathematician with many publications under their belt, and I would expect that it would be doubly difficult for someone who doesn’t have formal training in mathematics at a level equivalent to at least a PhD, let alone an undergraduate degree.
But as I said, the lack of a degree by itself would not be a problem for me.
Whether Springer or any other publisher wants to publish any given book is entirely up to them. I'm sure that plenty of books are published by people without degrees.
But note that first you need to attract an "acquisitions editor" who makes a preliminary determination whether it is worth their time to work with you. Convince them that you have the necessary background and writing skill: perhaps with a sample chapter. One determination is whether the material "fits" into what they think of as good things to be associated with.
Once you get into the system, your manuscript will almost certainly be sent to a few "reviewers" who will make comments on what you write and make a recommendation to the editor. If their judgments are favorable, then you will probably get published. But expect several rounds of review and re-edit before you get to the production phase.
As to your personal title, you can simply say independent researcher, which is probably better than hobbyist, assuming that you need any "title" at all.
But before you bother to submit anything, spend some time trying to understand the sorts of things they like to publish in your field.
Will lacking a degree in mathematics or physics prevent my book from being published?
It won't outright prevent your book from being published, but it will be a disadvantage. This stems from two things:
- Without a degree, unless you can demonstrate exceptional achievement, you are less of an authority.
- Without a degree, like it or not, your books will simply sell less. By extension, that means they're less profitable.
One can complain about these being unfair, of course, but the fact remains that given two books on chess theory, one written by a former world champion and the other written by an amateur nobody has heard of, it's practically guaranteed that the first book will sell much better regardless of content.
That said, not having a degree does not disqualify your book from being published. That still comes down to the publisher. Things they might take into account include "can I expect this book to sell regardless?", or "how much will it cost us to publish this manuscript", or even "do I need to maintain good relations with this author?" (if you can potentially publish more projects with them).
Since your manuscript is already mostly written, you have nothing to lose by contacting Springer. I would suggest doing it. Sample chapters accelerate the pace at which they can come to a decision as well.
Would "hobbyist" be an appropriate title for the section labeled "Title"?
No, use Mr., Mrs., or Ms. as applicable (along with your profession).