Here's my two cents: No and Yes.
Technically, the answer is no. You won't need any kind of degree (or even any kind of education) to publish in most journels. I assume some of you have heard of the case of a team of MIT computer programmers who developed a program that could 'write' gibberish, making it sound like academic gobblygook? They submitted one these gibberish articles to a reviewed humanities journal, only to have it accepted. The reviewers never understood the gobblygook, so instead of rejecting it (and admitting they didn't understand it), they accepted it.
The point is, there are countless journals out there. Some are meant for expository articles, while others are meant for cutting edge research. Some are reviewed by experts in the field, others are only briefly reviewed, and even others are not even reviewed at all. I even remember a mathematics journel published from somewhere in the US (Cornell?) that accepts papers that other journals have rejected!
Thus, there is nothing stopping you from finding a not-so-rigoroulsy reviewed journal, and publishing whatever the hell you want.
The second answer to your question is Yes. In practice, to be able to publish in a respected journal, you have to have a degree in your field (undergrad, at the very least, but read on). Academic research is very rarely done in secluded basements. It's done in collaboration with others and with the field as a whole. You need to know what to research and what the current topics are. You need to know who the big honchos in your field are (they will undoubtedly review your papers before accepting). This is why it's usually the case researchers are active members of a department -- not flying solo.
Moreover, gone are the days where scientific and mathematical results are 'obvious'. Today, everything obvious has been done, and the not-so-obvious things are way too difficult. Today, research is done in very specific domains and studies very specific problems. How do you know about these problems and domains? You spend years (often during your Ph.D.) researching them. Most mathematicians, by the time they've completed the first year or so of their Ph.D. only know the results of a handful of fields (say, 4) up to the early 20th century. By the time they've finished their Ph.D., they'll know the current research of perhaps only a single field up to the 21st century (minus 10-20 years).
The growth of mathematics is startling. A well-known figure for the number of theorems published yearly is 200,000(!). Usually, to understand just one of these theorems requires you to understand the specific field in which it's written, in addition to all the previous theorems and development it's based on. Science (and mathematics) is like a tree. Each branch leading to more and more branches. Admittedly, if you're brilliant, you can latch right onto a leaf and gobble that right up. But most of us (and for all intents and purposes, you can say all of us), need to start on the bottom and move our way up. Most of us can't do that without help. We've depended on teachers, professors, supervisors, and later, colleagues.
Gone is the romantic notion that a file clerk or an accountant can produce an original result.
