undergraduate - Having the knowledge of a graduate student but not having the degree requirements for graduate school?

Let's say that you're a 27 year old autodidact who has gained enough knowledge in a particular field of the formal sciences - like mathematics, for instance - to be able to take the highest course number from the selection of graduate courses for math.

Now this autodidact wants to enter university for the first time. However, from his perspective, he thinks that it would be a waste of time to restudy what he already knows.

I'm personally wondering if there is a way to demonstrate that he is capable of taking graduate courses so that he could apply for a graduate degree without having to study all the courses at the undergraduate level again.

Also, I'd like to add that this question is really hypothetical. It's just something I pondered on my way to school today.

Answer

If you're asking about graduate school in the US, probably the single most significant thing you can do is to take the GRE subject test for math (not the math section of the general test!), and score well on it. This test covers a broad spectrum of material that is taught in a typical undergraduate math curriculum, and so if you get a good score, it strongly supports your claim that you have the level of mathematics knowledge required to enter grad school. Without that key piece of evidence (i.e. a good GRE subject score), graduate admissions committees are likely to look at your statement that you have the knowledge to take graduate courses, contrast it with your lack of an undergraduate degree, and conclude that you're full of hot air, so to speak.

Now, of course there is more that has to be done to actually get yourself admitted. In my own field of theoretical physics, even a 990 (the top score) on the GRE isn't enough to get you into a good grad school by itself. I would imagine the same is true for the top schools in math, though perhaps at a less competitive school, it might be. But I think to be safe, you should assume that you'll have to present some other sort of evidence of accomplishment that could be viewed as equivalent to an undergraduate transcript. You'll also need recommendation letters and various sorts of essays and forms. But a lot of that can vary from school to school, and is more likely to be negotiable if you talk to someone in the department. The GRE subject score is the one thing you really need to get your foot in the door, so to speak.