Certain disciplines are very code driven to be able to practice them one can probably get by using a design guide. Examples would be common building type work for structural, HVAC, or Electrical Power to a building. The majority of that design can be done simply by a designer, and it's just law that it has to be stamped by a P.E. However, what happens when one is asked to design something very innovate e.g. that is not covered in any code and does not have a ridiculous factor of safety or is driven buy "rules of thumb" ? Then one has to rely on fundamental principles learned in school, statics, dynamics, strength of materials, fluids, thermodynamics, and yes mathematics. In this case the non-degreed engineer might run into trouble. Granted in most of these situations where one would run into this an P.E. would probably not be necessary (as a plan checker might not even understand what is involved in coming up with the design).
Once I was asked to design a drainage system for a storage tank in a refinery that would allow for rain water to drain into the tank when the level got high enough, yet close and be vapor-tight under normal conditions to prevent hydrocarbons from venting out of the tank. At the time I was a fresh graduate, not a P.E., and none of the P.E.s in my then large company could figure it out. The calculation was iterative and involved triple integrals, ect. I had to solve the equations in excel to avoid rounding errors. The design worked perfectly.
Another example is when Solar photo voltaic were not covered by the National Electrical Code.... someone had to use their best judgement (and fundamental engineering principles) in designing them to be safe. These lessons learned ultimately became code and now they are available for everyone to use. Now even an electrician can design them. In California licensed electrical and mechanical contractors can design the systems which they will install themselves without needing a P.E.... Why? Well simply because codes make it like a recipe for design which someone with sufficient understanding can follow and execute.
lusone: There are certain things one can learn in a classroom that can't be necessarily learned from work experience. Conversely, there are certain things on can learn from actual work experience that isn't taught in a classroom. Both attributes make for a competent and well-rounded engineer in my opinion.
