All science, and all philosophy, are enlightened common sense.

--Karl Popper, Objective Knowledge

A moment diagram with an understandable error.

Moment diagrams are rather abstract and mathematical, physical intuition helps with some things (like tension on the top for a cantilever), but not with shapes and slopes.

Here is a sketch of the corresponding cable shape, where the error in the moment diagram is carried through.

The error here is easier to catch, since the shape of a loaded string is much more intuitive than the shape of a moment diagram. A string would only have a kink like that if there were something pushing up.

Mathematics and physics are important tools to enlighten your common sense. We've seen that there are cases where intuition can be misleading, or can be overwhelmed by complexity. In those situations, mathematics is useful.

But which should you trust more: math or intuition? The answer is to always use both, and treat both with critical skepticism. Use one to question the other.

Before starting any anlaysis or calculation, always try to use your intuition and common sense to guess the general form of the answer before you start calculating, and compare the analysis result with that guess.

If common sense and math agree, then your common sense is probably becoming enlightened and your intuition maturing. If they disagree, you've either found an error in the math (which is good), or have discovered a lucky opportunity to further develop your intuition as you figure out why you guessed wrong (even better). Remember, your intution will never develop if you don't make that pre-anlaysis guess. The next time you encounter a similar situation, you'll probably guess right.