You can stop worrying now: what you're afraid of doesn't actually happen, though I sort of see how someone who hasn't spent their life studying mathematics might worry that it does. Since you have to back up your ideas with proofs, you can't in the long run hoodwink people into accepting false statements.

You also seem to worry about mathematicians accepting perfectly consistent sets of ideas even when those ideas contradict "inuititive observation of reality". To that I can only say that mathematics is not a subject where intuitive observation of reality plays any major role. What decides whether some piece of math is good or not is whether it is logically consistent, found interesting by people, and useful, either in other parts of mathematics or in applications to the real world. Notice in particular, that if the applications work no-one cares if part of the math leading to them contradicts any given person's intuition. For example, physics uses real numbers a lot and your intuition might tell you they don't make sense because there can't be uncountably many different things of any kind. But physics work extremely well and the math it uses is consistent, so we use it even if it doesn't sit well with a few people.