Why you can't prove a single mathematical thing beyond a shadow of a doubt
I'm an amateur philosopher and I'd like to start up an interesting debate. The topic will be mathematics (and, in general, everything) and how it is not logically possible to prove such a system. This is my informal argument:
1. Humans live in a "reality" governed by laws, axioms, and such.
2. This reality is defined as no more than our perceptions, for our understanding of it depends on our mental processes.
3. This reality is always changing as new studies, etc. change or break certain things.
4. All that is required to make something not 100% certain is a conceivable counter-example.
5. Since humans are not in a position to judge the truth of the world absolutely and without doubt, they must rely on themselves.
6. To borrow Descartes' thought-experiment, if a maleficent being is controlling the world and making sure things abide mathematically only when people are looking, or through some manipulation manages to make the "untruth" that 2+2=4 the truth, how are we to say that mathematics is 100% certain if this random counter-example is even conceivable?
I'd like to hear some thoughts on this. Our own logic dictates that it is not possible to say our logic is 100% sound.