"One of the tenets that most people hold is that mathematics is proven..."
Math is just a language. This assumption that it is some kind of observation is comparable to claiming that English, or Chinese, has been proven. It's just a way to discuss things, either to support theory or counter it with observation, data, and reason. The scientific method isn't math. Math, because it is a language designed to describe physical reality quantitatively, while other languages are general purpose means of discussion, attains to this patina of evidentiary quality that it doesn't actually have.
It is why word problems exist. All the things that can be expressed mathematically can be expressed in English or Chinese prose, or even rhyme, if the writer is good enough. Math is much more concise in expressing physical statements, but it's just a language, and, just like in every other language, you can lie with math. In fact I contend that all my knowledge is partial at best (if not outright wrong) and therefore necessarily is partially wrong because it cannot encompass the totality of knowledge that actually is potential, the whole of information regarding the universe, one interconnected thing. That totality necessarily affects every part, some of which I have some knowledge of, but can not know all that affect those bits, and therefore my knowledge, failing to incorporate those affects, is wrong to that degree. This includes what I express mathematically, because math also only states what is known, and does not include what is not known, and therefore necessarily is inaccurate.
Positive proof cannot exist, and this is the fundamental basis of the scientific method, why hypotheses are tested and the experimental results observed are then compared to the predictions of the hypotheses to disprove those that cannot be possible because the observations contradict predictions. Then the truth is only potential to other hypotheses that have not been disproved.
That is as close to proof that the scientific method can attain. What isn't true can be disproved.
So, 'math is wrong' is a fundamental misstatement of math. Math can only be wrong to the degree it discusses only such knowledge as exists, which is necessarily affected by what is unknown to an unknown degree. The vast majority of people do not have any idea at all of this actual condition of knowledge. 2+2=4 is only conditionally true, despite in our experience it is always true, and this is because we only exist in specific conditions, can only conceive of specific conditions that 2+2=4 is always true in. 2+2=4 is not always true, but it is only false in inconceivable conditions.
We can't change that. We are infinitesimal specks of goo, and nothing can ever enable us to encompass the totality of the universe conceptually. There is no point in speculating we can.
"Even order of operation and symbols are not defined"
This is just false. As maths were developed to discuss reality, order of operations, the grammar of math, developed rigorously to enable math to be communicated consistently. Always work from left to right, first do the operations inside parentheses and then use their products in the order of operations. It's very simple, and always produces the same result. The late claims to change that grammar is similar to the claim that 2+2=4 and 2x2=4 are the same statement, that 1x1=2 and not 1. It's a fundamental misstatement - a lie - told with math.
Thanks!