i think one assumption made with this is fundamentally misunderstood.

.1 .... is not actually equal to 1/9. it's treated as if it is 1/9 because .1 .... is the closest decimal approximation to 1/9 that we can have. but technically they aren't equal and 1/9 > .1 ....

the rest of the argument then just expands from that slight difference in what 1/9 is and how we attempt to represent it.

when did we switch to base 2?

0.1... is not an approximation, is precisely 1/9.

you need to express all mathematics on this thread in terms of base sqrt(2)!

you can prove math without science. Math is entirely a set of rules made by humans which needs not have any basis in reality. Philosophy is a way of thinking of which math is a part.
In philosophy (and math) you make a set of rules and you "prove" things by showing that any new statements obey those sets of rules and/or other "proven" statements.

In science you never truely prove anything. You make observations and come up with a set of rules which best approximates these observations. When we observe things that controdicts our "set of rules" we revise the rules (ie the theory) to fit our revised view of the world. This is the complete reverse of mathematics. We use mathematics as a tool to help create rules in science but we don't always need to do so. There are large aspects of science that don't really depend on mathematics at all (or to a limited extent like counting things). We use statistics in science as a more objective way of determining if things are "different" to compensate for our limited ability to observe everything.