Well, the form of mathematical reasoning that's prevalent was actually first developed during the Islamic Golden Age. Most of Western math is actually imported from the East. Symbolic logic, also is of Islamic origin.
If there is anything to the claim that it is “western”, it is in the way that “western” people regard it (possibly compared to how others regard it).
I don’t see how the idea that mathematical reasoning is inherently “western” (as a property of mathematical thinking, as opposed to as a property of “western”) could possibly stand up to scrutiny.
Sure, the symbols being used may be due to particular cultures, as well as a number of conventions (e.g. infix notation vs whatever, some minor choices made in some definitions, etc.), but, these are not inherent to mathematical reasoning.