The answer to "can math be used to prove falsehoods" at a simple level is of course "yes, if you are talking to someone dumb who knows far less than you." But that's not interesting since that's basically a conjuring trick. You're asking if REAL magic is possible. I think the answer is no, but I cannot prove it. To understand why, you have to appreciate the complexity of what you are asking.

Roger Penrose, in the beginning of Road to Reality has a very thoughtful discussion about the relationship between mathematics and reality.

Mathematics and logic (the two cannot be separated really) appears to be a self-contained system of truths about a universe of objects that miraculously happens to be a good description language for reality as well. There may be concepts/ideas in mathematics that do not help model anything in reality (some esoterica in number theory may turn out to be this way), and there may be things in reality that may not be possible to model mathematically ("consciousness" may be one of these things).

When mathematics or logic is used as a language to express modeling ideas, you need a "theory of reference" that maps the mathematical concepts to bits of reality. For example, something called "modal logic" can be mapped to the idea of multiple realities or "many worlds." The rule of inference known as "modus ponens" appears to model physical "causality" somehow. These are examples from near the extremes of the philosophy of mathematics. There are more extreme things. For example, "the axiom of choice" in set theory APPEARS to model something basic and intuitive about reality, but huge problems crop up around assuming or not assuming it.

That theory of reference can be falsified by empirical data, but the model itself is not a falsifiable construct in the sense of being a mathematical construct (unless of course it violates the internal rules of mathematics itself).

You may want to try "From Frege to Godel" by van Heijenoort and/or "The Philosopy of Mathematics" by Benaceraff and Putnam. I've been gradually dipping into both for several years. Some of the most heavy-lift thinking you will ever do, if you choose to tackle it. Prepare to break down in tears on occasion.

The most interesting question in the field, IMO is related to the "may" statement I made earlier: are there really things in math that are not usable in any model, and things in reality that cannot be described by math?

The most interesting candidate is real numbers. There is a very strong argument being made in the last 10 years or so, that the real numbers model nothing in reality (i.e. that the whole universe is discrete/digital).

I've been converging to the thought that maybe they model "consciousness." :)