Mathematics and ordinary language cannot show that a set of signs is not mathematical or semantic. For example x+2=2x-2. This set of signs is assumed to be a mathematical expression by working with it, mathematically, to assess it for truth or falsity. The conclusion, that the set of signs (equation) is false, misses the target: the set of signs x+2=2x-2 is not false, but rather is not a legitimate mathematical expression. There are no false mathematical expressions. In working out the truth or otherwise of a mathematical or semantic expression we must make the assumption (A) that what looks mathematical or semantic is mathematical or semantic. As this assumption is not one we take up on behalf of either maths or ordinary language, it follows that neither math, nor ordinary language, nor indeed any system of signs, can legitimately show, according to its rules or axioms, that a set of signs is true or not. Here is an example of working with assumption A in practice, from a definition of proposition from Wiki: In logic and philosophy, the term proposition (from the word "proposal") refers to either (a) the "content" or "meaning" of a meaningful declarative sentence or (b) the pattern of symbols, marks, or sounds that make up a meaningful declarative sentence. The meaning of a proposition includes having the quality or property of being either true or false, and as such propositions are claimed to be truthbearers. The author of the above quote takes assumption A, on board, without knowing it, it seems. In b) a set of marks are offered as bearers (truth-trackers) of truth or falsity. Yet these marks cannot show themselves to be legitimate syntax without either assuming that what looks legitimate is legitimate, or else defaulting to position a). To reiterate, the legitimacy of a set of marks, signs or syntax (whether a syntax is mathematical, etc, or not) cannot be demonstrated from its truth or falsity. Mathematical truth and falsity do not mirror the actual legitimacy of a proposition or set of signs that look mathematical. And that is why Mathematics is a contingent element in the ad infinitum of legitimacy. Other elements are other systems of signs, and, as systems of signs are independent, are contingent. The ad infinitum refers to the fact that proofs for truth or falsity do not ultimately show that a set of signs belongs to a system - is legitimate.