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 is false because it leads to a contradiction, misses the target completely: 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 merely looks mathematical or semantic is in fact mathematical or semantic; otherwise, we are unable to work with it and find out if the signs yield up as a valid mathematical or semantic expression. There is no system that can, by its own rules, show that a particular set of signs actually belongs to it. First, we must make the assumption that a set of signs does belong to it, so that we can legitimately apply the rules and find out if the set of signs actually does, or does not, belong to the system. If it does not belong to the system then our assumption was wrong. Yet, without the assumption systems are impotent.

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 do not show themselves to be legitimate syntax without either assuming that what looks legitimate is legitimate, or else by defaulting to position a).

To reiterate, the legitimacy of a set of marks, signs or syntax - that a set of signs is a possible candidate in a system - arises prior to its truth or falsity (Proof is another matter. Proof is a description of elements, legitimacy refers to the system itself). Truth and falsity, then, are the means by which we assess legitimacy. This is achieved by comparing the behaviour of the elements of our candidate syntax with the rules of behaviour imposed by the system.

And that is why Mathematics is a contingent element in the ad infinitum of legitimacy. By elements, I mean "systems", a of which mathematics is one. As systems are independent of each other then their presence in the framework of legitimacy is contingent, ad infinitum.

"Ad infinitum of legitimacy" is the name we give to any idea. The idea is independent of legitimacy, and indeed is its very foundation. The idea formulates the styles of legitimacy, styles such as mathematics.

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