I'm not sure how to explain my question, but it's hopefully a suitable one nonetheless. I don't want to answer this for fear of biasing myself, so any help would be greatly appreciated.

Ok so bear with me. Firstly, when I say 3+1=4, it is considered a tautology within a certain logical system, call it what you will, number-value system, I don't really know. Now if I change the logical system to something different, but I still have this statement, 3+1=4, it is, let's assume, blatantly not a tautology. It is logically reducible, and it may now, not be true. Perhaps the 'values' are irrational in other logical systems. In the rules of the logical system, no relationship can be found 'objectively enough' to be considered rational. The 'grammar' of the variables are just not 'proper' when combining them in any way through the required set of rules and axioms. In math, the methods of combining the variables/symbols are the operators I'm assuming.

And these operators are meant for manipulating the values of the variables/numbers.

So this 'variable', it is really just a piece of context is it not? My big question is, can any variable have methods of the 'more objective' manipulation seen in math?

I understand that in math, because the variables are static in a sense, 1 is always 1, and only has as that variable, one value at one given moment, or at least if there is an uncertainty or probability, there is a parameter which is not subjectively irrational...or irrational in some other way. And, the parameters are universal throughout the system of variables being manipulated by the certain operators. So the operators are allowed as much potential/room to manipulate the variables as there is an objective leverage to the value of the variables. Perhaps that is not quite correct though, perhaps it is that the moment you get subjective irrationalities (meaning no gradation/proportionality). It would be like trying to apply operators to infinity; infinity + 1 doesn't work.

Are there any other variables besides numbers that can be manipulated the same way assuming the logical system changes? I'm not talking about logic itself, that's different than what I mean to get at. And again, could one take any sets of variables/contexts and manipulate them 'logically'? Is there a way of converting irrationalities to rationalities sort-to-speak, or perhaps a way of 'measuring' only to a certain degree of its parameters (if irrational) the variable?

Also, if there is any formal diction that could replace the words I'm using to explain this, I'd be glad to know them.


EDIT: I realized now that 'subjectively irrational' is something to be laughed at. I'm sorry, I think now the better term would be parameterless, or undefined parameter?