“Actually, I am trying to ask a third question ‘what is the proof that 1+1=2; is it empirical or not?”

Does it aid our discussion to note that an “empirical” proof (which, presumably, refers to a thing known by way of experience) itself requires the proposition (i.e., 1 + 1 = 2) that you want to have proven?

What I mean to say is this: At the very least, our notion of experience requires two things. Those things are: a subject/experiencer and an object to be experienced. (I will disregard the possibilities of other necessary factors for an experience – e.g., other people, more than one distinct object of experience, more than one moment of experience, etc.) This much seems necessary: that experience requires two distinct entities. Together they form the experience. If it were the case that either of these two were not existant, then an experience would not occur. Therefore, you have built into your notion of experience the concepts of a thing and another distinct thing which, taken together, form a whole.

So, if we wished, we may substitue our factors here with symbols. The “subject” may – taken as a distinct entity – be referred to as “1.” The same would go for the distinct “object.” Finally, the experience at large could be symbolized as “2” (i.e., the prescence of both factors.)

Now, even if it were necessary to prove empirically the proof of contingent/factual objects’ relationships to one another, we seem to have the assumption of 1 + 1 = 2 already found within our concept of experience. Empirical proof of anything requires the assumption of this propositon.

So, can we prove something which is already assumed within our method of proof?