> (I cannot edit my other comment any longer, but I want to add that it's far from my intention to sound as if I'm lecturing anyone, I'm well aware a lot of these are open questions -- possibly unresolvable -- and I don't consider myself an expert or well-read on this topic. I find it fascinating to discuss. All of my remarks/questions/disagreements with you are made in good faith.)

Thanks, I greatly appreciate your politeness and goodwill. Everything I say is in good faith too. I appreciate my ideas can seem odd, and sometimes I write in haste so do not take the time to explain things properly.

> it's understood that the string "2+2=4" requires additional context to have meaning, it's just that this context is often implicit (i.e. we're talking about arabic digits in base 10 notation, + is sum as defined in ..., etc).

I would distinguish the symbols from the concepts they represent. The string (or words, or handwritten notes) "2+2=4" is one thing; the concepts that it refers to are another. I could use binary instead of base-10, and write "10+10=100". The string would be different, but the concepts that the string referred to would be the same.

Everything I say, unless otherwise stated, refers to the concepts, not to the string.

>> Would 2+2=4 be correct, and 2+2=5 be incorrect, only if there were a human being to say so? > I think it's a question that only makes sense if there's a human asking it. "Correct" is always relative to something

This is true: correct is always relative to something (or, better, measured against something).

> in this case, the meaning a human attaches to that string, a string that only exists as a physical configuration of neurons.

But I disagree here. I would say it must be measured against something outside the mind, not the meaning a person gives something. If the correctness of arithmetic is measured against something inside the person's mind, then a madman who thought that 2+2=5 would be just as right as someone who thought that 2+2=4. Because there would be nothing outside the mind to measure against. One person can only be correct, and the other wrong, if there is something independent of both people to measure against. So if we say that arithmetic describes reality (which it clearly does: all physics, chemistry, engineering, computer science, etc etc assumes the reality of arithmetic), then we must say that there is something extra-mental to measure people's ideas against. It is this extra-mental measure that makes them correct or incorrect.

This is true not just of math, but of the empirical sciences. For example, somebody who thinks that a hammer and a feather will fall at different velocities in a vacuum is wrong, and somebody who thinks they fall at the same velocity is right. But these judgements can only be made by comparing against an extra-mental reality

So it seems to me when you say that

> the qualities of "right" or "wrong" only exist as physical configurations in the minds of humans.

you imply that arithmetic (and by extension, any subject) cannot describe reality, which must be false. It's also self-contradictory, because in this conversation each of us claims to be describing reality.