I can't help thinking that diagram looks more like a measure of semantic familiarity than math education. Isn't arguing definitions of terms a language issue more than a math issue?

*person who never went past Trig*

You think that because the concept of the number 1, as well as basic addition and counting, is hardwired into your brain. You know them because you were born knowing them, and they're instinctive. It's difficult to get a true appreciation for just what that sort of leap is.

For simplicity's sake, we take these things that we tend to take for granted, label them as such, and call them

postulates.

Sometimes you can prove what was previously assumed to be a postulate, and there is much rejoicing. Alternately, you can create new logical frameworks and try to work from there - ordinals, Boolean algebra, etc.

Okay. So.

Craft a valid algebra such that you do not have to explicitly declare that x=x, because that is cheating, yet you can still prove it to be so.

This can be rather difficult but it doesn't necessarily need to be that bad - take set theory

Define the number zero as the the empty set, ∅

Define n+1 as n U {n} - that is, n+1 is the set which is the union of n and its set.

Thus, 0 = ∅, 1 = {0}, 2 = {0, 1}

Define two sets to be equal if they contain the exact same contents.

Thus 1 = 1 breaks down into {0} = {0} and thus is equal.

It's been years since I've studied set theory, and there is quite a bit I've skipped over there.

It should be fairly obvious that we've only really moved the goal posts here, because we're defining a successor function to zero, declaring it to be a number and providing no real logical context for doing so other than declaring it to be so. This is where your confidence starts to waver because it makes it rather plain that our logic

*requires* such obvious assumptions. And per Gödel's incompleteness theorem, it's only ever going to be this way - you can try to make a system more 'complete' e.g. with set theory here, but it will never be complete if you are consistent.