The foundations of mathematics are deeper than that basic assumption. I can for example say 1 + 1 = 10 in base 2, but people imply the base when when saying 1 + 1 = 2 as well as a great deal of other details few people actually care about.
This is the agreement that happens and why it's not universal. We have to agree that "1 = 1" for any statements like "1 + 1 = 2" to even make sense, let alone be true. Per your point, we have to agree that we're talking about something other than base 2 in order for "1 + 1 = 2" to be true, despite my intentions for it to be a potential example of a universal truth. Even "math" isn't universally true.
The above is false, math isn’t independent of axioms so there is nothing to agree upon beyond conventions where Math is simply the result of axiom choices.
= 2 @ + 1 1 is a different notation, but the underlying math is unchanged. So if you explain the notation difference an alien that might use = 2 @ + 1 1 they would also agree that in an our notation 1 + 1 = 2. The same relationship is true of the choice of axioms.
> math isn’t independent of axioms so there is nothing to agree upon beyond conventions where Math is simply the result of axiom choices.
> 1 = 1 isn’t inherently true on it’s own
This sounds a whole lot like "math isn't a universal truth". I guess we need to take a step back, since it seems this is where the disagreement lies: what does it mean for something to be a universal truth?
My position is that a truth is universal if it is self-evident; that is, it does not require an agreement between observers. These axioms are that agreement and therefore Mathematics, which stems from such axioms, is not universal truth.
Axioms are just another aspect of Mathematics and changing them has utility for mathematicians. Euclidean geometry for example uses one set and a wide range of non Euclidean geometry systems use different sets.
The universal truth that in Euclidean geometry the interior angles of a triangle add up to 180 degrees is not diminished because the interior angles of a triangle can add up to different numbers in spherical geometry. It’s just two different systems but they all fall under Mathematics.
This is a mistaken understanding. Axioms define the logic of Mathematics.
E: I guess I may have misinterpreted. I mixed up “supersede” with “precede” in my head. I mean to say axioms are a precursor to logic systems. The axiom has to be agreed upon before any truthful statements can be made.
No, axioms are part of any truthful mathematical statements.
If A, B, C … then Y is a self contained true statement. You can’t simply say Y alone is true because without A, B, C, … it’s not true.
People tend to assume A, B, C etc when they say things like 1 + 1 = 2 but that’s just a quark of language. Words like him or they work because people can work out the specifics from context.
