The mathematics depend heavily on physics. If physics outside of our universe differ - the math wouldn't add up.
I can make 2 + 2 = 5 in a simulation. That doesn't mean 2 + 2 = 5 outside of the simulation.
I can make 2 + 2 = 5 in a simulation. That doesn't mean 2 + 2 = 5 outside of the simulation.
I'm not sure you can. Yes you can make a chest with 2 apples contain 5 apples if you put 2 more, but it's impossible (IMHO) to make it consistent. How many apples do you have if you place 2 chests besides 2 other chests and put 1 apple in each? What if you repeat the process 2 times and then another 2 times?
Also - when some things in physics aren't intuitive (QM for example, but even adding velocities in relativity) we don't think math is wrong - we think math doesn't apply in that way to this particular phenomena.
Taking it further, what if rules and logic, and by extension, math as we know it, is just a construct that is not applicable in the "real" universe. What if, in that "real" space, 7+7=5, or mathematical operations are not possible. What if causality and time dont exist?
And if sometimes its 5 - they will do what we did with relativity when it turned out velocities doesn't add up like they "should". We haven't changed addition - we changed the equations used in physics.
I guess you could hand-made an intelligent life that is designed never to discover the true math.
The act of putting things in boxes (after first perceiving and classifying them as discrete entities) and the abstract notion of addition are only as related as you need them to be for a given purpose.
Yes. That was my point. How would you change the abstract notion invented by creatures in your simulation if not by changing reality to shape their abstractions? And how do you change enough instances that their math develops in such way that 2+2=5?
I think our basic math (logic, set theory, integer math) with some exceptions (choice theorem) is the only useful consistent basic math there is.
I do not believe that to be true. While physics is heavily dependent on math, the reverse is not true.
> I can make 2 + 2 = 5 in a simulation.
No, I am fairly confident that you absolutely can not. You can make a simulation where putting 2 oranges in a box and adding two more oranges results in 5 oranges in the box (rather like putting together certain nucleons together into a nucleus results in a mass different than the sum of their individual masses). But in that simulation it would be a peculiar property of oranges, not a property of mathematics -- just as we don't claim addition is invalidated by the behavior of nucleon masses.