# If intuitionism were true, could a mathematical system exist that is incompatible to our system?

If mathematical intuitionism were true, could there be a System that

1. Describes reality accurately and consistently like current maths do
2. Consists of equations that cannot be described by current maths or vice versa
3. Calculates solutions that cannot be represented by any number of our existing number system or vice versa

We can easily "translate" equations and solutions from binary to decimal. (I know that they are both part of our mathematical system, I'm just using this to clarify my point) I want to know if it would be theoretically possible to have a System S that is completely incompatible to math as we know it, but can still be used to calculate events in the real world. It would be impossible to "translate" anything from System S to math, but it would also be impossible to "translate" anything from math to System S.

If mathematical intuitionism were true, could System S exist?

• Intuitionistic math is not incompatible with "reality": computations with naturals and rationals (i.e. with the only available way to approximate real numbers) give the same result for "classical" math and intuitionistic one. Commented Jan 16, 2020 at 15:48
• Most philosophers would say that a system S can exist, but that is completely orthogonal to intuitionism. A species that does not interact with any matter we can sense would have a system untranslatable into ours, indeed, we won't even be able to communicate with them, let alone translate anything. There could even be a race we can sense but with an intellect so alien that we can not grasp its workings. But if we can understand somebody else's language that effectively means that we can translate it into ours (there is no reason to call it "language" otherwise, as Davidson pointed out). Commented Jan 20, 2020 at 7:35