Otherwise, of course a larger environment can outsmart you mathematically.
No, not of course. For example, suppose P were equal to PSPACE. Then even though a larger environment could fundamentally outsmart you mathematically (say by solving the halting problem), it couldn’t prove to you that it was doing so. In other words, the situation with polynomial-time computation would be more-or-less the same as it is with unlimited computation: superintelligent machines could only prove their superintelligence to other superintelligent machines.
That the situation with efficient computation appears to be different—i.e., that it appears superintelligent machines can indeed prove their superintelligence to fundamentally dumber machines—is (if true) a profound fact about the world that seems worth calling attention to. Sure, of course you can nullify it by assuming away all complexity considerations, but why? :-)
Otherwise, of course a larger environment can outsmart you mathematically.
No, not of course. For example, suppose P were equal to PSPACE. Then even though a larger environment could fundamentally outsmart you mathematically (say by solving the halting problem), it couldn’t prove to you that it was doing so. In other words, the situation with polynomial-time computation would be more-or-less the same as it is with unlimited computation: superintelligent machines could only prove their superintelligence to other superintelligent machines.
That the situation with efficient computation appears to be different—i.e., that it appears superintelligent machines can indeed prove their superintelligence to fundamentally dumber machines—is (if true) a profound fact about the world that seems worth calling attention to. Sure, of course you can nullify it by assuming away all complexity considerations, but why? :-)