It would need to be able to generate any set of surroundings. This goes beyond being a procedurally generated universe, as it would have to generate this one. — fdrake
It is possible to construct a square with an area arbitrarily close to that of a given circle. If a rational number is used as an approximation of pi, then squaring the circle becomes possible, depending on the values chosen. However, this is only an approximation and does not meet the constraints of the ancient rules for solving the problem. Several mathematicians have demonstrated workable procedures based on a variety of approximations.
So whether or not we can square a circle isn't open to empirical investigation? Then how do we determine that we can't? — Michael
So, there are two Pis. Let's call our Pi x and a simulated Pi y. Let's also assume that y is equal to the first 3 decimal places of x. IE y=3.141 — fdrake
If you want to frame it in terms of visual sense data, it's still possible: in a simulated universe, no one could see a valid proof of Pi is transcendental. — fdrake
When a circle is thought, its area will be equal to its universe's Pi times the radius squared. This simulated Pi cannot be our Pi, since our Pi requires an infinite amount of memory to store. Moreover, the simulated Pi will be a fraction. Rest of argument goes from there. — fdrake
But I suppose if you were to measure circle diameters and circumferences, you would observe a different value for Pi than the one we get. — fdrake
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