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
Reality does. It's Pi here, it's an approximation of Pi in simulated universes with finite memory. — fdrake
Why would you suppose reality has a value for pi? — Andrew M
We do have a mathematical value for PI which is irrational and cannot be computed (in full). A circle's definition is determined by the full value of PI, mathematically speaking. — Marchesk
Given the same axioms and definitions, whatever deductively follows in the real world will deductively follow in a simulated world. — Michael
I have no idea how that addresses my point, which is that given our axioms and definitions, that Pi is irrational deductively follows. Unless you want to say that a simulated world can defy logic, the same is true in a simulated world. — Michael
Whatever you want to call it -real, ideal, virtual, a platonic object, other things-, our mathematics has a specific value of Pi. A computer simulating our mathematical capabilities would also have to simulate their associated ideational structures, irrelevant of their ontological status in the final analysis. It may be that Pi isn't 'real' in the same sense as the ideal circles (and other things) it concerns; nevertheless it must be simulated. — fdrake
We do have a mathematical value for PI which is irrational and cannot be computed (in full). A circle's definition is determined by the full value of PI, mathematically speaking. — Marchesk
I think fdrake is arguing that the simulation would have to compute us coming up with irrational numbers and other things which aren't computable, such as transfinite numbers. Or the halting problem. — Marchesk
Yes. But as long as they are understood to be idealizations and not actualized, then I don't see the problem. As an analogy, we have a concept of infinity. It doesn't follow that the universe is necessarily infinite. Similarly for the simulation. — Andrew M
You don't think it would be a problem for the simulation computing our coming up with those idealizations? — Marchesk
Here's an interesting question. Could a simulation learn about the halting problem? — Marchesk
The value of pi is defined as the ratio of a circle's circumference to its diameter. That is sufficient to construct actual circles that approach the ideal circle or to approximately calculate pi.
The computer simulation can have the same mathematical definition of a circle and pi as us. But in both cases they are mathematical idealizations and can only be actualized approximately.
Conditional: brain in vats and memory editing. Should be believed? Nah. Kind of thing people can get therapy for. — fdrake
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