I really don't understand how this is relevant. Why do you think the ability to draw a circle in general is relevant to the computer's capacity to square a circle? The computer would be able to draw a very special square for lots of circles such that the square has the same area as the given circle.

Such a computer could not be simulating our universe. Our circles cannot be squared.

You said this:

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

So you're saying that if I were to draw a circle with a circumference of 100cm and a computer were to draw a circle with a circumference of 100cm then the two circles would have a different diameter?

Yes. Since they have different values for Pi.

Yes. Since they have different values for Pi. — fdrake

My hand doesn't have any value for Pi. It just draws a circle.

Reality does. It's Pi here, it's an approximation of Pi in simulated universes with finite memory. We could go round in circles playing the 'frame what the other person is saying as incorrect but do nothing to address their arguments' game, but I'd rather not.

How would a computer get around needing a specific value for Pi? How would a computer get around needing to know whether a digit is correct or incorrect? Without committing the whole thing to memory, in either case. Or, why is this an irrelevant or misguided question? Why doesn't the computer need a value for Pi when it would have circles with circumferences and diameters like ours? Ones which would presumably have a constant ratio.

Didn't realize that BIV discussion would result in a debate on whether a computer can simulate squaring PI.

The computer doesn't need to "store" a value of Pi. It just needs to follow a circle-drawing algorithm. Circles, like other shapes, drawn by computers are going to be more accurate than hand-drawn ones.

So I fail to see how there is necessarily an empirical difference between circles and squares drawn in the real world and circles and squares drawn in a simulation. Thus this "squaring a circle" defence can't help us show that we're not in a simulation.

Assume someone has drawn a circle in the real world, and there's a circle in the simulated world. Both have the same diameter. Do they have the same circumference? No, because Pi is different in each context. They're both circles, but they have a different circumference.

Can you please explain why you think drawn circles in the simulated reality and the real one somehow undermine what I'm saying? And also why this eschews the computer's responsibility to have a value for Pi? The algorithmically drawn circle in the computer will still have an implicit Pi in it. To the extent that raster circles are real circles anyway.

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? There are no perfect circles in reality, just approximations of them, like the rim of my coffee cup. There is a Planck length limit to how precise an actual circle can be. But pi can be calculated to as many decimal places as desired (within physical limits) without requiring an actual circle (e.g. 4*(1-1/3+1/5-1/7+1/9-...)).

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.

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.

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.

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

The issue I have is that the maths we use to show that Pi is irrational is the same maths that a simulated person can use to show that Pi is irrational. Given the same axioms and definitions, whatever deductively follows in the real world will deductively follow in a simulated world.

Given the same axioms and definitions, whatever deductively follows in the real world will deductively follow in a simulated world. — Michael

At the end of Carl Sagan's Contact book, a human computer finds a binary representation of a circle inside PI created by aliens who shaped our universe in a way such that PI would have that value so that any sufficiently advanced race who evolved could find out they were inside a created universe.

I always wondered how the value of PI could be modified by the shape of space, but Sagan put that in his story. If it can, then that might have bearing on the value of PI computed inside a simulation.

I always thought PI had to be the same value regardless, but it seems some disagree with this.

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.

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

Just pointing out that Sagan though the exact value of PI could be determined by the shape of space one exists in. It's not exactly the same, but it goes to the notion that PI's value might be calculated to be different depending on the world one lives in.

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

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.

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 would put it the other way around. The meaning of pi is based on the definition of a circle (e.g., a round plane figure whose boundary (the circumference) consists of points equidistant from a fixed point (the centre)).

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.

I agree with you @fdrake, but there is something about the BiV scenario that weasels out of your proof -- the evil scientist can also edit memories. So we may have memories of transcendental numbers, when in fact we have already squared circles, but those memories are erased as soon as we do so.

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?

Here's an interesting question. Could a simulation learn about the halting problem?

You don't think it would be a problem for the simulation computing our coming up with those idealizations? — Marchesk

It doesn't have to, the real people that are being fed the simulation will do that (assuming BIV or matrix-style simulation). They will observe their simulated surroundings and develop language and logic accordingly.

All the computer simulation has to do is simulate what is real (e.g., coffee cups with approximately-circular rims). The people in the simulation will come up with the ideas.

Here's an interesting question. Could a simulation learn about the halting problem? — Marchesk

I don't know, but the people in the simulation could which is sufficient. It's just a question of logic. No infinite storage or time is required to understand it.

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.

I agree. Talking about drawing circles is a bit of a red herring. What matters is whether someone could square a circle in the simulation.

I think that's a difference between BIV and the daemon. The daemon has a malicious streak and doesn't even grant us the capacity to think or feel as normal. Being a BIV does, that's what makes it interesting. Regardless, though, there are proofs that Pi is transcendental - do you think that these are fabricated memories or we're forced into believing them?

In reality? No. I'm not a skeptic at all. :D At least not a radical skeptic.

Merely pointing out that were we a BiV, then we could have fabricated memories implanted. It could even be algorithmic -- anytime someone squares a circle, then erase said memory and replace it with the impression that they proved that Pi is transcendental.

Conditional: brain in vats and memory editing. Should be believed? Nah. Kind of thing people can get therapy for.

Conditional: brain in vats and memory editing. Should be believed? Nah. Kind of thing people can get therapy for. — fdrake

Maybe the evil demon created us as BIVs 5 minutes ago with false memories of a past. But the evil demon itself is a simulation, so he has to trick us into thinking we know about transcendental numbers.

It's a daemon's power fantasies playing out in a holodeck in which he creates BIVs impregnated with the idea that they're all in the Matrix.

The skeptical scenario isn't proposed as something that should be believed, though. There's not much of a difference between the BiV and the demon -- the BiV was just meant to be a modern update to the demon, so that it came across as more plausible. It's just supposed to be a defeater to particular claims of knowledge, usually of the external world.

It amuses me that it's OK to entertain the possibility and the concurrent belief that entertaining the possibility destroys all knowledge, but it's not OK to sincerely believe it.

it seems to me that it's more consistent with the skeptical position, though. If the skeptic claimed the world we experience is the result of an evil scientist who has put our brain in a vat, then that would sort of belie the whole skeptical position -- that we do not have knowledge of the world.

It's the possibility of radical error which gives reason for doubt, and based on that doubt out goes knowledge of the external world. (at least, so goes a way of putting the argument)

That's a lot more words to say the same thing. It's a more plausible delusion to think that a mad scientist has made us into a brain in a vat than one of those unsophisticated, backwards medieval demons is tricking all our perceptions and memories using the thoroughly irrational and unscientific black magic.

Plausible? No. That's why I said there's not much of a difference.

It's more an issue of palate than reason. The skeptical challenge remains the same in both scenarios.