I still can't imagine a holodeck the size of the universe. — fdrake

It would only need to be the size of your immediate surroundings.

That's fascinating if it's correct. I'm a bit skeptical that math can be used to prove something metaphysical, but if it can, that's very deep stuff.

So you're saying that if a simulated person used a simulated compass and straightedge to draw a circle and a square then there would be a perceptible difference between what they see and what we would see were we to do it for real?

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.

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 wouldn't have to generate the whole universe, though. It would only have to generate the things that you will actually see.

It wouldn't have to generate the whole universe, though. It would only have to generate the things that you will actually see. — Michael

But it does have to potentially generate anything you or anyone could actually see, hear, etc.

It isn't a perceptual difference. It's akin to a law of nature.
Argument goes as follows:
Assume (A) we are in a simulation, then consider the following argument:

(1)Pi is transcendental.
(2) Transcendental numbers require an infinite amount of information to specify fully.
(3) If a transcendental number is not specified fully, it is approximated. (2)
(4) Approximations to transcendental numbers have terminating or repeating decimal expansions. (this is the definition of an approximation)
(5) Approximations to transcendental numbers are rational. (4)
(6) If Pi were rational, we could square the circle.

Supporting (6) from the Wiki page:

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.

(7) The computer running the simulation would have infinite memory if Pi were transcendental. (2)
(8) Computer memory must have non-zero size.
(9) The computer running the simulation would be infinitely large to contain infinite memory. (7, 8)
(10) The computer could not be infinitely large.
(11) The computer must use a finite approximation of Pi. (10,5)
(12) We could square the circle (6,11)

(13) We cannot square the circle.

Now we gotta follow the contradiction back:

(12) is false, so (11) is false since (6) has been proved independently, so (10) is false since (5) has been proved independently. So either the computer is infinitely large, or (8) is false, if (8) is false then either Pi is equal to an approximation (5) and then (1 or 2) is false. We know (1) and (2) and (5) are true, so either (A) is false or the computer is infinitely large.

The computer isn't infinitely large, so (A) is false.

It isn't a perceptual difference. — fdrake

So whether or not we can square a circle isn't open to empirical investigation? Then how do we determine that we can't?

Did you read the Wiki page on squaring the circle?

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

I think the argument is that in order for the simulation to make a circle non-squarable when we try to square it, it would have to compute the transcendental number PI, or we would be able to accomplish the task.

Yes, and it mentioned drawing a circle and a square, so I asked you if a simulated person would see something different when they try and you said that it isn't about what we see, hence my confusion.

This is what it means to be unable to square a circle: it is impossible to construct a square with exactly the same area as a given circle using only a ruler and compass. The 'impossibility' isn't something which is derived from perceiving squares, circles or generalising from how they look. The impossibility is derived through the way ('drawing using a ruler and compass' 'circles' 'squares', 'pi' and 'transcendental') relate to each-other as a mathematical composite; the first of which is something which can be done in a universe whenever those concepts make sense. The theorem states that there is no procedure - real, constructible - to make a square with exactly the same area as a given circle using only a ruler and compass.

Is it really that confusing to think that concepts (compass+straightedge construction and Pi being a transcendental number) relating to an activity can demarcate what is possible (the circle cannot be squared) within that activity? I think this is a pretty commonplace occurrence, and isn't confined to a-priori constructions either:

A rock could not become alive. (synthetic a-priori, probably)
A state in which no votes are ever cast could not be called democratic. (analytic)
A bachelor cannot currently be married. (analytic)
Radon is not chemically reactive at standard temperature and pressure [IE, no radon dioxide] (synthetic,empirical).

So what happens when a simulated person draws this circle and this square? If it is empirically indistinguishable from a real person drawing a circle and a square then this can't be used to show that we live in a real world and not a simulated world.

And they can repeat and use all the same equations that we do, so that can't be used either.

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

y is an approximation of x. This means that the decimal expansion of y stops at some point. Since the decimal expansion stops at some point, y is a fraction. In this case, y=3141/1000.

Now, that means* there can't be a proof that y is a transcendental number. IE, it cannot be shown that y is a solution to some polynomial equation with integer coefficients. This means there couldn't be any expression like:

(z-a1)(z-a2)(z-a3)...(z-aN) = 0
which has Pi as one of the solutions. A familiar polynomial might be z^2 - 1, others are z^3+2z+z, z^n + 3 where n is an arbitrary member of {0,1,2,3,...}. That kind of thing. x can't be a solution to one of those.

But y can. In fact, y is, 3141z - 1000 has y as a root (set it equal to 0 and solve it). This means y is not transcendental, since it is a root of a polynomial with integer coefficients. More generally, if y is a fraction a/b, then bz-a has y as a root. Thus, every approximation of y will not be transcendental.

If we are in a simulation, then y isn't equal to x, since the simulation can't be of infinite size (no infinite memory, infinite memory required to specify Pi exactly). But the reality has to be quite similar to the one with y=x, so y would be an approximation of x. And we're in the situation I described above.

This gives us a means of distinguishing simulated and non-simulated universes while we're in them. A sufficient condition for not being in a simulated universe is: there is a proof that Pi is transcendental. In our universe, there is such a proof.

I appreciate the attempt to frame the proof as being equivalent to some bundle of sense data, but it really isn't a bundle of sense data or a representational artifact. It's a fact, in our universe Pi is transcendental. The 'sense-data' of the proof isn't equivalent to the proof as a demonstration of a fact.

So proof isn't just a bundle of sense data, it's a demonstration that something is true about our universe. If 'pi being transcendental' isn't sufficient for this, surely 'no one can square a circle using a straightedge and compass' is, since it's well within the bounds of a simulation to allow people to play about with axiomatic systems in mathematics and for them to draw with compass and straightedge.

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. And conversely someone could see a construction in the simulated universe that allows squaring the circle. This still allows discrimination between simulated and non-simulated universes based off of the incapacity for anyone in a simulated universe to see a valid proof that Pi is transcendental, since such a proof couldn't exist*.

*unless the axioms of geometry were inconsistent, which they aren't.

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

Why are we assuming this? What I don't understand is why the person in the simulated universe can't use the same proof that we use (the Lindemann–Weierstrass theorem?) to show that Pi is a transcendental number. Is a simulation incapable of performing the mathematical steps that make up this proof?

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

So they won't be able to read Lindemann's proof? That seems an untenable claim.

It isn't that they wouldn't be able to read the proof. The strings could be arranged on the page in the same manner. It's just that that proof would no longer be valid since its result is false! So they wouldn't be seeing a valid proof.

Why would it be false? They start with the same axioms and perform the same calculations, coming to the same conclusions. That's all it means to be a valid proof.

I might as well say that Lindemann's proof isn't valid in our world because the conclusion is false, and that we live in a simulated world. And your argument against this would be?

Let's put it this way:

In the simulated universe, the area of a circle is yR^2, in our universe, the area of a circle is xR^2. They literally have different values for Pi since the value of Pi for our universe would have to be stored in a finite computer. In the simulated universe, y=a/b and so there is a counter example to the proof: bz-a has y as a root. So the proof is either valid and proves a false result, or it is invalid. We know it's not the former, so it's the latter. Therefore no one in a simulated universe could (modal, necessarily not) see a valid proof that Pi is transcendental. So, no one in a simulated universe could see the squaring of a circle (modal, necessarily not). Conversely, someone in a simulated universe could (modal, possible) see the squaring of a circle.

Why would the value of Pi need to be "stored" anywhere? That doesn't make any sense. Maybe this is just me not being a mathematical realist, but as far as I can see, the simulated person will perform the same measurements* and calculations that we do, coming to the same conclusions about the value of Pi and it being transcendental.

* Unless there is some empirical difference between simulated circles and real circles.

So I think you're just begging the question by saying that Pi is transcendental in our world but not in a simulated world. As I said before, I propose that Pi isn't irrational in our world, and that all the "proofs" we have are therefore invalid. How would such a proposal be addressed?

In our thoughts and mathematical operations, proofs: the whole ideational system of mathematics must be something the simulating computer does. This means it must devote a finite amount of computational resources - memory and processing - to doing any particular part of our mathematics.

When an algorithm is used to compute Pi, it will converge towards the value of Pi in the simulated universe. 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.

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

It doesn't need to "store" Pi. That's the bit that doesn't make sense. It only needs to draw a circle, which computers are quite capable of doing. We then measure it and make use of mathematical axioms, always coming to the conclusion that Pi is transcendental. The simulated person will do the same. Therefore this cannot be used to show that we're not in a simulated world.

It's the computer's ability to draw a square which has the same area as a given circle using compass and straight edge which is the problem in the first place. Anyway, how would the computer deal with Pi? There's one correct number for each digit, computer has to be able to access all of them for Pi to work. as it does in our reality. How can it do this without committing the whole of Pi to memory?

It's the computer's ability to draw a square which has the same area as a given circle using compass and straight edge which is the problem in the first place. — fdrake

So there will be an observable difference between a simulated circle and square and a real circle and square?

Modality man. Modality. The computer COULD do it, 'our' computer COULDN'T.

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. Edit: with sufficient measurement precision, anyway.

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

I'm pretty sure that computers can draw better circles than we can with pen and paper, which is presumably how we first calculated an estimated value of Pi?

I don't see the relevance of this. The computer could have a value for Pi so accurate that its error is below the Plank length for a circle with the diameter of the observable universe and there would still be no valid proof that it is transcendental - since it would be an approximation.

How would the computer deal with Pi without committing it to memory? It needs to know whether any digit is correct or not. See no way around it.

The point is that the computer doesn't need to store a value of Pi. It just needs to draw a circle.

Just as a piece of paper doesn't need to store a value of Pi; it just needs to have a circle drawn on it.