I was thinking along similar lines to a comment by Morten Bek in a recent thread about the simulation argument, after I read Nick Bostrom's simulation argument from 2003. I continued developing my counterargument and made it a little more general. I don't just conclude we're not living in a simulation, I conclude the universe we live in isn't essentially digital.

1. If something can be simulated on a digital computer, then it is represented by a series of bits, which could be in the form of an array in multiple dimensions. Mathematicians call a particular kind of array a matrix. (That explains only part of the connotations of the name of the movie The Matrix.)

2. All arrays of various sizes already exist in the mathematical sense of existence. They exist in the same way as all counting numbers exist, the same way an infinite number of primes exist, and the same way an answer exists for every addition problem of counting numbers. Also, all various possibilities of digital contents of arrays already exist mathematically. For example: [0] [1] [00] [01] [10] [11] ... [0000,0110,0110,0000] ...

3. If some array contains a description of part of our universe that is adequate to describe someone's conscious experience, then at least a certain number of the bits in that array are determined by that description. If a computer is producing a series of three-dimensional arrays that represents a fourth dimension, time, then if those arrays describe someone's world, what happens in someone's future is described by the rest of the series.

4. It isn't necessary for a computer actually to run for the arrays to exist. You don't have to ask a computer to calculate pi to exact precision and wait for the computer to output an infinite string of digits to know that pi is irrational and has an infinite number of digits in its digital form. You can know that pi is like that with certainty by mathematics. If you accept the same mathematical assumptions, one of the consequences is that all arrays exist.

5. So, somewhere in mathematical space is someone described digitally, whose future is every possible digital array that could follow, no matter how improbable. Some sort of laws of physics might appear to be followed in part of the arrays, but not in other parts.

6. We do not observe the world around us to contain such improbabilities. It appears to contain consistent laws of physics, not things popping into and out of existence on a massive scale and not edges of chaos that swallow up orderly existence.

7. Therefore, our conscious experience implies order that encompasses the material cosmos. That's not to say that we're conscious of how it implies order, just that it does. While all versions of us described digitally, which go through every imaginable digital future possibility and then some, exist mathematically, our conscious selves experience a reasonably orderly universe. The real universe is not digital. Q.E.D.

8. Some physicists believe in interpretations of quantum mechanics that seem to have something to do with this subject. The order we experience may be implied by our conscious existence only to a certain extent of orderliness, while leaving many variations to be determined by the irreducible randomness of standard quantum mechanics. If so, that's still more order than any sort of digital array simulation allows for. The physical things that are supposed to be random under that theory are so small that on the human scale the chance of anything highly unusual happening because of them averages out to about none.

9. The many-worlds interpretation of quantum mechanics is not compatible with this ontology. If all existent things exist on an equal footing, whether mathematical possibilities or some other sort of reality, then all branches of a many-worlds universe exist equally as much. However, in quantum physics most of the types of random events aren't binary choices and even when they are binary, don't have equal probability of going either way in real world situations, only in thought experiments of theoretically perfectly symmetrical measuring devices. How can we keep experiencing probabilities that are unequal, for example the blurring that happens at the magnification limit of a telescope in a smooth probability curve, if we live in a many-worlds universe where all possibilities are equally real?

I meant to get around to proving that nothing is random and something about our existential responsibilities, but I guess those are subjects for another night.

Looking at Morten's argument and mine side by side, I'd put the counterargument to the counterargument like this: To determine if the universe is orderly, if you're a rational skeptic and think by the anthropic principle that you may have arrived here despite the universe not being orderly, you can apply Bayes' theorem. Assign the odds of the universe actually being orderly 1 to 1 as your prior assumption. Then for every moment you continue to exist and don't change into a frog or whatever, double the first number in the odds. If you're not such a skeptic that you disbelieve there was similar existence in the past, then counting historical moments too gives you a way to instantly increase the odds to an absolutely astronomical number to 1, favoring the universe being and remaining orderly.

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## 8 comments:

A truly great post! I sincerely hope that all the readers of 'Cow's Blog' (and all the similar stories we've seen posted on digg/slashdot about matrices and simulated realities) will take the time to read and understand your points. Cudos for being able to formulate this in a clear and concise language. I especially like the 'counterargument to the counterargument' which I really had troubles putting down in writing. You've done a really excellent job on this - and I am very happy to to know that someone shares my view on all these articles that state that we 'definately live inside a simulation'.

Thanks for the response. I was questioning whether I was getting anywhere at learning to write and choosing interesting subjects, or it was just that I didn't have an audience. So I guess I should keep up what I've been developing in thinking and writing and try to get it out there to someone too.

I preferred your way of writing about it to my own, because yours was funny and casual, more like a stream of consciousness, less like a strained, pseudo-mathematically-formal philosophical proof. Your writing in English would work in a popular magazine or forum. It would just need a little editing for things like spelling, e.g. given your post here, an editor would correct or help you with the English for "kudos" and "definitely" and the idioms "had trouble" (unless you meant more kinds of trouble than usual) and "in clear and concise language" (unless you meant in English!?) and the typo "to to."

Substance and form. If we're typing on the Internet, and the form of sequences of letters is all we've got, then doesn't it have something to do with the substance to try to get the letters right or to consider every little bit of information that comes through as part of the human interaction? Then again, editors who fuss over the spelling variations in editions of Shakespeare are not his equal through doing that, even if they do it well.

You are most

definitelyright! :-) Thank you very much for your corrections. The idioms are bad translations of the Danish equivalents. For the spelling errors and the typo there is no excuse and I am right now beating my groin with a rusty cast iron radiator as punishment. It is great to see that at least someone on the web cares about (their) language!An interesting post, but I think your argument has a few holes - which you paper over.

"If something can be simulated on a digital computer, then it is represented by a series of bits". That's rather an assumption. Why should any simulation (created by an advanced civilisation) only be implemented on a digital computer? Why not an analog computer?

Some of your statements make big assumptions about controversial or unproven ideas. For example, "one of the consequences is that all arrays exist." You are very vague about what consitutes "existence". If the array "exists" then can you give it to me? No, you can't because it does not exist in physical reality. If you're talking about some notion of mathematical existence, ie., mathematical platonism then you're getting into quite controversial territory. My ideas on mathematical platonism (ie., that there is not actual "existence" of platonic objects) is given in a page on my blog entitled

The Mathematical Universe.

If, as I suggest in that page on my blog, there is no such thing as platonic existence (platonic existence is illusory) then you do, indeed, need some sort of computer program to give the equations (the simulation) physical reality. And your idea that such as program is unnecessary is wrong.

"If all existent things exist on an equal footing" - see the section "And now ... the Ultiverse!" on another page on my blog

The Anthropic Principle for more analysis of an

"everything goes" multiverse in which everything exists as you suggest. Again, I come to the conclusions that such an

arrangement would be logically inconsistent and so could not exist in physical reality.

Interesting post, though. Thanks.

Oops, forgot to say my main page about simulated reality can be found at The Big Brother Universe.

Why not an analog computer? I left out dealing with the scenario that reality is a simulation in an analog computer in an analog universe. I forgot that I left out analog scenarios and didn't disprove them in the argument posted. Yet I claimed in my introductory paragraph to have a disproof that reality is a simulation. Thanks for pointing out that hole in the argument.

Analog and not digital implies real numbers, otherwise it's easily reducible to digital. By Cantor's diagonal proof there are more than a countable infinity of real numbers. Yet there are only a countable infinity of numbers that can be specified in finite form. Such a series of finite forms could be programs, in a digital programming language, ordered by length, then alphabetically. So for something to be really analog and not just reducible to digital, there have to be real numbers that contain infinite information. Numbers that can never be specified, only approximated, no expression or formula for them ever being complete. Numbers of mystery. A theory of everything in mathematical form could never be complete in such an analog universe.

In the Max Tegmark document you link from your page The Mathematical Universe, the point where I see instability in a finite digital universe, that leads to me conclude we don't live in one, is what Tegmark calls "the measure problem." He only speculates there may be a reward for simplicity or a penalty for complexity in the description of a universe, that determine which of several digital or mathematical descriptions is more probably like the real universe we live in, to match the observations we make. That sounds like a restatement of Ockham's razor. If not multiplying entities without necessity leads to a more accurate description of the universe, then the idea of the universe being a simulation without evidence of simulation is massive multiplication with no necessity and inaccurate.

Do I assume Platonic existence of mathematics in the argument I posted? Not really. I deal with both senses of "exist" and imply that while mathematical descriptions of the universe exist mathematically, they don't exist as the real universe if they're digital.

I read the pages you linked before responding, and I enjoyed them, including the Java programs. It's a remarkable coincidence that the two movies you put stills from at the tops of the pages are the two movies I own on DVD. I guess that shows there's some overlap in the pop culture background that influenced our approach to these topics.

In response to your Ultiverse argument: Sets of axioms are valid if they don't contradict themselves. If there's an Ultiverse of "anything goes" some sets of axioms may try to refer to axioms outside their set and contradict them, but that doesn't mean they succeed. In one set of axioms, a symbol supposedly corresponding to the Ultiverse is "destroyed" but the Ultiverse could live on in other sets of axioms. So your idea that an Ultiverse or what Tegmark calls a level IV multiverse would be invalid because of too many axioms making paradoxes looks not demonstrated to me. I don't see where the unavoidable fatal paradox to the whole system is. If you've got it, it would be a disproof of formal mathematics, wouldn't it?

I'm inclined to wonder if the constraint that makes the universe logical and orderly, tenable and valid, is just that to think rationally about it, we have to accept the rational and reject the irrational. Scientific thinkers live in a universe describable by mathematics, within bounds of probability. But do they share that universe with others whose reality is paranormal and incompatible with science and reason? And if so, whose reality is the real one, or are both just as real?

It isn't necessary for a computer actually to run for the arrays to exist. You don't have to ask a computer to calculate pi to exact precision and wait for the computer to output an infinite string of digits to know that pi is irrational and has an infinite number of digits in its digital form. You can know that pi is like that with certainty by mathematics. If you accept the same mathematical assumptions, one of the consequences is that all arrays exist.You are confusing existence with conceptual possibility. With a few tweaks, God is conceptually possible. Does that make God exist?

Your arrays are merely conceptually possible.

There is an excellent book by Greg Egan,

Permutation City, which explains why a computer needs to actually "run" for a simulated consciousness to be self-aware.7. Therefore, our conscious experience implies order that encompasses the material cosmos. [...] While all versions of us described digitally, which go through every imaginable digital future possibility and then some, exist mathematically, our conscious selves experience a reasonably orderly universe. The real universe is not digital. Q.E.D.The conclusion you drew from your premise is not the only possible one.

The simulation can be of such nature that order rather chaos appears. This would explain why our conscious selves experience an orderly universe.

Thanks for the comment, Dan. It gave me another chance to work on the philosophical basics to support my position, although I felt close to giving up on the subject as ridiculous and beneath what I intuitively feel I should be working on. The point by point response I wrote follows:

There's a world of difference between mathematical existence and conceptual possibility. Mathematicians don't talk about how many primes in a certain range of numbers are conceptually possible, they talk about how many primes there

are, how manyexist. It doesn't matter whether a human mind can conceive of the specific primes, once the idea of primes is defined and the range of numbers is specified, it's just a factual, logical question whether the primes are there or not.Similarly, the number of different strings of n bits is 2 ^ n. It doesn't matter if some of those bit strings aren't conceptually possible for human beings or for any other sort of mind there might be.

What's conceptually possible doesn't necessarily exist mathematically, because it's possible to have concepts that are wrong or unclear. Some human concepts, such as deity, are so unclear that humans would never reach agreement on whether they have been instanced in a mathematical model.

I don't feel like arguing with characters in novels about this, but thanks for telling me what novel deals with this subject.

Yes, the conclusion I drew isn't the only possible one from the words I used. They are just words. You can reach any conclusion you like, including just restating the assumptions of the simulation argument as if I'd never made an argument. E.g. The simulation is X, that's why we experience the universe as X.

It doesn't matter what nature any one simulation has if there are different simulations with different futures that we could be in. If you believe in the simulation argument and thus a digital universe, you don't have a logical basis for expecting order, when the multiplicity of future digital universes that include some consciousness like yourself implies there is chaos.

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