The simulation argument

I've heard the simulation argument from time to time in different forms and decided to look into it a little bit more thoroughly. So I gave Nick Bostroms article from 2003 entitled "Are you living in a computer simulation?" a proper read through the other day.

So, what is the simulation argument anyway? Well, Bostrom summarizes it succinctly in the very last sentence of his paper: "Unless we are now living in a simulation, our descendants will almost certainly never run an ancestor-simulation" (p. 11). Not entirely clear? Let's look it a little more in depth.

First and foremost, the simulation argument is based on the notion that some form of functionalistic theory about consciousness is true. That is, it has to be true that humans, simulated in a computer, have some kind of qualia, that it actually feels a certain way to be one of those humans. This is somewhat of a leap of faith but, as I will come back to later, the beauty of the simulation argument isn't that the conclusion necessarily is true, but rather that the conclusion seems to follow given the premises.

Second, the simulation argument argues that future generations, given that slightly less future generations don't manage to start a global nuclear war or something equally unhealthy, will have a ridiculous amount of computing power available. If it is likely that somebody, or even just someone, would run a sufficient amount of simulations of their own prehistory, where the inhabitants would be unaware of the fact that they lived in a simulation, it would follow that most humans that have ever lived (simulated or real) would be simulated. And since there is no way of telling whether one lives in a simulation or not, wouldn't it be most probable that we actually lived in one of these simulations rather than in the "real" world?

Bostrom introduces a lot of twists and turns to this argument and the paper itself is well worth a read through just because it's pretty damn thought provoking. But, once again, what I find most fascinating with the simulation argument is that the conclusion actually seems to follow from the premises. Whether the premises are true is a different matter and here, I'm a bit skeptical.

The main question I find myself keep coming back to is the question of why anyone would want to run a simulation like this in the first place. In a posthuman society, capable of extremely demanding technological feats, people (or more likely cyborgs/robots) would probably be able to satisfy their needs directly without having to go through the fuss of changing the outer world to please themselves (this counterargument is based on the assumption that such a simulation would have no practical benefit but rather just be something created to entertain its creators, something which indeed could be questioned). Bostrom actually brings this up himself:

But perhaps many of our human desires will be regarded as silly by anyone who becomes a posthuman. Maybe the scientific value of ancestor-simulations to a posthuman civilization is negligible (which is not too implausible given its unfathomable intellectual superiority), and maybe posthumans regard recreational activities as merely a very inefficient way of getting pleasure – which can be obtained much more cheaply by direct stimulation of the brain’s reward centers. (p. 9)

However, it would be enough that just one individual would run one simulation, for half of the subjects ever to live to be simulations rather than the real deal. Bostrom points this out, and it's a good point indeed.

The simulation argument can be varied and applied to different situations. For example, imagine that you would want to live forever (in my experience, most people don't want that, something I find extremely strange, but let's not dwell on that right now). How would you achieve it? Well, one way would be to make sure that in the future, there would be a large number of simulations being run of your life, with the additional feature of an afterlife (assuming that no afterlife will be happening after you life). Given this, it would be more likely that you would live in one of these simulations than that you would be the original you.

In the end, I find the functionalistic assumption to be what makes the truth value of the conclusion of this argument doubtful. Show me that that assumption is true, and I will probably start saving up money for those computers I'll need to be immortal.

Dokum FTW!

In the summer of 2008, my friend Tobias had an idea: Let's go out on a weekend night, with a camera, a tripod and a microphone, interview a couple of people about their night and then, all of sudden, ask them some enormously complicated philosophical question. This was the birth of Dokum. Since that hot summer night, we've made three more episodes, inerviewing random people on the streets of Stockholm about more or less philosophical stuff.

Check it out! (All the interviews are in swedish)

It's fun to annoy people.

Daniel Dennett in Oslo

Dennett in Oslo (w/ english subtitles) from Speldosa on Vimeo.


Last Ocotber, Daniel Dennett visited Oslo and while I'm no big fan of him, I still had to go to see it. With me I brought my camera and my over enthusiastic friend Fredrik, who traveled to Oslo primarily to see Dennett and secondary to see me.

I am a strange loop

"I am a strange loop" is the second book by Douglas Hofstadter, the author whoose name I can't pronounce, , that I've read. The first one was "Gödel, Esher, Bach" (a.k.a GEB), his classic from around the time I was born. If there can be things like sequels in the world of philosophical litterature, "I am a strange loop" would stand in exactly that relation to GEB.

Let me start by saying that even though "I am a strange loop" picks up where GEB ends, this really is a different kind of book. While GEB was very broad in its approach to Hofstanders main thesis, "I am a strange loop" narrow it down and really focus on the main question. The question of what conscioussness is. To understand Hofstanders take on this question I have to present some of his basics thoughs about the relationship between high and low level explanation and his theory about the organisation of systems capable of executing adaptive behaviours.

Hofstander encourage the reader to imagine a machine consisting of a set of spring-loaded domino bricks, arranged in such a way that they can compute if a certain number is a prime or not (The computation is in itself pretty easy. If you want to know if n is a prime (where n being any integer), just try to divide it with every number from 2 to n-1. If you find a divisor, n is not a prime. If you don't, it is). All you have to do is to line up the number of bricks that represents the number you want to test and then let all hell break loose. Watching the turbulent process that follows, with dominos falling and rising, triggering different loops in different positions of the board, eventually, if the pattern of dominos find a divisor for the input number, a certain domino strech (Hofstander call this the "divisor strech") will fall, thous signaling to an observer that the number wasn't a prime.

Now, let's say we put 641 bricks in a row and start the machine. If someone were to watch the whole thing, he or she might point to a domino brick in the middle of the divisor strech and ask: "How come that domino there is never falling?".

Hofstader imagines there's two different answer to this question. The first one is that the domino brick doesn't fall because its predecessor didn't fall. However, he points out, this just means that you have to ask the same thing about the preceding brick, beggging the question. The second answer is "Because 641 is prime". I quote:

The point of this example is that 641's primality is the best explanation, perhaps even the only explanation, for why certain dominos did fall and certain other ones did not fall. In a word, 641 is the prime mover (p. 39).

Here Hofstadter is trying to concive an idea that even though a process can be given a low level explanation, the higher level alternative may be the only thing that really can explain what's going on.

The other basic thought Hofstadter is trying to mediate to his readers is the concept of symbols. To understand this, one must really know his distinction between perception and mere input. Imagine a toilet. The mechanism that makes it fill up after every flush is pretty simple. I encourage you to go check your toilets out right now if you're not familiar with the process. Basicly, when the tank empties, a floating part hits the bottom, pulling somekind of valve that starts the filling process. As the water adds upp, the floating part moves up, thereby stopping the onset of the water.

According to Hofstadter, this is not a perceptuall process. The toilet doesn't perceive that the water level has dropped or is rising. Its just a plain mechanical proecess.

What is an perceptuall process then? This is where the symbols make their entrence. A perceptuall process is when something in the world is being represented by a certain pattern and the representation can come about in many different ways. Take an object in your surronding that you can move. Look at it for a second and then move it so that you're seeing it from another perspective. Isn't there a feeling or experience that's reoccuring every time you look at it, no mather which perspective you're choosing? If I've understand it correct, this is what Hofstadter means with the concept of symbols.

The point is that even if I move my cup with tea around on my desk, I'll still know that I can drink from it and that I probably shouldn't push it of the edge. I have a symbol in my mind, or a pattern in my nerve cells, that somehow represents the concept of "cup".

Now, Hofstander proposes that conscioussness is something that comes about when a cetain system of symbols, a pattern of discrete parts if you will, start to have symbols of itself. When you somehow represent your own set of symbols in some kind of higher level symbols. If I seem to be a bit confused, that's because this is pretty abstract stuff.

So, what about the loopiness the title suggest? Well, if I've understood Hofstadters thoughs correctly, it comes about when the symbols of oneself are begining to be so advanced that they encompass themself. Since it infact are a part of the self, it has to include itself in itself to represent itself. But then it has to represent this representation with the representation of itself, thus beeing left one step behind again. It's like someone trying to catch her own shaddow. Or Gödel trying to say something about numbers with the use of numbers.

Let me make it absolute clear that Hofstadter is not a dualist. He firmly states that conscioussness is nothing but the physical pattern that constitues an organisms brain. That there is no distinction between me feeling something and my brain carrying out a certain process. It's only two different explanations of the same thing.

One objection to this might be that even if this is soo, why is it like that? Couldn't one think of a universe where everything looked exactly the same, but where people really were zombies, soulless creatures with no qualia? No, Hofstadter states.

To illustrate this, Hofstadter comes back to his discussion on low and high level explanation. Let's say that I reach out and pick up my cup to get a sip of tea. On the low level this could be explained by describing every single cell in my body and their changing states over time (probably we can't stop there since there's things like molecules, atoms etc.). If we on the other hand explain the behaviour in a high level fashion we can explain it by saying: "Martin was thirsty - that's why the current pattern came about". In the same way as you can't imagine a universe where the domino brick in the exact same pattern as previously descibed stood still and that 641 was a prime wasn't an explanation of this fact, Hofstadter states that you can't imagine a universe where the pattern of your brain is realised without you also feeling something.

One of the most bissare consequenses of Hofstadters theory is that a person isn't his or hers brain. A person is a specific pattern and the brain is just one of many mediums that can realize it. Just as it doesn't matter if a novels letters are printed in a book or illuminated on a computer screen when it comes to the meaing of those words. Josef K. gets procecuted no mather what (by the way, that book is horrible...there, I said it). This also means that we aren't bound to other bodies, and in one sence, not even our own patterns.

Hofstadter means that we construct symbols that represents others inside our own pattern. When I get to know a person, I start to build up a concept of what that person is like. I can store what she likes, what she doesn't like, how she responds in different situations, what she feels about other people and so on. In one sense I make a copy of that other persons pattern, but since the person isn't anything else then her pattern, what I really do is realising part of her pattern in another medium. She inhibits me. So if I die tomorrow, I'll partly live on in the people I know. It's not a full realisation of me, but it's still me. Sort of.

What I don't really understand with this position is this. Let's say that I'm out for a stroll in downtown Stockholm when a person comes up to me, put a gun to my head and says: "Do you want me to pull the trigger? I give a hundred bucks to charity if you say yes". I'll have to say that my answer would be no. Why? Well, I want to live on, that's why. Charity or not.

Let's say this person comply with my wish and walks away. The next day I walk down the same street and the same person comes up to me, put a gun to my head and says: "Do you want me to pull the trigger? I give a hundred bucks to charity if you say yes...and by the way. I've built an exact replica of you who's standing behind the corner.". If I would agree with Hofstadters ideas, what reason would there be for me to say "no"? Think about it. The perpitrator would give a hundred bucks to charity which would make me glad, and I would still live on in the body behind the corner. Why should I favourise a certain set of atoms that realizes my pattern rather than another? But would I say "yes"? Err...no. Why is that?

Hofstadter never explains how someone, in some sense, can be an explanation and I don't think he thinks that can be done either. To give a meta physical explanation on what the heck is going on is beyond our reach. But this also makes it hard to see why his thesis should be considered at all. Hofstadter himself considers this to be a good and consistant outline for an identity theory, but his main argument is really just him reciting a list of problems with the dualistic viewpoint, more or less ignoring all the problem that arise with the opposit view. However, that's an enormous discussion, something I won't go into now.

In sum, Hofstadter has written a quite ambitious book but somehow it feels that you already, in part, have to agree with him to fully understand it. It just feels that he constantly begs the question. I think GEB was better in the sense that even if you didn't agree with Hofstadters main idea, you could still learn a great deal from it. For example I learned many new things about the foundations of mathematics by reading that book. In "I am a strange loop" Hoefstadter really gets down to buisniess. Really esoteric buisniess.

But I guess someones got to do it.

The function of consciousness

When trying to discuss something as delicate and complex as consciousness, I think it's important to define the question at issue on a manageable level. If one tries to attack the core of the problem right away, there's a big chance to get lost in the infernal maze of metaphysics that follows.

One of the most intresting, and at least partly graspable, question is what role conscioussness plays in the big picture. It must be present for some reason. The question is what that reason is.

Before I start reasoning about this, let me just put in a disclaimer here. The following discussion takes for granted that there is something like an external material world. I admit that this is just an assumption I make, and I'll be comming back to this topic in future posts, both regarding to the problem of conscioussness and the more epistemic question about if we really can assume such a thing.

The way I see it, there really are two different ways you can fit in conscioussness into the whole process that the brain undertakes.

One possibility is that it's just a kind of parallel redundant process. In other words: The brain recives an input, make some calculations and then produces an output, and the conscioussness just follows the process without any manipulation of it. In this scenario, the conscioussness can be seen as a passive bystander. Whatever happends in the brain decides what we experience, but there is no feedback process back to the brain.

The other possibility is that conscioussness does play an active role in the process of thinking. The brain recives an input, make some calculations, then sends the information to the conscioussness which somehow manipulates it and then sends it back to the brain for further calculations and production of an output. Where the physical calculation fit in doesn't really matter for the theoretical outline. There could also be several steps of information exchange between the brain and the conscioussness but that doesn't really matter when discussing the general structure.

So, to summarize this in an illustrative way:

Model 1:
1.1 Input--->1.2 Calculations--->1.3 Output
Parallel process: 1.2 Consciousness

Model 2:
2.1 Input--->2.2 Calculations--->2.3 Consciousness (Calculations)--->2.4 Calculations--->2.5 Output

Let me state that it is my firm belief that even if the second model is the right one, the human behaviour doesn't require conscioussness. In other words: I see no reason why a computer theoreticaly shouldn't be able to simulate the exact same cognitive processes as we have, with or without the help of consciousness. Even if the conscioussness have some important calculating part in the animal mind, I see no reason why it couldn't be carried out in a strictly material way. Even if conscioussness is an unavoidable consequense of the pattern itself that constitutes human behaviour, I see no problem with posting a world where this wouldn't have to be the case.

Another thing that we can see right away is that the question "Why are we conscious?" have a single answer in the first model and two of them in the second one. Both explanation demands a physical reality that distinguish itself from our current view of the universe. In other words: No matter which model is the right one (if any), a metaphysical reason for consciousness must be present that answers the question how it is posible that something is consciousness at all. If the second model is the right one, we have to provide an evolutionary explanation for it, in addition to a metaphysical. Note that the first model still would have a genetical explanation, but since that proposal doesn't include some kind of function that would make consciousness affected by natural selection, it couldn't be seen in evolutionary terms.

Another thing we should note is that even if both models demands a metaphysical explanation, the explanations themselves can be totaly different. Model 1 only demands a physical reality that makes it posible for some kind of consciousness to arise. This consciousness doesn't have to do anything really. The only criteria is that it's correlated in an isomorphic way to what goes on in the material process that spawns it (for example a human brain).

Model 2 on the other hand demands, not only that it is posible for a consciousness to arise, but also that the mental realm in some way can affect the material realm. In one way or another, there has to be some kind of feedback process from the conscioussness or else we can't state that it have a function. An input signal that isn't returned doesn't do very much good for the overall process.

I think the best way to tackle this problem is to look at the demanded metaphysical realities that each model spawns, and somehow analyse them to see which one is most plausible. For example, the fact that our experiences seems to be coherent in respect to the situation. When my body is damaged, I feel unpleasant pain which I want to repelish, in the same way that my behaviour try to make it stop. If model 1 was the right one, there really wouldn't be any reason why I couldn't go around and feel pain all the time. The actual content of my feeling, really can't matter, if it isn't capable to affect the outcome of the situation. Let me try to illustrate this:

Let's say that we have person A and person B. Person A feels happy when his body isn't in danger of being damaged in any way, but feel sadness and pain once something happen. Person B feels sadness and pain all the time. Remeber that I only talk about the qualia here (the feeling itself). A and B will act in precisely the same way, since their consciousness can't affect their bodys in any kind of way. Therefore, it would't really matter if consciousness was constructed like in person A or B. Isn't it pretty strange then, that we seem to have this coherent mind that we at least think we have?

Another way to attack the problem is to examine what we are consciouss about. Needless to say, most of what goes on inside our brains, we aren't aware of. There are a lot of unconsciouss processes that takes place that can be messured in different kind of tests, althoug we ourself couldn't notice them even if a gun was put to our heads. One example is implicit knowledge. We can't explain how to ride a bike no mather how experienced we are with this activity. However, we have no problem actually riding it. It happends more or less automatic. Now, why would this be the case? Wouldn't it be posible to be consciouss about all the processes that happends inside the head? We already have the power to, to some extent, experience several things at once, so there shouldn't be any theoretical reason why a being subjectively couldn't take part of all the calculations that constitutes its behaviour. Why then is the human mind so restricted? If conscioussness in some way can affect the outcome in the physical world, then it makes sense. Conscioussness only experience the parts it can affect. It's there because of its power to change things, so why would it be present on any other level than its function? In the first model scenario, another account has to be given to why consciouss experience doesn't contain lower level processes.

Of course, all of this is pretty ungratefull work, since it's really hard to say something at all about what is plausible or not, when it comes to the construction of realities whose inner workings we know next to nothing about, and even more, realities that we are trapped inside ourself. I admit to not entierly escaping the very metaphysical maze that I warned about in the begining of this text. It's just really, really, really (understatement) hard to avoid it.

But rest assured. There will be more maze running in the future.

Gödel, Escher, Bach

I've finaly managed to get through Douglas R. Hofstader's quite thick book, "Gödel, Escher, Bach" (the library demanded it back so I had to rush through the last 150 pages this weekend). Even though Hofstader wrote this back in the 80´s, it still feels pretty modern, even though some of his thoughts have been proved uterly wrong in the last two decades.

So what is it about? Well, many people seem to have misunderstand this book. Why that is, I can't say. Sure, Hofstader talks about quite a few different topics, ranging from logic and mathematical history to molecular biology and the collevtive behaviour of ant collonies, but It´s always presented in a context of the main theme which is intelligence, thinking and conscioussness. However, allthough many surmises that the book tries to explain how conscioussness comes about, that isn't my understanding of it. Rather, I see it as a generall theory about the content of conscioussness, which is a completly different subject. In short, what Hofstader says, and I apoligize if I've completly missunderstood him, is this:

The human mind is a piece of machine which consist of billions of neurons. It can be understood on different levels, in the same way a computer can be. In the later case, you can interpret a program in terms of different functions and commands, but you could also break down every bit of these parts and get down to a much more basic level, physically describing every change that takes part in the systems hardware. Conscioussness is basicly the high level understanding of what goes on in our heads. For example: When you read this text, you don't have to concentrate on every single letter. Every letter is lumped togheter with the ones around it to create a word, every word is lumped together with the ones around it to create a sentence and every sentence is lumped togheter with the ones around it to create some kind of meaning. Now, every part of this process isn't conscioussly percieved. It's only the high level lumping that takes part conscioussly.

M.C. Escher's "Print Gallery"

So, what does Gödel, Escher and Bach has to do with anything? Well, the main theme Hofstader draws from these three different men is the notion of self reference. Gödel's theorem is posible thanks to self reference, Escher has created a lot of paintings where self reference is in focus (see the picture below) and Bach created music which often experimented with self reference. Combining all these mens work, Hofstader eventually tries to present the idea that conscioussness is something that forms when the mind starts to represent itself inside itself. When the perception is turned from the outside world, onto itself.

If Hofstader's goal with this book was to give a plausible metaphysical account of how conscioussness comes about, I'm not impressed. However, I don't think that that's what he tries to accomplish. Rather, he's done a quite good job on giving the reader an introductory set of tools and ideas for further investigations of the minds inner workings. I can't say that the book contained the huge climax I expected it to have, but hopefully it managed to equip me with directions to some new viewpoints that probably will prove to be fruitful in the future.

Incompleteness: The Proof and Paradox of Kurt Gödel

I recently finished Rebecca Goldstein's book "Incompleteness: The Proof and Paradox of Kurt Gödel" (Swedish translation (which I read): "Ofullständighet: Kurt Gödels bevis och paradox"). For those of you who aren't familiar with Gödel (I hardly was before I read this book), he was a logician, by many people though of as the greatest thinker of the last century along with Albert Einstein. This book tries to explain why, but often gets stuck in its vivid description of this somewhat strange and esoteric character. The fact that Kurt Gödel died of malnutrition due to paranoid delusions maybe explains why there's a lot of intresting anecdotes about this strange man well worth brought up, but it sometimes feel that the author is more excited about Gödels odd life than what he actually accomplished. With that said, the book is, in my opinion, a great and captivating introduction to his work.

The fact that everything isn't centered on hard scientific facts all the time makes this book a pretty easy and convenient read. More speculative philosophical discussions are blended with stories of Gödels life and his own ideas in respect to his logical claims. In the middle, there is a pretty technical, although pretty talkative, presentation of the work that made him almost immortal. The reader who wishes to really dig down into Gödels theorems - and at the same time spare oneself from those vauge discussions on different implications of it - will probably be pretty disapointed by this book though. However, if you know that you wouldn't last more than a couple of pages reading a book filled page up and page down with stringent logical arrangements, but still wants to grasp the main idea about Gödels theorems, look no further. This book will probably get you intrested enough to go on to the last page.

The fascinating thing about this book is how it's almost structured as good detective story. In the beginning, intresting implications of Gödels work is suggested, and when your half way through the book, you could recite all the passages about how great and novel his theorems was, without having read a single conclusive description about it. Sort of like this review.

***

Well, I feel I can't publish this post without at least taking a shot at trying to explain what Gödel's pioneering results were. This attempt, sprung from my megalomania, shouldn't be taken too seriously though. Ok, here we go.

Gödels proof of incompletness is really based on the old liar paradox. It goes something like this (there's countless different versions of it):

This sentence is false.

If you look at it, it soon becomes clear why it's a paradox. If the sentence is false as it claims, the statement that it's false is true, which makes the sentence true and hence contradicts itself. If the sentence is true then the statement that it's false isn't true, hence the sentence is false, which contradicts the fact that we said that is was true. As you can see, it becomes pretty complex, but the main idea should be pretty easy to grasp.

What Gödel does is to expand this paradox into the domain of mathematics. First, he construct a system, that translates logical statements to aritmetic statements. In this way, he can represent every logical statement with a specific number. Let's say for example that the statement "If A then B" in this system is represented by 10230. It doesn't really matter what the actual system is (although If it did, I couldn't give an account of it) but the important part is this. What is demanded from the system is that it is isomorphic with standard logic (which Gödel, a couple of years before his incompleteness theorem, prooved was complete). What this means is that there have to be a given set of rules to combine different numbers, depending on what rellation you want them to have to eachother, so that the output produces the correct logical statement. Let me try to illustrate this (since I don't know Gödels system, and since I don't really think I have the time nor the knowledge to create a coherent one myself, you have to ignore that this example is taking some liberties).

Let's say I want to make this series of statements (this is a so called syllogism):

(A) If A then B.
(B) A is true.
-----------------
(C) B is true.

Let's say that (A)=1, (B)=2 and (C)=3 after the translation. What we now want to do is to find some rule that we can use whenever we have two statements and want to know what the consequenses of those statements are. Well, in this example it's pretty easy. 1+2=3. So let's say that plus holds that function. The trick with Gödels system is to work out something that always is interchangeable with simple logic. In this case, you could clearly see that my model doesn't get very far. Let's try to state:

(A) If A then B.
(A) If A then B.

What's the result? 1+1=2. In other words:

(A) If A then B.
(A) If A then B.
-----------------
(B) A is true.

Hopefully not true. Well, let's move on.

Why did Gödel go through all this fuzz then? Well, his goal was to create an aritmetic system that could make claims about itself. If you can represent every logical statement with a number, then you can say something about the aritmetic system and then translate it into a number being part of an aritmetic system, hence make it say something about itself. It's almost like if you, everytime you said something about a pig, was turned in to one, and therefore said something about yourself (anyone who's ever been told by a parent that he or she eats like a pig, and witty responded that pigs usually have pigs for parents, know what I mean).

So, what's the point of doing this then (we're soon home free)? Well, this takes us back to the liars paradox. The reason why it is so effective is that it says something about itself. "This statement is false" get problematic because it refers back to itself, creating somewhat of an infinate loop (Yes, I'll get to "Gödel, Escher, Bach" in future posts).

Gödel now make this statement:

This statement is impossible to prove in an formal system capable of aritmetics.

He then translates this statement into a number and get something like, well, let's say:

52352359875239823987609238765987234987587237865982309589723

Yes, I just made that up. What the numbers above now is saying is this:

This statement is impossible to prove in a formal system capable of aritmetics (and this is a statement in a formal system capable of aritmetics).

What this statements says, given that it's true, is that there is true statements within a formal system capable of aritmetic, that still are impossible to prove. And what it says given that it's false, is that it actually can be proven. But since the statement is false, it could possible be proven, hence we have a contradiction.

So, what follows of all this? Well, when we construct a formal system, we don't want it to be able to contradict itself. That's really the worst thing that can happend. We can't have a system that says that 1+1 is 1 and that 1+1 is 2. That would mean that 1=2, which would make it possible to prove any kind of statement. 3+3=8? Sure.

1+1+1=3--->3*2+2=8--->3*1+2=8--->3*1+2+1-1=8--->
3+2+2-1=8--->3+2+1=8--->3+3=8

Since 1 and 2 are interchangeable at the same time as they have different meaning, we can prove whatever we want. We can't have that. Our only option is to accept the other outcome. There are true statements in formal systems capable of aritmetics that can't be proven. This made a huge impact on the mathematical world (to the extent that mathematicans were willing to accept it) when Gödel presented it in the the 30's. Untill then, the general idea was that every true statement in mathematics could be proved. That was by many though of the very analytic meaning of true. That it could be proven. Gödel showed that that wasn't the case.

***

Gödel's incompletness theorem spawns a lot of different implications in respect to how you choose to see it. From platonism to conscioussness, people have tried to apply Gödels famous work in a lot of different areas. I'll definitly come back to this topic in the future, but this will be all for now.

If anyone have any objections against how I've presented Gödels incompletness theorem, please tell me. I'm eager to improve.

For a kind of halfly vauge parallel to Gödels theorem, see my post titled "The epistemology of consciousness". What I try illustrate there is a (hypotetic) true statement that cannot be proved, although this one is not within a formal system. In you are a sceptic, you could come up with countless of examples of situations were this holds. I'll get to this too in due time.

The epistemology of consciousness

One of the biggest and most beautifull puzzles in the universe is, according to me, the question of consciousness. Therefore, I've decided to write a couple of texts about it, to see if I can shed some light on this mysterious phenomena. I'll start with an epistemological view of the subject, namely the fact that our inability to ever prove that consciouss exists in any other enteties than ourself spawns a sort of paradox.

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That I am conscioussness is self evident. It is something that I feel, and since a feeling is defined as something consciouss, I really can't be mistaken. Let's also assume that there is somebody else who's consciouss too. This person can also prove his or hers own conscioussness, but have no way of accessing mine, and can therefore not prove that it exists.

What we have here is a sort of a stand of. My knowledge proves my own conscioussness and the other persons knowledge proves his or hers. It's important to see here, that my knowledge isn't transferable in some sense. It isn't the content of my knowledge that is the important factor for the proof here, but rather the experience itself. The experience proves the phenomenon by beeing the phenomenon itself.

Let's say that for some reason, you don't know the answer to 7+5. You lack that knowledge. Now imagine I show up and tell you that the answer is 12. You gain that knowledge. But have you gained my knowledge? Well, no. You now have a replica of my knowledge. Its content is the same as mine, but it's still not the same experience. You have your experience, I have mine. They might post the exact same thing, but that doesn't mean that they aren't seperated. It's not the same knowledge. It's two different set of knowledge with the same content.

Most knowledge is of the type that can be combined and thereby extended itself to embrace more than it did on its own. If I know that Stockholm is the capital of Sweden and you know that Sweden is in Europe, we can combine our knowledge and get the statement that Stockholm is in Europe. What matters is the abstract contents of the knowledge, not the knowledge itself. I can give you all the reasons why I think that Stockholm is the capital of Sweden and once you got them, your knowledge is as good as mine.

The paradox, which is more of a dilemma rather than a paradox, is this: If it's the case that I am consciouss and know it, and you are consciouss and know it, we can't combine our knowledge and get the true statement that there exists more than one consciouss entity. I can't just tell you that I know that I'm consciouss, since the abstract content of the knowledge isn't the important factor for the proof here. To know that I'm consciouss, you actually have to be me. If there were some way to merge two minds togheter, we could merge our knowledge, but at the same time we would both share the same conscioussness, and thereby the statement that there exists two consciouss entities would't be true anymore. To prove the statement, you have to make the statement false, and thereby you have nothing left to prove. The catch 22 of consciousness.