# A paradox of existence

Introduction
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There is a question in philosophy "why is there something rather than nothing?" I have always thought of this question as completely impossible to answer: either it is a meaningless question, or at least there seems to be no way that we can ever begin to answer it. Yet it has always seemed very weird to me that there is such a thing as existence. I recently developed a paradox that has made this confusion less mysterious for me. The paradox is not about why something exists, but how it is possible that we *know* that we exist. Note, the question is still very confusing, but it has changed the level of confusion for me from something like "completely unsolvable ever" to *merely* roughly as confusing as the hard problem of consciousness. Maybe this idea already exists out there, or maybe I'm just being confused, but as far as I can tell, this paradox really seems to make my confusion around existence seem no longer completely intractable. 

First of all, note that we humans have a very strong sense that there is a “reality”, that “something exists” or that “I exist”, and that we actually know this fact (i.e. it is not thought of as merely a speculative hypothesis). We have uncertainty about this reality, e.g. perhaps the world as we perceive it is an “illusion” or a simulation, but we seem to know that something like a reality exists. It is not at all obvious that this is the case, i.e. that creatures in a reality know that they exist. It could just as well have been the case that (1) there is an existing reality within which intelligent lifeforms have evolved, but (2) at no point do these lifeforms notice that they exist, or that existence is even a thing, they just do the usual stuff of building spaceships and inventing the internet without ever reflecting on or noticing or even having the concept of “existence”.

The paradox
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So having said that, I'm going to state a paradox. By paradox I mean a set of observations, each of which *seems* (to the author and possibly the reader) to be true, but where it also *seems* that they cannot all be true together. There has to be a mistake somewhere, either one of the observations is wrong or confused, or the argument is wrong or confused, but I don’t know where the mistake is. The paradox is as follows:

1.  **We seem to know that there is a reality that exists**. Note, the claim is not just that there exists a reality at all, but that we as creatures in this reality *know* this. I don't want to explain too much what I mean by reality. I am just talking about the normal sense in which it very strongly seems to us that something exists that we are a part of. We seem to really *know* that this is the case, even if we don't know the exact nature of that reality (e.g. whether our experiences may be the result of a simulation). 
2.  **It seems to be the case that this reality is perfectly mathematically describable**. Here I am referring to the normal mainstream assumption/observation that runs through physics and natural sciences, that (1) there is a reality, and (2) there is a “fundamental theory of physics”, currently not fully known to us, thought to be specified by something like a dynamical system + initial state (modulo relativity), and this mathematical theory can in principle (modulo computational constraints) precisely predict everything about this reality. In particular, and this is where the core of the paradox will be, it seems to be that everything about our minds, *including observation 1 that we know that we exist*, is in principle derivable from this dynamical law and initial conditions. 
3.  **It seems that whether a mathematical universe exists/is real cannot be a mathematical property of that mathematical universe**. More precisely, it seems that we cannot see, purely from a mathematical description of a theory, whether that theory describes reality or not. I am referring here to the normal mainstream idea at the core of physical science; that (1) there is a reality, and a set of possible mathematical theories describing that reality, and (2) we can only know which theory describes reality by comparing the theory to reality, through empirical experiment, obviously not by merely looking at the theory itself. Even if it is somehow possible to use exotic tools, like introspecting our own minds and somehow drawing conclusions about reality from this, we are obtaining empirical information from reality as opposed to deriving a purely mathematical property of a theory. This idea is a very uncontroversial principle at the core of (natural) science. To summarize this in less precise terms: whether a mathematical universe exists (describes a reality) is not a mathematical property of that universe.

These three observations constitute the paradox for me, because it seems like they cannot be true at the same time:

**The paradox**: The fact that our reality exists is not a mathematical property of the fundamental mathematical theory that describes our reality, but an "extra fact", and yet, the exact states of our minds follow in principle purely mathematically from that fundamental mathematical theory, including the fact that we know that this universe “exists”. Assuming that our justification for thinking that we know of its existence is actually sound, this implies that the existence of our universe is in principle a mathematical implication of the fundamental theory that describes our universe. 

Note that I don’t consider this to be a solid logical argument. What makes a paradox a paradox is that there is some kind of mistake somewhere either in the argument or in the assumptions, and in this case I don’t see clearly where the mistake is. 

The hope I have is that the mistake is not in fact superficial, but points to a deeper inadequacy in the concepts I’ve used in this argument, and all of these concepts seem to me fairly fundamental and generally accepted. I will address this later sometime, but I also have some hope that this will shed some light on the hard problem of consciousness.

Discussion
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As I hinted at in the introduction, it is observation 1 that is most confusing to me. Why would creatures inside an existing reality somehow know that they exist? Descartes’ principle “cogito ergo sum” actually seems to me like it shouldn’t hold, because there presumably are a range of mathematical universes that are in principle definable but don’t describe any reality, that contain creatures which think (cogito). But those creatures don’t actually exist, which seems to show that the inference is wrong. Yet it somehow really does seem to be that we somehow know we exist by virtue of something like reflecting on our own experience and thoughts and so forth.

To go back to the paradox and hopefully to clarify it, we can make a slightly more concrete but stronger addition to it: Assume that in fact observations 1 and 2 are true. Then if we can actually define something like a pseudo algorithm that checks, given a mathematical description of some hypothetical mathematical reality, whether that mathematical reality exists. The algorithm won't be able to rule out that a theory describes an existing universe, but it can sometimes confirm it: 

**A seemingly impossible pseudo algorithm**. Take a mathematical universe, and unroll its dynamic law. Search throughout the universe for creatures, and check whether these creatures know that they are part of an existing universe (*). If so, then conclude that this mathematical universe is an existing reality. 

Note (*) that a lot hinges here on the ability to check whether a creature "knows that they exist". I cannot specify this precisely because observation 1 (and the notion of existence itself) is so confusing to me. In fact, observation 1 doesn’t directly imply that we can actually *check* from a mathematical description of a creature that the creature knows it exists, so this algorithm uses a slightly stronger assumption. 

Also, note that there being an algorithm that checks if a mathematical universe describes an existing reality is itself not surprising: We can define an algorithm that has an encoding of the true fundamental theory of physics and states that a theory describes an existing reality if and only if it exactly equals that fundamental theory. The weird part to me would be if it is possible to confirm existing realities on the basis of specifically the algorithm given above.

I want to end on a note about the hard problem of consciousness. I have recently been starting to think about the hard problem in terms of the "paradox of consciousness". I haven't yet written this down, but there seems to be a very strong similarity between the paradoxes of existence and consciousness as I've come to think about them. This very vaguely suggests to me that they might be related in an important way (but this might also be a false alarm, this is very speculative). 

As a final remark, note that I find this whole idea very confusing, and it might just be a straightforward confusion on my part. But it seems to me that this moves questions around existence, from something completely impervious to any kind of analysis, to something that actually seems to contain something like an inconsistency, and thus something that can be analyzed.