A paradox of existence

This is a hard concept to grasp. But if my understanding is correct, I think you have described a legitimate paradox, especially for physicalism. If everything is physical and nothing beyond, and physics can be explained by math (in terms of values of fundamental constants and various laws), then how come only one particular set of values are physical ("real"), while others are not. There seems to be a missing deciding factor not explained by math or physics. 

An obvious way out is of course to say "all mathematical possible universes ARE real. Physics is only trying to determine which particular universe we live in." Then the problem becomes how to define WE. Again, this to me appears an impossible task for physics. Imagine a complete physical description of dadadarren, it does not seem to cover the fact that he is me, or I am experiencing the world from that physical system's perspective. 

FWIW, I will take a swing at this paradox. 

We seem to know that there is a reality that exists. This is undeniable. But how do I know or believe there is a reality out there? Only from the interactions between me and the environment. Those interactions ultimately lead to various subjective experiences directly felt which form my belief in a "real world". 

(Conversely, if I question whether my experiences truly reflect what's out there, then I question reality. Like brain in a vat or similar skeptical arguments)

It seems to be the case that this reality is perfectly mathematically describable. Also undeniable. All interactions from the environment seem to be predictable/explainable using math (subject to inherent indeterminacies and computing power): If I let go of a ball, it would drop. If I look at the window I can see what's behind the glass as they are transparent. If I measure a spin of the election there is a certain probability for the outcome etc. 

However, if physics is the mathematics that explains those interactions, then it cannot describe everything in the universe. Most importantly its scope does not include me. But because I believe in reality and that you are real, I can imagine thinking from your perspective too. And it would be the same. Physics can explain the environment's action upon you but not you. However, now I am in the scope of physics from your perspective. And it doesn't have to be applied from a human being's viewpoint,  any physical system's perspective is just as valid. 

In this sense, the objectivity of physics does not mean it describes the entire universe with a "view from nowhere". But rather, those mathematical equations remain useful from a wide range of perspectives. Remember, whoever/whatever at the perspective's center is not described by physics. (IMO that is the domain of subjective experience and consciousness)

It seems that whether a mathematical universe exists/is real cannot be a mathematical property of that mathematical universe. I agree with this too. 

My solution of the "extra ingredient" which determines what is real versus what's merely mathematical is the first-person perspective. It is not something explainable by math or logic. It is based on subjective experience. There are other things in other mathematical universes that can "think" (using the term liberally). But only this body's subjective experience and consciousness are felt. So who's the first person is primitively clear. And by using expressions such as "the others" it is already clear which universe is "the real one". 

I like the Stephen Hawking quote: “Even if there is only one possible unified theory, it is just a set of rules and equations. What is it that breathes fire into the equations and makes a universe for them to describe?”

Max Tegmark's answer, I guess, is “nothing”—every possible mathematical object exists / manifests / has fire breathed into it / whatever, in exactly the same way as the fundamental laws of physics governing our own universe.

I think Tegmark's idea is intriguing, but I don't wholeheartedly endorse it. My general feeling on this topic is “confused and unsatisfied”. I haven't thought about it very much.

I think your mistake here is that:

It can be a property of a mathematical theory that particular objects, if they existed, would know they exist.

It cannot be a property of a mathematical theory that particular objects exist.

This renders your paradox moot.

Remember that mathematics is something we make up; mathematics isn't fundamental to, prior to, or indeed related to existence itself at all; mathematics is the process of formalizing rules and seeing what happens.  You can invent whatever rules you want, although the interesting stuff generally doesn't really happen unless the rules are consistent / satisfiable with respect to one another.

The fact that mathematics happen to be useful in describing reality doesn't imply that reality is fundamentally mathematical, except in the sense that reality does something like follow a consistent set of rules, and there may be a deep isomorphy between all sets of self-consistent rules - but it's also entirely possible that the mathematics we invent, we invent for a reason, and those reasons have an inherent relationship with the rules the universe itself follows.  Personally I lean towards "deep isomorphy", which would imply that there's only one "mathematical universe".

Let's consider inconsistent mathematics for a moment, however, because I will observe that your paradox does not depend upon the rules of the "mathematical universe" actually being consistent / satisfiable - the paradox doesn't depend on the idea that the universe being described can exist, only on the idea that an entity described by the mathematical framework can be aware of its own existence.  Suppose for a moment that there exists a mathematical system with one or more contradictions, which is still capable of "running" a dynamical system for a given set of limited parameters such as to give rise to an entity in that system which is "aware of its existence" (suppose for the sake of argument that the contradictions do not present a problem for that limited set of parameters, which includes a finite extent of time).  Does that entity "exist" for your purposes?

Note, as you consider this question, that one of the central claims of a religion which focuses largely on examining oneself is that you do not, in fact, exist.  Granted I think this claim is misleading, and I'd say the proper claim is something more like "The you that you think of as yourself is more like a mental image of yourself and is wholly imaginary", except that is also misleading, and "You don't exist" is actually somewhat closer to the true claim being made.  However, I think it is particularly applicable here, because the entity that makes the claim "Cogito ergo sum" is, in a particular sense, not actually real; or at least is real in the same sense that a mathematical entity which is examined to see whether or not it thinks it exists, which itself is real in the same sense as these words are real.

Are these words real?  They're embedded in physical hardware somewhere.  The act of running the dynamical system to see whether or not an entity thinks it exists, is also the act of embedding the entity being examined in physical hardware somewhere.  Unrolling the function to see whether or not an entity thinks it exists is equivalent to making that entity exist.

What if we could prove such an entity exists in a mathematical framework, without instantiating the specific entity by actually running the function?  Well, I suspect such a proof is impossible, but supposing it isn't; does that entity actually think that it exists?  It would have to exist in order to do so, no?  This no longer seems particularly paradoxical; I would analogize to a human being whose existence is contingent on my having sex with a particular person at a particular time.  They would think they exist, if they come to exist; this doesn't imply existence.

But supposing the paradox is still unresolved, I'd add the following considerations: Is the proof that such an entity would exist, if the function were run, itself a proof that the entity exists regardless of whether or not the function is run?  Does the proof cause the entity to exist, or does it exist regardless of whether or not any such proof is attempted?  (Do entities exist in the infinite possible encodings of the digits of irrational numbers?)

The physicist Arthur Eddington worked on a theory that would predict all of physics from mathematical principles. I think not much attention is paid to it these days.

Taking a shot at this, I think your mistake is that you think in terms of mathematical properties of a universe, rather than mathematical properties period. For example, I think the existence of something like a prime number is not a property of a universe but a logical truth that couldn't be false in any universe. More generally, I would say the same about any sentence in a formal language.

There may also be mathematical properties that are universe-specific (the best candidates here are natural constants), but the extent to which these exist is questionable -- perhaps understanding general math more would reveal that these properties are logically necessary. Even if not, you may still get far by only pondering non-universe-specific math.

To turn this into a suggestion, I think you should view mathematical insights as being independent of the universe they are made in, and independent of the agent who makes them. Then, if it is possible to argue from pure mathematics that only one universe makes sense, that has to be the one. Thus, I accept (1) and (2) but reject (3), and in particular, the sentence

we cannot see, purely from a mathematical description of a theory, whether that theory describes reality or not

I have high error bars around how credible LW will think this is, but personally, I think this video presents the most valuable ideas on the subject that I've ever heard, and I'd probably describe it as figuring out whether a theory is true purely from the description of the theory. (I've sadly seen similar ideas presented elsewhere in combination with extremely sloppy epistemology.) The idea is roughly that "nothing exists" is, in fact, as true as it is logically possible to be; our intuitions about what "nothing" means are just wrong.

Are you familiar with neutral monism?

but there seems to be a very strong similarity between the paradoxes of existence and consciousness as I've come to think about them

I'll admit I'm annoyed to see this because I'm working on a blog post that makes exactly this point. Now I feel unoriginal 😛. Below is the connection I see between the "paradoxes," copy-pasted from my notes.

The physical world, with nothing extra thrown in, necessitates that people will claim certain aspects of their experience to be non-material, and it gives a satisfying, purely physical explanation for why they do. 

The math, with nothing extra thrown in, necessitates that people will claim to be physically real, and it gives a satisfying, purely mathematical explanation for why they do.

Interesting paradox. 

As other commented, I see multiple flaws:

1. We believe to seem to know that there is a reality that exists. I doubt we can conceive reality, but only a vague understanding of it. Moreover we have no experience of "not existing", so it's hard to argue that we have a strong grasp on deeply understanding that there is a reality that exists.
2. Biggest issue is here imho  (this is a very common misunderstanding): math is just a tool which we use to describe our universe, it is not (unless you take some approach like the mathematical universe) our universe. The fact that it works well is selection bias. We use math that works well to describe our universe, we discard the rest (see e.g. negative solution to the equation of motion in newtonian mechanics). Math by itself is infinite, we just use a small subset to describe our universe.  Also we take insipiration from our universe to build math. 

I might have missed the point of this paradox, so here some points :

1. Is basically Descartes "cogito ergo sum". The observation of the tought process makes that we exists, even if in another form as what we perceived as real.
2. This point confuse me a bit. If we perceive something there must be rules that govern this perceptions, make it non chaotic to be perceived with order, granted. But those rules will only apply to this anthropocentric dimension of reality. What make it so that to survive we should be able to map all layers of reality ? like a shadow which, as a projection of a volumetric form, loose information to be mapped on its original dimension. Projections being non bijective by nature.
3. So why cannot a Larger Universe (unknown to our perceptions), could not include this one which only map the territory of human life organisation ? Mathematics are but a tool of our perceptions, like our eyes or our skin, limited to the broadband allowed to us by Nature.

So if I understand correctly a sufficiently complex mathematical equation describing a world with mermaids and unicorns and magical laws automatically becomes real if its simulated inhabitants believe it's real.

It seems to be the case that this reality is perfectly mathematically describable

Theres still a map territory distinction. However perfect the map, that is not a reason to think it is the same thing as the territory, or belongs in the same ontological category as the territory.

But without the premise that the territory is maths, the rest of the paradox doesn't follow.

Hi.  My comment is kind of long, so sorry about that up front.  

1. In regard to 
   
   “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”…this implies that the existence of our universe is in principle a mathematical implication of the fundamental theory that describes our universe.”
   
   The state of our minds and the existence of our universe doesn’t follow from a mathematical theory.  It follows from the reality that the mathematical theory describes.

2. In the first paragraph of the discussion, 

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.

mathematical universes exist in the mind of the person thinking about them in this universe.  Mathematical universes, or any possible constructs, that exist in the mind are different existent entities than what exists in actuality. 

3. I think the hard problem of consciousness is only hard because people seem to believe that thoughts and feelings are intangible, non-physical things.   If they exist in the brain, then the problem of consciousness, or what I think is more important self-awareness, isn’t as hard as we make it out to be.  It’s more just a biochemical, neuronal connection, engineering problem.

4. Suggestions that mathematical constructs, logic, etc. exist independent of the physical universe in some Platonic realm that no one can see, point out or experimentally demonstrate is not much different than religious faith to me.  It’s possible, but no one can really provide physical evidence for it.  It seems like creating a whole new Platonic realm to explain in addition to our universe makes things much harder than they need to be.

5. In regard to the main idea of "Why is there something rather than nothing?", others have suggested that the seeming insolubility of the question may not be because it's insoluble, but because there's a misunderstanding or a false assumption in the question.  I agree with this.  In the below, I'd like to challenge the assumption that what we usually consider to be "nothing" is in fact "nothing" and not an existent entity, or a "something".

Before beginning, it's very important to distinguish between the mind's conception of "nothing" and "nothing" itself, in which the mind would not be there.  When I use the term "nothing", I'm talking about "nothing" itself.  While one can't visualize this directly, it's important to try and get close and then extrapolate to what it might be like if the mind were not there.

    I think that to ever get a satisfying answer to the question "Why is there something rather than nothing?", we're going to have to address the possibility that there could have been "nothing", but now there is "something".  If this supposed "nothing” before the "something" was truly the lack of all existent entities, there would be no mechanism present to change, or transform, this “nothingness” into the “something” that is here now. But, because we can see that “something” is here now, the only possible choice if we start with "nothing" is that the supposed “nothing” we were thinking of was not in fact the lack of all existent entities, or absolute “nothing” but was in fact a "something".  Another way to say this is that if you start with a 0 (e.g., "nothing") and end up with a 1 (e.g., "something"), you can't do this unless somehow the 0 isn't really a 0 but is actually a 1 in disguise, even though it looks like 0 on the surface.  That is, in one way of thinking "nothing" just looks like "nothing".  But, if we think about "nothing" in a different way, we can see through its disguise and see that it's a "something". This then gets back around to the idea that "something" has always been here except now there's a reason why: because even what we think of as "nothing" is a "something".

    How can "nothing" be a "something"?  I think it's first important to try and figure out why any “normal” thing (like a book, or a set) can exist and be a “something”. I propose that a thing exists if it is a grouping.  A grouping ties stuff together into a unit whole and, in so doing, defines what is contained within that new unit whole.  This grouping together of what is contained within provides a surface, or boundary, that defines what is contained within, that we can see and touch as the surface of the thing and that gives "substance" and existence to the thing as a new unit whole that's a different existent entity than any components contained within considered individually.  This leads to the idea that a thing only exists where and when the grouping exists.  For instance, groupings can exist inside a person's mind or outside the mind.  For outside-the-mind groupings, like a book, the grouping is physically present and visually seen as an edge, boundary, or enclosing surface that defines this unit whole/existent entity. For inside-the-mind groupings, like the concept of a car (also, fictional characters like Sherlock Holmes, etc.), the grouping may be better thought of as the top-level label the mind gives to the mental construct that groups together other constructs into a new unit whole (i.e., the mental construct labeled “car” groups together the constructs of engine, car chassis, tires, use for transportation, etc.).  This idea of a unit whole or a unity as being related to why things exist isn't new, but I think its application in providing a reason why what has traditionally thought of as “nothing” is actually an existent entity

    Next, when you get rid of all matter, energy, space/volume, time, abstract concepts, laws or constructs of physics/math/logic, possible worlds/possibilities, properties, consciousness, and finally minds, including the mind of the person trying to imagine this supposed lack of all, we think that this is the lack of all existent entities, or "absolute nothing" But, once everything is gone and the mind is gone, this situation, this "absolute nothing", would, by its very nature, define the situation completely. This "nothing" would be it; it would be the all. It would be the entirety, or whole amount, of all that is present. Is there anything else besides that "absolute nothing"? No. It is "nothing", and it is the all. An entirety/defined completely/whole amount/"the all" is a grouping, which means that the situation we previously considered to be "absolute nothing" is itself an existent entity. It's only once all things, including all minds, are gone does “nothing” become "the all" and a new unit whole that we can then, after the fact, see from the outside as a whole unit. One might object and say that being a grouping is a property so how can it be there in "nothing"? The answer is that the property of being a grouping (e.g., the all grouping) only appears after all else, including all properties and the mind of the person trying to imagine this, is gone. In other words, the very lack of all existent entities is itself what allows this new property of being the all grouping to appear.

Some important points are:

1. The words "was" (i.e., "was nothing") and "then"/"now" (i.e., "then something") in the above imply a temporal change, time would not exist until there was "something", so I don't use these words in a time sense. Instead, I suggest that the two different words, “nothing” and “something”, describe the same situation (e.g., "the lack of all"), and that the human mind can view the switching between the two different words, or ways of visualizing "the lack of all", as a temporal change from "was" to "now".

2. Because  the mind's conception of "nothing" and "nothing" itself are two different things, our talking about "nothing" itself (which is derived from the mind's conception of "nothing") doesn't reify "nothing" itself.  Our talking about it has nothing to do with whether or not "nothing" itself exists or not.

3. It's very important to distinguish between the mind's conception of "nothing" and "nothing" itself, in which no minds would be there. These are two different things. Logically, this is indisputable. In visualizing "nothing" one has to try to imagine what it's like when no minds are there.  Of course, this is impossible, but we can try to extrapolate.

4. We think about and visualize “nothing” in our minds, which exist and are “somethings”.  In our existent minds, “nothing” just looks like nothing, but from this, we cannot then just assume that “nothing” itself, in which our minds are gone, would not then gain the property of being the all grouping and thus be a “something”.  I believe this unfounded assumptions is what has held us back from solving the question “Why is there something rather than nothing?”.

    If you've read this far, thanks for wading through my probably excess wording!