Looking for clarification on superposition I have always had a hard time accepting the concept of superposition from quantum mechanics. I know that the leading physicists say that the cat is both alive and dead until it is observed and that an electron is in multiple places at once (is a wave) until it is observed, but I sadly have never been convinced. I've taken a basic course on the subject and watched countless documentaries that explain this, but I have a hard time accepting that we can change the state of the world just by observing. Questions that have popped into my head are things like:


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*What if a brain dead person looks and doesn't comprehend?

*What if I take a picture and don't look at that picture for years later? Is the cat both alive and dead for years all because I didn't look at the picture?

*What if a friend and I flip a coin, and only I look? When asked which came up, I would say "Heads" and he would say "Both". Are we both right? Doesn't my absolute knowledge make him wrong?

*It seems like the trend in science has always been to show us how small and irreverent we are. From finding out that we are not at the center the universe, to finding out that we are just one planet of many, in one solar system of many, in one galaxy of many, and possibly even in one universe of many, to finding out that all life on earth came from the same place (making us less special then we thought), to finding out that our genetic code is filled with genetic bloat and the little left over is nearly identical to a banana. After all that, I find out that I am in fact so special that all matter in the universe ceases to exist when I close my eyes!


Currently I am one of those people that think that moon is there even when I'm not looking. Can someone point me a convincing argument for all this so that I can finally be convinced?
EDIT:
What I gathered from the replies so far is that superposition is a way of representing our uncertainty of an answer. By saying that a cat is in a state between life and death, we are actually saying that we don't know in a mathematically describable way. This would imply that our observation does not change reality, it only changes our certainty.
But what about electrons acting like waves (going through 2 slits at once) when we're not observing and like particles (being visible electrons) when we are observing in the double slit experiment? This seems to show that they are both in reality and indicates that our observation does change reality (which of the two it is at the moment).
This is the contradiction that I struggle with. (And if I am wrong about my interpenetration, please correct me.)
RE-EDIT:
I think I'm understanding this now. The only remaining issue is this:
The electrons in the double slit experiment that act like waves before we look and like particles after we look are exposed to the environment (gas, photons, etc.) the whole time. Shouldn't it be one or the other the whole time due to this exposure?
 A: I would say that you are suffering from a little bit of pop-science fatigue. You don't need to be convinced of these things because the great majority of working physicists don't think about quantum mechanics in this way.

  
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*What if a brain dead person looks and doesn't comprehend?
  

This is called "begging the question", I think. Except for a handful of people who have (in my opinion) gone off the rails, no working quantum physicist is going to tell you that our consciousness has anything to do with the results of measurements. The "collapse of the wave function" or whatever you want to call it is not a consequence of us consciously changing things. It is rather a consequence of the interaction/correlation between the measurement device and the system being measured. (Unless you are a quantum Bayesian and think that a quantum mechanical state is a person's state of knowledge, but then quantum mechanics is a theory about how we gain information about the universe rather than a description of how the universe works, and even within that subjective framework (which, let's be clear, is a minority opinion about the foundations of quantum mechanics), no one is suggesting that the lack of understanding about an object somehow changes the properties of those objects!


  
*What if I take a picture and don't look at that picture for years later? Is the cat both alive and dead for years all because I didn't look at the picture?
  

No. You don't know whether the cat is alive or dead, but the cat is either dead or alive, because it is a macroscopic object. As far as quantum mechanics goes, we don't say, "the electron is in two places at once." We say that it is in a superposition of different quantum states and that we cannot predict the outcome of a measurement made on that electron. Edit: and to be clear, the superposition of two quantum states is itself a quantum state, so it is never the case that anything is in two states at once.


  
*What if a friend and I flip a coin, and only I look? When asked which came up, I would say "Heads" and he would say "Both". Are we both right? Doesn't my absolute knowledge make him wrong?
  

Hopefully, your friend says "I don't know", not "Both", because "Both" is an absurd answer. Once again, superposition is not (in my opinion) about what-the-electron-is-doing-right-now, it's about the-results-of-measurements-are-unpredictable. (This is of course, not quite right, because the foundations of quantum mechanics are yet to be well understood, if that's even possible. I'm just trying to correct some of the misconceptions you are displaying.) Anyway, you don't have absolute knowledge that the coin is Heads; your brain could be lying to you.


  
*It seems like the trend in science has always been to show us how small and irreverent we are. From finding out that we are not at the center the universe, to finding out that we are just one planet of many, in one solar system of many, in one galaxy of many, and possibly even in one universe of many, to finding out that all life on earth came from the same place (making us less special then we thought), to finding out that our genetic code is filled with genetic bloat and the little left over is nearly identical to a banana. After all that, I find out that I am in fact so special that all matter in the universe ceases to exist when I close my eyes!
  

First of all, science says nothing about how small and irrelevant we are. We just happen to be a species that values our uniqueness for some reason, and we're not as unique as we think we are. Who cares that we're not at the center of the universe?
But as far as your last statement: NO. Physicists don't think that the universe ceases to exist if you're not observing it. This is some strange kind of pop bastardization of the principles of quantum mechanics that has somehow made it out into the world, due to (1) scientists not doing a good job of explaining what they're doing (which is hard, and they're not trained to do this kind of communication), (2) people not realizing that scientists often use analogies to get their point across and therefore confusing the analogy with the reality, and (3) people using the language of quantum mechanics to justify their own pet beliefs about things.
A: I think the fundamental misunderstanding of superposition has a lot to do with the popular interpretation of quantum mechanics. That is, how Schrödinger's cat is portrayed in popular science. When a quantum system is in a state of superposition, it means that the outcome of a measurement of some property of that system is uncertain. The wacky thing about quantum mechanics is that it behaves as if it were in both states at the same time. However, this is only true while the system is still coherent. What the popular science explanation calls "observations" has nothing to do with consciousness or an observer making a measurement. It has to do with a phenomenon called quantum decoherence. Basically, the system exists in a state of superposition until that systems interacts with the environment in a thermodynamically irreversible way. This interaction is what causes the wave function to collapse and Schrödinger's cat to be dead or alive. That interaction could be an electron striking a detector, or or could be random neutrinos passings through the system, or a photon interacting with it, or just about anything. This is why it's so hard to build quantum computer; you have to ensure that the system doesn't interact with the environment in any way to ensure that is stays coherent (in a state of superposition). So to answer your questions:
1. What if a brain dead person looks at it and doesn't comprehend? Observation by a human consciousness is not a requirement of decoherence. The wavefunction was already collapsed by the photons hitting it that allowed the brain dead person to see.
2. What if I take a picture and don't look at that picture for years later? Is the cat both alive and dead for years all because I didn't look at the picture? Same as above, the wavefunction collapsed into a determined state the moment that it interacts with photons of light.
3. What if a friend and I flip a coin, and only I look? When asked which came up, I would say "Heads" and he would say "Both". Are we both right? Doesn't my absolute knowledge make him wrong? First thing, a coin flip is not an example of quantum superposition. Let's suppose you "flipped" and electron so that it lands in either a spin up or spin down state, with 50% probability. When you make a measurement of the spin, the wavefunction collapses due to the interaction with whatever tool you are using to make the measurment. Then you could say "up" or "down" but your friend would say "I don't know". Your friend could then measure the spin of the same electron and he would get the same answer that you got. Once the wavefunction collapses the spin is no longer in a state of superposition. There isn't any way you could get "both up and down" with a single measurement.
4. Does matter continue to exist after you close your eyes? Yes. Again, having a human observer is not a requirement to having the wavefunction collapse. Also, quantum effects are generally not observable at the macroscopic level (with a few exceptions). Don't ask this questions of any philosophers, though.
The whole point of the Schrödinger's cat thought experiment was to show how absurd quantum effects would be on a macroscopic scale. To make it actually work in practice, the cat in the box would have to be completely isolated from the outside environment, or else the wavefunction would not stay in a state of superposition.
A: Just regarding question number 3:  
1)  If your friend knows anything at all about quantum mechanics, he will not say "Both", because quantum mechanics does not allow anything to be in two states at once, ever.  
2)  If your friend knows a small amount about quantum mechanics, he might make the mistake of saying "Neither", because, after all, most coins (at least most very very small coins) are in fact in neither of these two states (out of uncountably many other possibilities).  
3)  But if your friends really understands quantum mechanics, he will probably say "One or the other, but I don't know which", provided of course that he's aware that you've made your measurement, which changes the state of the coin.  (And provided that the state of the coin has not had an opportunity to evolve further since you've made your measurement --- in which case he will say "One of X or Y, but I don't know which", where X and Y are the time-evolutions of "Heads" and "Tails".)  (Assuming, of course, that he knows the relevant Hamiltonian and can calculate.)  
(Alternatively, he might take the view that you and the coin are in an entangled state.  But once again, you are in one and only one entangled state. )
Regarding Question 2 (and the general tone of all your questions):  It is (quite obviously, I should think) not possible for a cat to be both alive and dead.  It is possible, in principle, for it to be in one of a great many states other than alive and dead (e.g. 2*Live+3*Dead, or (3i+1)Alive-(6i)Dead), but it can only be in one state at a time.  There is nothing more fundamental to quantum mechanics than that. 
A: The process of "collapse" can almost entirely be handled just by including your measurement apparatus in a quantum description. It's the quantum interaction between observer and observed that causes collapse. For the mathematics of this process, you might want to look at my answer to Particle interactions which are NOT considered observations? for more on this. Now, even if your eyes are closed, everything is constantly interacting with everything else (gas molecules bumping, radiation, sound, you name it) so if you know the state of the coin (that is, if the quantum state of the coin has decohered and is now bound up with the state of the environment), then your friend "knows" too, because he's interacted with the coin - if just through gas molecules.  So if this quantum stuff is just a "regular" interaction, what's the big deal?
You've seen what happens to an electron. It causes real interference effects with itself, and therefore passes through both slits at once. 
You've (hopefully) seen what happens to a silver atom when passed through a Stern-Gerlach apparatus. Whereas a classical macroscopic magnet can have a magnetic moment in any direction, a silver atom will only have a magnetic moment along the direction you measure, of $+$ some unit or $-$ that unit. Weird, but maybe we're just not smart enough to figure out the completely classical rules that govern the silver atom. 
Hopefully you've heard (quote from wikipedia) "Feynman was fond of saying that all of quantum mechanics can be gleaned from carefully thinking through the implications of this single experiment." But maybe you're a tougher sell than that and want to hang on to classical mechanics for dear life.
There is more concrete evidence but it can be a long story. I recommend the treatment of Bell's inequality here: 


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*A great exposition of the problems involved by veritasium on youtube: https://www.youtube.com/watch?v=ZuvK-od647c

*A lecture, pretty advanced, by Sidney Coleman http://media.physics.harvard.edu/video/?id=SidneyColeman_QMIYF who illustrates clearly: If you're a curmudgeon who wants to use classical mechanics and hates quantum mechanics, you must violate the finite speed of light (local realism). This in turn violates either the principle of relativity (that there is no preferred reference frame) or causality (that events in the future follow from events in the past). Take your pick.

*a rigorous but advanced exposition on Bell's inequalities http://physics.ucsd.edu/~mcgreevy/w15/130C-2015-chapter01.pdf Importantly, the idea is: If you assume that the particle has both a position and momentum value simultaneously, you just don't know it, then you can run into contradictions. (Unless, of course, you use your curmudgeon relativity-violating theory).
A: At the moment, one observe a state, not the state does collapse, but it's our knowledge about this state, that collapses. This is right for cats in boxes with poison watches.
An other thing is, if you have to do with superposition of quantum states. In a Bose-Einstein condensate all involved atoms are in a superosition and a powerful enouth observation of an atom destroy the superposition of this atom in the BEC.
Edit
You edited your question and pointed out, that if one observe an electron, it acts like a particle. If they are in some special interaction (with slits {or one slit or an single edge}), the secondary observation of the fringes shows a wave like distribution.
To conclude from this stationary distribution about incoming waves has some weakness. For water waves, if one shot an image on this waves behind a double slit, there will be a intensity distribution with doubled amplitudes in some points and between points with wave extinction. But if one shot more images in an aperiodical manner, all the minimum and maximum points will be in different positions, they move. 

Comparing this behavior for water waves with electrons or photons, what we have to conclude?
I want to give a second answer (the first one would be, that in the case of electrons and photons the intensity patterns have to move too). If all the waves start from the same point, and this point is the very sharp edge (don't matter, is this a single edge or a slit or multi slits), then we see a stationary distribution. To get such a result we have to consider the interaction between the particles and the edge. Edges contain surface electrons with their electrical potential and sharp edges have a high potential. Electrons have an electric field too of course and photons have a oscillating electric field. It is a fact, that the fringes from electrons behind a slit widen the geometrical shadow, when for photons some fringes lay inside the geometrical shadow. Think a little bit longer about this fact.
The interaction of the electric fields of flying particles and edges is quantized and this is the reason for the intensity distribution behind edges.
