What if, to some extent, we could prove the existence of life after death through the implementation of some basic mathematical principles?   

Let's be straight; this question has some pretty serious implications.  But have you ever really thought about it in a logical manner?  George Shaffner presents some pretty compelling arguments in his book, The Arithmetic of Life.  I'll summarize them here:

1.  First off,  there is a concept called the "Rule of Always More."  What this means is that science has consistently proven, without exception, that there is always more beyond what we can detect or measure -- there are always more dimensions of complexity and scale.  Every time we make an advancement in science like a new powerful telescope or particle accelerator, more is discovered.  We are continuously discovering new species.  We have learned that matter is not what we thought and that there is something called dark matter.  Progressive math points to as many as 10 dimensions of existence.  And so, we are aware that there is always more to be discovered.

Can mankind be the one exception to this rule?  The 'rule of always more' maintains that there are always more levels of complexity and scale than we understand at present.  Considering that there are an infinite amount of scenarios in nature where the 'rule of always more' can be applied, we can safely say that the likelihood of mankind not having more levels of complexity and scale than we understand is equal to one divided by infinity.  This is as close to zero as you can get, without ever getting there.  While this does not prove life after death, it does prove that there is more to mankind than meets the eye -- and much more than we understand.

2.  The Law of Conservation states that energy can not be created (made from nothing), or destroyed (made to disappear to no-where) and that energy can be changed from one form to another (such as electrical energy into heat energy). 

As death represents a change of state, then all of our energy must be maintained or changed to a new form.  This includes all aspects of us -- what we can see, and what we can't see.  So, it can be assumed that our personality, which is something that we cannot see, is also maintained (from an energy standpoint) or transformed.  While this does not prove that our personality survives death, it does say that it must go somewhere, and that it continues to exist....in some form, or another.  At the very least, the law of conservation proves that our components (physical & non-physical) continue to exist in some form or another.  Why?  Because EVERYTHING is energy, and energy cannot be created or destroyed.

3.  But let's say that our identity, or personality is not maintained after death.  Then this would mean that every sighting of every apparition ever recorded in history (countless, no doubt) was either a lie, or a delusion.  Again, the odds of every single person being wrong or lying, without exception, are astronomical.  Say that there have been a billion sightings, and only 10 (.000001 %) were accurate.  Well then, that is still 10 times as many sightings as you would need to prove the existence of life after death.  It only takes one.

4.  In the same manner, let's count up every case of a past life that has ever been recorded through hypnosis, many of which reveal unexplainable historical facts that seem to prove their validity, by themselves.  Again, these cases are endlessly documented.  Can none of this information actually be stored in the brain from an earlier time?  And if it is, where does it come from?  Is it all spontaneously made up on the spot -- in every single instance?

Likewise, let's add up all of the cases of the near-death-experience -- of people who have nearly died and then returned with consistent stories of seeing light, loved ones, or other entities, on the other side.  Again, countless instances of similar stories.  In all of these cases, every single person would have to be lying or duped by some biological effect of the brain.  It is possible that there is some biological explanation, however, that seems rather wasteful (if there is nothing beyond death) -- and we have seen that nature wastes nothing. 

On the whole, when you add all of these cases together, the odds of every single person being mistaken or lying (without exception) are pretty damn low.   Because it only takes one person to be correct to prove the existence of life after death.

5.  And finally -- the big bang theory.  For a number of years, scientists were disturbed at the fact that mathematically (per Newton's theories) there did not appear to be enough matter in the universe to have allowed galaxies to form.  How is this possible?

This meant one of three things:

1.  That Newton was incorrect (despite 300 years of supportive evidence). 

2.  The Milky Way never really existed.

3.  That there was more matter out in the universe than scientists were able to detect.

The scientists chose option three, and created the concept of 'Dark Matter,' which cannot be seen, but is believed to compose 90% of the mass of the universe.   Yup, 90%.  In recent years, dark matter has been proven to be real, although we cannot see it.  Look it up.  Again, the 'rule of always more' comes into effect. 

When you combine all of these things, the evidence is pretty compelling.  We can reasonably assume that there is life after death.  Keep in mind that there are many more mathematical arguments that can be made as well.  This is just a nice overview of some ways of looking at the evidence. 

Does this make you view things a bit differently? 

To quote Shaffner in The Arithmetic of Life,

"Either our personalities are the only known exception to Nature's Law of Conservation, or there is life after death.  Either every past-life experience in the history of man has been an invention, or there is life after death.   Either every life-after-life experience has been an illusion, or there is life after death.  Either every paranormal experience has been a delusion, or there is life after death.  Either every religion has been a concoction, Jesus did not return from the dead, and the Dali Lama has never come back at all, much less thirteen times, or there is life after death.  Either mankind is the only known exception to the "Rule of Always More," or there is life after death."