Two other people share this checkerboard with you.
One claims there is a helium atom somewhere in your checkerboard world.
The other claims there is not.
Who is right? How do we determine it?
Let's assume you have a helium detector, which can detect a single atom of helium within one checkerboard square.
You could wander around checking a few squares at random. A traditional checkerboard consists of 8x8, or 64 squares, so in each square, you have a 1 in 64 (1.6%) chance of finding the helium atom - if it in fact exists. If there are two helium atoms, your odds are even better. Twice as good in fact.
Now if the helium atom is stationary, by the time you have checked 32 squares you have a 50% chance of finding the atom. But until you have checked EVERY square, you can't be certain that there is no helium. In other words, it is actually slightly easier to prove it does exist than to prove that it doesn't. To prove it exists, you need only search until you have found one instance. To prove that none exist, you must search every square.
But just testing each square is not sufficient either. Unless the helium atom is stationary. Let's say we set up the following pattern to check every square, by traversing side to side as we move forward across the board. It would be very easy for a moving helium atom to slip past on the left side of the board while we are over on the right side.
There are a few ways I can overcome this:
1) With enough people and sensors, You can place one sensor in each row, side by side, and they can move down the rows in unison, effectively creating a net to ensure the helium doesn't slip past.
2) You could use a net that stretches from one end to the other, and drag it across the board until I have left a single row, then I can traverse that row with my sensor.
3) You could design a sensor which will test an entire row at one time, then move across the rows from one end to the other.
4) You could design a sensor or sensor web which tests every square at once.
Now, what if instead of a 2 dimension checkerboard we are talking about a three dimension cube? Now instead of 8x8=64 squares to test, you have 8x8x8=512. Probability of finding the atom in one cube is 0.2%.
To apply the strategy in 1 you would need enough people and sensors for each row and column, you have gone from needing 8 of each to 64 of each.
To apply strategy 2, you would first need to drop a net from the ceiling to reduce the search area to an 8x8x1 space, then drag the end-to-end net across to get to the single 8x1x1 column necessary.
Strategy 3 would require the same 64 sensors as strategy 1
And strategy 4 would require 512 sensors.
What of the cube is significantly bigger? 1000 x 1000 x 1000? That would be 1 billion cubes!
What if it is much, much bigger? Say 91 billion light years across(the distance across the observable universe)? There aren't enough people, enough sensors capable of creating the necessary web, nor sensors capable of spanning that mind-mindbogglingly huge distance.
And if you have never actually seen a helium atom - know nothing of it's physical makeup - how can you be certain your helium detector actually detects helium?
The same logic applies to god(/s) detection. We have an enormous space, and insufficient resources to scan said space, for a being (or beings) of unknown composition. In fact, one could argue it is even worse, as we would also need to consider the space outside of observable space. Or the possibility that said being(/s) exist in an as yet undefined, unobserved dimension (yes, I know it sounds pretty sci-fi, but then so did spaceships and robots and glow in the dark sheep once upon a time).
Therefore, anyone who declares that there is a deity or deities, is doing so based on personal belief, and not by scientific analysis.
Any anyone who declares that there is no god is doing so based on personal belief and not by scientific analysis.
We simply do not have the means to prove or disprove the existence of deity through pure scientific, mathematical process.
What is left then is speculation based on observation and interpretation of available evidence.
More on that later.