We know how they look. They're not visible to the naked eye. We have a model (the Electron Cloud model) for how they are structured that is scientifically consistent.
That is the whole problem I am trying to make you aware of. They are scientifically consistent with observations we have made, which means that they are based those observations and not the true nature of the atom itself. Even if you could see it, you are still seeing our perception of it, and not the actual nature of the atom itself.
We know exactly what atoms are composed of. Protons, neutrons, and electrons; the precise number of each dictating what element they are and how they bond with other atoms.
This is a very rudimentary understanding of atoms that is taught to us in elementary school. Atoms are indeed composed of subatomic particles, protons, neutrons, and electrons. Those particles though are not the smallest subdivision of matter, there are smaller particles that compose those. http://en.wikipedia.org/wiki/Subatomic_particle if you feel like enriching yourself further.
"Why they exist"? They are literally everything. They can't not exist.
Or they could not exist, and then there would just be nothing. No good reason for this not to be the case, unless of course you believe in a God that created everything.
We have huge fields of study in them because they exist and can be researched. If we were just guessing, we couldn't find any facts based on them.
Anything based on induction can very well be likely to happen, but can never be considered truly fact. Did those observations actually happen? Yes, they did happen. That is fact, but trying to project those observations to future events can never be fact.
The idea that you can't be truly sure of anything that isn't visible is incredibly naive. Are we not sure that the atmosphere is made of various gases? Are we uncertain of where the drains in our sinks and toilets go? We need "blind faith" to prove to ourselves that whatever is behind us still exists when our head is turned?
Naive? I think your understanding of even the simplest induction is far more naive. Those things that are, such as the composition of the atmosphere can in fact be checked. And whatever is found can be considered fact, but only for that instance. You can't say that the composition of the atmosphere will always be the same simply because you observed it to be so on several other occasions. Its actually laughable that you would use the drains in our sinks as an example. Once again, you may observed it several times, but that never guarantees that it will always work that way. It simply proves that it worked that way in those observations, and it implies heavily that it will happen again; never guarantees.
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As for induction, they can generally be made slightly more specific in order to become provable. E.g. "All life depends on water" is induced, but "All life as we know it depends on water" is proven... Not having knowledge of everything doesn't make what we do know somehow untrue.
Do I need to repeat myself again? We are aware that every life form we have observed requires water. That is fact, but you cannot generalize all life forms based on these observations. Generalizing all life forms is induction, simply saying that all the life forms we have observed require water is not inductive reasoning, it is simply stating a fact.