At the core of
solipsism is the problem of other minds. Solipsism takes the stance that there is only one mind in the universe - your own. While there are no prominent
solipsists in the history of philosophy, it has been discussed often. Very few are willing to accept the
logical consequences of
solipsism - that of the fact that you are alone in the universe.
Solipsism takes two forms termed strong and weak. Strong solipsism says that the only thing in the universe is that of your mind. All the objects and constructs around it are images in your mind - there is no physical reality.
Weak solipsism takes the stance that there may be a physical reality that exists, however the only consciousness in the universe is mine. In essence, everything and everyone else is an empty shell.
How do you prove that another person has a mind?
With Descartes, he plunged through the method of doubt and doubted the existence of everything eventually getting to the well known phrase "I think therefore I am" and building from there to the existence of God. From the benevolent nature of God, he claimed that God would not deceive him as to the existence of things that seem to have minds but don't therefore there must be other minds. Some reject the 'God-bridge' from "I am alone in the universe" to "there are other minds" and solipsism still looms.
The weaker form is often coupled with skepticism that claim that the only thing that is known is the individual's mind and all other things (including other minds) are justifiably doubtable. Still, at its core, weak solipsism makes the same claim as strong solipsism and is just as flawed.
With Descartes the prospect of solipsism was thought to be plausible. The very claim "I am the only mind that exists" however causes solipsism to crumble. This statement requires the use of language - a shared idea. With the ability to think thoughts to be shared requires another entity to share it with. (see Private Language argument for more on this - a solipsist must speak a private language) Other approaches use the overwhelming burden of circumstantial evidence of other minds and if all of history involves "real people" or "characters in my history".
Solipsism will often claim that some creative genius is just a portion
of himself - a partition. By making this partition and sectioning his
conscious awareness from this other portion the solipsist now has created
two entities, both 'real' but without knowledge of the other's internal
states - another mind.
For an example of one possible "dialogue" with a solipsist, I encourage
you to read How the Sphere encouraged me in a Vision, chapter 20 from
Flatland: A Romance of Many Dimensions.
Don't try arguing with a solipsist - you will only waste your breath and be regarded as a figment of their imagination.
The person who doubts there is an external world does not need proof:
he needs a cure.
--Johann Fichte
http://www.cfh.ufsc.br/~mafkfil/solipsism.htm