Perhaps the question ought to be: can something exist even if you cannot prove that it exists?
But what will suffice as proof?
We tend to limit ourselves scientific (i.e. physical) proof, since this is the only kind of proof which we can 'digest' with our human senses. The downside of this is that the only things which we will accept as proven are those which can be physically sensed - which may imply that anything that isn't physical doesn't exist!
Indeed, we even try to explain non-physical things as physical phenomena:
Emma_85 wrote:I'd say that ideas really exist. They [...] are really there in your brain as electrical signals.
How can such a thing be proven? Scientifically, it cannot be. While conducting any research concerning thoughts (or any other invisible experience which a person can undergo, e.g. emotion), the scientist has to take it on trust
that the subject of the study really is thinking when he/she claims to be! How do you know that the person is telling the truth? (How do you know if the person can even correctly identify the sensation of thinking in him/herself?)
Alternatively, the scientist could conduct the study on him/herself... but then the test would not be reliable (as in, nobody else could repeat it under the exact same conditions), meaning that findings could not be generalised to people at large.
Even supposing that the person really is
thinking, you still cannot determine the relationship between the thought and the brain activity. How do you know that the thought isn't some immaterial thing, and that the brain activity is a result
of the thought?
An increase in temperature causes the mercury in the thermometer to rise, but the expansion of the mercury is a result
of temperature - not temperature itself!