This will probably be one of my weirdest posts
and not sure whether to post it because it might be complete idiocy, but what the hell.
So taking the ASA definition of personality:
Personality refers to individual differences in characteristic patterns of thinking, feeling and behaving. The study of personality focuses on two broad areas: One is understanding individual differences in particular personality characteristics, such as sociability or irritability. The other is understanding how the various parts of a person come together as a whole.
Evidently if someone has a mental illness it will affect their personality. Removing their mental illness will affect their personality.
Lets just speculate wildly and suggest that a specific individuals anxiety is caused by a structural variation in their brains. This will affect pretty much all aspects of their personality, and we can't really say that their brain physiology isn't 'them'. How their brain is built is the very core of what makes them themselves. If you were to alter the brain and remove the anxiety you have a completely different person. Its like changing someones genes to create the same effect.
So the answer to your question imo, is going to be investigating how much of mental illness is a biological phenomenon. I would look into:
1)Biological evidence for mental illnesses
2)What happens to the brain when therapies are successful (i.e. does a therapeutic improvement correlate with a change in the brain in some way).
So, if you can 'remove' a mental illness without physically altering the brain in any way, then mental illness can
be a separate thing to 'who they are'. If you can't then mental illness is core to the individual.
From what I know about mental illness, the latter seems almost certainly true. It would be pretty surprising to me if a mental illness could be removed without the brain being physiologically altered in some way.
(what follows might sound completely insane btw, its just how I try to understand it in my own mind).
So imagine a brain is a mass of connections between various points in three dimensions. An individual thinking is those connections flipping around and joining other connections (this is probably an awful way to try to picture it, but its the best I can come up with). What is a mental illness? is it a variation in the physical properties of the connections (their length, their number, their potential connections, their material), or is it simply a result of which connections connect to which connections? (is it just 'thoughts'?)
What happens when an individual is conditioned by their environment? are we talking about certain connections becoming a preference? or does a structural change force
the brain into new connections (and ways of thinking)?
I personally can't believe this to be 'just thoughts'. Something structural must happen during conditioning that makes the thoughts and feelings change, otherwise the person could just immediately alter their thinking. Depression is forced
upon the individual, as is anxiety, so whether you have the change brought about by genetics (the structure is different to begin with) or environment (the structure is modified) the result is that a mental illness has to have some corresponding physiological alteration, in some way.
If you make a physical alteration to a brain, you have a different person (imo).