How can brain disorders tell us more about brain function?
The most fundamental reason is this: There are vastly more ways to be wrong than to be right.
For example, if I ask you, "what is 6 times 7?" there are literally infinitely many possible wrong answers you could give me, but only one right one: 42.
The human brain has been optimized by millions of years of evolution. In the vast majority of circumstances---particularly those similar to our ancestral environment on the African savannah 200,000 years ago---it gives the right answer.
But giving the right answer doesn't tell you how the underlying mechanics work; all it tells you is that they work... well. There are many different possible architectures that could yield that same correct response: both I and a calculator can give you the answer "42", but we work in radically different ways.
In order to understand the underlying mechanics of the brain, you need to see it work wrong. You need to watch it take shortcuts (heuristics, one of the main things I study in my own research), or see what happens when it gets damaged (neuropsychology, the subject of this question), unless of course you can actually look inside the brain and see what it is doing (neuroimaging, which due to the technology required is a field of study no older than I am).
Before we had brain-scanning equipment or clever experimental methods to detect heuristic judgments, we had people with mental disorders and brain injuries. Indeed, we've always had people with mental disorders and brain injuries, though our ability to treat them has improved dramatically over time.
Yet simply by observing what happens to people's behavior as a result of injuries to particular parts of the brain, we were able to learn a great deal about what those particular parts do.
Imagine if you didn't know what the various parts of a computer did, but you took a bunch of computers, and hit various pieces with sledgehammers. This would give you at least some sense of how computers work: The one with a smashed hard drive lost a bunch of data, so the hard drive must store data. The one with a smashed screen doesn't display things properly, but will still perform computations just fine; so the screen is involved in display but not computation. The one with a smashed processor doesn't do anything at all, so that must be a really important component.
This is basically what neuropsychology does, only with human brains.
To use a particularly famous example, Phineas Gage infamously took a railroad nail to the prefrontal cortex, and then became impulsive and aggressive; so the prefrontal cortex must be involved in inhibiting impulsive behaviors.
Other brain structures discovered this way include Broca's area and Wernicke's area, both discovered due to particular speech problems (aphasia) that people with damage to each area have. People with Broca's aphasia can understand language but have trouble speaking it, and often speak primarily in swear words because (for whatever reason) those are the last to go. People with Wenicke's aphasia can speak language fluently and eloquently, but can't understand it very well and often speak in utter nonsense. Thus, we know that Broca's area is involved in the production of speech and Wernicke's area is involved in the comprehension of speech.
The kind of knowledge you can get this way is limited, and studying more requires you to either get "lucky" (the patients don't feel so lucky) and find patients with particular types of brain damage to study, or else intentionally induce brain damage (usually in cats or monkeys). The former is improbable, the latter questionably ethical. As a result this was more of a way for us to find out some of the basics of brain function about a century ago before we discovered cognitive science methods and invented neuroimaging technology.
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