It cannot be a complete coincidence that several outstanding logicians of the twentieth century found shelter in asylums at some point in their lives: Cantor, Zermelo, Gödel and Post are some.
Elsewhere, he hints that the rate of mental illness among logicians is much higher than the general population.
Do logicians suffer from mental illness at significantly higher rates than the population at large (or other similar groups, such as academics)? If so, is there an accepted explanation for this association?
I think it might be quite credible to imply that outstanding logicians would or could tend towards having serious mental health issues at some point in their lives.
A person's state of mind (and categorically, a person's sanity as well) can be considered to be 'floating points' - kept aloft moment to moment, and which can be broken down.
This brings into question some of the complex concepts of what sanity is and whether a person can be said to be entirely sane - as it is gradually being realized that people can experience 'flickers' of insanity (which do or do not materialize as events, but that's another story) which begins to ask whether these moments inflict a permanent change (and further tendency) in future mindsets.
It can also be said that everyone trends towards a breakdown much like entropy is guaranteed. To many minds, this goes hand in hand with an enhanced perception and definition of reality over time. Life gets more complex as we age and those with greater blissful ignorance often find themselves happier. Others grasp too tightly on some of the 'realities' of life, or have a greater need to do so, those who adventure onto concepts that cannot be grasped, or which are outstanding - many minds of these intelligent types lose cohesion at some point during their run.
In other ways, there are genetic considerations. For example, some presupposed (and credible) links between schizoidism and creativity.
Also stress-related considerations - work/life balance and finding happiness (the hierarchy of needs etc).
Finally, the link between intelligence and clinical depression, which touches neurochemically upon the previous couple of paragraphs.
To elaborate briefly (and crudely): The differences between minds are mostly composed of the pathways with varying and chemically variable levels of resistance. The electrostatic self pulses among this, and our wider consciousness is composed of regional 'constellations'. Some constellations receive more attention electrically through varied levels of chemicals squirted from the deeper brain. Some people develop over-extensive neuronal networks, in particular logicians, theoreticians and crazy folk, and the fundamental chemicals which allow us to be emotionally stable (or abundant) are quite frankly spread too thinly. Sometimes, a part of the brain has received so much attention and feedback that the 'neuronal map' is deviant towards it and the paths of least resistance become embedded - you become the result of your own feedback. As a result of this, people with more exposure to information coupled with the ability (or inherent) inability to process it, are more likely to have a problem (or more optimistically, a solution).
So basically what I'm saying is that a more intelligent type is more likely to have a problem. (Admittedly, this person is in a better position to see them. And if you look, you shall find). More accurately, these types are more likely to have more to juggle and find it harder to balance the weights and woes (and joys) of life.
There is indeed is a fine line between genius and madness - and often they pair up.