So, what is it like to be a genius at math? Ironically, it cannot be quantified, and mathematicians usually lack the qualitative/literary skill to express themselves mythsemantically. But this one is quite clear, and many of the skills he describes are generalizable to other fields, say, medical diagnosis. A skilled diagnostician can filter out all sorts of noise that may distract and deceive the novice, and hone in on subtle clues missed by the uninitiated.
Thus, "You are often confident that something is true long before you have an airtight proof for it." We call this "intuition," but that doesn't mean it isn't rooted in a kind of real perception (or perception of reality).
Rather, "you have a large catalogue of connections between concepts, and you can quickly intuit that if X were to be false, that would create tensions with other things you know to be true, so you are inclined to believe X is probably true to maintain the harmony of the conceptual space."
This is one reason why it can be frustrating to try to explain a higher or deeper truth to someone who isn't as intelligent or perceptive as you are. You may not be a particularly systematic thinker, so you may not even be able to articulate the implicit steps that led you to your conclusion. You could say that it's a "feeling," but it is much deeper than that.
For example, I always have a particular "feeling" when I read Schuon, but it's not merely an emotion. It's very hard to describe, because it is what it is and not something else. But it includes a kind of deep and expansive pneuma-cognitive satisfaction, almost comparable to how we are somehow satisfied by a musical piece. Why should music satisfy us at all, and what is being satisfied? There's something about the logic of the piece, as if everything about it is inevitable and complete.
When I read Schuon, there is a strong feeling that he goes as far as thought can take us. In that regard, it is very "satisfying." Again, what is being satisfied? Well, being that man is made to know, it must be a kind of comprehensive satisfaction of that need.
I suppose scientists and mathematicians experience something similar, but it's hard for me to imagine anything approaching total satisfaction from those fields. So much is left out of even their most sophisticated models, that they would leave me hungry for more.
I have read any number of biologists who talk about the deep intellectual satisfaction that accompanies their appreciation of natural selection. Well, yes. I am well aware of that feeling myself. Maybe I just have a bigger appetite, but I consider it only a first course. Metaphysics is the dessert. And theology is the after dinner cigar.
Polanyi was a "meta-scientist," as it were. He basically examined the logic of scientific discovery, and built a more general theory of knowledge based upon it. Thus, "imagination sets actively before us the focal point to be aimed at, but it is intuition that supplies our imagination with the organization of subsidiary clues to accomplish its focal goal, as well as the initial assessments of the feasibility of this goal. Intuition thus guides our imagination" (Prosch).
And all of this action takes place beneath the surface of consciousness. We are always guided by we-know-not-what. But it doesn't mean we aren't being pulled by this invisible gradient of meaning, AKA a nonlocal attractor.
Indeed, many if not most of the things we believe are due to prior non-conscious "commitments" to ideas, principles, concepts, and conclusions of various kinds. For Polanyi, these can never be rendered fully explicit, but we can in a sense know them by that to which they point. So, if, say, an atheist tells you that you cannot specify all the reasons why you believe in God, it is no different for the atheist: nor can he specify all of the implicit and subsidiary clues that led him to his conclusion.
A key thoughtlet occurs to me: if you believe all of your evidence can be rendered explicit, you have to be a pretty shallow individual. This was the philosophy of positivism, and although it has been discarded by philosophy as such, there are still a lot of crude scientistic positivists running around.
I recently evaluated an intelligent young woman who said she had discarded her Catholicism for Science. The problem is, you can superimpose Science on yourself from above, but it won't do anything to speak to all of those implicit clues demanding an answer. Therefore, you have to dismiss them as irrelevant, which can lead to a kind of existential pain that doesn't seem to have any "cure."
Here is another example of what it's like to be a math wiz: "You develop a strong aesthetic preference for powerful and general ideas that connect hundreds of difficult questions, as opposed to resolutions of particular puzzles."
Now, such ideas can obviously mislead, and we have to be cautious about indulging this ability. For example, Marxism is a quintessential case of a "powerful and general idea that connects hundreds of difficult questions." It is also utterly false, but that doesn't diminish its appeal to the susceptible.
To be perfectly accurate, we all have this susceptibility to total explanation by virtue of being human. It comes with the standard package. But is a total explanation even possible without God at the top? No, it is not. For starters, if there is no God, then there is no top or bottom at all, just as Darwin says. And there is no reason in the world we should pay attention to the pompous declamations of a randomly evolved talking monkey.
The piece concludes with a good one: "You are humble about your knowledge because you are aware of how weak math is, and you are comfortable with the fact that you can say nothing intelligent about most problems."
However, this particular mathematician just said a number of intelligent things about human problems. But it wasn't via math per se, rather, a meta-mathematical analysis of the mathematician's cognitive abilities. Quality trumps quantity every time, for the same reason that semantics can never be reduced to syntax. Math has no meaning that isn't ultimately self-referential. It requires a deeper insight in order to extricate oneself from that closed loop and make contact with the transcendent real, AKA God.
God is like a good accountant: he's surely adept at math, but can also tell you what to do to avoid an audit.