The mind of the enlightenment era mathematician might have been the ultimate tool ever devised for creating pointless and convoluted connections between pi and everything else.
i - Secret based religion
There are certain Buddhist traditions, e.g. Dzogchen, in which "enlightenment" or some other desired state or status is predicated upon knowing some hidden knowledge. This is most popular in Buddhist faiths but by no means confined to them, some gnostic traditions are also fund of secret knowledge.
This begs a question for the would-be believer:
Why not readily give out this knowledge?
If all I have to do to /reach enlightenment/attain nirvana/understand the nature of God/ is to read a few sentences, why not readily give them to everyone?
The answer to this has something to do with "mind preparedness", one is not ready to understand until they have some prerequisite baggage. But still, why not give them to everyone first, then tell them to go get the prerequisite knowledge. After all, the prerequisites might be different for everyone, this way, as soon as they have them, things will instantly click, and they won't have to /meditate/chant/pray/ for longer than necessary.
To which the crazier believers answer something like:
Because God will blindeth the unworthy who lookedth uponith thy sacredest texts.
But the saner ones say something like:
Look, this is the most profound knowledge on Earth, but if we give it to you before you are prepared, you won't see that. You will get used to it and thus it will forever lose importance, it will become a banality in your mind. Only reading with fresh eyes makes it have the power it does, and you can only do that once, so I'd better be after you learnt enough to grasp its value".
Sound like a load of rubbish? Ok, I agree, but the methodology these sects invented for causing a feeling of revelation might be quite generic and ingenious. Let me give a brief summary of how this goes:
- Have some hidden knowledge, vaguely tell the student about it, but don't focus on it. The hidden knowledge is usually simple, somewhat circular and somewhat abstract: "Evil is a construct of man, to understand God is to understand that even under great pain, the default condition of the mind is happiness", "If the nature of the world is perfect emptiness that also implies perfect connectedness", "There is no centre to experience in your /hear/head/body/"... etc.
- Give the student some problems or paradoxes to think about, the problems or paradoxes are solved or dissolved by the hidden knowledge.
- If the student figures out some version of the hidden knowledge along the way, that's fine, upon the knowledge being revealed they will be "enlightened" because "they had the answer all along"
- If the student doesn't figure out some version of the hidden knowledge, it will feel like a satisfactory answer to the problem they were ruminating on for so long, on top of that it will seem so simple that they will be awestruck they didn't think of it. Thus the student will be "enlightened".
Also, it's worth noting that in many of these religions "enlightenment" seems to be something like "a feeling of supreme insight without any associated insight". Granted, the religious people would say "there is associated insight, we just can't explain it in words, or actions, to anyone but those that are enlightened", but I've taken large enough doses of acid not to fall for that one, there are ways to generate "a feeling of supreme insight" while revealing nothing of value about the world.
So the above, I would claim, is a tried and tested pattern for generating something that feels like awe or understanding or enlightenment or insight or something equally blissful and important.
ii - High school math
Did you ever ponder some mathematic formula or problem and had a moment when things click and felt "Ohhh, ****, that makes total sense" and your worldview slightly but notably shifted forever. Maybe something around some properties of a circle or of
e. Maybe something around edge-cases for probabilities (sailor problem, sleeping beauty problem, Monty hall problem are common).
I had this feeling a bunch of time, the one I most vividly remember is when I figured out how to compute the probability of an outcome happening once, for multiple independent events that have the same range of outcomes (e.g. what's the change of rolling 6 at least once if you roll 6 6-sided dices). Usually, it's something simplistic, because most of us aren't really able to "get" complex math on an intuitive enough level to deduce things enough inferential steps away for them to be insightful.
Most of these things we already "know", we are told about them in school, usually much earlier than we could figure them out on our own. The only reason we can "relearn" them is, I assume, that we forget them due to how horribly they are "taught" to us.
So I wonder if teaching mathematics should take a page out of the mystic handbook and focus on giving people exercises and letting them try to figure useful properties and theorems out by themselves. Or, at most, teaching certain theorems only when they are needed, i.e. once a student hits their head enough against an exercise so that the answer to their problems, so simple and clear, seem revealing, inspires awe, insight.
Mind you, I'm not sure this would be "good" in absolute terms, or good for the current (lacklustre) value structure of society, but I think it would make more people love math a lot more. Two anecdotes lead to the solidification of this idea as worth investigating:
1. I and many other people I know that are decent at math, but not exceptionally good at it, seemed to have really loved math up until somewhere around the 3rd to 7th grade. Incidentally, this seems to be the place where math "picks up speed" and it's almost impossible to stay "ahead of the teacher" and discover stuff for yourself. It's easy to reason your way to insights when learning Euclidean math, there's a reason Greek rationalists thought everyone "knew" it inherently. Not only is Euclidean math simple, you've got 6 years to learn it, and you don't even cover most of it.
It's much more difficult to even start fathoming the need for calculus, let alone "discover" stuff about it on your own, yet we are thrust into it right after Euclidean math in a much more rapid way, with 1000+ years of in-between mathematics glanced over. So losing interest in math at the start of or soon before high-school might be the fact that we go from a bunch of moments when "things click", when we feel insight, to being told facts we have no way of appreciating or "getting" at a level where they seem important.
... but maybe it's testosterone, that could be it too, it also checks out in terms of timing.
2. Being very smart (at least as far as an IQ test can measure) is heavily correlated with being a good mathematician and, seemingly, with loving math. The extent of this is surprising since other fields of thought seem less reliant on intelligence and much more reliant on luck and "hard work". Even closely related fields like computer science. Even more so, "loving" any given field doesn't seem predicated on how smart people are, except for mathematics, where the people that actually "love" math are usually smart to the point of associated mental illness.
The "love" bit is confounded by people social signalling that "they love math" a lot more than they social signal love for other sciences. But I'd protest that this is an argument for the "very smart people love math" hypothesis since math is essentially useless in most jobs and areas of life, but intelligence isn't, so social signal for intelligence are very useful. Thus I'd propose that people try signalling "I am smart" by saying "I love math" is actually the result of a shared observation of a reality (very smart people usually love math and only very smart people can love math).
Granted, this is all heavily anecdotal. But if true, one explanation for it would be that very smart people are able to "stay ahead of the teacher" when it comes to math all the way through school and high-school, and thus get exponentially more "eureka" moments than the just-slightly-dumber people that are almost always one step behind the teacher.
... or maybe it's just that, like many other things, mathematical intelligence has exponential returns. Von Neuman gets to invent most of what we know as "anything of scientific relevance since the 40s" and the next 100,000 smartest people get to invent "most of the scraps that were left to them by Von Neuman"... maybe a slight exaggeration, but you get the point, there's certainly a case to be made for exponential returns on intelligence. I strongly protest against that case, but that's an argument for another day.
Still, I would very much like to know if this style of "letting kids figure it out for themselves" has been tried with mathematic education and what the results are, once controlled for confounders.