R. Hasen says: >When you say "how to best ground the formalities" what you really mean is how to reach students unaware of the formalities, but the formalities themselves are grounded in one thing only, formal thinking (is that better than "reasoned thought"?)
I do believe you really mean formal logic, and I'll add that in my view (among others) that is merely a subset of computation. But to your point, if there were no correspondence, nobody at all would much care in times past, the subject would never have arisen as important, or at very best gathered many followers in the first place. The 20th century is pure anomaly, in that more than a very few people can make a living investigating meaningless formal systems (not that they are easy to separate). Chalk it all up to cheap energy and a population boom. The clock is ticking.
Again, R.H. says: >but the formalities themselves are grounded in one thing only, formal thinking
My but you do repeat that thought. Its absurd. To give you some credit though, and as I have have mentioned, sometimes, (please understand-- sometimes) formal logic or symbolic, syntactic concerns lead the way. That's the whole point of me bringing up something abstruse like Martinez's book. But if you read it, you'll also note that things like accounting rose alongside the mathematicians formalities, which is really what kept things like "negative numbers" alive -- practical concerns. Today is really no different, even if the thing keeping some obtuse formality alive is the Pentagon funding men who stare at goats -- OPM.