"Moreover, reason cannot start absent faith, -- theorem cannot be proven absent axiom. But faith can start absent reason, as we see in children."
Beautifully stated.
I remember going through a bit of a "crisis" as I got into the higher levels of calculus in college.
Things had changed from a neat world where, for instance, one could multiply length x width of a rectangle and have an "exact" area (of course the italics are there because one can never have an exact area of an actual rectangular surface due to the limitations of measuring the lengths and determining the precision of a right angle) -- to a much messier world where one was dealing with irregular objects.
At the very least, with a theoretical rectangle, one can calculate an exact area in a way that the average mind can visualize. I remember talking to the professor at the time, asking him if the problems we were solving really gave an exact calculation of the area of a surface with irregularly curved edges, or whether it was just an estimation -- even in a theoretical, perfect world situation.
He thought a minute, and then replied: "What is important is not that it is exact. What is important is that it is consistent." In other words, he was differentiating clearly between consistent reproducibility and truth, even in a discipline as "hard" as mathematics. (He was a Christian -- a conservative Catholic, as I recall -- incidentally.)
And of course, even the consistency of reproducibility depends on the conditions, even as specified in theoretical problems. Newtonian physics is perfectly servicable for ordinary everyday life (and according to Newtonian physics it is, of course, impossible for a man to walk on water), but it breaks down as a tool for explaining other situations -- thus the development of quantum mechanics, relativity, etc... And of course the highest levels of theoretical physics today almost resemble philosophy and metaphysics more than they resemble what we think of as a reproducible science.
Certainly the evolutionary biology that I learned demanded incredible amounts of axiomatic faith. The only way one could be attuned to seeing the faith involved in evolutionary biology was if one had a sense of the authority of revelation that was powerful enough to force one to step back and look at it more objectively.
You are exactly right that reason, and even observation for that matter, require faith (i.e. working assumptions or axioms) as a starting point. The fact that these axioms produce working results reinforces the wisdom of trusting them, but by definition an axiom is an unprovable proposition that one chooses to trust in.
Likewise, ones direct experience of God together with the historical experience of the Church reinforces the wisdom of trusting the starting axiom or working assumption of Christianity -- that is, the deposit of faith once delivered, Holy Tradition in its entirety, that we in faith accept and believe.
There is something known as conditioning that does not require reason, but simply programs a living creature nor matter how high or low to a set of responses. It's not a matter of reason or faith, but simply of rewards. Feels good; feels bad.
With all due respect to your professor, an area of any shape can be calculated if proper expressions are integrated. An expression is broken every-diminishing derivatives and an exact area is theoretically possible in a theoretical world. In the real world, the precision is carried to practical limits. So, the answer is -- theoretically -- yes; practically no. And his observation is realistically correct: consistency is much more important for most applications.