I noticed this book "Quantum calculus" http://www.amazon.com/exec/obidos/ASIN/0387953418/ref=nosim/ericstreasuretro
It's about doing calculus without taking limits. I'm curious if it might have some ideas relevant for your "delta-calculus" and "lambda-calculus" distinction.
It came up for me because I'm studying the q-analogue of simplexes. I have a combinatorial interpretation (giving successive vertices weights 1, q, q2, q3... and giving all the edges weight 1/q). I found an algebraic version of this at this page: http://mathworld.wolfram.com/q-BinomialCoefficient.html where they cite the "Quantum calculus" book above.
I'm studying different combinatorial interpretations of the q-analogue for binomial coefficients. I hope soon to share a long letter I've been writing to Foundations of Math group about "implicit math" vs. explicit math, with results regarding simplexes, the "field with one element", polytopes, etc.