Conventional wisdom, fooled by our misleading "physical intuition", is that the real world is "continuous", and that discrete models are necessary evils for approximating the "real" world, due to the innate discreteness of the digital computer.

Mathematics my foot! Algorithms are mathematics too, and often more interesting and definitely more useful.

Let me also remind you that zero, like all of mathematics, is fictional and an idealization. It is impossible to reach absolute zero temperature or to get perfect vacuum. Luckily, mathematics is a fairyland where ideal and fictional objects are possible.

The real work of us mathematicians, from now until, roughly, fifty years from now, when computers won't need us anymore, is to make the transition from human-centric math to machine-centric math as smooth and efficient as possible.

The 'lowly' finite is MUCH more beautiful than any 'infinite'

No Victor, you got it backwards, you should evaluate these integrals non-rigorously if you can, and rigorously if you must.

When a problem seems intractable, it is often a good idea to try to study "toy" versions of it in the hope that as the toys become increasingly larger and more sophisticated, they would metamorphose, in the limit, to the real thing.