As soon as an Analytical Engine exists, it will necessarily guide the future course of the science. Whenever any result is sought by its aid, the question will then arise — by what course of calculation can these results be arrived at by the machine in the shortest time?
Perhaps the most surprising and powerful aspect of place-value arithmetic is how it reduces any calculation to a set of purely abstract symbolic manipulations. In principle, I suppose, one could even be trained to perform such symbol-jiggling procedures without any comprehension whatever of the underlying meaning. We could even (if we can possible imagine being so cruel) force young children to memorize tables of symbols and meaningless step-by-step procedures, and then reward or punish them for their skill (or lack thereof) in this dreary and soulless activity. This would help protect our future office workers from accidentally gaining a personal relationship to arithmetic as a craft or enjoying the perspective that outlook would provide. We could turn the entire enterprise into a rote mechanical process and then reward those who show the most willingness to be made into reliable and obedient tools. I wonder if you can imagine such a nightmarish, dystopian world? Let's try not to think about it.