I always feel that I come away with something new whenever I read Ludwig Wittgenstein's Remarks on the Foundations of Mathematics - likely because I understood so little on each previous read. In the book, one thing he tries to get at is what we mean by words "mathematics" and "calculation," and in doing so he asks questions that are so basic that they call into question our implicit assumptions about what these words mean. One of these sets of questions ask about whether our mental state and attitude in any way influences whether or not we are actually "doing mathematics."
"Imagine the geometry of four-dimensional space done with a view to learning about the living conditions of spirits. Does that mean that it is not mathematics?... Could people be imagined, who in their ordinary lives only calculated up to 1000 and kept calculations with higher numbers for mathematical investigations about the world of spirits?"Does it matter what we think we are doing when we are doing math? As long as we are moving the symbols around correctly does it still count as mathematics?
Elsewhere he asks "What would happen, if we rather often had this: we do a calculation and find it correct; then we do it again and find it isn't right; we believe we overlooked something before - then we go over it again and our second calculation doesn't seem right, and so on. Now should I call this calculating, or not?"
Does calculation require a social convention - if one person performed something once, could it be considered an algorithm? "What about this consensus - doesn't it mean that one human being by himself could not calculate? Well, one human being could at any rate not calculate just once in his life."
Some of the most fascinating thought experiments that Wittgenstein proposed (way back in 1942-1944) were about (what we would now call) computers or "mobile devices":
"Does a calculating machine calculate? Imagine that a calculating machine had come into existence by accident; now someone accidentally presses its knobs (or an animal walks over it) and it calculates the product 25 x 20."Has any calculation happened in this case? Later he suggests a scenario that now seems quite familiar:
"Imagine that calculating machines occurred in nature, but that people could not pierce their cases. And now suppose that these people use these appliances, say as we use calculation, though of that they know nothing. Thus e.g. they make predictions with the aid of calculating machines, but for them manipulating these queer objects is experimenting. These people lack the concepts which we have, but what takes their place?"Very (unintentionally) prescient Ludwig! We are actually now living in a reality which closely resembles this thought experiment - and an environment that sounds like the classroom of the future as imagined by Computer Based Math. What will replace current concepts of number once our experience with calculation is mediated entirely by machines whose cases cannot be pierced? And will we even notice that they have been replaced once they are gone?