[eximius]

Calculus would likely not be invented without infinity. Many theorems, identities, and techniques may be approximated but ultimately rely on proofs using infinities - I doubt we would have discovered them quickly or at all without infinity. After all, the very concept of a limit evokes the concept of an infinite sequence.

[tgflynn]

That is true historically but I doubt that it is true necessarily. I mean I suspect that starting from what we now know it should be possible to reconstruct calculus (at least for all practical purposes) without reference to infinities or infinitesimals.

One argument in favor of this belief is that neither practical computations nor analytic intuition require actual infinitesimals.

The later, at least, has been my experience. I think I have a fairly decent practical intuition for calculus based on imagining dx becoming smaller and smaller until it's small enough, but I don't think my brain has any actual representation of "true infinitesimals" and my intuition breaks down completely if I try to imagine things like the relationship between the rational and irrational numbers. Maybe that's due to my intellectual limitations, but I wonder if it isn't because these concepts might be over-elaborate abstractions that don't really exist in our world.

[Enginerrrd]

>One argument in favor of this belief is that neither practical computations nor analytic intuition require actual infinitesimals.

Don't derivatives count? That's a pretty important and trivial calculation. Sure, you can approximate it when it's nicely behaved, but they aren't always. There's also lots of verrrrrrry slowly converging series that can't be easily computed numerically.

[tgflynn]

So what practical applications do these functions have whose derivatives can't be approximated numerically ?

[eximius]

Probably lots in physics, aerospace, electrical engineering.

And, again, coming up with derived theorems that are useful.