Yep — I’m a big proponent of explaining the intuition; I even go for intuition first, because it’s only by having a notion of what you want to discuss that you can make a meaningful formalism. After all, how do you know what the right formalism would be without an intuitive notion of what you’re trying to model?
But when you want to resolve a question like “what can we say about continuity and differentiation?” suddenly the answer depends intimately on your formalism. And the only way to resolve that we have conflicting ideas about the intuition for “continuous” and “differentiable” is for us both to formalize that intuition in a model — then compare the two to see where we disagree.
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As an aside, you can formalize the notion of infinitesimals, but it requires a lot more machinery which introduces its own quirks. And it was only because we formalized calculus (and tried to formalize all of math) that we had sufficiently advanced model theory. I won’t claim to be an expert, but the topic is Nonstandard Analysis. I believe similar constructions show up in game theory and quantum mechanics.
You can introduce nonstandard analysis, but it's not necessarily incredibly intuitive either, even if you skip a formal construction (which requires either ultrafilters or model theory). I think that's still a bit removed from just giving ad-hoc reasoning about "infinitely small" quantities without specifying precisely what you can and cannot do with them.
But when you want to resolve a question like “what can we say about continuity and differentiation?” suddenly the answer depends intimately on your formalism. And the only way to resolve that we have conflicting ideas about the intuition for “continuous” and “differentiable” is for us both to formalize that intuition in a model — then compare the two to see where we disagree.
- - - - -
As an aside, you can formalize the notion of infinitesimals, but it requires a lot more machinery which introduces its own quirks. And it was only because we formalized calculus (and tried to formalize all of math) that we had sufficiently advanced model theory. I won’t claim to be an expert, but the topic is Nonstandard Analysis. I believe similar constructions show up in game theory and quantum mechanics.
https://en.wikipedia.org/wiki/Nonstandard_analysis