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I think I'd also prefer the infinitesimal version of calculus, but the idea of limits is applicable to many other areas of math, not just calculus (evaluating i

by dvt 3y ago

I think I'd also prefer the infinitesimal version of calculus, but the idea of limits is applicable to many other areas of math, not just calculus (evaluating infinite series, for example). So learning limits is probably a better pathway to higher mathematics.

JadeNB 3y ago
> I think I'd also prefer the infinitesimal version of calculus, but the idea of limits is applicable to many other areas of math, not just calculus (evaluating infinite series, for example). So learning limits is probably a better pathway to higher mathematics. I'd say that limits in the sense that you mean (as opposed to category-theoretic limits) are precisely the domain of calculus or, if one wishes so to call it (because one is proving things!), analysis. For example, many US universities, mine included, regard the computation of infinite series as part of Calculus II.
pfdietz 3y ago
And non-standard analysis is useful elsewhere in math as well. Here's an example: https://discreteanalysisjournal.com/article/87772-a-simple-combinatorial-proof-of-szemeredi-s-theorem-via-three-levels-of-infinities https://discreteanalysisjournal.com/article/87772-a-simple-c...
ykonstant 3y ago
More importantly for this audience, the idea of limits lies at the heart of numerical analysis. Explicating the quantifiers of the definition of a limit is the first step in obtaining control over any estimator of your data. This, among other things, is why I am perpetually baffled at the people in this audience who say that limits is something "they will never need". Limits, their algebra (which subsumes all of high school algebra and inequalities) and their delicate analysis is what makes a ton of numerical algorithms work; Higham's classic has it all, and is perfectly clear about it all.
btilly 3y ago
I think that Big O / little o are both more approachable and provide a richer understanding than limits. See, for example, https://micromath.wordpress.com/2008/04/14/donald-knuth-calculus-via-o-notation/ https://micromath.wordpress.com/2008/04/14/donald-knuth-calc... to see Donald Knuth agreeing. As an example of the conceptual richness, pick up a Calculus book and flip to the problem section for L'Hôpital's rule. Without using any special rules at all, attempt to write them out in o-notation and observe that you generally don't need L'Hôpital's rule to work them out. It is possible to produce examples that can be calculated by L'Hôpital's rule, but not by simply understanding o-notation. But it isn't easy, and you're unlikely to find them in textbooks. It is probably true that as you go on, limits are more useful in higher mathematics than o-notation. But o-notation is far more useful in most subjects that use mathematics. Given how easy it is to master limits if you know o-notation, why not teach o-notation first?

by dvt 3y ago

JadeNB 3y ago

> I think I'd also prefer the infinitesimal version of calculus, but the idea of limits is applicable to many other areas of math, not just calculus (evaluating infinite series, for example). So learning limits is probably a better pathway to higher mathematics. I'd say that limits in the sense that you mean (as opposed to category-theoretic limits) are precisely the domain of calculus or, if one wishes so to call it (because one is proving things!), analysis. For example, many US universities, mine included, regard the computation of infinite series as part of Calculus II.

pfdietz 3y ago

And non-standard analysis is useful elsewhere in math as well. Here's an example: https://discreteanalysisjournal.com/article/87772-a-simple-combinatorial-proof-of-szemeredi-s-theorem-via-three-levels-of-infinities https://discreteanalysisjournal.com/article/87772-a-simple-c...

ykonstant 3y ago

More importantly for this audience, the idea of limits lies at the heart of numerical analysis. Explicating the quantifiers of the definition of a limit is the first step in obtaining control over any estimator of your data. This, among other things, is why I am perpetually baffled at the people in this audience who say that limits is something "they will never need". Limits, their algebra (which subsumes all of high school algebra and inequalities) and their delicate analysis is what makes a ton of numerical algorithms work; Higham's classic has it all, and is perfectly clear about it all.

btilly 3y ago

I think that Big O / little o are both more approachable and provide a richer understanding than limits. See, for example, https://micromath.wordpress.com/2008/04/14/donald-knuth-calculus-via-o-notation/ https://micromath.wordpress.com/2008/04/14/donald-knuth-calc... to see Donald Knuth agreeing. As an example of the conceptual richness, pick up a Calculus book and flip to the problem section for L'Hôpital's rule. Without using any special rules at all, attempt to write them out in o-notation and observe that you generally don't need L'Hôpital's rule to work them out. It is possible to produce examples that can be calculated by L'Hôpital's rule, but not by simply understanding o-notation. But it isn't easy, and you're unlikely to find them in textbooks. It is probably true that as you go on, limits are more useful in higher mathematics than o-notation. But o-notation is far more useful in most subjects that use mathematics. Given how easy it is to master limits if you know o-notation, why not teach o-notation first?