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> Some of my smart classmates could not really understand the concept of limit intuitively. that's because the way it's presented is silly. limits aren't the k

by a-dub 4y ago

> Some of my smart classmates could not really understand the concept of limit intuitively. that's because the way it's presented is silly. limits aren't the key idea, the continuity of the reals is.

strikelaserclaw 4y ago
but limits are a key idea, their invention allowed a whole class of mathematical problems to be solved
- a-dub 4y ago
  sure. and despite all the trouble people have, their definition is quite simple. the crux of the problem is understanding continuity. if you truly understand the continuity of the reals (which is the "real" invention), limits can be derived almost effortlessly. in most calculus teaching, continuity is quickly stated and then there's a bunch of rules and these seemingly contradictory discussions involving delta and epsilon. they're not contradictory if continuity is understood and internalized.
  - denial 4y ago
    Do you prefer the Dedekind cut construction of the real numbers over the identification of Cauchy sequences of rational numbers with common limit? In ZFC, they coincide, but in constructive mathematics, one can build a model wherein they differ (see https://mathoverflow.net/questions/128569/a-model-where-dedekind-reals-and-cauchy-reals-are-different https://mathoverflow.net/questions/128569/a-model-where-dede... ).
hintymad 4y ago
I used limit as an example. It could be other concepts, simple or sophisticated. I don’t think it’s about how professors teach, but about that people simply can’t get some abstractions. Have you seen kids in high school who can’t do algebra even if you find the best teachers for them?
- deleted 4y ago
  [deleted]
codethief 4y ago
I disagree, I would say limits are the key idea. 1. How do you even express the idea of continuity if not through limits? (Sure, there's the topological definition and the epsilon-delta criterion but those are merely an abstraction or a different way of phrasing the same idea.) 2. The key difference between the rationals and the reals is that the latter are Cauchy-complete. How do you even define (in an intuitive way) what that means without referring to limits? 3. The reals are typically constructed as (equivalence classes of) Cauchy sequences, i.e. by their very definition they are the limits of their Cauchy sequences.
- a-dub 4y ago
  i think, for the purpose of introducing the continuity of the reals, for the purpose of introducing limits, one could skip any discussion that involves constructing the rationals or building the reals from them, at least to start. instead go directly from integers to reals by introducing countable vs. uncountable infinities.

by a-dub 4y ago

> Some of my smart classmates could not really understand the concept of limit intuitively. that's because the way it's presented is silly. limits aren't the key idea, the continuity of the reals is.

strikelaserclaw 4y ago

but limits are a key idea, their invention allowed a whole class of mathematical problems to be solved

a-dub 4y ago
sure. and despite all the trouble people have, their definition is quite simple. the crux of the problem is understanding continuity. if you truly understand the continuity of the reals (which is the "real" invention), limits can be derived almost effortlessly. in most calculus teaching, continuity is quickly stated and then there's a bunch of rules and these seemingly contradictory discussions involving delta and epsilon. they're not contradictory if continuity is understood and internalized.
- denial 4y ago
  Do you prefer the Dedekind cut construction of the real numbers over the identification of Cauchy sequences of rational numbers with common limit? In ZFC, they coincide, but in constructive mathematics, one can build a model wherein they differ (see https://mathoverflow.net/questions/128569/a-model-where-dedekind-reals-and-cauchy-reals-are-different https://mathoverflow.net/questions/128569/a-model-where-dede... ).

hintymad 4y ago

I used limit as an example. It could be other concepts, simple or sophisticated. I don’t think it’s about how professors teach, but about that people simply can’t get some abstractions. Have you seen kids in high school who can’t do algebra even if you find the best teachers for them?

deleted 4y ago
[deleted]

codethief 4y ago

I disagree, I would say limits are the key idea. 1. How do you even express the idea of continuity if not through limits? (Sure, there's the topological definition and the epsilon-delta criterion but those are merely an abstraction or a different way of phrasing the same idea.) 2. The key difference between the rationals and the reals is that the latter are Cauchy-complete. How do you even define (in an intuitive way) what that means without referring to limits? 3. The reals are typically constructed as (equivalence classes of) Cauchy sequences, i.e. by their very definition they are the limits of their Cauchy sequences.

a-dub 4y ago
i think, for the purpose of introducing the continuity of the reals, for the purpose of introducing limits, one could skip any discussion that involves constructing the rationals or building the reals from them, at least to start. instead go directly from integers to reals by introducing countable vs. uncountable infinities.