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1. We introduce limits. Which are blindingly artificial, and so it is hard for us to form a concept about why people want this. I don't see limits as artificia

by Fargren 15y ago

1. We introduce limits. Which are blindingly artificial, and so it is hard for us to form a concept about why people want this. I don't see limits as artificial as all. For example, Zeno's Paradox[1] is resolved thanks to the limit of a succession. And the concept of the infinitesimal, which was something I had thought about before I learnt about limits, is another rather intuitive thing that is explained by limits. Derivatives are only one of the most common uses of limits, they are definetely not the only reason they are useful. [1]http://en.wikipedia.org/wiki/Zeno%27s_paradoxes#Achilles_and_the_tortoise http://en.wikipedia.org/wiki/Zeno%27s_paradoxes#Achilles_and...

psykotic 15y ago
The limit concept was introduced by Cauchy but lacked a rigorous definition until Weierstrass came along. In the absence of such a definition, Cauchy committed some infamous errors involving nested limits, e.g. he ascribed properties to continuous functions that actually require the stronger assumption of uniform continuity. Take into account that Cauchy's treatment of calculus was by far the most rigorous for its time, so he was hardly careless in these matters. If everything was as straightforward and obvious as you imply, neither Cauchy's nor Weierstrass's work on the foundations of calculus would have served any meaningful purpose. The more you learn, the more you come to realize that the nature of the real numbers is deeply mysterious and not to be treated lightly. That's not to say you cannot convey many useful intuitions to students without the somewhat abstruse formal machinery.
- Fargren 15y ago
  That's not to say you cannot convey many useful intuitions to students without the somewhat abstruse formal machinery. That's more or less what I was trying to say. There are some deeply intuitive concepts for which limits are, at the very least, a very good aproximation. It may occur in some cases that the actual case and the intuition that one has about it doesn't match, but that doesn't mean that the model is arbitrary or artificial.
btilly 15y ago
I'm sorry, but every part of your response betrays a severe lack of understanding of the history. People understood the solution to Zeno's paradox centuries before we had a rigorous definition of limit. And limiting processes were understood both before, and after, without necessarily referring to limits. (In fact in computer science people prefer the more flexible Big-O/little-o notation for that purpose. Knuth has in fact suggested replacing limits in introductory Calculus with that notation and I'd be quite interested to see how well that would work.) The idea of infinitesmals were thought up well before limits. In fact the idea of a limit came out of Cauchy's work. Cauchy tried to define an infinitesmal as a function that went to 0. (His idea of an infinitesmal became what we now call a Cauchy sequence.) When limits came they did not explain infinitesmals - they replaced them. It was not until Abraham Robinson came up with non-standard analysis in the 1950s that anyone came up with a sound explanation of infinitesmals and properly explained the connection with limits. And finally I was trying to explain what things are like from the perspective of the student.)

by Fargren 15y ago

psykotic 15y ago

The limit concept was introduced by Cauchy but lacked a rigorous definition until Weierstrass came along. In the absence of such a definition, Cauchy committed some infamous errors involving nested limits, e.g. he ascribed properties to continuous functions that actually require the stronger assumption of uniform continuity. Take into account that Cauchy's treatment of calculus was by far the most rigorous for its time, so he was hardly careless in these matters. If everything was as straightforward and obvious as you imply, neither Cauchy's nor Weierstrass's work on the foundations of calculus would have served any meaningful purpose. The more you learn, the more you come to realize that the nature of the real numbers is deeply mysterious and not to be treated lightly. That's not to say you cannot convey many useful intuitions to students without the somewhat abstruse formal machinery.

Fargren 15y ago
That's not to say you cannot convey many useful intuitions to students without the somewhat abstruse formal machinery. That's more or less what I was trying to say. There are some deeply intuitive concepts for which limits are, at the very least, a very good aproximation. It may occur in some cases that the actual case and the intuition that one has about it doesn't match, but that doesn't mean that the model is arbitrary or artificial.

btilly 15y ago

I'm sorry, but every part of your response betrays a severe lack of understanding of the history. People understood the solution to Zeno's paradox centuries before we had a rigorous definition of limit. And limiting processes were understood both before, and after, without necessarily referring to limits. (In fact in computer science people prefer the more flexible Big-O/little-o notation for that purpose. Knuth has in fact suggested replacing limits in introductory Calculus with that notation and I'd be quite interested to see how well that would work.) The idea of infinitesmals were thought up well before limits. In fact the idea of a limit came out of Cauchy's work. Cauchy tried to define an infinitesmal as a function that went to 0. (His idea of an infinitesmal became what we now call a Cauchy sequence.) When limits came they did not explain infinitesmals - they replaced them. It was not until Abraham Robinson came up with non-standard analysis in the 1950s that anyone came up with a sound explanation of infinitesmals and properly explained the connection with limits. And finally I was trying to explain what things are like from the perspective of the student.)