Timeline for Is there a reason why integrals are so much easier to evaluate than sums?
Current License: CC BY-SA 2.5
6 events
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| Feb 18, 2011 at 17:05 | comment | added | Theo Johnson-Freyd | @Harry: Absolutely. There are guaranteed to be sequences with nice sums, and those sequences are of independent interest. But they're hard to write in the vocabulary of a very basic calculator. @Sandor: I'm trying to say something different. A piecewise function with steps at the integers is the same data as a sequence, but it's not the same thing. A sequence is a function on the integers, and the basic operations you can do to them include shifting, and hence differencing and summing. These are algebraic operations, that are worse-behaved than their smooth cousins. | |
| Feb 18, 2011 at 14:24 | comment | added | Kaveh Khodjasteh | Can one put this also in terms of the difference between first and second order logic? | |
| Feb 18, 2011 at 10:29 | comment | added | Martin Brandenburg | So perhaps the idea is: In the context of derivatives we have $\epsilon^2=0$, but in the context of differences not. | |
| Feb 18, 2011 at 7:03 | comment | added | Sándor Kovács | Aren't you essentially saying that differentiable functions are better behaved than piecewise linear functions (whose integrals are what we also call sums)? | |
| Feb 18, 2011 at 6:50 | comment | added | Harry Altman | Of course, $n\mapsto \binom{n}{k}$ has nice differences but not nice sums, at least if we stick to that basis... | |
| Feb 18, 2011 at 2:04 | history | answered | Theo Johnson-Freyd | CC BY-SA 2.5 |