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I was reading this question. Im not mathematician but I like a lot and in spare time I try to learn something.

My question is: reading the previous question of the link I understand (if Im not wrong) that summation and integral are the same thing but in different type or measure, countable measure and uncountable measure or so.

So the mechanic is the same the only that changes is measure. So, why use different notation or simbology for the same generalized thing? Thank you.

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Historically, they were not considered the same (though the symbol for integral is initially an $S$ for sum), the link has become much clearer with Lebesgue integral (the sum is the integral wrt a discrete measure). Apart from that, I agree with 5xum, no need to introduce at lower levels a notation that is explained by much higher mathematics. –  Jean-Claude Arbaut Sep 2 at 7:36
    
The integral is quite a specific sort of summation, so instead of writing out a long summation (including limits too actually) to describe an integral, why not just use a shorter notation? Especially since integrals are all over the place in mathematics. –  Sheheryar Zaidi Sep 2 at 7:36
    
@SheheryarZaidi Actually, if you introduce the correct measure, integrating wrt that measure is exactly the same as summing. No limits involved. I believe that is what OP is asking. –  5xum Sep 2 at 7:39
    
$$\int_a^bf(x)~dx~ = ~\lim_{n\to\infty}~\frac1n\cdot\sum_{k=0}^{(b-a)n}f\bigg(a+\frac kn\bigg)~ = ~f^{(-1)}(b)-f^{(-1)}(a).$$ –  Lucian Sep 2 at 19:08

2 Answers 2

up vote 4 down vote accepted

Shortest answer: Because mathematics is useful.


Short answer: I see no reason to use a unified notation just because there exists some highly theoretical framework in which two things are identical.


Longer answer:

The purpose of mathematical notation is an optimal way of conveying information from the author to the reader. Since people generally start to learn sums and integrals long before they learn measure theory, it is (at that point) completely natural that two notations are introduced.

Later, when the two things are shown to be two sides of one coin, you can simply weight the pros and cons of changing notation:

Pros:

  • The notation is now more in tune with measure theory

Cons:

  • The notation is harder to learn
  • The new notation requires mathematitians to "unlearn" the old notation from their subconsciousness (or else it will cause them to "think" more slowly since they will have to translate)
  • The new notation confuses engineers and other professions which use mathematics without going to the deep theoretical stuff. It also reinforces the view by these people that mathematics is mental masturbation for its own purpose.

Another point:

Do you really think using only integral notation will be any better? Let us say I have a sequence $a_n$ and a function $f(x)$. If I want to integrate the function and sum the sequence, I write

Let $a_n$ be a sequence and $f$ a real function. Then we calculate $$\int_a^b f(x)dx + \sum_k^l a_n$$

And it is immediatelly clear what is happening. In your case, I would have to say

Let $a$ be a function from $\mathbb N$ to $\mathbb R$ and $f$ be a function from $\mathbb R$ to $\mathbb R$ and let $\mu$ be the Lebesgue measure on $\mathbb R$ and $\lambda$ the counting measure on $\mathbb R$. Then we calculate $$\int fd\mu + \int ad\lambda$$

Is the second way really better than the first?

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Ok, I understand the practical issues with tradition, but tradition isnt exactly science, more the opposite to science. Comes to my mind the thing that happen with roman numbers. I know is a difference but I dont think that defend a tradition maybe good in any way. –  Masacroso Sep 2 at 7:43
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My argument has nothing to do with tradition. It has to do with practicality. You cannot teach measure theory to 15 year olds, so you must introduce two notations. It is then impractical to unify the two theories as it brings more harm than good. –  5xum Sep 2 at 7:44
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@Masacroso From time to time, there are people who try to teach crazy things to young pupils. There have been "mathématiques modernes" in France, following more or less the same in the USA. It was a huge mistake, with dramatic consequences on math education. But that won't prevent (some) people to think that a given subject from higher mathematics should be taught early. It's simply that mathematicians are not necessarily good teachers (I mean, they are usually good at their level, not for the young) –  Jean-Claude Arbaut Sep 2 at 7:51
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@Masacroso See my last edit to my answer. Would you say the first or the second way is easier to understand? –  5xum Sep 2 at 7:59
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@Masacroso Sure, I agree there may be some case where using integral notation for summing is useful. But in most real world cases, using two notations is much more practical. –  5xum Sep 2 at 8:07

If you need to, you can think of the sum notation as a convenient notation for some common cases in which the full machinery of integration and measure theory is unnecessary, and where a more elementary approach is possible.

The fact that these elementary ideas are also encompassed within the framework of integration, and that the ideas of measure can unify the treatment, for example, of discrete, continuous and mixed distributions is an illustration of the power of the more general point of view. But this does not invalidate the elementary approach for those cases in which it works.

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