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Urs has added Euler integration prompted by Tom’s post at nCafe; I wanted to do that and will contribute soon. I noticed there is no entry integral in $n$Lab, but it redirects to integration. I personally think that integral as a mathematical object is a slightly more canonical name for a mathematical entry than integration, if the two are not kept separated. Second, the entry is written as an (incomplete) disambiguation entry and with a subdivision into measure approach versus few odd entries. I was taught long time ago by a couple of experts in probability and measure theory that a complete subordination to the concept of integral to a concept of measure is pedagogically harmful, and lacks some important insights. This has also to do with the choice of the title: integration points to a process, and the underlying process may involve measure. Integral is about an object which is usually some sort of functional, or operator, on distributions which are to be acted upon.
Thus I would like to rename the entry into integral (or to create a separate entry from integration) and make it into a real entry, the list of variants being just a section, unlike in the disambiguation only version. What do you think. Then I would add some real ideas about it.
On the other hand, Lie integration does not belong into a version of an integral, so maybe it makes sense to have two highly overlapping entries ??
P.S. I disagree that Euler integration should be classified as a version of integration in combinatorics, as it is under integration. It is developed historically mainly for integral geometry and its applications (tomography, geometrical probability etc.) and also it prompted the development of motivic integration. Combinatorics designation is not that primary there. Of course, discreteness of constructive sheaves gives it prone to combinatorial treatment and applications, but this is not primary.
I disagree that Euler integration should be classified as a version of integration in combinatorics,
Right, I wondered a while where to put it, and then went for “combinatorics”, but hesitantly. Feel free to make another choice!
Urs, after being overwhelmed by a sheer vision of the task and the range of the subject involved, I made the following strategy: let us put more stuff first to extend the scope of both entries and then do tidying once the more complete picture appears ! I have now two entries
which differ a bit in scope. Both have now an idea section, and I envision that both the links and the idea sections will grow in a useful way! I hope you agree.
Zoran, I like what you’ve written. But I don’t understand why anything on one page shouldn’t be on the other. I would support a combined page at integral.
Well, Toby, though I had exactly the same thought at start, after lots of thought I decided that it may be slight better to keep them separate. I am still open to discussion on this (though it is maybe better first to accumulate diverse material to get the feeling of full scope before deciding).
While many integration notions involve integrals, “Lie integration” and similar notions are not about or involving integrals, though the name is motivated by the integration of differential equations which in very special cases does involve quadratures. So, I can not put Lie integration under integral, which suggests a bit different range of contexts.
On the dual side, there are also special cases of integrals like Euler and Selberg integral which are just static integral expressions and do not refer to some special integration procedure. I mean I would not like to talk about “Selberg integration” as a notion. In fact, there is also Euler integral producing the Euler beta function while Euler integration has nothing to do with it – it is about using appropriately Euler characteristic instead of measure, as we just learn these days at cafe.
Hence some one third of cases seem to clearly favor one or the other notion.
P.S. The matter will complicate even more once one gets to study integrals in and on Hopf algebra like objects (an integral in a Hopf algebra and integral on a Hopf algebra are different notions!). They came from the generalizations of functionals like invariant integration on a compact group. I hope to cover them with appropriate entries soon.
Whether the entry is called integral or integration, all of these examples can be included, because the entry will cover both terms. (The opening header “Integrals” that I put ast the top of integral would be “Integrals and integration”.)
It’s not the lists of examples that make me want to combine these, despite the heavy potential overlap on those lists. It’s the general material. I want to copy the introduction of integral to integration, because it’s a good explanation. I want to copy the discussion about integration and differentiation at integration to integral, because it’s about integrals. Etc.
OK; I will do a careful merger within a day or so.
Some basic terminology, especially definite integral.
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