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I noticed that the entry analysis is in a sad state. I now gave it an Idea-section (here), which certainly still leaves room for expansion; and I tried to clean up the very little that is listed at References – General
The mathematical field of analysis is concerned with the concept of convergence of sequences (or more generally: of nets), in particular as concerns the infinitesimal analysis that gives rise to the theory of differentiation and integration (“calculus”).
I think I’d phrase that a little differently, insofar as infinitesimal analysis is not a primary concern of classical analysis (of the type founded on the reals as the unique complete ordered field) – in fact the two are incompatible.
One possible rephrasing might be:
“In mathematics, analysis usually refers to any of a broad family of fields that deals with a general theory of limits (convergence of sequences or more generally of nets), particularly those fields that pursue developments that originated in “the calculus”, i.e., the theory of differentiation and integration of real and complex-valued functions. The classical foundation of this general subject is usually based on the idea that the real number system is describable as the (essentially unique) complete ordered field, with limits ultimately referring back to limits of sequences of real numbers.
Analysis can also refer to other responses to the problem of founding these developments, especially “infinitesimal analysis” which admits infinitesimal quantities not found in the classical real number system and which takes various forms, for example the nonstandard analysis first introduced by Abraham Robinson, or “synthetic differential analysis” whose rigorous foundations were largely introduced by Bill Lawvere and other category theorists who, following the example of Grothendieck, consider nilpotent infinitesimals (instead of invertible ones à la Robinson) as a basis for understanding differentiation.”
Thanks, Todd!
I have merged that into the Idea-section here. Please feel invited to edit right there.
I was thinking of “infinitesimal analysis” as a synonym for “infinitesimal calculus”, but I suppose it’s right that it came to carry a more specific connotation.
Oh, I hadn’t noticed the article calculus before!
I largely agree with the spirit of the beginning of calculus, but I did have some questions. The term that I learned to refer to the calculus of differentials and integrals (as opposed to sequent calculus, etc.) is “the calculus” – the “the” signifying a usual or default meaning. I can’t remember ever hearing anyone utter the words “infinitesimal calculus” unless they really meant to refer to infinitesimal quantities. Then again, I don’t have experience outside “anglosaxon universities”; is some cognate of “infinitesimal calculus” used in Francophone or German universities? Also, “anglosaxon” (unhyphenated and lower case!) looks a little odd to me – is that a common locution I am unaware of? Could we have just as well said “Anglophone”?
I’m going to perform some minor edits, including adding this MO discussion to the references.
The bulk of the entry calculus is due to Zoran, rev 1 from March 2013. He seems not to be active here anymore, but I suppose since you are the native speaker, you should feel invited to adjust the terminology where appropriate.
Regarding “infinitesimal calculus”: Yes, at German schools they think that they are teaching Infinitesimalrechnung (see also these Google results)
It’s too bad that Zoran’s level of participation has dropped off recently, but I suppose in this instance he wouldn’t mind my making some small adjustments, which I’ll perform in a moment.
Wikipedia writes, “Calculus has historically been called “the calculus of infinitesimals”, or “infinitesimal calculus”.” But later in the article it says, “Following the work of Weierstrass, it eventually became common to base calculus on limits instead of infinitesimal quantities, though the subject is still occasionally called “infinitesimal calculus”.” – which I take to mean that this term is no longer much in fashion. I think “the differential/integral calculus” or simply “the calculus” is much more common (although that latter locution might also be obsolescent – I’m not sure).
Thanks, Todd. It’s good to have the now common use of terminology accurately reflected.
On general grounds though I suppose it is not entirely unreasonable to acknowledge implicit infinitesimals. Every formalization of the concept of differentiation knows about the infinitesimal in some sense, since, after all, the differential of a function is its infinitesimal rate of infinitesimal change. In this sense every incarnation of differentiation theory is some kind of computation with the infinitesimal, and in this sense may provide a calculus of the infinitesimal, even if isolated/explicit infinitesimal quantities are not part of the formalism.
In this sense it seems not unreasonable (even if uncommon) to use “infinitesimal analysis” to distinguish that part of analysis that concerns itself with differentiation/integration from the vast rest of analysis that still considers limits of sequences, just not those of the very special kind that compute infinitesimal rates of infinitesimal change.
That’s a very good point, Urs. I added a footnote to that effect (but please feel free to rewrite it if you wish).
I never liked just saying ‘calculus’ (although ‘the calculus’ is slightly better, and I often resign myself to saying ‘Calculus’, which the capital letter warning people that this is some sort of special name), because there are so many other calculuses in the world, even in mathematics. So saying ‘the infinitesimal calculus’ is a disambiguation. And since Calculus (as opposed to analysis!) is a practical method of doing calculations, it's not relevant whether one rigorously founds Calculus on infinitesimals, epsilontics, or indeed anything but gossamer and moonbeams; what's relevant is that it's a method of calculating with things (specifically, $\mathrm{d}x$ and the like) that one imagines as being infinitely small.
So Calculus is always the infinitesimal calculus, even though analysis is not always infinitesimal analysis.
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