Talk:Mathematical analysis

Proposed link update for vector analysis

The current "Vector analysis" section links to book articles. It would be more useful to link to Vector calculus which provides a broader overview of the field. What do you think? mw (talk) 12:08, 7 August 2024 (UTC)

Clearly, vector calculus is the correct {{main article}} of this stub section. I have fixed this.

By the way, I have removed the text-book style explanation of what is a vector, and the basic examples that do not belong to this section. D.Lazard (talk) 16:06, 7 August 2024 (UTC)

Famous textbooks

Famous is not a WP criterion, WP:Notability is. The following have been removed:

• Foundation of Analysis: The Arithmetic of Whole Rational, Irrational and Complex Numbers, by Edmund Landau
• Introductory Real Analysis, by Andrey Kolmogorov, Sergei Fomin
• Differential and Integral Calculus (3 volumes), by Grigorii Fichtenholz
• The Fundamentals of Mathematical Analysis (2 volumes), by Grigorii Fichtenholz
• A Course Of Mathematical Analysis (2 volumes), by Sergey Nikolsky
• Mathematical Analysis (2 volumes), by Vladimir Zorich
• A Course of Higher Mathematics (5 volumes, 6 parts), by Vladimir Smirnov
• Differential And Integral Calculus, by Nikolai Piskunov
• A Course of Mathematical Analysis, by Aleksandr Khinchin
• Mathematical Analysis: A Special Course, by Georgiy Shilov
• Theory of Functions of a Real Variable (2 volumes), by Isidor Natanson
• Problems in Mathematical Analysis, by Boris Demidovich
• Mathematical Analysis: A Modern Approach to Advanced Calculus, by Tom Apostol
• Real Analysis: Measure Theory, Integration, and Hilbert Spaces, by Elias Stein
• Complex Analysis: An Introduction to the Theory of Analytic Functions of One Complex Variable, by Lars Ahlfors
• Complex Analysis, by Elias Stein
• Functional Analysis: Introduction to Further Topics in Analysis, by Elias Stein
• Analysis (2 volumes), by Terence Tao
• Analysis (3 volumes), by Herbert Amann, Joachim Escher
• Real and Functional Analysis, by Vladimir Bogachev, Oleg Smolyanov
• Real and Functional Analysis, by Serge Lang

References

An article can be written about any notable textbook in this list. Then it is eligible for inclusion in section Notable textbooks. Rgdboer (talk) 21:41, 24 August 2025 (UTC)

Semi-protected edit request on 18 August 2026

Typo: change "opeerators" to "operators" in the Harmonic Analysis subsection ~2026-45451-58 (talk) 18:03, 18 August 2026 (UTC)

 Done RudolfRed (talk) 18:32, 18 August 2026 (UTC)

"used to study equations" in the sciences

@Sławomir Biały your version of the lead has:

Analysis has remained closely connected with applications in the sciences, where it is used to study equations, approximate one object by another, and estimate the accuracy of such approximations.

I feel like "used to study equations" is misleading / missing the point. It would perhaps be better to say that it is used to model processes or systems, possibly with a wikilink to Mathematical model. (I'm not sure about the correct phrasing though.) The phrasing "approximate one object by another" is also kind of clunky and ambiguous, since it's not obvious what kind of "objects" we are talking about. I wonder if we can find an external source somewhere with a more elegant description. –jacobolus (t) 19:25, 10 September 2026 (UTC)

I suggest

Analysis has remained closely connected with applications in the sciences, where it is used to model phenomena that involve continuous variations.

D.Lazard (talk) 21:15, 10 September 2026 (UTC)

I like this one. Go for it. –jacobolus (t) 21:27, 10 September 2026 (UTC)

This leaves put approximation and error, which is the point at opposed to "continuous phenomena". Mathematocal analysis is also used in discontinuous modeling, such as a Poisson process. How about "where it is used to study mathematical models, construct approximations, and estimate their accuracy." Sławomir Biały (talk) 05:55, 11 September 2026 (UTC)

Discontinuous modeling, approximations, and accuracy estimations are methods for dealing with models of phenomena involving continuous variations. So, there are not excluded by the suggested formulation. On the other hand, your formulation is misleading by excluding a large part of the relationship between analysis and applications. A prominent example is distribution theory that is fundamental for quantum mechanics and has nothing to do with your formulation. D.Lazard (talk) 09:02, 11 September 2026 (UTC)

This reverses my point. My wording is deliberately inclusive. Yours focuses only on continuous variations. But much of mathematical analysis is not about continuous variations. Examples include measure theory, probability theory, symbolic dynamics. I have no idea what the example of quantum mechanics is supposed to illustrate, but if you think that operator estimates and approximations play no role in the theory of distributions or in quantum mechanics, then you are very very wrong. And, although it's not applied, most of number theory is not concerned with continuous quantities. For example, estimating the prime counting function is famously done using mathematical analysis, even though it is integer-valued. Sławomir Biały (talk) 09:10, 11 September 2026 (UTC)

Your wording seems too broad to me. We use all sorts of other parts of mathematics to "study models".

The Princeton Companion has some possibly relevant bits.

Thus, as a first approximation, one might say that a branch of mathematics belongs to analysis if it involves limiting processes, whereas it belongs to algebra if you can get to the answer after just a finite sequence of steps. However, here again the first approximation is so crude as to be misleading, and for a similar reason: if one looks more closely one finds that it is not so much branches of mathematics that should be classified into analysis or algebra, but mathematical techniques.

[...] As this example suggests, although analysis often involves limiting processes and algebra usually does not, a more significant distinction is that algebraists like to work with exact formulas and analysts use estimates. Or, to put it even more succinctly, algebraists like equalities and analysts like inequalities.

–jacobolus (t) 09:28, 11 September 2026 (UTC)

This is, in essence, what I had hoped to convey with the wording: that analysis is concerned with approximations and controlling errors in models. The sentence should be read as a whole: approximation and error are the sine qua non technique of analysis, not whether the models are discrete or continuous. Sławomir Biały (talk) 09:35, 11 September 2026 (UTC)

It is numerical analysis not mathematical analysis that is concerned with "approximations and controlling errors in models". For giving another example than distributions, let recall that differential calculus is a foundational part of analysis that has been introduced by Newton for modeling gravitation, not for providing approximations and their estimations. Confusing numerical analysis and mathematical analysis and restricting mathematical analysis to numerical analysis is definitively misleading. D.Lazard (talk) 11:13, 11 September 2026 (UTC)

I strongly dispute that the concept of an estimate or an approximation "is numerical analysis". Differential calculus is a case in point: the derivative comes from the best linear approximation, its definition requires a little o estimate of the error in that approximation. Taylor's theorem gives big O control of the same error. The basic concept in functional analysis is a bounded linear operator, whose defining property is that perturbations are controlled by a linear estimate. In differential equations, an a priori estimate controls uniqueness, stability, and senstitivity of a problem. These are not features of "numerical analysis", but are fundamental techniques of these subjects. Most of analytic number theory, too, is about approximations and error estimates. Surely analytic number theory is not "numerical analysis"? Estimates, approximation, and error are woven in to every major area of analysis, not merely the "numerical" aspects. The Princeton Companion precisely makes this broader methodological point. Sławomir Biały (talk) 11:46, 11 September 2026 (UTC)

I like Trefethen's definition of numerical analysis (which he has written about in various places, among them the Princeton Companion): "Numerical analysis is the study of algorithms for solving the problems of continuous mathematics". –jacobolus (t) 18:38, 11 September 2026 (UTC)

History section

Hi @Sławomir Biały, I reverted your change of "The modern foundations of mathematical analysis were established in 17th century Europe" to "The precursors ...". I think your disagreement with whoever wrote this part is largely a matter of opinion about what the phrase "foundations of mathematical analysis" means, and perhaps what "modern" means in this context ("modern" is often used in contrast to "ancient", so everything in 17th century Europe would be "modern", but sometimes is instead used to mean "since the 20th century"), but if you're going to change this, it should be more systematic, since a few sentences later the article claims that "Descartes's publication of La Géométrie in 1637 [...] is considered to be the establishment of mathematical analysis". Reliable sources, especially in discussions of history, are probably not going to agree on the precise scope of "modern mathematical analysis", so it might also be worth trying to discuss how that should be chosen/presented to readers. –jacobolus (t) 16:07, 12 September 2026 (UTC)

I agree this section should be rewritten. The first sentence is particularly wrong: I can find no evidence that this view is even remotely supported by the cited source. Instead, it cuts the other way: foundations were not a concern until the 18th century, and the foundations of modern analysis were not to come until the 19th century. "Modern analysis" refers to post-19th century foundations. The cited source uses the term this way many times. If it is used another way, it requires another source. Sławomir Biały (talk) 16:44, 12 September 2026 (UTC)

I think you are interpreting the word "foundations" as a specific jargon word (as in Foundations of mathematics), while the meaning intended in this context by whoever wrote this passage is something closer to the plain-language meaning of the word. (But I agree that is confusing/ambiguous and we should try to be more careful and explicit.) –jacobolus (t) 17:54, 12 September 2026 (UTC)

Apparently this phrasing was introduced by user:Rurik the Varangian in 2014 in special:diff/601899943. –jacobolus (t) 18:02, 12 September 2026 (UTC)

If the intended meaning in this context is some other meaning than the one used in the actual subject, the answer is to use a different word, not abuse a word with a multiplicity of meaning in that context. Sławomir Biały (talk) 18:06, 12 September 2026 (UTC)