Are there any pedagogical/tutorial articles presenting the historical development of modern real analysis?

Context and some information on my background.

I am currently self-studying Understanding Analysis by Stephen Abbott (2015), with a view to moving to Principles of Mathematical Analysis by Walter Rudin (1976) at some point. This is my first exposure to a more rigorous, abstract, and at times beautiful, style of mathematics; most of my previous exposure to mathematics has been skewed towards applied/computational areas. Hence the type of article I am looking for is at the 1st year undergraduate level.

Article style.

Primarily, I am soliciting recommendation on tutorial articles which contextualise mathematical developments of standard topics in analysis within a historical perspective, as a supplement to some of the epilogues in Abbott.

I have previously read articles aimed at mathematically literate, but not necessarily deeply technically proficient audiences whereby the author introduces the topic in layman's terms, then some mathematical formalism, then summarises some crowning achievements of the field. All of this is interwoven with a historical narrative of how these tools developed through attempts to resolve seminal problems/paradoxes, together with broader debates in mathematics at the time.

As an example, in probability and statistics, I really enjoyed Part II: A tutorial on probability, measure and the laws of large numbers (page 56 onwards) of the following article by Richard Mauldin in a special edition of Los Alamos Science dedicated to the mathematician Stanislaw Ulam:

Mauldin, R. D. (1987). Probability and nonlinear systems. Los Alamos Science 15, Special Issue 1987 15. 52-90.


I've had experience with both formal education and self-study in economics, machine learning and statistics. Generally I've found that in the case of formal education, time constraints generally mean that the courses are taught with a view to getting students up to speed quickly on a battery of techniques/proofs/computational skills. In the case of mathematical monographs, I've often found that historical developments are often treated as "literature reviews" at the end of a chapter. This is not a deficiency as the constraint of fluent mathematical flow often does not allow for interwoven comments on historical developments. What can be lost however, is valuable historical context.

As a final example, when I studied time-series techniques as part of formal economics training using Time-Series Analysis by Hamilton, and during self-study using Introductory Time Series Analysis by Brockwell and Davis, ergodicity and ensembles of time series are presented without much discussion. It was only when I skimmed some of the communications engineering roots of time series e.g. the work of Norbert Wiener, did it become apparent that modern persentations have decontextualised these methods from their original statistical mechanics overtones (it seems that Norbert Wiener discussed time series in relation to Birkhoff's ergodic theorem).

Any suggestions from the community here would be greatly appreciated.

  • 1
    $\begingroup$ The Calculus Gallery by Dunham is a really good book. $\endgroup$
    – littleO
    Jul 11 at 19:51

The book Analysis by Its History by E. Hairer and G. Wanner provides a thorough and instructive description of historical development in chronological order. I appreciate the wealth and quality of the graphics as well as the nicely selected examples of citations from historical sources.


Perhaps you would find A Radical Introduction to Real Analysis interesting. https://www.maa.org/press/maa-reviews/a-radical-approach-to-real-analysis is a review by the MAA.


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