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I've browsed many math history books, but I've never read too deep into any single one. I always find myself reading the about the same facts and same people over and over -- the set of topics doesn't seem to reflect the richness of mathematical history.

I want to learn how the techniques and topics in math have changed over time, and what role and applications math played in various societies. I find knowing the historical development of more modern techniques really helpful for understanding math from school. As well, I like learning about the different ways different societies think of mathematical concepts.

These are many broad ideas, so I guess I am interested in any historical material (books, websites, etc) that doesn't follow the pervasive pattern I see -- where the exposition is crowded with random biographical details and uncritical praise of mostly upper-class white men, with little attention to the broad trends, and women and non-european mathematics barely mentioned. I feel that many math history books implicitly push the idea that math is primarily the result of individual genius, but I don't think that is true at all -- certainly there are visionaries but I believe the bulk of mathematical knowledge is created and recreated by innumerable people, most of whom create very little new research.

PS for context I am almost finished an undergraduate degree and have that level of math understanding. I especially like algebra and topology.

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  • $\begingroup$ Your observation that pop sources just echo earlier pop sources, including cultural/gender biases, is certainly correct... as in many spheres of human activity. I do not have a remedy for this... $\endgroup$ Mar 16 at 2:13
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    $\begingroup$ "mostly upper-class white men" Is that accurate? As I recall a number of famous mathematicians like Gauss and Fourier had working-class roots, others like Bessel had a middle-class background. If you wish to explore the female element in Western mathematics, you might want to start with Maria Gaetana Agnesi, who wrote one of the first textbooks on calculus (in the vernacular). $\endgroup$
    – njuffa
    Mar 17 at 18:57
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    $\begingroup$ This is exactly what I am talking about -- various developments in math attributed solely to a few men, making deities of of them and stripping those discoveries from their context. Would we have complex analysis without Cauchy? It may have developed at a later time, with different names for the theorems, but I have no doubt other people would do that work. Would anyone have the audacity to talk about schemes without Grothendieck? Most surely, but at a later time. $\endgroup$
    – Max
    Mar 18 at 21:04
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    $\begingroup$ Attributing entire branches of mathematics "almost" entirely to one man is a grand erasure of everything else that went into those discoveries, namely the foundations on which (for example) Cauchy already had, the capable teachers he had throughout his education, and his contemporaries who no doubt provided him with ideas and inspiration. Without any one of those components there would be no Cauchy integral formula. As well, it is not lost on me that you chose to cite only men, with the vision of Emmy Noether and many others within easy reach. $\endgroup$
    – Max
    Mar 18 at 21:06
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    $\begingroup$ This question could be strengthened considerably if the OP shared the material they've referenced. $\endgroup$
    – DJohnson
    Mar 28 at 17:18

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"I've browsed many math history books" -- I'm not sure what this means. And "I've never read too deep into any single one" is not any more encouraging. A book on the history of mathematics may have a focus: What is the history of the current mathematics -- that is, what led up to it? Maybe it will stop some decades or even a century before "now," whatever time "now" was when the book was written. Or maybe a book will be about the mathematical activity in human cultures throughout time and the world. A book may focus on a topic or certain topics; "calculus" is likely to be a somewhat popular choice in the current state of affairs. I have not seen one focused on an imperialistic point of view, that is, how empires, and lets broaden it to include trade, promulgated learning throughout an empire and its trading partners, and whether this was bad or good (no doubt a question with nuanced answers).

Of the first kind, which deals with the history of the mathematics we study in universities around the world, the books I've read have been classics: Morris Kline, Dirk Struik, Carl Boyer. In one way they're a bit dissatisfying: they more or less tell the same story. It would be worse, however, if they told vastly different stories. The main thread of these books traces the development of mathematics from some starting points to Europe from the Renaissance through the 19th century, at which point it begins to take root almost everywhere. The starting points are not in Europe, but in Egypt, Babylon, Persia, India, China, and Japan. At some point mathematics starts to take hold in Greece. From there, the next flourishing is in Alexandria and then to the Arabs. (Rome, while having a highly educated class, did not contribute much to mathematical research, it seemed.) From the Arabs, mathematics takes root again in Europe. China and Japan were somewhat isolated from Europe, and it seems the discoveries of each did not influence each other much. Or such is the broad arc of the story. If you're interested in how we arrived at the mathematics that people study through the first or second year an undergraduate program, this is it. The European flourishing brought modern versions of algebra, analysis, and statistics, which contain the principal subjects studied. Naturally, the people involved were European. I'd assume these Europeans were white, if ever I thought about it, but the races of the mathematicians were not mentioned (or I don't remember it). Most of them were male, which reflects inequities in the society at the time, not bias in the historian; women were mentioned by name, nonetheless, for the mere offense of having done some significant mathematics (a worthy reason, by the way, which the sarcasm is meant to highlight), but only very few were mentioned. I'm told that even "Men of Mathematics" features a female mathematician (one classic I have not read). I'd assume (again) that very few women had the opportunity to do significant mathematics, although there is some evidence that it could be done privately and not be published (be "silenced" if you prefer). It is important to recognize why only few women are mentioned -- and omitting to do so may reflect bias, or at least a blind spot, in the historian -- because historically, and even currently, the lesson drawn has been either that women are incapable or that they should not be allowed to do it, which is wrong. So wrong that it needs to be pointed out.

That "math is primarily the result of individual genius" is an impression one sometimes gets. Kline's book often shows how an idea evolves due to the work of several people, but I've usually preferred his book for its focus on ideas rather than on people, although it does contain biographical tidbits that the OP complains about. I think an in-depth history is a complicated business, and books with a broad scope like the ones I'm discussing do not take the space to go into the details. For a in-depth look at, say, Cauchy's contribution to single-variable calculus, see Judith Grabiner, The Origins of Cauchy's Rigorous Calculus. At some length, she details what Cauchy had to work with, how many of the ideas he put together had been thought either clearly or obscurely by others, and why what he did was significant and in what sense original. She also argues her thesis when it diverges from other historians', and an interested reader could follow up her references and try to sort it out. Alas, I don't believe any of the historical personages are female (perhaps I don't know everyone's gender) or people of color. I assume that is historically accurate and that she has not erased them from history. She does mention Algarotti's Newtonianism for the Ladies, so there is at least a nod to half the world's population. I think it shows how difficult it is to write about the history of an idea and satisfy a reader who wants it to be a history of disfranchised segments of the population.

If you want a glimpse at how mathematical ideas are built, some original books or papers explain what they are building on. For instance, Lagrange's Fonctions Analytiques or Riemann's Ueber die Darstellbarkeit.... The first few pages of each show what the author says he is responding to and building on.

If you want a different sort of focus, perhaps you would like Marcia Ascher's Ethnomathematics or other books. Whether they are historical or anthropological, or what the distinction is, is open to debate (one I'm unqualified to engage in, btw), but they have a non-western focus.

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You can do a search on Arab or Chinese mathematicians and you'll find lots to read. Gender, up until 150 years, was a strong barrier. But certainly you should check out the Marquise de Chatelet.

My advice is to ignore the mathematicians and focus on the math. Get a copy of Wallis' or Euler's Algebra or Descartes Géométrie and try to make sense of them. You'll learn a lot.

Or, if you want something a bit curated, look at Whiteside's edition of Newton's mathematical papers or Stillman Drake's or Paulo Palmieri's recreation of Galileo's experiments and calculations. Working through stuff like that will stand you in good stead when you discover the contributor none of the rest of us know about - but should.

Good luck with it!

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  • $\begingroup$ I would also add Indian maths in that first paragraph. $\endgroup$
    – Mauricio
    Mar 18 at 15:30
  • $\begingroup$ Thanks!! I'll check those out $\endgroup$
    – Max
    Mar 18 at 21:10
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You're right, many popular expositions of math and its history tend to round up the usual suspects over and over again.

Try browsing through the journal Historia Mathematica. Volumes more than four years old are open access. It is a grab bag of articles, references and book reviews about all aspects of math history. Perhaps five percent of the articles are about non-European math history. It's searchable.

archive.org is a repository of historical texts, some going back to the 1600's. You said that "knowing the historical development of more modern techniques really helpful for understanding math from school.", and here you can find a lot of that. Often modern treatments can be overly formal and abstract, whereas the historical texts are more down to earth. Just search by subject matter and mathematicians.

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One way to get around always reading about the same facts is to read more technical facts: look at books that delve into mathematics at a level beyond what would be considered suitable for a general audience. The following books are probably only going to be appreciated by someone who has some familiarity with the areas of math that they are about.

Bottazzini & Gray, Hidden Harmony - Geometric Fantasies: The Rise of Complex Function Theory

Dieudonne, History of Algebraic Geometry

Dieudonne, A History of Algebraic and Differential Topology: 1900-1960

Dieudonne, History of Functional Analysis

Kolmogorov and Yushkevich, Mathematics of the 19th Century Vol. 1: Mathematical Logic, Algebra, Number Theory, Probability Theory

Kolmogorov and Yushkevich, Mathematics of the 19th Century Vol. 2: Geometry, Analytic Function Theory

Kolmogorov and Yushkevich, Mathematics of the 19th Century Vol. 3: Function Theory According to Chebyshev, Ordinary Differential Equations, Calculus of Variations, Theory of Finite Differences

van der Waerden, A History of Algebra: From al-Khwārizmī to Emmy Noether

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  • $\begingroup$ Cool!! Thanks for the recommendations! $\endgroup$
    – Max
    Apr 10 at 21:53
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There are more general books eg History of Number which do give a fair look at non-Euro eg Egypt, Babylon, Sanskrit ... worth a look

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