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I really like History and this includes the history of Mathematics. Personally, I have dabbled into many old Math works. Since my base knowledge is only up to Calculus 2, I can only appreciate the history of calculus. I have read "Introduction to analysis of the infinite" by Euler, some chapters of "Foundations of Differential Calculus", some passages from "A Treatise on Fluxions" by Collins Maclaurin (particularly on his derivation of the Taylor series), and many papers of Euler, D'Alembert and Lagrange.

I find them, apart from Euler, conceptually difficult to understand. Obviously, they don't rely on concepts that are taught in modern curriculum, and the antiquated notations do not help. For Maclaurin, for example, he relies a lot on geometric concepts, just like Newton.

For example, the origin of the now essential power rule for the integral of a polynomial function, $$\int x^{n} \,dx=\dfrac{x^{n+1}}{n+1}+C$$

is found during the XVII and XVIII century, but they are exceedingly difficult to follow the original line of thoughts, because the method being employed is geometric in nature, the so called the "Method of Indivisibles".

So my question is, do you think the field of History of Mathematics underdeveloped? I personally think so. I have a few reasons for why such is the state of being of the matter:

  1. It is challenging to re-learn Math concepts and arguments of past mathematicians. Obviously, this increases as the time period recedes into distant past.
  2. It is unnecessary, because nobody is going to learn much from old and distant Math because new ideas have been developed. People can learn a lot more from compacted version of modern textbooks and modern curricula.
  3. Along that line of reasoning, mathematicians and students who will become future mathematicians are only interested in solving unsolved problems. Advance in mathematics is about looking forward, not backward. There are no reasons to go back further, except if the history of a mathematical concept or problem is directly related to the problem you are trying to solve.

I think out of all fields of Mathematics, Calculus are studied the most. Apart from this, little are written on the history of Linear Algebra, for example, or Topology. The only book that I have read that is concerned with a fairly modern mathematical concept is "Lebesgue's Theory of Integration: Its Origins and Development" by Thomas Hawkin.

If you look at the field of history of Mathematics, it is a small overlap between History and Mathematics, as a discipline. It is very insignificant because few modern readers are interested in the history of Mathematics as compared to the fields of political history, cultural and economic history.

Do you think there are any values in relearning old Math? Perhaps as a hobby, for modern mathematicians have too many concerns than to learn Math from the past.

DMcMor
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    I doubt that historians of mathematics would agree with you. https://en.wikipedia.org/wiki/Category:Historians_of_mathematics

    Not sure that a lifetime would be enough to read their work.

    –  Jan 26 '21 at 20:54
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    Take this to https://hsm.stackexchange.com/ . – kimchi lover Jan 26 '21 at 20:58
  • I think you have pointed out a fallacy I have when thinking about this topic. I think there is a correlation between books being published and the development of the field. Obviously I am wrong here. – James Warthington Jan 26 '21 at 20:58
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    Are you familiar with John Stillwell's (characteristically excellent) Mathematics and its History? It spends a great deal of time talking about the origins of ideas from all walks of mathematics, and translates a lot of original arguments into modern language. – HallaSurvivor Jan 26 '21 at 20:58
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    There's quite a lot written on history of mathematics, actually. You may for example be interested in Plato's Ghost (one of my favorites), but that's just one of a countless number. You are underestimating how much work on history of mathematics is out there, to put it mildly. – Noah Schweber Jan 26 '21 at 21:03
  • There are also academic journals devoted solely to history of mathematics, see e.g. here. – Noah Schweber Jan 26 '21 at 21:05
  • Grossly ignorant is a fitting description of my understanding of this field. :) – James Warthington Jan 26 '21 at 21:05
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    And some (necessarily recent) books on the relevatively recent history of mathematics can be found at this mathoverflow question. – Noah Schweber Jan 26 '21 at 21:07
  • Even though I think the answer to the question in the title is no, the question seems by nature opinion-based and unanswerable in the current format. One could also ask whether maths as a field is underdeveloped, and though the volume of publication is great, one could argue it could be greater given the current underfunding problem in academia as a whole and that many in our society shy away from math due to prejudice. I'd suggest the OP could ask for instance about resources that explore and explain 18th century math, or books on the history of modern algebra, which would be answerable. – cesaruliana Jan 26 '21 at 22:35
  • See at least Archive for History of Exact Sciences. But there are many others... – Mauro ALLEGRANZA Jan 27 '21 at 09:25
  • Although the comments do a great job answering the question in the title, it appears to me that the "reasons" from the OP are actually answering a different question: "Why do people generally not learn mathematics from a historical perspective?" I can't speak authoritatively, but for myself (1) is much closer to true than the others. As you have noticed, standards of proof and exposition do change— to say nothing of notation! Also, while nobody has ever set out to write a bad paper, results are often streamlined by future expositors, who have the benefit of newer results and broader context. – Eric Nathan Stucky Feb 18 '21 at 07:58
  • An aside: "Since my base knowledge is only up to Calculus 2, I can only appreciate the history of calculus. [...] I think out of all fields of Mathematics, Calculus are studied the most. Apart from this, little are written on..." This is an extremely common intellectual trap— we think that the things we know the most about, are the best understood/most important. At the risk of being too cutesy, that's because "We know what we know, and we don't know what we don't know!" This is of course not limited to math or even academia; you (and we all!) would do well to internalize how wrong it can be. – Eric Nathan Stucky Feb 18 '21 at 08:00

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