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In contrast with such lucid, pedagogical, inspiring books such as Visualizing Complex Analysis by Needham and Introduction to Applied Mathematics by Strang, I've had the pleasure of coming across the non-rigorous, thought-provoking/stimulating, somewhat quirky works of Heaviside on operational calculus and divergent series; of Ramanujan on series, in particular, his use of his master theorem/formula (as explicated by Hardy); and the relatively unknown posthumous notes of Bernard Friedman on distributions and symbolic/operational calculus Lectures on Applications-Oriented Mathematics.

These works just blindside you. You think, "What the hey?" and slowly they grow on you and you start to understand them after further research using other texts, translating the terminology/concepts, and working out details. You're left with a deeper understanding and appreciation of the originality and applicability of the work--much like Hardy professed the day after he received Ramanujan's letter, no doubt.

(Friedman's Applied Mathematics, in contrast, is of a very different nature and an immediately enlightening intro to its topics.)

Any similar experiences with other mathematical works?

(This question is not research-level per se and may be more appropriate for MSE, but certainly the works cited have inspired and continue to inspire much advanced research into the related topics, and the question falls in a similar category to MO-Q1 and the MO-Qs that pop up in the Related section of that question).

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    $\begingroup$ Rota may have had a similar experience. He speaks of "the eerie witchcraft" of the umbral calculus as originally presented by Blissard and sucessors, before being tamed by Rota and his colleagues. $\endgroup$
    – Tom Copeland
    Commented Jan 28, 2020 at 0:00
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    $\begingroup$ You consider Strang's and Needham's books "rigorous" ?? $\endgroup$
    – Alexandre Eremenko
    Commented Jan 28, 2020 at 0:01
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    $\begingroup$ Strang's books that I know (and use in my teaching) lack not only rigorous proofs but even clear, unambiguous definitions. $\endgroup$
    – Alexandre Eremenko
    Commented Jan 28, 2020 at 0:30
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    $\begingroup$ @AndreasBlass’s “Seven Trees in One” is a quirky paper which draws inspiration from “the style of eighteenth-century analysis, where meaningless computations (e.g. manipulating divergent series as though they converged absolutely and uniformly) somehow gave correct results.” arxiv.org/abs/math/9405205 $\endgroup$
    – user44143
    Commented Jan 28, 2020 at 3:36
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    $\begingroup$ Anthing in QFT would qualify $\endgroup$
    – Piyush Grover
    Commented Jan 28, 2020 at 12:38

4 Answers 4

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Andrew Granville and Jennifer Granville's "Prime Suspects" is a comic book which despite being phrased as a murder mystery does a surprisingly lucid job connecting ideas involving primes with ideas involving permutations. This paper Andrew Granville outlining the basic idea is pretty non-rigorous but fascinating.

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    $\begingroup$ Nice. (The paper is "THE ANATOMY OF INTEGERS AND PERMUTATIONS" by Andrew Granville. // Links often are broken, so ... .) $\endgroup$
    – Tom Copeland
    Commented Jan 28, 2020 at 13:43
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To elaborate on Piyush Grover's comment, large swaths of theoretical physics can be considered "non-rigorous yet inspiring." The adjective "quirky" might not be so apposite, though. But perhaps somewhat in the spirit of the question is Richard Feynman's semi-popular book QED: The Strange Theory of Light and Matter. There are still some open questions about the mathematical consistency of QED.

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    $\begingroup$ Quarky, not so quirky. $\endgroup$
    – Ben McKay
    Commented Jan 28, 2020 at 17:07
  • $\begingroup$ From Wikipedia (in agreement with recall of readings many years ago): Historically, as a book-keeping device of covariant perturbation theory, the graphs were called Feynman–Dyson diagrams or Dyson graphs, because the path integral was unfamiliar when they were introduced, and Freeman Dyson's derivation from old-fashioned perturbation theory was easier to follow for physicists trained in earlier methods. Feynman had to lobby hard for the diagrams, which confused the establishment physicists trained in equations and graphs. $\endgroup$
    – Tom Copeland
    Commented Jan 28, 2020 at 19:36
  • $\begingroup$ (Cont.) From "Julian Schwinger: Nuclear Physics, the Radiation Laboratory, Renormalized QED, Source Theory, and Beyond" by Kimball Milton: The formal solution of Schwinger’s differential equations was Feynman’s functional integral; yet while the latter was ill-defined, the former could be given a precise mean- ing, and for example, required the introduction of fermionic variables, which initially gave Feynman some difficulty. $\endgroup$
    – Tom Copeland
    Commented Jan 28, 2020 at 19:41
  • $\begingroup$ (Cont.) It may be fair to say that while the path integral formulation to quantum field theory receives all the press, the most precise exegesis of field theory is provided by the functional differential equations of Schwinger resulting from his action principle. $\endgroup$
    – Tom Copeland
    Commented Jan 28, 2020 at 19:42
  • $\begingroup$ Quirky, as initially received historically , seems à propos. (Familiarity, in this case, doesn't breed contempt.) $\endgroup$
    – Tom Copeland
    Commented Jan 28, 2020 at 19:46
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This book is very elementary, but has a lot of excellent material.

@BOOK{Polya54,
  author =       {Polya, G.},
  title =        {Mathematics and Plausible Reasoning. Vol.1: Induction and Analogy in Mathematics. Vol 2. Patterns of Plausible Inference},
  publisher =    {Princeton University Press},
  year =         {1954},
}
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    $\begingroup$ OK, it's a fine book, but is it quirky? is it non-rigorous? does it blindside you? $\endgroup$
    – Gerry Myerson
    Commented Feb 2, 2020 at 11:08
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    $\begingroup$ Lots of it is non rigours, indeed the whole section on analogy. Polya was very keen on guessing and pattern spotting. Some bits did blindside me when I first read it. I commend the book, particularly for less experienced mathematics students. $\endgroup$
    – Chris Sangwin
    Commented Feb 2, 2020 at 18:49
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I enjoyed reading the book 'Quantization, Classical and Quantum Field Theory and Theta Functions' by Andrej Tyurin very much. It is certainly not rigorous but it was very inspiring for me.

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