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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).

Harry Gindi
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Tom Copeland
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    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. – Tom Copeland Jan 28 '20 at 00:00
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    You consider Strang's and Needham's books "rigorous" ?? – Alexandre Eremenko Jan 28 '20 at 00:01
  • @AlexandreEremenko, not in the sense that they begin with axiomatics and give full proofs but in the sense that the operations they build upon are standard/mainstream in analysis, geometry, and linear algebra. – Tom Copeland Jan 28 '20 at 00:11
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    E.g., I know mathematicians who still balk at using umbral notation or fractional calculus, taking the derivative of the Heaviside step function, or are unfamiliar with the utility of divergent series. – Tom Copeland Jan 28 '20 at 00:21
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    Strang's books that I know (and use in my teaching) lack not only rigorous proofs but even clear, unambiguous definitions. – Alexandre Eremenko Jan 28 '20 at 00:30
  • Okay, let's put it this way. I doubt that Strang's or Needham's books inspired the construction of cottage industries to explain them in contrast to work on the Laplace transform, distributions, convolutional algebras, etc. for Heaviside operational calculus, whole theories in analysis to explain Fourier's claims, Ramanujan's, Feynman's (ongoing), etc. – Tom Copeland Jan 28 '20 at 00:47
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    @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.” https://arxiv.org/abs/math/9405205 –  Jan 28 '20 at 03:36
  • Cartier's "Mathemagics"? – Tom Copeland Jan 28 '20 at 04:55
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    Anthing in QFT would qualify – Piyush Grover Jan 28 '20 at 12:38
  • @PiyushGrover, physics is mathematics' muse. – Tom Copeland Jan 28 '20 at 13:29
  • Although Dyson in a review (https://physicstoday.scitation.org/doi/abs/10.1063/1.3062600?journalCode=pto) of "Introduction to Fourier Analysis and Generalized Functions" by Lighthill claims it is fully rigorous, I found the style refreshingly distinct from the dry, axiomatic, approach of Schwartz with many surprising (to me on first reading) connections made. For me, in its uniqueness among books on distribution theory, it lies somewhere between the two categories of books I've tried to characterize. – Tom Copeland Jan 28 '20 at 14:31
  • @MattF. nice paper. Why not write it up as an answer? – Tom Copeland Jan 28 '20 at 19:57
  • @TomCopeland, it doesn't exactly answer the question. Neither Lawvere nor Blass looked at Euler's work and sought to understand or translate or work out details; they were just inspired by the (anti-rigorous or absent) methodology when exploring a different area. –  Jan 28 '20 at 20:08
  • Most of Blass's paper however is simply brimming with meaningful and rigorous combinatorial (and later, logical) considerations. I think of it as being more "fun" than actually "quirky". – Todd Trimble Feb 02 '20 at 12:25
  • Related https://mathoverflow.net/questions/115032/non-rigorous-reasoning-in-rigorous-mathematics?rq=1 – Tom Copeland Feb 02 '20 at 21:57
  • Newton's Philosophiæ Naturalis Principia Mathematica must have provoked similar reactions historically concerning the derivative (fluxions) and infinitesimals with various approches to understanding, summarized by Thurston in "On proof and progress in mathematics" on page 3 (https://arxiv.org/abs/math/9404236). – Tom Copeland Feb 02 '20 at 22:58
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    Would "Goedel, Escher, Bach" count? It's been several decades since I've read it but I remember it as being 'quirky', 'non-rigorous', 'inspiring' (to me anyway), and at least math-adjacent. – JCK Feb 09 '20 at 18:11
  • @JCK, maybe so. From Wiki: Douglas Hofstadter's books, especially Metamagical Themas and Gödel, Escher, Bach, play with many self-referential concepts and were highly influential in bringing them into mainstream intellectual culture during the 1980s. – Tom Copeland Feb 10 '20 at 15:22
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    Personally, I learnt from "Group Theory" (i.e. Birdtracks) by Predrag Cvitanovic more about Lie algebra than from all textbooks in my math lib (and I think the potential of graphical methods hasn't even scratched the surface) - but then, I'm a living quirk either. – Hauke Reddmann Aug 06 '20 at 10:37
  • @HaukeReddmann, thanks I wasn't aware of that book. I'll give it a look. You reminded of a book by Cvi and his colleagues that I've always wanted to get back to--the Chaos Book, or Chaos: Classical and Quantum . And, it has a fitting quote from Alice: (for those with ambiguity tolerance or delight in ambiguity) – Tom Copeland Aug 06 '20 at 23:05
  • It seems very pretty,” she said when she had finished it, “but it’s rather hard to understand!” (You see she didn’t like to confess, even to herself, that she couldn’t make it out at all.) “Somehow it seems to fill my head with ideas — only I don’t exactly know what they are!” Lewis Carroll, Through the Looking Glass – Tom Copeland Aug 06 '20 at 23:07

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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.

JoshuaZ
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    Nice. (The paper is "THE ANATOMY OF INTEGERS AND PERMUTATIONS" by Andrew Granville. // Links often are broken, so ... .) – Tom Copeland Jan 28 '20 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.

Todd Trimble
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Timothy Chow
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    Quarky, not so quirky. – Ben McKay Jan 28 '20 at 17:07
  • 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. – Tom Copeland Jan 28 '20 at 19:36
  • (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. – Tom Copeland Jan 28 '20 at 19:41
  • (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. – Tom Copeland Jan 28 '20 at 19:42
  • Quirky, as initially received historically , seems à propos. (Familiarity, in this case, doesn't breed contempt.) – Tom Copeland Jan 28 '20 at 19:46
  • Feyman's popularizing book QED is a lucid jewel. – Tom Copeland Jan 28 '20 at 19:51
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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},
}
Chris Sangwin
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    OK, it's a fine book, but is it quirky? is it non-rigorous? does it blindside you? – Gerry Myerson Feb 02 '20 at 11:08
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    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. – Chris Sangwin Feb 02 '20 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.

Ulrich Pennig
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Sebastian
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