Accounts of modular forms say that they were studied in the early 19th century, but then define modular forms using terminology that didn't exist until the 20th century. How did the earliest mathematicians to investigate modular forms define them? What motivated their exploration?
[Math] How did Gauss and contemporaries think of modular forms
ho.history-overviewmodular-forms
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The St. Andrews site is an invaluable resource. From that list, I picked (usually) at most one great mathematician born in each year from 1860 to 1910:
$\textbf{EDIT: By popular demand, the list now extends from 1849 to 1920.}$
1849: Felix Klein, Ferdinand Georg Frobenius
1850: Sofia Vasilyevna Kovalevskaya
1851: honorable mention: Schottky
1852: William Burnside
1853: honorable mentions: Maschke, Ricci-Curbastro, Schoenflies
1854: Henri Poincare
1856: Emile Picard (honorable mention: Stieltjes)
1857: honorable mention: Bolza
1858: Giuseppe Peano (honorable mention: Goursat)
1859: Adolf Hurwitz (honorable mention: Holder)
1860: Vito Volterra
1861: honorable mention: Hensel
1862: David Hilbert
1864: Hermann Minkowski
1865: Jacques Hadamard (honorable mention: Castelnuovo)
1868: Felix Hausdorff
1869: Elie Cartan
1871: Emile Borel (honorable mentions: Enriques, Steinitz, Zermelo)
1873: honorable mentions: Caratheodory, Levi-Civita, Young
1874: Leonard Dickson
1875: Henri Lebesgue (honorable mentions: Schur, Takagi)
1877: Godfrey Harold Hardy
1878: Max Dehn
1879: honorable mentions: Hahn, Severi
1880: Frigyes Riesz
1881: Luitzen Egbertus Jan Brouwer
1882: Emmy Amalie Noether (honorable mentions: Sierpinski, Wedderburn)
1884: George Birkhoff, Solomon Lefschetz
1885: Hermann Weyl (honorable mention: Littlewood)
1887: Erich Hecke (honorable mentions: Polya, Ramanujan, Skolem)
1888: Louis Joel Mordell (honorable mention: Alexander)
1891: Ivan Matveevich Vinogradov
1892: Stefan Banach
1894: Norbert Wiener
1895: honorable mention: Bergman
1896: Carl Ludwig Siegel (honorable mention: Kuratowski)
1897: honorable mention: Jesse Douglas
1898: Emil Artin, Helmut Hasse (honorable mentions: Kneser, Urysohn)
1899: Oscar Zariski (honorable mentions: Bochner, Krull, Ore)
1900: Antoni Zygmund
1901: Richard Brauer
1902: Alfred Tarski (honorable mention: Hopf)
1903: John von Neumann (hm's: Hodge, Kolmogorov, de Rham, Segre, Stone, van der Waerden)
1904: Henri Cartan (honorable mentions: Hurewicz, Whitehead)
1905: Abraham Adrian Albert
1906: Kurt Godel, Andre Weil (honorable mentions: Dieudonne, Feller, Leray, Zorn)
1907: Lars Ahlfors, Hassler Whitney (honorable mentions: Coxeter, Deuring)
1908: Lev Pontrjagin
1909: Claude Chevalley, Saunders Mac Lane (honorable mentions: Stiefel, Ulam)
1910: Nathan Jacobson (honorable mention: Steenrod)
1911: Shiing-shen Chern (honorable mentions: Birkhoff, Chow, Kakutani, Witt)
1912: Alan Mathison Turing (honorable mentions: Eichler, Zassenhaus)
1913: Samuel Eilenberg, Paul Erdos, Israil Moiseevich Gelfand (dis/honorable mention: Teichmuller)
1914: honorable mentions: Dantzig, Dilworth, Kac
1915: Kunihiko Kodaira (honorable mentions: Hamming, Linnik, Tukey)
1916: Claude Elwood Shannon (honorable mention: Mackey)
1917: Atle Selberg (honorable mentions: Iwasawa, Kaplansky)
1918: Abraham Robinson
1919: honorable mention: Julia Robinson
1920: Alberto Calderon
I agree with the commentators that the question is rather too broad, but here's an attempt to answer it anyway.
Readers of MO will likely have less familiarity with non-mathematical logic, so it might help to begin by skimming the tables of contents of the 18-volume (!) Handbook of Philosophical Logic to get some feeling for what people mean by "philosophical logic." [Edit: The preceding link no longer works; one can find some content using Google Books and the Wayback Machine.] It includes many topics that will likely be unfamiliar to mathematicians, such as temporal logic, multi-modal logic, non-monotonic reasoning, labelled deductive systems, and fallacy theory.
Roughly speaking, philosophical logic is the general study of reasoning and related topics. As in other areas of philosophy, this study is not necessarily formal. However, the success of formal methods in mathematical logic has led philosophers to try to formalize many other kinds of reasoning. Formalized modal logics are perhaps the best known of these. These are not always classified as "mathematical logic" because in mathematics one does not typically reason formally about concepts such as possibility, necessity, belief, etc. On the other hand, once a system of logic has been made sufficiently formal, it can of course be subject to mathematical study. Thus the boundary between (for example) formal modal logic and traditional mathematical logic is somewhat blurry. A notable example of the cross-fertilization that is possible here is Fitting and Smullyan's book on Set Theory and the Continuum Problem, which develops the (highly mathematical) subject of forcing from the perspective of modal logic, providing a fresh and completely rigorous approach to a now-classical mathematical subject.
If I had to summarize in one sentence, I would say that mathematical logic is the subfield of philosophical logic devoted to logical systems that have been sufficiently formalized to admit mathematical study. This is a slightly broader definition of mathematical logic than is customary, but I think it's a good definition in the context of this MO question, which tacitly seems to be asking if mathematicians have anything to learn from so-called "philosophical logic."
Best Answer
There is also the book of F. Klein, Development of Mathematics in XIX century, vol. I, which has a large chapter on Gauss which describes his work on modular forms. This was written in XX century, but Klein was essentially a XIX century mathematician, so you can see from this book "how did they think".