¹ Frege, Gottlob, “What May I Regard as the Result of My Work?”, in Frege, , Posthumous Writings, ed. Hermes, Hans, Kambartel, Friedrich, and Kaulbach, Friedrich, trans. Peter Long and Roger White (Oxford, 1979), 184Google Scholar. I cite standard English translations, except for Frege, Begriffsschrift und andere Aufsätze (Hildesheim, 1964), which I have preferred to the standard English translation Conceptual Notation and Related Articles, ed. and trans. Terrell Ward Bynum (Oxford, 1972). Unattributed translations are my own.

² Margarete's illness is unknown, as is much about her and her relationship with Gottlob; see Kreiser, Lothar, Gottlob Frege: Leben—Werk—Zeit (Hamburg, 2001), 496CrossRefGoogle Scholar. As Michael Beaney has observed, Frege's work seems to have been made possible by the “very supportive domestic environment” created in turn by his mother Auguste, Margarete, and his housekeeper Meta Arndt, though we know unfortunately little about these women. Beaney, Michael, “Gottlob Frege: The Light and Dark Sides of Genius,” British Journal for the History of Philosophy 12/1 (2004), 159–68, at 167Google Scholar. On Frege's illness and leave see Kreiser, Gottlob Frege, 512–13.

³ Frege, “What May I Regard as the Result of My Work?” 184; translation modified.

⁴ Frege, Gottlob, Begriffsschrift, eine der arithmetischen nachgebildete Formelsprache des reinen Denkens (Halle, 1879)Google Scholar; Begriffsschrift und andere Aufsätze retains the original pagination. I reserve italics to distinguish Begriffsschrift the book from Begriffsschrift the notation.

⁵ Gottlob Frege, “Introduction to Logic,” in Frege, Posthumous Writings, 185–96.

⁶ Quine, W. V., “On Frege's Way Out,” Mind 64/254 (1955), 145–59, at 158Google Scholar. On Quine's role in the emergence of analytic philosophy in America see Isaac, Joel, “W. V. Quine and the Origins of Analytic Philosophy in the United States,” Modern Intellectual History 2/2 (2005), 205–34CrossRefGoogle Scholar.

⁷ On Frege's centrality to the analytic tradition see Beaney, Michael, “What Is Analytic Philosophy?” in Beaney, , ed., The Oxford Handbook of The History of Analytic Philosophy (Oxford, 2013), 3–29CrossRefGoogle Scholar; Reck, Erich H., “Wittgenstein's ‘Great Debt’ to Frege: Biographical Traces and Philosophical Theme,” in Reck, ed., From Frege to Wittgenstein: Perspectives on Early Analytic Philosophy (Oxford, 2002), 3–38CrossRefGoogle Scholar. Frege's correspondence with Husserl appears in Frege, Gottlob, Philosophical and Mathematical Correspondence, ed. Gottfried, Gabriel, Hermes, Hans, Kambartel, Friedrich, Thiel, Christian, Veraart, Albert, and McGuinness, Brian, trans. Hans Kaal (Chicago, 1980), 60–71Google Scholar. ’s critical “Review of E. G. Husserl, Philosophie der Arithmetik,” Zeitschrift für Philosophie und philosophische Kritik 103 (1894), 313–32, is translated in Frege's, Collected Papers on Mathematics, Logic, and Philosophy, ed. McGuinness, Brian (Oxford, 1984), 195–209Google Scholar. See also Føllesdal, Dagfinn, Husserl und Frege (Oslo, 1958)Google Scholar, trans. in Haaparanta, L., ed., Mind, Meaning and Mathematics (Dordrecht, 1994), 3–47CrossRefGoogle Scholar; Mohanty, J. N., Husserl and Frege (Bloomington, 1982)Google Scholar.

⁸ van Heijenoort, Jean, “Historical Development of Modern Logic,” Modern Logic 2/3 (April 1992), 242–255, at 242Google Scholar.

⁹ E.g. Putnum, Hilary, “Peirce the Logician,” Historia Mathematica 9/3 (1982), 290–301CrossRefGoogle Scholar. Quine, W. V. reconsidered his aforementioned designation of Frege as the father of modern logic in “Peirce's Logic,” in Quine, , Selected Logic Papers, enlarged ed. (Cambridge, MA, 1995), 258–65Google Scholar. On Bertrand Russell's role (via Van Heijenoort) in promoting the eventually widespread exaggeration of Frege's importance to turn-of-the-century logic see Anellis, Irving H., “Peirce Rustled, Russell Pierced,” Modern Logic 5/3 (1995), 270–328Google Scholar.

¹⁰ Recently some philosophers have reconsidered Begriffsschrift's merits: Macbeth, Danielle, Frege's Logic (Cambridge, MA, 2005)CrossRefGoogle Scholar; Landini, Gregory, Frege's Notations: What They Are and How They Mean (New York, 2012)CrossRefGoogle Scholar; Schlimm, Dirk, “On Frege's Begriffsschrift Notation for Propositional Logic: Design Principles and Trade-Offs,” History and Philosophy of Logic 39/1 (2018), 53–79CrossRefGoogle Scholar.

¹¹ The impulse to conceptualize scientific practice as “puzzle-solving” comes from Thomas Kuhn's “normal science,” a useful notion but one insufficiently grounded in concrete skills. Kuhn, Thomas, The Structure of Scientific Revolutions, 4th edn (Chicago, 2012), 35–42CrossRefGoogle Scholar. Warwick's influential study of the Cambridge Mathematical Tripos situated puzzle-solving techniques in training regimes for problem-based written exams: Warwick, Andrew, Masters of Theory: Cambridge and the Rise of Mathematical Physics (Chicago, 2003)CrossRefGoogle Scholar, “practice-ladenness” at 168. Kaiser, David, Drawing Theories Apart: The Dispersion of Feynman Diagrams in Postwar Physics (Chicago, 2005)CrossRefGoogle Scholar, has demonstrated that Feynman diagrams, far from being unambiguous formal representations of quantum phenomena, were subject to different uses at different sites based on local practitioners’ specific skills. Seth, Suman, Crafting the Quantum: Arnold Sommerfeld and the Practice of Theory, 1890–1926 (Cambridge, MA, 2010), 2CrossRefGoogle Scholar, has shown how attending to a “physics of problems” illuminates research cultures that reproduce researchers more successfully than cultures prioritizing an epistemologically sensational “physics of principles.”

¹² Shapin, Steven and Schaffer, Simon, Leviathan and the Air-Pump: Hobbes, Boyle, and the Experimental Life (Princeton, 2011), 60Google Scholar. How mathematical proving creates conviction in practice is a vexing question. Rotman argues that writing constitutes mathematics itself, “creating [mathematical] reality through the very language which claims to ‘describe’ it.” Rotman, Brian, Mathematics as Sign: Writing, Imagining, Counting (Stanford, 2000), 37Google Scholar. Netz, Reviel, The Shaping of Deduction in Greek Mathematics: A Study in Cognitive History (Cambridge, 1999), 163CrossRefGoogle Scholar, has analyzed Euclidean geometric proofs as “post-oral, but pre-written” exercises in socially situated, necessity-preserving persuasion. The classic presentation of proof as a socially negotiated process is Lakatos, Imre, Proofs and Refutations: The Logic of Mathematical Discovery, ed. Worrall, John and Zahar, Elie (Cambridge, 1976)CrossRefGoogle Scholar. Livingston, Eric, The Ethnomethodological Foundations of Mathematics (London, 1986)Google Scholar; and Rosental, Claude, Weaving Self-Evidence: A Sociology of Logic, trans. Catherine Porter (Princeton, 2008)Google Scholar, examine specific instances of proving sociologically. On the controversial emergence of mechanized proving without witnesses see MacKenzie, Donald, Mechanizing Proof: Computing, Risk, and Trust (Cambridge, MA, 2001), 101–49CrossRefGoogle Scholar; on hybrid human–computer proving practices see Dick, Stephanie, “AfterMath: The Work of Proof in the Age of Human–Machine Collaboration,” Isis 102/3 (2011), 494–505CrossRefGoogle ScholarPubMed.

¹³ Frege, Gottlob, Basic Laws of Arithmetic: Derived Using Concept-Script, trans. Philip A. Ebert, Marcus Rossberg, and Crispin Wright (Oxford, 2013), viiiGoogle Scholar.

¹⁴ Daston, Lorraine, “The Empire of Observation, 1600–1800,” in Daston, Lorraine and Lunbeck, Elizabeth, eds., Histories of Scientific Observation (Chicago, 2011), 81–113Google Scholar. Daston and Lunbeck, Histories of Scientific Observation, shows observation to be an amalgam of personal techniques and social coordination emerging out of medieval roots and evolving in disparate directions. For a dispute exemplifying the debate around visual apprehension's role in mathematics see Daston's, “The Physicalist Tradition in Early Nineteenth Century French Geometry,” Studies in History and Philosophy of Science Part A 17/3 (1986), 269–95CrossRefGoogle Scholar.

¹⁵ Daston, Lorraine and Galison, Peter, Objectivity (New York, 2010), 253Google Scholar.

¹⁶ On the diversity of representational methods and social contexts that give them meaning see Lynch, Michael and Woolgar, Steve, eds., Representation in Scientific Practice (Cambridge, MA, 1990)Google Scholar; Coopmans, Catelijne, Lynch, Michael, Vertesi, Janet, and Woolgar, Steve, eds., Representation in Scientific Practice Revisited (Cambridge, MA, 2014)CrossRefGoogle Scholar. Catarina Dutilh Novaes has theorized formal notations as technologies that expand a human agents’ cognitive capabilities by displacing the process of proving onto external symbols without semantic meaning. Novaes, Catarina Dutilh, Formal Languages in Logic: A Philosophical and Cognitive Analysis (Cambridge, 2012)CrossRefGoogle Scholar. Dutilh Novaes's characterization of formal systems is convincing in general, but Frege's case demands a special focus on the visual, hence my invocation of the admittedly imprecise category of “diagram.” Dominique Tournés, “Diagrams in the Theory of Differential Equations (Eighteenth to Nineteenth Centuries),” Synthese 186/1 (2012), 257–88, at 285, identifies two major categories of diagram in eighteenth- and nineteenth-century mathematics: physically accurate graphs and qualitative aids to intuition—a distinction less applicable to Begriffsschrift, which does not describe anything physical, but does not seem therefore qualitative. Macbeth, Danielle, “Diagrammatic Reasoning in Frege's ‘Begriffsschrift,’” Synthese 186/1 (2012), 289–314CrossRefGoogle Scholar, argues that reading Begriffsschrift diagrammatically is the key to Frege's claim that his proofs produce new knowledge despite following from given premises. Amirouche Moktefi has recently argued that diagrams in mathematics and logic should literally be considered scientific instruments, a view that finds support in the present study. Moktefi, Amirouche, “Diagrams as Scientific Instruments,” in Benedek, A. and Veszelszki, A., eds., Virtual Reality—Real Visuality: Visual, Virtual, Veridical (Frankfurt am Main: Peter Lang, 2017), 81–9Google Scholar.

¹⁷ Gottlob Frege, “Boole's Logical Calculus and the Concept-Script,” in Frege, Posthumous Writings, 9–46, at 12. Frege's invocation of “logic itself” raises the question of Platonism. For an argument that Frege was a Platonist see Burge, Tyler, “Frege on Knowing the Third Realm,” Mind 101/404 (1992), 633–50CrossRefGoogle Scholar; for the contrary view see Carl, Wolfgang, “Frege: A Platonist or a Neo-Kantian?”, in Beaney, Michael and Reck, Erich H., eds., Gottlob Frege: Critical Assessments of Leading Philosophers, 4 vols. (Abingdon, 2005), 1: 409–24Google Scholar.

¹⁸ In contrast, Wittgenstein would eventually come to see logical demonstration as a deeply social process. Wittgenstein, Ludwig, Remarks on the Foundations of Mathematics, ed. von Wright, G. H., Rhees, R., and Anscombe, G. E. M., trans. Anscombe, G. E. M., revised edn (Cambridge, MA, 1978), e.g. §§153–6Google Scholar. Drawing on Wittgenstein, Bloor, David, Knowledge and Social Imagery, second edn (Chicago, 1991), 97–9Google Scholar, argued that Frege's own notion of the objective is realized precisely by social institutions—an intriguing proposal but certainly anathema to Frege's own position.

¹⁹ See especially Warwick, Andrew and Kaiser, David’s programmatic “Conclusion: Kuhn, Foucault, and the Power of Pedagogy,” in Kaiser, David, ed. Pedagogy and the Practice of Science: Historical and Contemporary Perspectives (Cambridge, MA, 2005), 393–409Google Scholar. Important studies in this vein include Olesko, Kathryn M., Physics as a Calling: Discipline and Practice in the Königsberg Seminar for Physics (Ithaca, 1991)CrossRefGoogle Scholar; Warwick, Masters of Theory; Kaiser, Drawing Theories Apart; Seth, Crafting the Quantum.

²⁰ The four-volume Beaney and Reck, Gottlob Frege: Critical Assessments, provides a broad and relatively recent orientation to the literature.

²¹ Probably the most widely circulating use of the word “Begriffsschrift” before Frege's was in Adolf Trendelenburg's 1867 essay on Leibniz: “Ueber Leibnizens Entwurf einer allgemeinen Charakteristik,” in Trendelenburg, Historische Beiträge zur Philosophie, vol. 3 (Berlin, 1867), 1–47, at 4, which Frege cited in Begriffsschrift, v. On Frege and Leibniz see Kluge, Eike-Henner W., “Frege, Leibniz et alii,” Studia Leibnitiana 9/2 (1977), 266–74Google Scholar; Kluge, , “Frege, Leibniz, and the Notion of an Ideal Language,” Studia Leibnitiana 12/1 (1980), 140–54Google Scholar; Korte, Tapio, “Frege's Begriffsschrift as a Lingua Characteristica,” Synthese 174/2 (2010), 283–94CrossRefGoogle Scholar.

²² Frege, Begriffsschrift, iv.

²³ For a technically focused history of logicism see Grattan-Guinness, I., The Search for Mathematical Roots, 1870–1940: Logics, Set Theories and the Foundations of Mathematics from Cantor through Russell to Gödel (Princeton, 2000)Google Scholar, the coining of “logicism” at 500–2. Competing theories during logicism's heyday were formalism (mathematics as game-like manipulation of symbols) and intuitionism (mathematics as product of the human mind). On the Kantian context of Frege's probing the nature of mathematical knowledge see Sluga, Hans, Gottlob Frege (London, 1980)Google Scholar; and Gabriel, Gottfried’s essays “Frege als Neukantianer,” Kant-Studien 77 (1986), 84–101Google Scholar; “Frege, Lotze, and the Continental Roots of Early Analytic Philosophy,” in Reck, Frege to Wittgenstein, 39–51; “Frege and the German Background to Analytic Philosophy,” in Beaney, History of Analytic Philosophy, 280–97.

²⁴ Frege, Begriffsschrift, iv.

²⁵ Ibid.

²⁶ Ibid., v.

²⁷ Ibid.

²⁸ Ibid.

²⁹ Ibid., vi.

³⁰ Chad Wellmon has dubbed disciplinarity “the last technology of the Enlightenment,” interpreting the research university as a solution to the problem of “information overload.” Wellmon, Chad, Organizing Enlightenment: Information Overload and the Invention of the Modern University (Baltimore, 2015), 7Google Scholar.

³¹ For a comprehensive survey of mathematical notations see Cajori, Florian, A History of Mathematical Notations, 2 vols. (Mineola, NY, 1993)Google Scholar. For a concise overview of this long mathematical tradition emphasizing its relevance to modern logic see Dutilh Novaes, Formal Languages in Logic, 66–89. Frege specified that his notation “touch[ed] upon that of arithmetic most directly in its manner of employing letters.” Frege, Begriffsschrift, iv. In contrast to the many elements of novelty in the Begriffsschrift, such a use of letters in logic in fact went back to Aristotle—a usage that Reviel Netz has suggested Aristotle likely adapted from geometry. Netz, Shaping of Deduction in Greek Mathematics, 48–9.

³² For an interpretation of nineteenth-century chemistry foregrounding visual imagination see Rocke, Alan J., Image and Reality: Kekulé, Kopp, and the Scientific Imagination (Chicago, 2010)CrossRefGoogle Scholar. On the role of notations construed as “paper tools” in experimental practice see Klein, Ursula, Experiments, Models, Paper Tools: Cultures of Organic Chemistry in the Nineteenth Century (Stanford, 2003)Google Scholar.

³³ For brevity, I limit my discussion to the basic propositional connectives, omitting such elements as functions and generality that, while important, do not weigh on my argument regarding Begriffsschrift's visual layout.

³⁴ Frege, Begriffsschrift, 6.

³⁵ Ibid., 2.

³⁶ Ibid., 7–10, on Aristotelian modes at 9.

³⁷ Ibid., 10.

³⁸ In later work Frege introduced even more inverted, rotated, and creatively diacriticized symbols pilfered from disparate realms of academic print culture, including the recently published International Phonetic Language and the marks used to indicate poetic meter. Green, J. J., Rossberg, Marcus, and Ebert, Philip A., “The Convenience of the Typesetter: Notation and Typography in Frege's Grundgesetze der Arithmetik,” Bulletin of Symbolic Logic 21/1 (2015), 15–30CrossRefGoogle Scholar.

³⁹ To read the symbolism in figure 5 as “A and B” can be counterintuitive. The key is that Frege's conditional is noncausal: “If A then B” means more precisely “It cannot be that A is true but B false.” To negate a conditional is, then, to assert precisely the one scenario the conditional ruled out: the negation of “If A then B” is “A and not B.”

⁴⁰ E.g. Frege, Begriffsschrift, 6, 11, 12, 13.

⁴¹ Ibid., 11.

⁴² Frege's rhetoric, not intended in an optical sense, nonetheless bears the imprint of the world-class optics industry then booming in Jena, led by Frege's mentor Ernst Abbe, who co-owned an optical manufacturing company. As Tappenden, Jamie, “A Primer on Ernst Abbe for Frege Readers,” Canadian Journal of Philosophy supplementary vol. 38 (2008), 31–118CrossRefGoogle Scholar, has argued, Abbe's sophisticated investigations into the limits of microscopic vision invite us to read Frege's analogy as freighted with awareness that microscopic vision was neither passive nor unmediated. Frege's immersion in his optically oriented milieu was not purely intellectual—Abbe used his considerable fortune to support the sciences in Jena, making Frege a principal beneficiary; see Stelzner, Werner, “Ernst Abbe und Gottlob Frege,” in Gabriel, Gottfried and Kienzler, Wolfgang, eds., Frege in Jena: Beiträge zur Spurensicherung (Würzburg, 1997), 5–32Google Scholar.

⁴³ Vilkko, Risto, “The Reception of Frege's Begriffsschrift,” Historia Mathematica 25 (1998), 412–22CrossRefGoogle Scholar (reviews listed at 414), argues that this initial reception was less hostile than some later readers suggested.

⁴⁴ Ernst Schröder, review of Frege, Begriffsschrift, Zeitschrift für Mathematik und Physik 25, Historical Literary Section (1880), 81–94.

⁴⁵ Schröder, “Begriffsschrift,” 81.

⁴⁶ Ibid. Peckhaus, Volker, Logik, Mathesis universalis und allgemeine Wissenschaft: Leibniz und die Wiederentdeckung der formalen Logik im 19. Jahrhundert (Berlin, 1997), 14Google Scholar, argues that Leibniz's bearing on nineteenth-century logic had less to do with influence than with the authority his name invoked for mathematicians encroaching on the traditionally philosophical domain of logic. For an argument that Leibniz conceived of mathematical and philosophical notation in visual terms see Jones, Matthew L., The Good Life in the Scientific Revolution: Descartes, Pascal, Leibniz, and the Cultivation of Virtue (Chicago, 2006), 192–5CrossRefGoogle Scholar.

⁴⁷ Schröder, “Begriffsschrift,” 82.

⁴⁸ Schröder made no accusation of plagiarism, confident that Frege had arrived at Boolean achievements “all too independently.” Ibid., 84. For an argument that Schröder was correct about Frege's ignorance of Boole see Terrell Ward Bynum, “On the Life and Work of Gottlob Frege,” in Frege, Conceptual Notation, 1–54, at 15–24.

⁴⁹ Boole introduced his logic in Boole, George, Mathematical Analysis of Logic, Being an Essay towards a Calculus of Deductive Reasoning (Cambridge, 1847)Google Scholar; and refined it in Boole, An Investigation of the Laws of Thought, on Which Are Founded the Mathematical Theories of Logic and Probabilities (London, 1854).

⁵⁰ Schröder, “Begriffsschrift,” 89.

⁵¹ Ibid., 91.

⁵² Ibid., 88 n.

⁵³ Ibid., emphasis in the original.

⁵⁴ Venn, John, review of Frege, Begriffsschrift, Mind 5 (1880), 297Google Scholar.

⁵⁵ Tannery, Paul, review of Frege, Begriffsschrift, Revue philosophique de la France et de l’étranger 8 (1879), 108–9, at 108Google Scholar.

⁵⁶ Frege, “Boole's Logical Calculus,” 11. The lost original manuscript consisted of “103 closely written sides of quarto.” Quoted in ibid., 9 n.

⁵⁷ Ibid., 12.

⁵⁸ Ibid.

⁵⁹ Ibid., 46.

⁶⁰ Ibid., 12, original emphasis. This purported difference of purpose provided the starting point for Heijenoort, Jean van’s influential distinction between “Logic as Calculus and Logic as Language,” Synthese 17/3 (1967), 324–30CrossRefGoogle Scholar. The distinction between two major traditions in mathematical logic is useful, but rather overstated by Van Heijenoort. See Sluga, Hans, “Frege against the Booleans,” Notre Dame Journal of Formal Logic 28/1 (1987), 80–98CrossRefGoogle Scholar; Peckhaus, Volker, “Calculus Ratiocinator versus Characteristica Universalis? The Two Traditions in Logic, Revisited,” History and Philosophy of Logic 25 (2004), 3–14CrossRefGoogle Scholar.

⁶¹ Frege, “Boole's Logical Calculus,” 13.

⁶² Ibid., 24.

⁶³ Ibid., 12, emphasis added.

⁶⁴ Ibid., 13, translation modified.

⁶⁵ Frege, Gottlob, “Über die wissenschaftliche Berechtigung einer Begriffsschrift,” Zeitschrift für Philosophie und philosophische Kritik 81 (1882), 48–56Google Scholar, reprinted in Frege, Begriffsschrift und andere Aufsätze, 106–14, at 111.

⁶⁶ Gottlob Frege, “Über den Zweck der Begriffsschrift,” Jenaische Zeitschrift für Naturwissenschaft 16, supplement (1882), 1–10, reprinted in Frege, Begriffsschrift und andere Aufsätze, 97–106, at 103–4.

⁶⁷ Frege, Gottlob, Die Grundlagen der Arithmetik: Eine logisch mathematische Untersuchung über den Begriff der Zahl (Breslau, 1884)Google Scholar, trans. as Frege, , The Foundations of Arithmetic: A Logico-mathematical Enquiry into the Concept of Number, trans. Austin, J. L., second revised edn (New York, 1960)Google Scholar. Frege targeted psychologist and formalist attempts to ground the nature of numbers in the human mind or the manipulation of symbols respectively. His case against psychologism is often considered a decisive refutation, but for an argument that the widespread rejection of psychologism was due instead to a professional hostility toward experimental psychology on the part of “pure” philosophers see Kusch, Martin, Psychologism: A Case Study in the Sociology of Philosophical Knowledge (London, 1995)Google Scholar.

⁶⁸ The only substantive review was Giuseppe Peano, Revista di Matematica 5 (1895), 122–8.

⁶⁹ Bynum, “Life and Work,” 34 n. 1.

⁷⁰ Kreiser, Gottlob Frege, 280–84, provides a chart listing every course Frege ever offered, with enrollment numbers through the summer semester 1907, after which we lack enrollment records.

⁷¹ Scholem, Gershom, Walter Benjamin: The Story of a Friendship, trans. Harry Zohn (Philadelphia, 1981), 48–9Google Scholar.

⁷² Carnap, Rudolf, “Intellectual Autobiography,” in Schilpp, Paul Arthur, ed., The Philosophy of Rudolf Carnap (La Salle, 1963), 1–84, at 5Google Scholar.

⁷³ Gabriel, Gottfried, “Introduction: Frege's Lectures on Begriffsschrift,” in Frege, Gottlob, Frege's Lectures on Logic: Carnap's Student Notes, 1910–1914, ed. Gottfried Gabriel, trans. Erich H. Reck and Steve Awodey (Chicago, 2004), 1–15, at 11Google Scholar.

⁷⁴ Carnap, “Intellectual Autobiography,” 4.

⁷⁵ Carnap's notes, held at the Archive of Scientific Philosophy at the University of Pittsburgh, have been translated as Frege, Frege's Lectures on Logic.

⁷⁶ On Frege's modification of his system over time see Christian Thiel, “‘Not Arbitrarily and out of a Craze for Novelty’: The Begriffsschrift 1879 and 1893,” in Beaney and Reck, Gottlob Frege: Critical Assessments, 2: 13–28; for a comparison of Carnap's notes with Frege's published work see Gabriel, “Introduction,” 2–10.

⁷⁷ Frege defined a function as an “unsaturated” expression, e.g. “2x ³ + x,” where x is not part of the function but merely a placeholder for any definite argument that could complete the function. He defined a concept as a function whose value for any argument is either true or false; thus “x ² = 1,” a function whose value is “true” for 1 or –1 and “false” for all other arguments, is the concept we call “square root of 1.” He also distinguished between a sign's Sinn and its Bedeutung: in the case of a proper name, the Bedeutung is the object the name designates, the Sinn the objective sense of the expression used to name it. The names “1 + 1” and “5 – 3,” for example, express objectively different senses (Sinne) but mean the same object, the number 2. The distinction applies also to concept words and propositions; in particular, for Frege a proposition's Sinn is the thought it expresses while its Bedeutung is its truth value. See Frege's prose essays “Function and Concept,” in Frege, Collected Papers, 137–56; and “On Sense and Meaning” in ibid., 157–77.

⁷⁸ Frege, Lectures on Logic, 119, original emphasis. The proposition proved (reproduced here in Begriffsschrift in figure 8) was “If both a and b are limits as the argument goes to positive infinity, then a and b coincide.” Ibid. 103. Carnap's notes on the proof are in ibid., 103–19.

⁷⁹ Strategies of abbreviation are common in performances of proofs at the blackboard, a practice central to professional mathematics since the early nineteenth century. On the history of the blackboard in mathematical education see Kidwell, Peggy Aldrich, Ackerberg-Hastings, Amy, and Roberts, David Lindsay, Tools of American Mathematics Teaching, 1800–2000 (Washington, DC, 2008)Google Scholar, chap. 2, “The Blackboard: An Indispensable Necessity.” For an ethnographic account of mathematicians’ actual use of blackboards see Michael J. Barany and Donald MacKenzie, “Chalk: Materials and Concepts in Mathematics Research,” in Coopmans et al., Representation in Scientific Practice Revisited, 107–29. Begriffsschrift's function as a paper trail of unproved assumptions resembles Richard Feynman's original bookkeeping function for his diagrams in quantum electrodynamics—a function the diagrams fairly quickly lost in their employment by other physicists at far-flung locations. Kaiser, Drawing Theories Apart, 43–51.

⁸⁰ Gottlob Frege to Anton Marty, 29 Aug. 1882, in Frege, Correspondence, 99–102, at 102.

⁸¹ Frege, Lectures on Logic, 121, translation modified.

⁸² Carnap, “Intellectual Autobiography,” 5.

⁸³ Ibid.

⁸⁴ Flitner, Wilhelm, Erinnerungen 1889–1945 (Gesammelte Schriften 11) (Paderborn, 1986), 127Google Scholar.

⁸⁵ Frege, Foundations, xvi.

⁸⁶ Gershom Scholem to Christian Thiel, 10 March 1978, in Scholem, Gershom, Briefe III: 1971–1982, ed. Shedletzky, Itta (Munich, 1999), 177Google Scholar.

⁸⁷ The diary was first published as Gottlob Frege, “[Tagebuch],” ed. Gottfried Gabriel and Wolfgang Kienzler, Deutsche Zeitschrift Für Philosophie 42/6 (1994), 1067–98; trans. in Mendelsohn, Richard L., “Diary: Written by Professor Dr Gottlob Frege in the Time from 10 March to 9 April 1924,” Inquiry 39/3–4 (1996), 303–42CrossRefGoogle Scholar. Milkov, Nikolay, “Frege in Context,” British Journal for the History of Philosophy 9/3 (2001), 557–70CrossRefGoogle Scholar, has suggested that in light of Frege's earlier liberalism, the diary can be explained as “the product of a terminally ill man who was also facing financial ruin.” Mendelsohn, “Diary,” 304, disagrees, contending that “the views Frege expresses in this diary reflect a more deeply entrenched outlook.”

⁸⁸ Frege, Frege's Lectures on Logic, 67. Frege stands in contrast to Third Reich efforts to define an intrinsically German, anti-Jewish mathematics; see Mehrtens, Herbert, “Ludwig Bieberbach and ‘Deutsche Mathematik,’” in Phillips, Esther R., ed., Studies in the History of Mathematics (Washington, DC, 1987), 195–241Google Scholar.

⁸⁹ Frege, Basic Laws of Arithmetic, xi.

⁹⁰ Women were prohibited from enrolling in courses at the University of Jena until 1906–7. Kreiser, Gottlob Frege, 286.

⁹¹ Flitner, Erinnerungen 1889–1945, 126–7.

⁹² Following Seth, we might understand Carnap's inventiveness as a lighthearted move toward what Foucault called “initiatory time,” an apprenticeship model of training that allows for the creativity absent in the highly regimented “disciplinary time” that Foucault saw as defining modern pedagogy; accepting the disciplinary analysis of numerous modern institutions, Seth shows how initiatory time nonetheless persists in the higher reaches of scientific education. Seth, Crafting the Quantum, 64–70; Foucault, Michel, Discipline and Punish: The Birth of the Prison, trans. Alan Sheridan (New York, 1979), 156–62Google Scholar.

⁹³ Bertrand Russell to Gottlob Frege, 16 June 1902, in Frege, Correspondence, 130–31, at 130.

⁹⁴ Russell, Bertrand, The Philosophy of Logical Atomism (La Salle, 1985), 132Google Scholar.

⁹⁵ Gottlob Frege to Bertrand Russell, 22 June 1902, in Frege, Correspondence, 131–3, at 132.

⁹⁶ Ibid., 133.

⁹⁷ On Frege's unsuccessful revision see Quine, “On Frege's Way Out.”

⁹⁸ Gottlob Frege, “On Schoenflies: Die logischen Paradoxien der Mengenlehre,” in Frege, Posthumous Writings, 176–83, at 176.

⁹⁹ In late unpublished writings Frege doubted logic's adequacy as a foundation for arithmetic and looked to geometry instead; Gottlob Frege, “Numbers and Arithmetic,” in Frege, Posthumous Writings, 275–7.