Element contents
Mathematical Notations
Published online by Cambridge University Press: 30 January 2025
Summary
This Element lays the foundation, introduces a framework, and sketches the program for a systematic study of mathematical notations. It is written for everyone who is curious about the world of symbols that surrounds us, in particular researchers and students in philosophy, history, cognitive science, and mathematics education. The main characteristics of mathematical notations are introduced and discussed in relation to the intended subject matter, the language in which the notations are verbalized, the cognitive resources needed for learning and understanding them, the tasks that they are used for, their material basis, and the historical context in which they are situated. Specific criteria for the design and assessment of notations are discussed, as well as ontological, epistemological, and methodological questions that arise from the study of mathematical notations and of their use in mathematical practice.
Keywords
- Type
- Element
- Information
- Online ISBN: 9781009472128Publisher: Cambridge University PressPrint publication: 30 January 2025
References
Alexander, J. W. (1928). Topological invariants of knots and links. Transactions of the American Mathematical Society, 30(2):275–306.CrossRefGoogle Scholar
Alexander, J. W. and Briggs, G. B. (1926). On types of knotted curves. Annals of Mathematics, 28(1/2):562–586.CrossRefGoogle Scholar
Allwein, G. and Barwise, J., editors (1996). Logical Reasoning with Diagrams. Oxford University Press, Oxford.CrossRefGoogle Scholar
Avron, A. (1988). The semantics and proof theory of linear logic. Theoretical Computer Science, 57:161–184.CrossRefGoogle Scholar
Bauer, F. L. (2002). From the stack principle to ALGOL. In Broy, M. and Denert, E., editors, Software Pioneers, pages 26–42. Springer, New York.CrossRefGoogle Scholar
Bellucci, F., Moktefi, A., and Pietarinen, A.-V. (2018). Simplex sigillum veri: Peano, Frege, and Peirce on the primitives of logic. History and Philosophy of Logic, 39(1):80–95.CrossRefGoogle Scholar
Blanchette, P. (2012). Frege’s Conception of Logic. Oxford University Press, Oxford.CrossRefGoogle Scholar
Boole, G. (1854). An Investigation of the Laws of Thought, on Which Are Founded the Mathematical Theories of Logic and Probabilities. Walton and Maberly, London.CrossRefGoogle Scholar
Bos, H. J. M. (2001). Redefining Geometrical Exactness: Descartes’ Transformation of the Early Modern Concept of Construction. Springer, New York.CrossRefGoogle Scholar
Brown, J. R. (2008). Philosophy of Mathematics: A Contemporary Introduction to the World of Proofs and Pictures. Routledge, New York, 2nd edition.Google Scholar
Cajori, F. (1925). Leibniz, the master-builder of mathematical notations. Isis, 7(3):412–429.CrossRefGoogle Scholar
Cajori, F. (1928a). A History of Mathematical Notations, volume I: Notations in Elementary Mathematics. Open Court Publishing Company, Chicago.Google Scholar
Cajori, F. (1928b). Uniformity of mathematical notations: Retrospect and prospect. In Fields, J. C., editor, Proceedings of the International Congress of Mathematicians, Toronto, August 11–16, 1924, volume 2, pages 924–936, University of Toronto Press, Toronto.Google Scholar
Cajori, F. (1929). A History of Mathematical Notations, volume II: Notations Mainly in Higher Mathematics. Open Court Publishing Company, Chicago.Google Scholar
Carnap, R. (1929). Abriss der Logistik, mit besonderer Berücksichtigung der Relationentheorie und ihrer Anwendungen, volume 2 of Schriften zur wissenschaftlichen Weltauffassung. Springer Verlag, Vienna.Google Scholar
Carter, J. (2010). Diagrams and proofs in analysis. International Studies in the Philosophy of Science, 24(1):1–24.CrossRefGoogle Scholar
Carter, J. (2024). Introducing the Philosophy of Mathematical Practice. Cambridge University Press, Cambridge, UK.CrossRefGoogle Scholar
Cauchy, A. (1846). Mémoire sur les fonctions de variables imaginaires. Comptes rendus des Séances de l’Académie des Sciences. Paris, 23:271–275. Reprinted in Oeuvres Completes, series 2, vol. 23, pp. 405–435, Gauthier-Villars, 1932.Google Scholar
Chase, W. G. and Simon, H. A. (1973). Perception in chess. Cognitive Psychology, 4:55–81.CrossRefGoogle Scholar
Cheng, P. C.-H. (2020). Truth diagrams versus extant notations for propositional logic. Journal of Logic, Language and Information, 29(2): 121–161.CrossRefGoogle Scholar
Chrisomalis, S. (2010). Numerical Notation: A Comparative History. Cambridge University Press, Cambridge, UK.CrossRefGoogle Scholar
Chrisomalis, S. (2020). Reckonings: Numerals, Cognition, and History. Massachusetts Institute of Technology Press, Cambridge, MA.CrossRefGoogle Scholar
Couturat, L., Ladd-Franklin, C., and Peirce, C. S. (1902). Symbolic logic. In Baldwin, J. M., editor, Dictionary of Philosophy and Psychology, pages 640–651. The Macmillan Company, London.Google Scholar
De Cruz, H. and De Smedt, J. (2010). Mathematical symbols as epistemic actions. Synthese, 190(1):1–17.Google Scholar
De Morgan, A. (1847). Formal Logic, or The calculus of inference, necessary and probable. Taylor and Walton, London.Google Scholar
De Toffoli, S. and Giardino, V. (2014). An inquiry into the practice of proving in low-dimensional topology. Boston Studies in the History and the Philosophy of Science, 308:315–336.CrossRefGoogle Scholar
Dedekind, R. (1888). Was sind und was sollen die Zahlen? Vieweg, Braunschweig. English translation by Wooster W. Beman, revised by William Ewald, in Ewald (1996), pp. 787–833.Google Scholar
Dehaene, S. (1992). Varieties of numerical abilities. Cognition, 44:1–42.CrossRefGoogle ScholarPubMed
Durham, J. W. (1992). The introduction of “Arabic” numerals in European accounting. The Accounting Historians Journal, 19(2):25–55.CrossRefGoogle Scholar
Dutilh Novaes, C. (2012). Formal Languages in Logic: A Philosophical and Cognitive Analysis. Cambridge University Press, Cambridge, UK.CrossRefGoogle Scholar
Dutz, J. and Schlimm, D. (2021). Babbage’s guidelines for the design of mathematical notations. Studies in History and Philosophy of Science, 88:92–101.CrossRefGoogle ScholarPubMed
Elkind, L. D. C. and Zach, R. (2023). The genealogy of ‘
.’ Review of Symbolic Logic, 16(3):862–899.CrossRefGoogle Scholar
.’ Review of Symbolic Logic, 16(3):862–899.CrossRefGoogle ScholarEnderton, H. B. (2001). A Mathematical Introduction to Logic. Harcourt Academic Press, San Diego, 2nd edition.Google Scholar
Ewald, W. (1996). From Kant to Hilbert: A Source Book in Mathematics. Clarendon Press, Oxford. Two volumes.Google Scholar
Foster, J. E. (1947). A number system without a zero-symbol. Mathematics Magazine, 21(1):39–41.CrossRefGoogle Scholar
Frege, G. (1879). Begriffsschrift. Eine der arithmetischen nachgebildete Formelsprache des reinen Denkens. Verlag Louis Nebert, Halle a/S. English translation: Begriffsschrift, A Formula Language, Modeled upon That for Arithmetic, in van Heijenoort (1967), pp. 1–82.Google Scholar
Frege, G. (1884). Die Grundlagen der Arithmetik. Verlag von Wilhelm Koebner, Breslau. English translation by J. L. Austin: The Foundations of Arithmetic, Basil Blackwell, Oxford, 1953.Google Scholar
Frege, G. (1984). Logical investigations. In Collected Papers on Mathematics, Logic, and Philosophy, pages 351–406. Basil Blackwell, Oxford. Edited by McGuinness, Brian.Google Scholar
Gentzen, G. (1935). Untersuchungen über das logische Schließen. Mathematische Zeitschrift, 39:176–210, 405–431. English translation: Investigations into logical deduction, in Szabo (1969), pages 68–131.CrossRefGoogle Scholar
Goldfarb, W., editor (1971). Jacques Herbrand: Logical Writings. D. Reidel Publishing Company, Dordrecht.Google Scholar
Goodman, N. (1968). Languages of Art: An Approach to a Theory of Symbols. Bobbs-Merrill, Indianapolis.Google Scholar
Greenberg, J. H. (1978). Generalizations about numeral systems. In Greenberg, J. H., Ferguson, C. A., and Moravcsik, E. A., editors, Universals of Human Language, volume 3: Word Structure, pages 249–295. Stanford University Press, Stanford, CA.Google Scholar
Grosholz, E. R. (2007). Representation and Productive Ambiguity in Mathematics and the Sciences. Oxford University Press, Oxford.CrossRefGoogle Scholar
Haffner, E. (2021). The shaping of Dedekind’s rigorous mathematics: What do Dedekind’s drafts tell us about his ideal of rigor? Notre Dame Journal of Formal Logic, 62(1):5–31.CrossRefGoogle Scholar
Haigh, T. and Ceruzzi, P. E. (2021). A New History of Modern Computing. Massachusetts Institute of Technology Press, Cambridge, MA.CrossRefGoogle Scholar
Heath, T. L. (1910). Diophantus of Alexandria: A Study in the History of Greek Algebra. Cambridge University Press, New York, 2nd edition.Google Scholar
Heyting, A. (1930). Die formalen Regeln der intuitionistischen Logik. Sitzungsberichte der Preussischen Akademie der Wissenschaften, pages 42–56.Google Scholar
Hutchins, E. (1995). Cognition in the Wild. Massachusetts Institute of Technology Press, Cambridge, MA.CrossRefGoogle Scholar
Jevons, W. S. (1896). Studies in Deductive Logic: A Manual for Students. Macmillan, New York, 3rd edition.Google Scholar
Kirsh, D. (1990). When is information explicitly represented? The Vancouver Studies in Cognitive Science, pages 340–365.Google Scholar
Kirsh, D. (2003). Implicit and explicit representation. In Nadel, L., editor, Encyclopedia of Cognitive Science, pages 478–481. Nature Publishing, London.Google Scholar
Kirsh, D. (2010). Thinking with external representations. AI & Society, 25:441–454.CrossRefGoogle Scholar
Klein, U. (2001). Paper tools in experimental cultures. Studies in the History and Philosophy of Science, 32(2):265–302.CrossRefGoogle Scholar
Klein, U. (2002). Experiments, Models, Paper Tools: Cultures of Organic Chemistry in the Nineteenth Century. Stanford University Press, Stanford, CA.CrossRefGoogle Scholar
Knobloch, E., editor (2016). Gottfried Wilhelm Leibniz. De quadratura arithmetica circuli ellipseos et hyperbolae cujus corollarium est trigonometria sine tabulis. Springer Spektrum, Berlin.CrossRefGoogle Scholar
Krämer, S. (2003). Writing, notational iconicity, calculus: On writing as a cultural technique. Modern Language Notes: German Issue, 118(3):518–537.Google Scholar
Kripke, S. (2023). Wittgenstein, Russell, and our concept of the natural numbers. In Posy, C. and Ben-Menahem, Y., editors, Mathematical Knowledge, Objects, and Applications: Essays in Memory of Mark Steiner, pages 137–156. Springer, Cham.CrossRefGoogle Scholar
Lakatos, I. (1976). Proofs and Refutations. Cambridge University Press, Cambridge, UK. Edited by Worrall, John and Zahar, Elie.CrossRefGoogle Scholar
Landy, D., Allen, C., and Zednik, C. (2014). A perceptual account of symbolic reasoning. Frontiers in Psychology, 5(275):1–10.CrossRefGoogle ScholarPubMed
Larkin, J. H. and Simon, H. A. (1987). Why a diagram is (sometimes) worth ten thousand words. Cognitive Science, 11:65–99.CrossRefGoogle Scholar
Lengnink, K. and Schlimm, D. (2010). Learning and understanding numeral systems: Semantic aspects of number representations from an educational perspective. In Löwe, B. and Müller, T., editors, Philosophy of Mathematics: Sociological Aspects and Mathematical Practice, pages 235–264. College Publications, London.Google Scholar
Lorayne, H. (1992). Miracle Math: How to Develop a Calculator in Your Head. Barnes & Noble, New York.Google Scholar
Manders, K. (1996). Diagram contents and representational granularity. In Seligmann, J. and Westerstahl, D., editors, Logic Language and Computation, pages 389–404. Stanford University Press, Stanford, CA.Google Scholar
Manders, K. (2008). The Euclidean diagram (1995). In Mancosu, P., editor, The Philosophy of Mathematical Practice, pages 80–133. Oxford University Press, Oxford.CrossRefGoogle Scholar
Menary, R. and Gillett, A. (2022). The tools of enculturation. Topics in Cognitive Science, 14:363–387.CrossRefGoogle ScholarPubMed
Miller, G. A. (1956). The magical number seven, plus or minus two: Some limits on our capacity for processing information. Psychological Review, 63(2):81–97.CrossRefGoogle ScholarPubMed
Miller, K. F., Kelly, M., and Zhou, X. (2005). Learning mathematics in China and the United States: Cross-cultural insights into the nature and course of preschool mathematical development. In Campbell, J. I. D., editor, Handbook of Mathematical Cognition, pages 163–178. Psychology Press, Hove, UK.Google Scholar
Nicod, J. (1917). A reduction in the number of the primitive propositions of logic. Proceedings of the Cambridge Philosophical Society, 19:32–41.Google Scholar
Norman, D. (2013). The Design of Everyday Things. Massachusetts Institute of Technology Press, Cambridge, MA, revised and expanded edition.Google Scholar
Nuerk, H.-C., Moeller, K., and Willmes, K. (2015). Multi-digit number processing. overview, conceptual clarifications, and language influences. In Cohen Kadosh, R. and Dowker, A., editors, The Oxford Handbook of Numerical Cognition, chapter 7, pages 106–139. Oxford University Press, Oxford.Google Scholar
Nuerk, H.-C., Wegner, U., and Willmes, K. (2001). Decade breaks in the mental number line? putting the tens and units back in different bins. Cognition, 82(1):B25–B33.CrossRefGoogle Scholar
Palmer, S. E. (1978). Fundamental aspects of cognitive representation. In Rosch, E. and Lloyd, B., editors, Cognition and Categorization, pages 259–303. Lawrence Elbaum Associates, Hillsdale, NJ.Google Scholar
Pantsar, M. (2021). Cognitive and computational complexity: Considerations from mathematical problem solving. Erkenntnis, 86:961–997.CrossRefGoogle Scholar
Pantsar, M. (2024). Numerical Cognition and the Epistemology of Arithmetic. Cambridge University Press, Cambridge, UK.CrossRefGoogle Scholar
Peirce, C. S. (1870). On an improvement in Boole’s calculus of logic. Proceedings of the American Academy of Arts and Sciences, 7:250–261. Reprinted in Peirce (1966), III, pp. 27–98.Google Scholar
Peirce, C. S. (1885). On the algebra of logic: A contribution to the philosophy of notation. American Journal of Mathematics, 7(2):180–196. Reprinted in Peirce (1966), III, pp. 210–249.CrossRefGoogle Scholar
Peirce, C. S. (1960–1966). Collected Papers. Harvard University Press, Cambridge, MA. Four volumes. Reprint of 1931–1958, eight volumes. Edited by Charles Hartshorne and Paul Weiss.Google Scholar
Polya, G. (1954). Mathematics and Plausible Reasoning, volume 1: Induction and Analogy in Mathematics. Princeton University Press, Princeton, NJ.Google Scholar
Ramachandran, V. S. and Hubbard, E. M. (2001). Synaesthesia: A window into perception, thought and language. Journal of Consciousness Studies, 8(12):3–34.Google Scholar
Ramsey, F. P. (1927). Facts and propositions. Proceedings of the Aristotelian Society, Supplementary Volumes, vol. 7, Mind, Objectivity and Fact (1927), pp. 153–170.CrossRefGoogle Scholar
Saidan, A. S., editor (1978). The Arithmetic of Al-Uqlʙ;disʙ;. D. Reidel Publishing Company, Dordrecht.Google Scholar
Saxe, G. B. (1981). Body parts as numerals: A developmental analysis of numeration among the Oksapmin in Papua New Guinea. Child Development, 52(1):306–316.CrossRefGoogle Scholar
Schlimm, D. (2013). Axioms in mathematical practice. Philosophia Mathematica (Series III), 21(1):37–92.CrossRefGoogle Scholar
Schlimm, D. (2018). Numbers through numerals: The constitutive role of external representations. In Bangu, S., editor, Naturalizing Logico-Mathematical Knowledge: Approaches from Philosophy, Psychology and Cognitive Science. Routledge, Abingdon-on-Thames, UK.Google Scholar
Schlimm, D. (2021). Peano on symbolization, design principles for notations, and the dot notation. Philosophia Scientiæ, 25(1):139–170.Google Scholar
Schlimm, D. (2022). Tables as powerful representational tools. In Giardino, V., Linker, S., Burns, R., Bellucci, F., Boucheix, J.-M., and Viana, P., editors, Diagrammatic Representation and Inference. 13th International Conference, Diagrams 2022, Rome, Italy, September 14–16, 2022, Proceedings, pages 185–201, Springer, New York.Google Scholar
Schlimm, D. and Neth, H. (2008). Modeling ancient and modern arithmetic practices: Addition and multiplication with Arabic and Roman numerals. In Sloutsky, V., Love, B., and McRae, K., editors, Proceedings of the 30th Annual Meeting of the Cognitive Science Society, pages 2097–2102, Austin, TX. Cognitive Science Society.Google Scholar
Schlimm, D. and Skosnik, K. (2011). Symbols for nothing: Different symbolic roles of zero and their gradual emergence in Mesopotamia. In Cupillari, A., editor, Proceedings of the 2010 Meeting of the Canadian Society for History and Philosophy of Mathematics, May 29–31, Montreal, volume 23, pages 257–266.Google Scholar
Schlimm, D. and Waszek, D. (2020). Multiple readability in principle and practice: Existential graphs and complex symbols. Logique & Analyse, 251:231–260.Google Scholar
Sheffer, H. M. (1913). A set of five postulates for Boolean algebras, with application to logical constants. Transactions of the American Mathematical Society, 14(4):481–488.CrossRefGoogle Scholar
Shimojima, A. (2015). Semantic Properties of Diagrams and Their Cognitive Potentials. CSLI Publications, Stanford, CA.Google Scholar
Siegmund-Schultze, R. (1991). Mathematics and ideology in fascist Germany. In Woodward, W. R. and Cohen, R. S., editors, World Views and Scientific Discipline Formation, pages 89–95. Kluwer Academic Publishers, Berlin.CrossRefGoogle Scholar
Sigler, L. (2002). Fibonacci’s Liber Abaci. A Translation into Modern English of Leonardo Pisano’s Book of Calculation. Springer, New York.Google Scholar
Stamm, E. (1911). Beitrag zur Algebra der Logik. Monatshefte für Mathematik und Physik, 22(1):137–149.CrossRefGoogle Scholar
Stenning, K. (2000). Distinctions with differences: Comparing criteria for distinguishing diagrammatic from sentential systems. In Anderson, M., Cheng, P., and Haarslev, V., editors, Theory and Application of Diagrams, volume 1889 of Lecture Notes in Artificial Intelligence, pages 132–148, Springer, Berlin.Google Scholar
Stenning, K. (2002). Seeing Reason: Image and Language in Learning to Think. Oxford University Press, Oxford.CrossRefGoogle Scholar
Stjernfelt, F. (2007). Diagrammatology: An Investigation of the Borderlines of Phenomenology, Ontology, and Semiotics. Springer, Dordrecht.CrossRefGoogle Scholar
Strickland, L. and Lewis, H. R., editors (2022). Leibniz on Binary: The Invention of Computer Arithmetic. Massachusetts Institute of Technology Press, Cambridge, MA.CrossRefGoogle Scholar
Szabo, M. E., editor (1969). The Collected Papers of Gerhard Gentzen, North-Holland, Amsterdam.Google Scholar
Tolchinsky, L. (2003). The Cradle of Culture and What Children Know About Writing and Numbers Before Being Taught. Erlbaum Associates, Mahwah, NJ.CrossRefGoogle Scholar
Tufte, E. R. (2001). The Visual Display of Quantitative Information. Graphics Press, Cheshire, CT, 2nd edition.Google Scholar
van Heijenoort, J. (1967). From Frege to Gödel: A Source Book in Mathematical Logic, 1879–1931. Harvard University Press, Cambridge, MA.Google Scholar
Vold, K. and Schlimm, D. (2020). Extended mathematical cognition: External representations with non-derived content. Synthese, 179(9):3757–3777.CrossRefGoogle Scholar
Wagner, R. (2017). Making and Breaking Mathematical Sense: Histories and Philosophies of Mathematical Practice. Princeton University Press, Princeton.Google Scholar
Warde, B. (1955). The crystal goblet, or Printing should be invisible. In Jacob, H., editor, The Crystal Goblet. Sixteen Essays on Typography, pages 11–17. The Sylvan Press, London.Google Scholar
Waszek, D. (2024). Informational equivalence but computational differences? Herbert Simon on representations in scientific practice. Minds and Machines, 34:93–116.CrossRefGoogle Scholar
Wege, T., Batchelor, S., Inglis, M., Mistry, H., and Schlimm, D. (2020). Iconicity in mathematical notation: Commutativity and symmetry. Journal of Numerical Cognition, 6(3):378–392.CrossRefGoogle Scholar
Whitehead, A. N. and Russell, B. (1910). Principia Mathematica, volume 1. Cambridge University Press, Cambridge.Google Scholar
Widom, T. R. and Schlimm, D. (2012). Methodological reflections on typologies for numeral systems. Science in Context, 25(2):155–195.CrossRefGoogle Scholar
Wittgenstein, L. (1922). Tractatus Logico-Philosophicus. Kegan Paul, Trench, Trubner & Co., London.Google Scholar
Zhang, J. and Norman, D. A. (1995). A representational analysis of numeration systems. Cognition, 57:271–295.CrossRefGoogle ScholarPubMed
This Element is free online from 30th January 2025 - 27th February 2025