##### This article has been cited by the following publications. This list is generated based on data provided by CrossRef.

*Philosophy of mathematical practice: a primer for mathematics educators*. ZDM, Vol. 52, Issue. 6, p. 1113.

Published online by Cambridge University Press:
**08 January 2018**

The deviation of mathematical proof—proof in mathematical practice—from the ideal of formal proof—proof in formal logic—has led many philosophers of mathematics to reconsider the commonly accepted view according to which the notion of formal proof provides an accurate descriptive account of mathematical proof. This, in turn, has motivated a search for alternative accounts of mathematical proof purporting to be more faithful to the reality of mathematical practice. Yet, in order to develop and evaluate such alternative accounts, it appears as a necessary prerequisite to first possess a clear picture of what the deviation of mathematical proof from formal proof consists in. The present work aims to contribute building such a picture by investigating the relation between the elementary steps of deduction constituting the two types of proofs—mathematical inference and logical inference. Many claims have been made in the literature regarding the relation between mathematical inference and logical inference, most of them stating that the former is lacking properties that are constitutive of the latter. Such differentiating claims are, however, usually put forward without a clear conception of the properties occurring in them, and are generally considered to be immediately justified by our direct acquaintance, or phenomenological experience, with the two types of inferences. The present study purports to advance our understanding of the relation between mathematical inference and logical inference by developing a detailed philosophical analysis of the differentiating claims, that is, an analysis of the meaning of the differentiating claims—through the properties that occur in them—as well as the reasons that support them. To this end, we provide at the outset a representative list of the different properties of logical inference that have occurred in the differentiating claims, and we notice that they all boil down to the three properties of formality, generality, and mechanicality. For each one of these properties, our analysis proceeds in two steps: we first provide precise conceptual characterizations of the different ways logical inference has been said to be formal, general, and mechanical, in the philosophical and logical literature on formal proof; we then examine why mathematical inference does not appear to be formal, general, and mechanical, for the different variations of these notions identified. Our study results in a precise conceptual apparatus for expressing and discussing the properties differentiating mathematical inference from logical inference, and provides a first inventory of the various reasons supporting the observations of those differences. The differentiating claims constitute thus a set of data that any philosophical account of mathematical inference and proof purporting to be more faithful to mathematical practice ought to be able to accommodate and explain.

Type

Research Article

Information

Copyright

Copyright © Association for Symbolic Logic 2018

Aberdein, A. (2006). The informal logic of mathematical proof. In Hersh, R., editor. 18 Unconventional Essays on the Nature of Mathematics. New York: Springer, pp. 56–70.CrossRefGoogle Scholar

Avigad, J. (2008). Understanding proofs. In Mancosu, P., editor. The Philosophy of Mathematical Practice. Oxford: Oxford University Press, pp. 317–353.CrossRefGoogle Scholar

Azzouni, J. (2004). The derivation-indicator view of mathematical practice. Philosophia Mathematica, 12(3), 81–105.CrossRefGoogle Scholar

Azzouni, J. (2009). Why do informal proofs conform to formal norms? Foundations of Science, 14(1–2), 9–26.CrossRefGoogle Scholar

Azzouni, J. (2013). The relationship of derivations in artificial languages to ordinary rigorous mathematical proof. Philosophia Mathematica, 21(2), 247–254.CrossRefGoogle Scholar

Bonnay, D. (2006). Qu’est-ce qu’une Constante Logique? Ph.D. Thesis, Université Paris I.Google Scholar

Bonnay, D. (2008). Logicality and invariance. Bulletin of Symbolic Logic, 14(1), 29–68.CrossRefGoogle Scholar

Bundy, A., Atiyah, M., Macintyre, A., & MacKenzie, D. (2005). The nature of mathematical proof [special issue]. Philosophical Transactions of the Royal Society A, 363(1835), 2329–2461.Google Scholar

Bundy, A., Jamnik, M., & Fugard, A. (2005). What is a proof? Philosophical Transactions of the Royal Society A, 363(1835), 2377–2391.CrossRefGoogle Scholar

Cellucci, C. (2008). Why proof? What is a proof? In Lupacchini, R. and Corsi, G., editors. Deduction, Computation, Experiment: Exploring the Effectiveness of Proof. Milan: Springer, pp. 1–27.Google Scholar

Church, A. (1956). Introduction to Mathematical Logic, Vol. 1. Princeton: Princeton University Press.Google Scholar

Corcoran, J. (1973). Gaps between logical theory and mathematical practice. In Bunge, M., editor. The Methodological Unity of Science. Dordrecht: Reidel, pp. 23–50.CrossRefGoogle Scholar

Corcoran, J. (2014). Schema. In Zalta, E. N., editor. The Stanford Encyclopedia of Philosophy *(*Spring 2014 Edition*)*. Available at: http://plato.stanford.edu/archives/spr2014/entries/schema/.Google Scholar

Curry, H. B. (1950). A Theory of Formal Deducibility. Notre Dame Mathematical Lectures, Number 6. Notre Dame, Indiana: University of Notre Dame.Google Scholar

Detlefsen, M. (1992a). Poincaré against the logicians. Synthese, 90(3), 349–378.CrossRefGoogle Scholar

Detlefsen, M. (2009). Proof: Its nature and significance. In Gold, B. and Simons, R. A., editors. Proof and Other Dilemmas: Mathematics and Philosophy. Washington, D.C.: The Mathematical Association of America.Google Scholar

Dutilh Novaes, C. (2011). The different ways in which logic is (said to be) formal. History and Philosophy of Logic, 32(4), 303–332.CrossRefGoogle Scholar

Feferman, S. (1979). What does logic have to tell us about mathematical proofs? The Mathematical Intelligencer, 2(1), 20–24.CrossRefGoogle Scholar

Feferman, S. (1999). Logic, logics, and logicism. Notre Dame Journal of Formal Logic, 40(1), 31–54.Google Scholar

Feferman, S. (2012). And so on…: Reasoning with infinite diagrams. Synthese, 186(1), 371–386.CrossRefGoogle Scholar

Frege, G. (1893/1964). The Basic Laws of Arithmetic: Exposition of the System. Berkeley: University of California Press.Google Scholar

Frege, G. (1897/1984). On Mr. Peano’s conceptual notation and my own. In McGuiness, B., editor. Collected Papers on Mathematics, Logic, and Philosophy. New York: Basil Blackwell, pp. 234–248.Google Scholar

Gödel, K. (1931). Über formal unentscheidbare Sätze der Principia Mathematica und verwandter Systeme I. Monatshefte für Mathematik und Physik, 38(1), 173–198.CrossRefGoogle Scholar

Gödel, K. (193?/1995a). Undecidable diophantine propositions. In Feferman, S., Dawson, J. W. Jr., Goldfarb, W., Parsons, C., and Solovay, R. M., editors. Collected Works, Vol. III: Unpublished Essays and Lectures. Oxford: Oxford University Press, pp. 164–175.Google Scholar

Gödel, K. (1995b). The present situation in the foundations of mathematics. In Feferman, S., Dawson, J. W. Jr., Goldfarb, W., Parsons, C., and Solovay, R. M., editors. Collected Works, Vol. III: Unpublished Essays and Lectures. Oxford: Oxford University Press, pp. 45–53.Google Scholar

Goethe, N. B. & Friend, M. (2010). Confronting ideals of proof with the ways of proving of the research mathematician. Studia Logica, 96(2), 273–288.CrossRefGoogle Scholar

Goldfarb, W. (2001). Frege’s conception of logic. In Floyd, J. and Shieh, S., editors. Future Pasts: The Analytic Tradition in Twentieth-Century Philosophy. New York: Oxford University Press, pp. 25–41.CrossRefGoogle Scholar

Hales, T. (2012). Dense Sphere Packings: A Blueprint for Formal Proofs. London Mathematical Society Lecture Notes Series, Vol. 400. Cambridge: Cambridge University Press.CrossRefGoogle Scholar

Hardy, G. H. & Wright, E. M. (1975). An Introduction to the Theory of Numbers (Fourth Edition). Oxford: Oxford University Press.Google Scholar

Kitcher, P. (1984). The Nature of Mathematical Knowledge. New York: Oxford University Press.Google Scholar

Kreisel, G. (1967). Informal rigour and completeness proofs. In Lakatos, I., editor. Problems in the Philosophy of Mathematics. Amsterdam: North-Holland, pp. 138–186.CrossRefGoogle Scholar

Kreisel, G. (1970). The formalist-positivist doctrine of mathematical precision in the light of experience. L’Âge de la Science, 3, 17–46.Google Scholar

Kreisel, G. (1981). Neglected possibilities of processing assertions and proofs mechanically: Choice of problems and data. In Suppes, P., editor. University-Level Computer-Assisted Instruction at Stanford: 1968–1980. Stanford, CA: Stanford University, Institute for Mathematical Studies in the Social Sciences, pp. 131–148.Google Scholar

Larvor, B. (2012). How to think about informal proofs. Synthese, 187(2), 715–730.CrossRefGoogle Scholar

Leitgeb, H. (2009). On formal and informal provability. In Linnebo, Ø. and Bueno, O., editors. New Waves in Philosophy of Mathematics. New York: Palgrave Macmillan, pp. 263–299.CrossRefGoogle Scholar

Lycan, W. G. (1989). Logical constants and the glory of truth-conditional semantics. Notre Dame Journal of Formal Logic, 30(3), 390–400.CrossRefGoogle Scholar

Mac Lane, S. (1986). Mathematics: Form and Function. New York: Springer-Verlag.CrossRefGoogle Scholar

MacFarlane, J. G. (2000). What does it Mean to Say that Logic is Formal? Ph.D. Thesis, University of Pittsburgh.Google Scholar

MacFarlane, J. G. (2002). Frege, Kant, and the logic in logicism. Philosophical Review, 111(1), 25–65.CrossRefGoogle Scholar

MacFarlane, J. G. (2014). Logical constants. In Zalta, E. N., editor. The Stanford Encyclopedia of Philosophy *(*Spring 2014 Edition*)*. Available at: http://plato.stanford.edu/archives/sum2014/entries/logical-constants/.Google Scholar

MacKenzie, D. (2001). Mechanizing Proof: Computing, Risk, and Trust. Cambridge, MA: MIT Press.Google Scholar

Marciszewski, W. & Murawski, R. (1995). Mechanization of Reasoning in a Historical Perspective. Poznaǹ Studies in the Philosophy of the Sciences and the Humanities, Vol. 43. Amsterdam: Rodopi.Google Scholar

Marr, D. (1982). Vision: A Computational Investigation into the Human Representation and Processing of Visual Information. New York: W. H. Freeman and Company.Google Scholar

McGee, V. (1996). Logical operations. Journal of Philosophical Logic, 25(6), 567–580.CrossRefGoogle Scholar

Myhill, J. (1960). Some remarks on the notion of proof. The Journal of Philosophy, 57(14), 461–471.CrossRefGoogle Scholar

Poincaré, H. (1894). Sur la nature du raisonnement mathématique. Revue de Métaphysique et de Morale, 2, 371–384.Google Scholar

Prawitz, D. (2012). The epistemic significance of valid inference. Synthese, 187(3), 887–898.CrossRefGoogle Scholar

Prawitz, D. (2013). Validity of inferences. In Frauchiger, M., editor. Reference, Rationality, and Phenomenology: Themes from Føllesdal. Heusenstamm: Ontos Verlag, pp. 179–204.Google Scholar

Rav, Y. (1999). Why do we prove theorems? Philosophia Mathematica, 7(3), 5–41.CrossRefGoogle Scholar

Rav, Y. (2007). A critique of a formalist-mechanist version of the justification of arguments in mathematicians’ proof practices. Philosophia Mathematica, 15(3), 291–320.CrossRefGoogle Scholar

Robinson, J. A. (1991). Formal and informal proofs. In Boyer, R. S., editor. Automated Reasoning: Essays in Honor of Woody Bledsoe. Automated Deduction Series, Vol. 1. London: Kluwer Academic Publishers, pp. 267–282.CrossRefGoogle Scholar

Robinson, J. A. (1997). Informal rigor and mathematical understanding. In Gottlob, G., Leitsch, A., and Mundici, D., editors. Computational Logic and Proof Theory: Proceedings of the 5th Annual Kurt Gödel Colloquium, August 25–29, 1997. Lecture Notes in Computer Science, Vol. 1289. Heidelberg & New York: Springer, pp. 54–64.CrossRefGoogle Scholar

Robinson, J. A. (2000). Proof = guarantee + explanation. In Hölldobler, S., editor. Intellectics and Computational Logic. Applied Logic Series, Vol. 19. Dordrecht: Kluwer Academic Publishers, pp. 277–294.CrossRefGoogle Scholar

Robinson, J. A. (2004). Logic is not the whole story. In Hendricks, V., Neuhaus, F., Scheffler, U., Pedersen, S. A., and Wansing, H., editors. First-Order Logic Revisited. Berlin: Logos Verlag, pp. 287–302.Google Scholar

Russell, B. (1913). The philosophical importance of mathematical logic. The Monist, 22(4), 481–493.CrossRefGoogle Scholar

Sher, G. (1991). The Bounds of Logic: A Generalized Viewpoint. Cambridge, MA: MIT Press.Google Scholar

Sieg, W. (2009). On computability. In Irvine, A., editor. Handbook of the Philosophy of Mathematics. North-Holland: Elsevier, pp. 535–630.CrossRefGoogle Scholar

Sjögren, J. (2010). A note on the relation between formal and informal proof. Acta Analytica, 25(4), 447–458.CrossRefGoogle Scholar

Stenning, K. & Van Lambalgen, M. (2008). Human Reasoning and Cognitive Science. Cambridge, MA: MIT Press.Google Scholar

Sundholm, G. (2012). “Inference versus consequence” revisited: Inference, consequence, conditional, implication. Synthese, 187(3), 943–956.CrossRefGoogle Scholar

Suppes, P. (2005). Psychological nature of verification of informal mathematical proofs. In Artemov, S., Barringer, H., d’Avila Garcez, A., Lamb, L. C., and Woods, J., editors. We Will Show Them: Essays in Honour of Dov Gabbay, Vol. 2. London: College Publications, pp. 693–712.Google Scholar

Tanswell, F. (2015). A problem with the dependence of informal proofs on formal proofs. Philosophia Mathematica, 23(3), 295–310.CrossRefGoogle Scholar

Tarski, A. (1936/2002). On the concept of following logically. History and Philosophy of Logic, 23, 155–196.CrossRefGoogle Scholar

Tarski, A. (1986). What are logical notions? History and Philosophy of Logic, 7(2), 143–154.CrossRefGoogle Scholar

Thurston, W. P. (1994). On proof and progress in mathematics. Bulletin of the American Mathematical Society, 30(2), 161–177.CrossRefGoogle Scholar

Van Benthem, J. (1989). Logical constants across varying types. Notre Dame Journal of Formal Logic, 30(3), 315–342.CrossRefGoogle Scholar

Weir, A. (2016). Informal proof, formal proof, formalism. The Review of Symbolic Logic, 9(1), 23–43.CrossRefGoogle Scholar

Full text views reflects PDF downloads, PDFs sent to Google Drive, Dropbox and Kindle and HTML full text views.

Total number of HTML views: 1

Total number of PDF views: 223 *

View data table for this chart

* Views captured on Cambridge Core between 08th January 2018 - 25th February 2021. This data will be updated every 24 hours.

1 Cited by

Cited by

Crossref Citations

Hamami, Yacin
and
Morris, Rebecca Lea
2020.
*Philosophy of mathematical practice: a primer for mathematics educators*.
ZDM,
Vol. 52,
Issue. 6,
p.
1113.

To send this article to your Kindle, first ensure no-reply@cambridge.org is added to your Approved Personal Document E-mail List under your Personal Document Settings on the Manage Your Content and Devices page of your Amazon account. Then enter the ‘name’ part of your Kindle email address below. Find out more about sending to your Kindle. Find out more about sending to your Kindle.

Note you can select to send to either the @free.kindle.com or @kindle.com variations. ‘@free.kindle.com’ emails are free but can only be sent to your device when it is connected to wi-fi. ‘@kindle.com’ emails can be delivered even when you are not connected to wi-fi, but note that service fees apply.

Find out more about the Kindle Personal Document Service.

MATHEMATICAL INFERENCE AND LOGICAL INFERENCE

- Volume 11, Issue 4

To send this article to your Dropbox account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your <service> account. Find out more about sending content to Dropbox.

MATHEMATICAL INFERENCE AND LOGICAL INFERENCE

- Volume 11, Issue 4

To send this article to your Google Drive account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your <service> account. Find out more about sending content to Google Drive.

MATHEMATICAL INFERENCE AND LOGICAL INFERENCE

- Volume 11, Issue 4

×
####
Submit a response