Hostname: page-component-cd9895bd7-8ctnn Total loading time: 0 Render date: 2024-12-26T15:49:01.618Z Has data issue: false hasContentIssue false
  • English
  • Français

MATHEMATICAL INFERENCE AND LOGICAL INFERENCE

Published online by Cambridge University Press:  08 January 2018

YACIN HAMAMI
YACIN HAMAMI*
Affiliation:
Centre for Logic and Philosophy of Science, Vrije Universiteit Brussel
*
*CENTRE FOR LOGIC AND PHILOSOPHY OF SCIENCE VRIJE UNIVERSITEIT BRUSSEL BRUSSELS B-1050, BELGIUM E-mail: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

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.

Keywords

00A30mathematical inferencelogical inferencemathematical proofformal proofformalitygeneralitymechanicality
Type
Research Article
Information
The Review of Symbolic Logic , Volume 11 , Issue 4 , December 2018 , pp. 665 - 704
Copyright
Copyright © Association for Symbolic Logic 2018 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

BIBLIOGRAPHY

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. (2006). Mathematical method and proof. Synthese, 153(1), 105–159.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
Barnes, J. (2007). Truth, etc. Oxford: Oxford University Press.Google 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
Bourbaki, N. (1970). Théorie des Ensembles. Paris: Hermann.Google 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
Burgess, J. P. (2015). Rigor and Structure. Oxford: Oxford University Press.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. (editor) (1992b). Proof, Logic and Formalization. London: Routledge.Google 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
Frege, G. (1979). Posthumous Writings. Chicago: University of Chicago Press.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
Goldfarb, W. (2003). Deductive Logic. Indianapolis: Hackett Publishing.Google 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. (1981). Mathematical rigor–Who needs it? Noûs, 15(4), 469–493.CrossRefGoogle Scholar
Kitcher, P. (1984). The Nature of Mathematical Knowledge. New York: Oxford University Press.Google Scholar
Kleene, S. C. (1952). Introduction to Metamathematics. New York: van Nostrand.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
Lewis, D. (1988). Relevant implication. Theoria, 54(3), 161–174.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
Ryle, G. (1954). Dilemmas. Cambridge: Cambridge University Press.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
Yablo, S. (2014). Aboutness. Princeton: Princeton University Press.Google Scholar