Skip to main content
×
×
Home

MATHEMATICAL INFERENCE AND LOGICAL INFERENCE

  • YACIN HAMAMI (a1)
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.

Copyright
Corresponding author
*CENTRE FOR LOGIC AND PHILOSOPHY OF SCIENCE VRIJE UNIVERSITEIT BRUSSEL BRUSSELS B-1050, BELGIUM E-mail: yacin.hamami@gmail.com
References
Hide All
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. 5670.
Avigad, J. (2006). Mathematical method and proof. Synthese, 153(1), 105159.
Avigad, J. (2008). Understanding proofs. In Mancosu, P., editor. The Philosophy of Mathematical Practice. Oxford: Oxford University Press, pp. 317353.
Azzouni, J. (2004). The derivation-indicator view of mathematical practice. Philosophia Mathematica, 12(3), 81105.
Azzouni, J. (2009). Why do informal proofs conform to formal norms? Foundations of Science, 14(1–2), 926.
Azzouni, J. (2013). The relationship of derivations in artificial languages to ordinary rigorous mathematical proof. Philosophia Mathematica, 21(2), 247254.
Barnes, J. (2007). Truth, etc. Oxford: Oxford University Press.
Bonnay, D. (2006). Qu’est-ce qu’une Constante Logique? Ph.D. Thesis, Université Paris I.
Bonnay, D. (2008). Logicality and invariance. Bulletin of Symbolic Logic, 14(1), 2968.
Bourbaki, N. (1970). Théorie des Ensembles. Paris: Hermann.
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), 23292461.
Bundy, A., Jamnik, M., & Fugard, A. (2005). What is a proof? Philosophical Transactions of the Royal Society A, 363(1835), 23772391.
Burgess, J. P. (2015). Rigor and Structure. Oxford: Oxford University Press.
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. 127.
Church, A. (1956). Introduction to Mathematical Logic, Vol. 1. Princeton: Princeton University Press.
Corcoran, J. (1973). Gaps between logical theory and mathematical practice. In Bunge, M., editor. The Methodological Unity of Science. Dordrecht: Reidel, pp. 2350.
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/.
Curry, H. B. (1950). A Theory of Formal Deducibility. Notre Dame Mathematical Lectures, Number 6. Notre Dame, Indiana: University of Notre Dame.
Detlefsen, M. (1992a). Poincaré against the logicians. Synthese, 90(3), 349378.
Detlefsen, M. (editor) (1992b). Proof, Logic and Formalization. London: Routledge.
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.
Dutilh Novaes, C. (2011). The different ways in which logic is (said to be) formal. History and Philosophy of Logic, 32(4), 303332.
Feferman, S. (1979). What does logic have to tell us about mathematical proofs? The Mathematical Intelligencer, 2(1), 2024.
Feferman, S. (1999). Logic, logics, and logicism. Notre Dame Journal of Formal Logic, 40(1), 3154.
Feferman, S. (2012). And so on…: Reasoning with infinite diagrams. Synthese, 186(1), 371386.
Frege, G. (1893/1964). The Basic Laws of Arithmetic: Exposition of the System. Berkeley: University of California Press.
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. 234248.
Frege, G. (1979). Posthumous Writings. Chicago: University of Chicago Press.
Gödel, K. (1931). Über formal unentscheidbare Sätze der Principia Mathematica und verwandter Systeme I. Monatshefte für Mathematik und Physik, 38(1), 173198.
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. 164175.
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. 4553.
Goethe, N. B. & Friend, M. (2010). Confronting ideals of proof with the ways of proving of the research mathematician. Studia Logica, 96(2), 273288.
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. 2541.
Goldfarb, W. (2003). Deductive Logic. Indianapolis: Hackett Publishing.
Hales, T. (2012). Dense Sphere Packings: A Blueprint for Formal Proofs. London Mathematical Society Lecture Notes Series, Vol. 400. Cambridge: Cambridge University Press.
Hardy, G. H. & Wright, E. M. (1975). An Introduction to the Theory of Numbers (Fourth Edition). Oxford: Oxford University Press.
Kitcher, P. (1981). Mathematical rigor–Who needs it? Noûs, 15(4), 469493.
Kitcher, P. (1984). The Nature of Mathematical Knowledge. New York: Oxford University Press.
Kleene, S. C. (1952). Introduction to Metamathematics. New York: van Nostrand.
Kreisel, G. (1967). Informal rigour and completeness proofs. In Lakatos, I., editor. Problems in the Philosophy of Mathematics. Amsterdam: North-Holland, pp. 138186.
Kreisel, G. (1970). The formalist-positivist doctrine of mathematical precision in the light of experience. L’Âge de la Science, 3, 1746.
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. 131148.
Larvor, B. (2012). How to think about informal proofs. Synthese, 187(2), 715730.
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. 263299.
Lewis, D. (1988). Relevant implication. Theoria, 54(3), 161174.
Lycan, W. G. (1989). Logical constants and the glory of truth-conditional semantics. Notre Dame Journal of Formal Logic, 30(3), 390400.
Mac Lane, S. (1986). Mathematics: Form and Function. New York: Springer-Verlag.
MacFarlane, J. G. (2000). What does it Mean to Say that Logic is Formal? Ph.D. Thesis, University of Pittsburgh.
MacFarlane, J. G. (2002). Frege, Kant, and the logic in logicism. Philosophical Review, 111(1), 2565.
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/.
MacKenzie, D. (2001). Mechanizing Proof: Computing, Risk, and Trust. Cambridge, MA: MIT Press.
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.
Marr, D. (1982). Vision: A Computational Investigation into the Human Representation and Processing of Visual Information. New York: W. H. Freeman and Company.
McGee, V. (1996). Logical operations. Journal of Philosophical Logic, 25(6), 567580.
Myhill, J. (1960). Some remarks on the notion of proof. The Journal of Philosophy, 57(14), 461471.
Poincaré, H. (1894). Sur la nature du raisonnement mathématique. Revue de Métaphysique et de Morale, 2, 371384.
Prawitz, D. (2012). The epistemic significance of valid inference. Synthese, 187(3), 887898.
Prawitz, D. (2013). Validity of inferences. In Frauchiger, M., editor. Reference, Rationality, and Phenomenology: Themes from Føllesdal. Heusenstamm: Ontos Verlag, pp. 179204.
Rav, Y. (1999). Why do we prove theorems? Philosophia Mathematica, 7(3), 541.
Rav, Y. (2007). A critique of a formalist-mechanist version of the justification of arguments in mathematicians’ proof practices. Philosophia Mathematica, 15(3), 291320.
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. 267282.
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. 5464.
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. 277294.
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. 287302.
Russell, B. (1913). The philosophical importance of mathematical logic. The Monist, 22(4), 481493.
Ryle, G. (1954). Dilemmas. Cambridge: Cambridge University Press.
Sher, G. (1991). The Bounds of Logic: A Generalized Viewpoint. Cambridge, MA: MIT Press.
Sieg, W. (2009). On computability. In Irvine, A., editor. Handbook of the Philosophy of Mathematics. North-Holland: Elsevier, pp. 535630.
Sjögren, J. (2010). A note on the relation between formal and informal proof. Acta Analytica, 25(4), 447458.
Stenning, K. & Van Lambalgen, M. (2008). Human Reasoning and Cognitive Science. Cambridge, MA: MIT Press.
Sundholm, G. (2012). “Inference versus consequence” revisited: Inference, consequence, conditional, implication. Synthese, 187(3), 943956.
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. 693712.
Tanswell, F. (2015). A problem with the dependence of informal proofs on formal proofs. Philosophia Mathematica, 23(3), 295310.
Tarski, A. (1936/2002). On the concept of following logically. History and Philosophy of Logic, 23, 155196.
Tarski, A. (1986). What are logical notions? History and Philosophy of Logic, 7(2), 143154.
Thurston, W. P. (1994). On proof and progress in mathematics. Bulletin of the American Mathematical Society, 30(2), 161177.
Van Benthem, J. (1989). Logical constants across varying types. Notre Dame Journal of Formal Logic, 30(3), 315342.
Weir, A. (2016). Informal proof, formal proof, formalism. The Review of Symbolic Logic, 9(1), 2343.
Yablo, S. (2014). Aboutness. Princeton: Princeton University Press.
Recommend this journal

Email your librarian or administrator to recommend adding this journal to your organisation's collection.

The Review of Symbolic Logic
  • ISSN: 1755-0203
  • EISSN: 1755-0211
  • URL: /core/journals/review-of-symbolic-logic
Please enter your name
Please enter a valid email address
Who would you like to send this to? *
×

Keywords

Metrics

Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed