A metasemantic challenge for mathematical determinacy. (English) Zbl 1475.03058
Summary: This paper investigates the determinacy of mathematics. We begin by clarifying how we are understanding the notion of determinacy (Sect. 1) before turning to the questions of whether and how famous independence results bear on issues of determinacy in mathematics (Sect. 2). From there, we pose a metasemantic challenge for those who believe that mathematical language is determinate (Sect. 3), motivate two important constraints on attempts to meet our challenge (Sect. 4), and then use these constraints to develop an argument against determinacy (Sect. 5) and discuss a particularly popular approach to resolving indeterminacy (Sect. 6), before offering some brief closing reflections (Sect. 7). We believe our discussion poses a serious challenge for most philosophical theories of mathematics, since it puts considerable pressure on all views that accept a non-trivial amount of determinacy for even basic arithmetic.
MSC:
| 03A05 | Philosophical and critical aspects of logic and foundations |
| 00A30 | Philosophy of mathematics |
References:
| [1] | Balaguer, M., Towards a nominalization of quantum mechanics, Mind, 105, 418, 209-226 (1996) |
| [2] | Beltrami, E., Saggio di interpretazione della geometria non-euclidea, Giornale di Mathematiche, VI, 285-315 (1868) · JFM 01.0275.02 |
| [3] | Beltrami, E., Teoria fondamentale degli spazii di curvatura costante, Annali di Matematica Pura ed Applicata, 2, 232-255 (1868) · JFM 01.0208.03 |
| [4] | Benacerraf, P., God, the Devil, and Gödel, The Monist, 51, 9-32 (1967) |
| [5] | Benacerraf, P., Mathematical truth, Journal of Philosophy, 70, 661-680 (1973) |
| [6] | Boolos, G., On “seeing” the truth of the Gödel sentence, Behavioral and Brain Sciences, 13, 4, 655-656 (1990) |
| [7] | Button, T.; Smith, P., The philosophical significance of Tennenbaum’s theorem, Philosophia Mathematica (III), 00, 1-8 (2011) |
| [8] | Carnap, R., The logical syntax of language (1934), London: Routledge & Kegan Paul, London · JFM 63.0820.05 |
| [9] | Chalmers, Dj, Constructing the world (2012), Oxford: Oxford University Press, Oxford |
| [10] | Clarke-Doane, J., What is absolute undecidability?, Noûs, 47, 3, 467-481 (2013) · Zbl 1328.03004 |
| [11] | Cohen, P., The independence of the continuum hypothesis I, Proceedings of the National Academy of Sciences of the United States, 50, 6, 1143-1148 (1963) · Zbl 0192.04401 |
| [12] | Cohen, P., The independence of the continuum hypothesis II, Proceedings of the National Academy of Sciences of the United States, 51, 1, 105-110 (1964) · Zbl 0192.04401 |
| [13] | Dean, W. (2002). Models and recursivity. Replacement posted in doi:10.1.1.136.446 |
| [14] | Field, H., Science without numbers (1980), Princeton: Princeton University Press, Princeton · Zbl 0454.00015 |
| [15] | Field, H., Can we dispense with space-time, PSA: Proceedings of the Biennial Meeting of the Philosophy of Science Association, 1984, 2, 33-90 (1984) |
| [16] | Field, H., Realism, mathematics and modality (1989), Oxford: Blackwell, Oxford · Zbl 1098.00500 |
| [17] | Field, H.; French, P.; Uehling, T.; Wettstein, H., Are our mathematical and logical concepts highly indeterminate, Midwest studies in philosophy (1994), New York: Blackwell, New York |
| [18] | Field, H.; Schirn, M., Do we have a determinate conception of finiteness and natural number, The philosophy of mathematics today (1998), Oxford: Oxford University Press, Oxford · Zbl 0924.03005 |
| [19] | Field, H.; Dales, Hg; Olivari, G., Which undecidable mathematical sentences have determinate truth values, Truth in mathematics (1998), Oxford: Oxford University Press, Oxford · Zbl 0946.03009 |
| [20] | Field, H., Truth and the absence of fact (2001), Oxford: Oxford University Press, Oxford · Zbl 0984.03003 |
| [21] | Gaifman, H.; Enayat, A.; Kossak, R., Non-standard models in a broader perspective, Contemporary mathematics: Nonstandard models of arithmetic and set theory, 1-22 (2004), Providence: American Mathematical Society, Providence · Zbl 1068.03007 |
| [22] | Gödel, K., On formally undecidable propositions of principia mathematica and related systems I, Monatshefte für Mathematik und Physik, 38, 173-198 (1931) · Zbl 0002.00101 |
| [23] | Gödel, K., Consistency proof for the generalized continuum-hypothesis, Proceedings of the National Academy of Sciences, 25, 4, 220-224 (1939) · JFM 65.0185.02 |
| [24] | Gödel, K.; Benacerraf, P.; Putnam, H., What is Cantor’s continuum problem, Philosophy of Mathematics: Selected Readings (1964), New York: Cambridge University Press, New York |
| [25] | Goodstein, R., On the restricted ordinal theorem, Journal of Symbolic Logic, 9, 33-41 (1944) · Zbl 0060.02306 |
| [26] | Hale, B., Abstract objects (1987), Oxford: Basil Blackwell, Oxford |
| [27] | Hale, B.; Wright, C., The reason s proper study (2001), Oxford: Oxford University Press, Oxford · Zbl 1005.03006 |
| [28] | Horwich, P., Truth (1998), Oxford: Clarendon Press, Oxford |
| [29] | Isaacson, D.; Novak, Z.; Simonyi, A., The reality of mathematics and the case of set theory, Truth, reference and realism, 1-76 (2011), Budapest: Central European University Press, Budapest |
| [30] | Jech, T., Non-provability of Souslin’s hypothesis, Commentationes Mathematicae Universitatis Carolinae, 8, 291-305 (1967) · Zbl 0204.00701 |
| [31] | Kirby, L.; Paris, J., Accessible independence results for Peano arithmetic, Bulletin of the London Mathematical Society, 14, 4, 285 (1982) · Zbl 0501.03017 |
| [32] | Koellner, P. (2010). On the question of absolute undecidability. In K. Gödel, S. Feferman, C. Parsons & S. G. Simpson (eds.), Philosophia mathematica. Association for symbolic logic (pp. 153-188). · Zbl 1113.03011 |
| [33] | Lavine, S. (Unpublished). Skolem was wrong. |
| [34] | Leeds, S., Theories of reference and truth, Erkenntnis, 13, 1, 111-129 (1978) |
| [35] | Lucas, Jr, Minds, machines and Gödel, Philosophy, 36, 112-127 (1961) |
| [36] | Maddy, P., Naturalism in mathematics (1997), Oxford: Oxford University Press, Oxford · Zbl 0931.03003 |
| [37] | Maddy, P., Second philosophy: A naturalistic method (2007), Oxford: Oxford University Press, Oxford · Zbl 1190.03009 |
| [38] | Malament, D., Science without numbers by Hartry Field, Journal of Philosophy, 79, 9, 523-534 (1982) |
| [39] | Mcgee, V., We turing machines aren’t expected-utility maximizers (even Ideally), Philosophical Studies, 64, 1, 115-123 (1991) |
| [40] | Mcgee, V., How we learn mathematical language, Philosophical Review, 106, 35-68 (1997) |
| [41] | Odifreddi, P., Classical recursion theory: The theory of functions and sets of natural numbers (1989), Amsterdam: Elsevier, Amsterdam · Zbl 0661.03029 |
| [42] | Paris, J.; Harrington, L.; Barwise, J., A mathematical incompleteness in Peano arithmetic, Handbook of mathematical logic, 1133-1142 (1977), Amsterdam: Elsevier, Amsterdam |
| [43] | Parsons, C. (2001). Communication and the uniqueness of the natural numbers. The Proceedings of the First Seminar of the Philosophy of Mathematics in Iran, Shahid University. |
| [44] | Penrose, R., The emperor’s new mind (1989), New York: Oxford University Press, New York |
| [45] | Penrose, R., Shadows of the mind (1994), Oxford: Oxford University Press, Oxford |
| [46] | Putnam, H.; Schoenman, R., The thesis that mathematics is logic, Bertrand russell: Philosopher of the century, 273-303 (1967), London: Allen and Unwin, London |
| [47] | Putnam, H., Mathematics without foundations, Journal of Philosophy, 64, 1, 5-22 (1967) |
| [48] | Putnam, H., Models and reality, Journal of Symbolic Logic, 45, 3, 464-482 (1980) · Zbl 0443.03003 |
| [49] | Quine, Wvo, Two dogmas of empiricism, Philosophical Review, 60, 1, 20-43 (1951) |
| [50] | Rosser, Jb, Extensions of some theorems of Gödel and Church, Journal of Symbolic Logic, 1, 230-235 (1936) |
| [51] | Schechter, J., The reliability challenge and the epistemology of logic, Philosophical Perspectives, 24, 437-464 (2010) |
| [52] | Shapiro, S., Foundations without foundationalism: A case for second-order logic (1991), Oxford: Oxford University Press, Oxford · Zbl 0732.03002 |
| [53] | Shapiro, S., Incompleteness, mechanism, and optimism, Bulletin of Symbolic Logic, 4, 3, 273-302 (1998) · Zbl 0921.03005 |
| [54] | Shapiro, S., Mechanism, truth, and Penrose’s new argument, Journal of Philosophical Logic, 32, 1, 19-42 (2003) · Zbl 1022.03001 |
| [55] | Solovay, Rm; Tennenbaum, S., Iterated Cohen extensions and Souslin’s problem, Annals of Mathematics, 94, 2, 201-245 (1971) · Zbl 0244.02023 |
| [56] | Suslin, M., Probléme 3, Fundamenta Mathematicae, 1, 223 (1920) |
| [57] | Tennenbaum, S., Souslin’s problem. U.S.A, Proceedings of the National Academy of Sciences, 59, 60-63 (1968) · Zbl 0172.29503 |
| [58] | Warren, J., Conventionalism, consistency, and consistency sentences, Synthese, 192, 5, 1351-1371 (2015) · Zbl 1369.03085 |
| [59] | Warren, Jared, Epistemology versus non-causal realism, Synthese, 194, 5, 1643-1662 (2016) · Zbl 1382.03035 |
| [60] | Warren, J. & Waxman, D. (Unpublished). Reliability, explanation, and the failure of mathematical realism. |
| [61] | Weir, A., Truth through proof: A formalist foundation for mathematics (2010), Oxford: Clarendon Press, Oxford · Zbl 1243.03002 |
| [62] | Weston, T., Kreisel, the continuum hypothesis and second order set theory, Journal of Philosophical Logic, 5, 2, 281-298 (1976) · Zbl 0341.02053 |
| [63] | Williamson, T., Vagueness (1994), London and New York: Routledge, London and New York |
| [64] | Woods, J. (2016). Mathematics, morality, and self-effacement. Noûs. · Zbl 1436.03090 |
| [65] | Wright, C., Frege’s conception of numbers as objects (1983), Aberdeen: Aberdeen University Press, Aberdeen · Zbl 0524.03005 |
| [66] | Zermelo, E., Über stufen der quantifikation und die logik des unendlichen, Jahresbericht der Deutschen Mathematiker-Vereinigung (Angelegenheiten), 31, 85-88 (1930) · JFM 58.0060.04 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
