Theory of Quantum Computation and Philosophy of Mathematics. Part II

• Authors: Wójtowicz Krzysztof
• Languages of publication: EN

Abstracts

EN

In the article, the philosophical significance of quantum computation theory for philosophy of mathematics is discussed. In particular, I examine the notion of “quantum-assisted proof” (QAP); the discussion sheds light on the problem of the nature of mathematical proof; the potential empirical aspects of mathematics and the realism-antirealism debate (in the context of the indispensability argument). I present a quasi-empiricist account of QAP’s, and discuss the possible impact on the discussions centered around the Enhanced Indispensabity Argument (EIA).

Keywords

EN

quantum computation; quantum-assisted proofs; indispensability argument; mathematical realism; quasi-empiricismfs; quasi-empiricism

• Publisher: Uniwersytet Mikołaja Kopernika (Nicolaus Copernicus University in Toruń)
• Journal: Logic and Logical Philosophy
• Year: 2019
• Volume: 28
• Issue: 1
• Pages: 173-193

Dates

published

2019-03-15

Contributors

author

Wójtowicz Krzysztof

• Institute of Philosophy Warsaw University, Poland

References

• Aaronson, S., 2005, “NP-complete problems and physical reality”, Electronic Colloquium on Computational Complexity, Report no. 26. arXiv:quant-ph/0502072v2.
• Aaronson, S., 2013, Quantum Computing Since Democritus, Cambridge University Press, Cambridge, New York. DOI: http://dx.doi.org/10.1017/CBO9780511979309
• Andréka, H., I. Németi and P. Németi, 2009, “General relativistic hypercomputing and foundation of mathematics”, Natural Computing 8 (3): 499–516. DOI: http://dx.doi.org/10.1007/s11047-009-9114-3
• Appel, K., and W. Haken, 1977, ”Every planar map is four colorable. Part I: discharging”, Illinois Journal of Mathematics 21: 429–490.
• Appel, K., W. Haken and J. Koch, 1977, “Every planar map is four colorable. Part II: reducibility”, Illinois Journal of Mathematics 21: 491–567.
• Baker, A., 2005, “Are there genuine mathematical explanations of physical phenomena?”, Mind 114 (454): 223–238. DOI: http://dx.doi.org/10.1093/mind/fzi223
• Baker, A., 2008, “Experimental mathematics”, Erkenntnis 68: 331–344. DOI: http://dx.doi.org/10.1007/s10670-008-9109-y
• Baker, A., 2009, “Mathematical explanation in science”, British Journal for the Philosophy of Science 60 (3): 611–633. DOI: http://dx.doi.org/10.1093/bjps/axp025
• Baker, A., and A. Colyvan, 2011, “Indexing and mathematical explanation”, Philosophia Mathematica 19: 232–224. DOI: http://dx.doi.org/10.1093/philmat/nkr026
• Balaguer, M., 1998, Platonism and Anti-Platonism in Mathematics, Oxford University Press, New York, Oxford.
• Bangu, S., 2013, “Indispensability and explanation”, British Journal for the Philosophy of Science 64 (2): 255–277. DOI: http://dx.doi.org/10.1093/bjps/axs026
• Baron, S., 2014, “Optimisation and mathematical explanation: doing the Lévy Walk”, Synthese 191: 459-479. DOI: http://dx.doi.org/10.1007/s11229-013-0284-2
• Bender, C.M., D.C. Brody and M.P. Müller, 2017, “Hamiltonian for the zeros of the Riemann zeta function”, Physical Review Letters 118, 130201. DOI: http://dx.doi.org/10.1103/PhysRevLett.118.130201 (available as: href="https://journals.aps.org/prl/pdf/10.1103/PhysRevLett.118.130201">https://journals.aps.org/prl/pdf/10.1103/PhysRevLett.118.130201)
• Bennett, C., E. Bernstein, G. Brassard and U. Vazirani, 1997, “Strengths and weaknesses of quantum computing” , SIAM J. Comput. 26 (5): 1510–1523; arXiv:quant-ph/9701001. DOI: http://dx.doi.org/10.1137/S0097539796300933
• Berry, M.V., and J.P. Keating, 1999, “H = xp and the Riemann zeros”, pages 355–367 in J.P. Keating, D.E. Khmelnitski and I.V. Lerner (eds.), Supersymmetry and Trace Formulae: Chaos and Disorder, Kluwer Academic/Plenum, New York. DOI: http://dx.doi.org/10.1007/978-1-4615-4875-1_19
• Colyvan, M., 1999, “Confirmation theory and indispensability”, Philosophical Studies 96: 1–19.
• Colyvan, M., 2001, The Indispensability of Mathematics, New York, Oxford University Press. DOI: http://dx.doi.org/10.1093/019513754X.001.0001
• Colyvan, M., 2008, “The ontological commitments of inconsistent theories”, Philosophical Studies 141: 115–123. DOI: http://dx.doi.org/10.1007/s11098-008-9266-5
• Chihara, C., 1990, Constructibility and Mathematical Existence, Clarendon Press, Oxford. DOI: http://dx.doi.org/10.1093/0198239750.001.0001
• Daly, C., and S. Langford, 2009, “Mathematical explanation and indispensability arguments”, The Philosophical Quarterly 59: 641–658. DOI: http://dx.doi.org/10.1111/j.1467-9213.2008.601.x
• Detlefsen, M., and M. Luker, 1980, “The four color-problem and mathematical proof‘”, Journal of Philosophy 77: 803–820. DOI: http://dx.doi.org/10.1111/10.2307/2025806
• Field, H., 1980, Science Without Numbers, Basil Blackwell, Oxford. DOI: dx.doi.org/10.1093/acprof:oso/9780198777915.001.0001
• Feynman, R.P., 1982, “Simulating physics with computers”, International Journal of Theoretical Physics 21 (6/7): 467–488. DOI: http://dx.doi.org/10.1007/BF02650179
• Hales, T.C., 2000, “Cannonballs and honeycombs”, Notices of the American Mathematical Society 47 (4): 440–449.
• Hales, T.C., 2005, “A proof of the Kepler conjecture”, Annals of Mathematics. Second Series 162 (3): 1065–1185. DOI: http://dx.doi.org/10.4007/annals.2005.162.1065
• Harrow, A., A. Hassidim and S. Lloyd, 2009, “Quantum algorithm for linear systems of equations”, Phys. Rev. Lett. 15 (103): 150502, arXiv:0811.3171. DOI: http://dx.doi.org/10.1103/PhysRevLett.103.150502
• Hellman, G., 1989, Mathematics Without Numbers, Clarendon Press, Oxford. DOI: http://dx.doi.org/10.1093/0198240341.001.0001
• Krakowski, I., 1980, “The four-color problem reconsidered”, Philosophical Studies 38: 91–96. DOI: http://dx.doi.org/10.1007/BF00354531
• Lange, M., 2013, “What makes a scientific explanation distinctively mathematical?”, British Journal for the Philosophy of Science 64 (3): 485–511. DOI: http://dx.doi.org/10.1093/bjps/axs012
• Levin, M.A., 1981, “On Tymoczko’s argument for mathematical empiricism”, Philosophical Studies 39: 79–86. DOI: http://dx.doi.org/10.1007/BF00354815
• Liggins, D., 2014, “Abstract expressionism and the communication problem”, British Journal for the Philosophy of Science 65: 599-620.
• Lipton, P., 2004, “What good is an explanation”, pages 1-21 in J. Cornwell (ed.), Explanations. Styles of Explanation in Science, Oxford: Oxford University Press. DOI: http://dx.doi.org/10.1007/978-94-015-9731-9_2
• Lyon, A., 2012, “Mathematical explanations of empirical facts, and mathematical realism”, Australasian Journal of Philosophy 90 (3): 559–578. DOI: http://dx.doi.org/10.1080/00048402.2011.596216
• Lyon, A., and M. Colyvan, 2008, “The explanatory power of phase spaces”, Philosophia Mathematica 16 (2): 227–243. DOI: http://dx.doi.org/10.1093/philmat/nkm025
• Montanaro, A., 2015, “Quantum algorithms: an overview”, https://www.nature.com/articles/npjqi201523 (also: arXiv:1511.04206v2). DOI: http://dx.doi.org/10.1038/npjqi.2015.23
• Németi, I., and G. Dávid, 2006, “Relativistic computers and the Turing barrier”, Journal of Applied Mathematics and Computation 178 (1): 118–142. DOI: http://dx.doi.org/10.1016/j.amc.2005.09.075
• Nielsen, M.A., and I.L. Chuang, 2000, Quantum Computation and Quantum Information, Cambridge University Press. DOI: http://dx.doi.org/10.1017/CBO9780511976667
• Quine, W.V.O., 1953, “Two dogmas of empiricism”, pages 20–46 in From a Logical Point of View, Harvard University Press, Cambridge, Mass.
• Quine, W.V.O., 1981, “Things and their place in theories”, pages 1–23 in Theories and Things, The Belknap Press of Harvard University Press, Cambridge, Mass.
• Rav, Y., 1999, “Why do we prove theorems?”, Philosophia Mathematica 7: 5–41. DOI: http://dx.doi.org/10.1093/philmat/7.1.5
• Shor, P., 1994, “Algorithms for quantum computation: Discrete logarithms and factoring”, pages 124–134 in Proc. 35th Annual Symposium on Foundations of Computer Science, IEEE. DOI: http://dx.doi.org/10.1109/SFCS.1994.365700
• Swart, E.R., 1980, “The philosophical implications of the four-color problem”, American Mathematical Monthly 87: 697–707. DOI: http://dx.doi.org/10.2307/2321855
• Teller, P., 1980, “Computer proof”’, The Journal of Philosophy 77: 797–803. DOI: http://dx.doi.org/10.2307/2025805
• Tymoczko, T., 1979, “The four-color problem and its philosophical significance”, The Journal of Philosophy 76 (2): 57–83. DOI: http://dx.doi.org/10.2307/2025976
• Yablo, S., 2005, “The myth of the seven”, pages 90–115 in M.E. Kalderon (ed.), Fictionalism in Metaphysics, Oxford, Oxford University Press. DOI: http://dx.doi.org/10.1093/acprof:oso/9780199266487.003.0010
• Yablo, S., 2012, “Explanation, extrapolation, and existence”, Mind 121 (484): 1007–1029. DOI: http://dx.doi.org/10.1093/mind/fzs120
• Wójtowicz, K., 2009, “Theory of quantum computation and philosophy of mathematics. Part I”, Logic and Logical Philosophy 18 (3–4): 313–332. DOI: http://dx.doi.org/10.12775/LLP.2009.016
• Wójtowicz, K., 2015, “Could empirical facts become mathematical truths?”, pages 213–230 in J. Ladyman, S. Presnell, G. McCabe, M. Eckstein and S.J. Szybka (eds.), Road to Reality with Roger Penrose, Copernicus Center Press, Kraków.