Abstract
This paper sets out to show how mathematical modelling can serve as a way of ampliating knowledge. To this end, I discuss the mathematical modelling of time in theoretical physics. In particular I examine the construction of the formal treatment of time in classical physics, based on Barrow’s analogy between time and the real number line, and the modelling of time resulting from the Wheeler-DeWitt equation. I will show how mathematics shapes physical concepts, like time, acting as a heuristic means—a discovery tool—, which enables us to construct hypotheses on certain problems that would be hard, and in some cases impossible, to understand otherwise.
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Notes
- 1.
See Polya (1954), Hanson (1958), Lakatos (1976), Laudan (1977), Simon (1977), Nickles (1980a, b), Simon et al. (1987), Gillies (1995), Grosholz and Breger (2000), Grosholz (2007), Abbott (2004), Darden (2006), Weisberg (2006), Magnani (2001, 2013), Magnani and Li (2007), Nickles and Meheus (2009), Magnani et al. (2010), Gertner (2012), Cellucci (2013), Ippoliti (2011, 2014).
- 2.
That is the strings of symbols representing objects and operations and assembled according to certain syntactic rules.
- 3.
See e.g. Devito (1997) for a non-linear account of time.
- 4.
Note that this would also imply the replacement of the differential equations with difference equations, so the time derivative would be replaced by a finite difference.
- 5.
See Hagar (2014), pp. 64–75, for a careful analysis of the issue.
- 6.
Formally this means that the space is invariant under rotations in space, since rotations can map any direction onto any other direction.
- 7.
Diffeomorphism invariance is a way of mathematically expressing general covariance—the invariance of the form of physical laws under arbitrary differentiable coordinate transformations and then the background independence of a theory.
- 8.
Where \(\partial^{2}\) is the second derivative w.r.t. x; x the position; \(\Psi\) the Shroindger wave function; E the energy; V the potiantial energy.
- 9.
- 10.
See for instance the answer provided by the termal time hypothesis in Connes and Rovelli (1994).
- 11.
- 12.
See e.g. Ippoliti (2006) for a critical examination of the notion of relevance.
- 13.
- 14.
For instance Feynman integrals can be regarded as an altered version of the notion of integral—see Ippoliti (2013).
- 15.
See Kvasz (2008) for a deep account of this point.
- 16.
- 17.
The “machinery” is the term used by Feynman to express the mechanism that explains why, but especially how, certain processes take place—see also Morrison (2000, p. 3) on this point.
References
Abbott, A. (2004). Method of discovery. New York: W.W. Norton & Company Inc.
Barrow, I. (1660). Letiones geometricae (J. M. Child, Trans., 1916). The geometrical lectures of isaac barrow. London: The Open Court Publishing Company.
Bolzano, B. (1817). Rein analytischer Beweis des Lehrsatzes dass zwischen je zwey Werthen, die ein entgegengesetztes Resultat gewaehren, wenigstens eine reele Wurzel der Gleichung liege. Prague. (S. B. Russ “A Translation of Bolzano’s Paper on the Intermediate Value Theorem”, Trans.) History of Mathematics, 7, 156–185, 1980.
Brown, H. (2015). Against interpretation in mathematical physics. Draft.
Bunge, M. (1981). Analogy in quantum theory: From insight to nonsense. The British Journal for the Philosophy of Science, 18(4), 265–286.
Butterworth, B. (1999). The mathematical brain. London: Macmillan.
Carazza, B., & Kragh, H. (1995). Heisenberg’s lattice world: The 1930 theory sketch. American Journal of Physics, 63, 595.
Cellucci, C. (2013). Rethinking logic. New York: Springer.
Cellucci, C. (2015). Naturalizing the applicability of mathematics. Paradigmi Rivista di critica filosofica, 2, 23–42.
Connes, A., & Rovelli, C. (1994). Von neumann algebra automorphisms and time-thermodynamics relation in general covariant quantum theories. Classical and Quantum Gravity, 11, 2899–2918.
Darden, L. (Ed.). (2006). Reasoning in biological discoveries: Essays on mechanisms, inter-field relations, and anomaly resolution. New York: Cambridge University Press.
Devito, C. L. (1997). A non-linear model for time. In W. G. Tifft & Cocke, W. J. (Eds.), Modern mathematical models of time and their applications to physics and cosmology (pp. 357–370). Springer.
Devlin, K. (2000). The math gene: How mathematical thinking evolved and why numbers are like gossip. New Tork: Basic Books.
DeWitt, B. S. (1967). Quantum theory of gravity. I. The canonical theory. Physical Review, 160, 1113.
Everett, H. (1957). Relative state formulation of quantum mechanics. Reviews of Modern Physics, 29, 454–462.
Fabri, E. (2005). Insegnare relatività nel XXI secolo. Bollettino trimestrale dell’Associazione Italiana per l’insegnamento della fisica. XXXVIII (2). Supplemento.
Feynman, R. (1963). Six easy pieces. New York: Basic Books.
Feynman, R. (1977). Lectures on physics. Addison Wesley.
Gertner, J. (2012). The idea factory. New York: Penguin Press.
Gillies, D. (1995). Revolutions in mathematics. Oxford: Oxford University Press.
Grosholz, E. (2007). Representation and productive ambiguity in mathematics and the sciences. Oxford: Oxford University Press.
Grosholz, E., & Breger, H. (Eds.). (2000). The growth of mathematical knowledge. Dordercht: Springer.
Hagar, A. (2014). Discrete or continuous? The quest for fundamental length in modern physics. New York: Cambridge University Press.
Hanson, N. (1958). Patterns of discovery: An inquiry into the conceptual foundations of science. Cambridge: Cambridge University Press.
Heisenberg, W. (1930). The self-energy of the electron. In A. Miller (Ed.), Early quantum electrodynamics (pp. 121–128). Cambridge: Cambridge University Press 1994.
Ippoliti, E. (2006). Demonstrative and non-demonstrative reasoning by analogy. In C. Cellucci & P. Pecere (Eds.), Demonstrative and non-demonstrative reasoning in mathematics and natural science (pp. 309–338). Cassino: Edizioni dell’Università di Cassino.
Ippoliti, E. (2011). Between data and hypotheses. In C. Cellucci, E. Grosholz, & E. Ippoliti (Eds.), Logic and knowledge. Newcastle Upon Tyne: Cambridge Scholars Publishing.
Ippoliti, E. (2013). Generation of hypotheses by ampliation of data. In L. Magnani (Ed.), Model-based reasoning in science and technology (pp. 247–262). Berlin: Springer.
Ippoliti, E. (Ed.). (2014). Heuristic reasoning. London: Springer.
Johnson, M. (1988). Some constraints on embodied analogical understanding. In D. H. Helman (Ed.), Analogical reasoning: Perspectives of artificial intelligence, cognitive science, and philosophy. Dordrecht: Kluwer.
Kvasz, L. (2008). Patterns of change. Linguistic innovations in the development of classical mathematics. Basel: Birkhäuser.
Lakatos, I. (1976). Proofs and refutations: The logic of mathematical discovery. Cambridge: Cambridge University Press.
Lakoff, G., & Nuñez, R. (2001). Where mathematics come from: How the embodied mind brings mathematics. New York: Basics Books.
Laudan, L. (1977). Progress and its problems. Berkeley and LA: University of California press, California.
Magnani, L. (2001). Abduction, reason, and science. Processes of discovery and explanation. New York: Kluwer Academic.
Magnani, L. (2013). Model-based reasoning in science and technology. Berlin: Springer.
Magnani, L., & Li, P. (Eds.). (2007). Model-based reasoning in science, technology, and medicine. London: Springer.
Magnani, L., Carnielli, W., & Pizzi, C. (Eds.). (2010). Model-based reasoning in science and technology: Abduction, logic, and computational discovery. Heidelber: Springer.
Morrison, M. (2000). Unifying scientific theories. New York: Cambridge University Press.
Nickles, T. (Ed.). (1980a). Scientific discovery: Logic and rationality. Boston: Springer.
Nickles, T. (Ed.). (1980b). Scientific discovery: Case studies. Boston: Springer.
Nickles, T., & Meheus, J. (Eds.). (2009). Methods of discovery and creativity. New York: Springer.
Parentani, R. (1997). Interpretation of the solutions of the Wheeler-DeWitt equation. Physical Review D, 56(8), 4618–4624.
Peres, A. (1999). Critique of the Wheeler-DeWitt equation. In A. Harvey (Ed.), On Einstein’s path (pp. 367–379). New York: Springer-Verlag.
Piaget, J., & Cook, M. T. (1952). The origins of intelligence in children. New York: International University Press.
Polya, G. (1954). Mathematics and plausible reasoning. Princeton: Princeton University Pres.
Rovelli, C. (2001). Notes for a brief history of quantum gravity. arXiv:gr-qc/0006061v3.
Simon, H. (1977). Models of discovery. Dordrecht: Reidel.
Simon, H., Langley, P., Bradshaw, G., & Zytkow, J. (Eds.). (1987). Scientific discovery: Computational explorations of the creative processes. Boston: MIT Press.
Turner, M. (1988). Categories and analogies. In D. H. Helman (Ed.), Analogical reasoning: Perspectives of artificial intelligence, cognitive science, and philosophy. Dordrecht: Kluwer.
Turner, M. (2005). The literal versus figurative dichotomy. In S. Coulson & B. Lewandowska-Tomaszczyk (Eds.), The literal and nonliteral in language and thought (pp. 25–52). Frankfurt: Peter Lang.
Tzanakis, C. (2002a). On the relation between mathematics and physics in undergraduate teaching. In I. Vakalis, et al. (Eds.), 2nd international conference on the teaching of mathematics (at the undergraduate level) (p. 387). New York, NY: Wiley.
Tzanakis, C. (2002b). Unfolding interrelations between mathematics and physics, in a presentation motivated by history: Two examples. International Journal of Mathematical Education in Science and Technology, 30(1), 103–118.
Weisberg, R. (2006). Creativity: Understanding innovation in problem solving, science, invention, and the arts. Hoboken (NJ): John Wiley & Sons.
Windred, G. (1933a). The history of mathematical time: I. Isis, 19(1), 121–53.
Windred, G. (1933b). The history of mathematical time: II. Isis, 20(1), 192–219.
Windred, G. (1935). The interpretation of imaginary mathematical time. The Mathematical Gazette, 19(235), 280–290.
Acknowledgments
I would like to thank Sergio Caprara, Angleo Vulpiani and the other friends at the Dept. of Physics—Sapienza University of Rome for the fruitful dialogues about technical as well as theoretical issues. I would also like to thank the two anonymous referees for their comments and suggestions that helped me to improve the paper.
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Ippoliti, E. (2016). Mathematical Models of Time as a Heuristic Tool. In: Magnani, L., Casadio, C. (eds) Model-Based Reasoning in Science and Technology. Studies in Applied Philosophy, Epistemology and Rational Ethics, vol 27. Springer, Cham. https://doi.org/10.1007/978-3-319-38983-7_7
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