Abstract
The paradox of phase transitions raises the problem of how to reconcile the fact that we see phase transitions happen in concrete, finite systems around us, with the fact that our best theories—i.e. statistical-mechanical theories of phase transitions—tell us that phase transitions occur only in infinite systems. In this paper we aim to clarify to which extent this paradox is relative to the mathematical framework which is used in these theories, i.e. classical mathematics. To this aim, we will explore the philosophical consequences of adopting constructive instead of classical mathematics in a statistical-mechanical theory of phase transitions. It will be shown that constructive mathematics forces certain ‘de-idealizations’ of such theories: talk of actually infinite systems is meaningless, there are no discontinuous functions, and—in a sense which will be clarified—constructive real numbers reflect our imperfect methods of determining the values of physical quantities. As such, so it will be argued, constructive mathematics offers a means to gain insight in the idealizations introduced in classical theories and the philosophical issues surrounding them.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
Notes
Examples are mean field theories, Lee-Yang theory, Landau’s theory, and the renormalization group method.
The phase transitions with which we will be concerned are the so-called ‘first-order’ phase transitions. These are phase transitions where the system goes from one phase to another crossing a so-called ‘coexistence line’ or ‘phase boundary’, which corresponds to a discontinuity in a macroscopic (thermodynamic) property, which in turn corresponds to a discontinuity in the first derivative of the free energy of the system (see below). There are also ‘continuous’ phase transitions, in which a system changes phase without crossing a phase boundary (for details, see Goldenfeld 1992). Both kinds of phase transitions are philosophically interesting, but given that the physics of continuous phase transitions is much more complicated than that of the first-order ones, we will focus on first-order phase transitions here.
I agree with Shech (2013, 2015), that no paradox can be found in these facts per se, and that, in order to obtain a paradox, assumptions have to be made concerning the representative relationship between the infinite (abstract) and the finite (concrete) systems. However, this issue will not concern us here.
There are various kinds of free energy, each appropriate in different contexts. We will be concerned with Helmholz free energy \(F = U - TS\) (where U is the internal energy of the system, T the absolute temperature, and S the entropy), which is the maximum amount of work that a system can do at a constant volume and temperature.
For more details, see e.g. McComb (2004).
Cfr. e.g. Norton (2012, 2014), Shech (2013). If I understand him correctly, then Batterman takes his ‘physical discontinuities’ (at phase transitions and in other cases) to show that macrolevel phenomena in concrete systems cannot be explained in terms of the behavior of the system at the microlevel (2005). However, as I see it, this argument is unconvincing as long as it is not clear exactly in which sense these ‘physical discontinuities’ are discontinuous.
This argument presupposes that it is possible to define ‘observed qualitative distinctions’ in such a way that only (2) phase transitions—and no (1) changes within a phase—involve observed qualitative distinctions. It is not clear to me that this can be done, but we will ignore this issue here.
For an overview, see Mainwood (2005).
There are several versions of constructivism, and the notion of infinity and its potentiality takes in each of these theories a (slightly) different shape. For now we will disregard these differences, but of course it is an important part of my proposal to determine which constructive theory is the best for our purposes.
This was convincingly shown—in a context a little different from ours—by Mancosu (2009).
Note that the criterion for Cauchy convergence, too, differs in the constructive from that in the classical framework: In classical mathematics, a sequence of rationals \(\{r_n\}\) satisfies the Cauchy criterion for convergence if for any positive real number \(\epsilon \), there exist a natural number \(N (\epsilon )\), such that for every \(n > N (\epsilon )\) and every \(p > 0\), \(|a_{n + p} - a_n| < \epsilon \). In constructive mathematics, on the other hand, a sequence \(\langle r_n \rangle \) of rationals satisfies the Cauchy criterion for convergence if for every natural number k, we can effectively construct a natural number N(k), such that for every \(n > N (k)\) and every \(p > 0\), \(|a_{n + p} - a_n| < \frac{1}{k}\).
See Schechter (2001) for an example.
Thanks to an anonymous referee for pointing out that this was in need of clarification.
It must be noted though, that in his Cambridge lectures, Brouwer also considered ‘measurable functions’: functions that are defined almost everywhere on \({\mathbb {R}}\), but need not be continuous, and thus could be used as a way of modeling discontinuous phenomena (Dalen 1981). Thanks to an anonymous referee for making me aware of this.
This objection is raised in Shech (2013).
Note that a proof by contradiction can only be obtained from a proof of negation by plugging in \(\lnot p\) in place of p, if double negation elimination—or LEM—is assumed. Hence this trick works only classically.
I thank an anonymous referee for this suggestion.
I thank Vincent Ardourel for making me aware of the work of Ye.
Ye shows that all the logical constants defined in strict finitism follow the intuitionistic logical laws, as well as the axiom of choice (Ye 2011, p. 56, Theorem 2.11). Thus, strict finitism can be seen as an interpretation of Bishop-style constructive mathematics, with additional philosophical commitments.
With the possible exception of Brouwer’s measurable functions, which might provide a means to formulate discontinuous functions—see footnote 14. As I see it, it is more natural and useful for our purposes to opt for a constructive systems which forces towards continuity, but even considering a constructive theory which does not do so might still be interesting, for within such a framework it will still be the case that talk of actually infinite systems is meaningless and the constructive conception of real numbers fits well with measurement practice in physics.
References
Batterman, R. (2005). Critical phenomena and breaking drops: Infinite idealizations in physics. Studies in the History and Philosophy of Modern Physics, 36, 225–244.
Bauer, A. (2012). Intuitionistic mathematics and realizability in the physical world. In H. Zenil (Ed.), A computable universe. Singapore: World Scientific.
Bauer, A. (n.d.) Proof of negation and proof by contradiction. On: Mathematics and Computation (blog): http://math.andrej.com/2010/03/29/proof-of-negation-and-proof-by-contradiction/.
Beeson, M. (1985). Foundations of constructive mathematics: Metamathematical studies. Berlin: Springer.
Bridges, D., & Dedui, L. (1997). Paradise lost, or paradise regained? In: CDMTCS research report series.
Brouwer, L. (1927/1967) On the domains of definitions of functions. In van Heijenoort, J. (ed.), From Frege to Gödel: A Source Book in Mathematical Logic, 1879–1931. Cambridge: Harvard University Press.
Butterfield, J. (2011). Less is different: Emergence and reduction reconciled. Foundations of Physics, 41, 1065–1135.
Callender, C. (2001). Taking thermodynamics too seriously. Studies in History and Philosophy of Modern Physics, 32(4), 533–539.
van Dalen, D. (1981). Brouwer’s Cambridge lectures on intuitionism. Cambridge: Cambridge University Press.
Döring, A., & Isham, C. (2008). A topos foundation for theories of physics: I Formal languages for physics. Journal of Mathematical Physics, 49, 053515.
Dummett, D. (2000). Is time a continuum of instants? Philosophy, 75, 497–515.
Dummett, M. (1977). Elements of intuitionism. Oxford: Clarendon Press.
Fletcher, P. (2002). A constructivist perspective on physics. Philosophia Mathematica, 10, 26–42.
Fletcher, P. (2007). Infinity. In D. Jacquette (Ed.), Philosophy of logic, handbook of the philosophy of science (Vol. 5, pp. 523–585). Amsterdam: Elsevier.
Goldenfeld, N. (1992). Lectures on phase transitions and the renormalization group. Boulder: Westview Press.
Iemhoff, R. (2008). Intuitionism in the philosophy of mathematics. In The Stanford Encyclopedia of Philosophy (winter 2016 edn).
Isham, C., & Butterfield, J. (2000). Some possible roles for topos theory in quantum theory and quantum gravity. Conference on Towards a New understanding of Space, Time and Matter (pp. 1707–1735). Dordrecht: Kluwer.
Lanford, O. (1975). Time-evolution of large classical systems. In J. Moser (Ed.), Dynamical systems, theory and applications: Lecture notes in theoretical physics (Vol. 38). Heidelberg: Springer.
Mainwood, P. (2005). Phase transitions in finite systems.
Mancosu, P. (2009). Measuring the size of infinite collections of natural numbers: Was Cantor’s theory of infinite number inevitable? The Review of Symbolic Logic, 2(4), 612–646.
McComb, W. (2004). Renormalization methods: A guide for beginners. Oxford: Clarendon Press.
Norton, J. (2012). Approximation and idealization: Why the difference matters. Philosophy of Science, 79, 207–232.
Norton, J. (2014). Infinite idealizations. In F. Stadtler (Ed.), European philosophy of science—Philosophy of science in Europe and the Viennese heritage: Vienna Circle Institute Yearbook. Dordrecht: Springer.
Pincock, C. (2014). How to avoid inconsistent idealizations. Synthese, 191, 2957–2972.
Ruelle, D. (1999). Statistical mechanics: Rigorous results. London: Imperial college Press.
Ruelle, D. (2004). Thermodynamic formalism. Cambridge: Cambridge University Press.
Schechter, E. (2001). Constructivism is difficult. The American Mathematical Monthly, 108(1), 50–54.
Shech, E. (2013). What is the paradox of phase transitions? Philosophy of Science, 80, 1170–1181.
Shech, E. (2015). Scientific misrepresentation and guides to ontology: The need for representational code and content. Synthese, 192, 3463–3485.
Stanley, H. (1971). Introduction to Phase Transitions and Critical Phenomena. Oxford: Oxford University Press.
Ye, F. (2011). Strict finitism and the logic of mathematical applications. Dordrecht: Springer.
Acknowledgements
This paper profited from discussions with Charlotte Werndl, Giovanni Valente, Erik Curiel, Patricia Palacios, and Sam Sanders on the subject, and from comments on drafts of this paper by the first three, and two anonymous referees. Thank you all very much, you have been very helpful. Further, the author wishes to thank the organizers and audiences of the Infinite Idealizations in Science conference, the Work in Progress seminar, and Yet Another Great Workshop on Explanation, Causation, and Idealization at the Munich Center for Mathematical Philosophy, and of the Philosophy Workshop and the Irreversibility Workshop at the University of Salzburg, for their interest and helpful questions and suggestions. Last but certainly not least, a special thanks to Michele Ginammi for helpful discussions and comments on drafts of this paper, and especially for his continuous support.
Author information
Authors and Affiliations
Corresponding author
Additional information
The author greatfully acknowledges that this paper is partly written while visiting the Munich Center for Mathematical Philosophy, supported by their Junior Visiting Fellowship, and partly while visiting the University of Salzburg.
Rights and permissions
About this article
Cite this article
van Wierst, P. The paradox of phase transitions in the light of constructive mathematics. Synthese 196, 1863–1884 (2019). https://doi.org/10.1007/s11229-017-1637-z
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11229-017-1637-z