Abstract
In the previous four chapters we have considered certain issues of theory reduction in the specific subjects of statistical mechanics, macroscopic phenomena, chaos, quantum mechanics and chemistry, which are united by the presence of singular limits and emergent properties. Nevertheless, these subjects appear to be almost completely independent of each other, and of interest to scholars only as separate fields.
Cause when the goin’ gets tough... The tough get goin’!
J.B. Blutarsky 1978.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Notes
- 1.
We characterise our approach as non-reductionist rather than anti-reductionist.
- 2.
In general, a single dimension is not enough to fully characterise such measures, which typically have a multifractal nature; infinitely many dimensions (the Renyi dimensions) are necessary for that. This technical issue is not important for our purpose.
- 3.
There are cases in which emergent features are almost trivial consequences of smoothing mechanisms. For instance the bouncing on the asphalt of a small bead of, say, half a centimetre radius, is quite irregular, because the roughness of the asphalt is of a size comparable to that of the bead. By contrast, a tennis ball bounces regularly, as if the asphalt were smooth, because its diameter is significantly larger than the scale of the asphalt roughness.
- 4.
The Boltzmann-Gibbs law:
$$\begin{aligned} \rho (\mathbf {X})= Const. \, e^{-\beta H(\mathbf {X})} \end{aligned}$$describes the statistical features of a system which exchanges energy with an external environment at given temperature \(1/\beta \), and can be obtained from the microcanonical distribution in the large-\(N\) limit.
- 5.
Critical phenomena can be treated by means of the powerful renormalisation group method, which has played an important role in the physics of the second half of the twentieth century, both as a conceptual and a computational tool. Remarkably, the renormalisation group has a probabilistic interpretation as a generalisation of the central limit theorem to the case of non independent variables. This fact has been understood in the mid-seventies by Bleher and Sinai, and has been widely investigated by Jona-Lasinio (2001).
- 6.
Logical empiricists went certainly too far in this direction, but they appreciated this fundamental point. Some contemporary scientists and philosophers (notably postmodernists) seem very far from getting the message.
- 7.
Understanding the existence of the laws of nature was a great achievement brought about by the scientific thought. The path towards such an important step has been long and tortuous, often affected by theological views, such as in Leibniz’s philosophy.
- 8.
Such a view of the science is shared by many scientists in the positivism or neopositivism currents. As an interesting exception, we may recall Born (1948), who ironically noted: if we want to economise thinking, the best way would be to stop thinking at all, and then the expression “economy of thinking” may have an appeal to engineers or others interested in practical applications, but hardly to those who enjoy thinking for no other purpose than clarify a problem.
- 9.
Wigner (1963) considers the understanding of the distinction between laws and initial conditions as the most important contribution that Newton made to science. He considers this even more important than the laws of gravitation and dynamics.
- 10.
For instance, in the case of the protocol \(\alpha (n) = \alpha (n-1)+\delta _\alpha \), the sequences can be compressed. For intrisically random processes, they cannot.
- 11.
Such a difficulty is well known in statistical physics; it has been stressed e.g. by Onsager and Machlup (1953) in their seminal work on fluctuations and irreversible processes, with the caveat: how do you know you have taken enough variables, for it to be Markovian? In a similar way, Ma (1985) notes that: the hidden worry of thermodynamics is: we do not know how many coordinates or forces are necessary to completely specify an equilibrium state.
References
Atkins, P.: Galileo’s Finger: the Ten Great Ideas of Science. Oxford University Press, Oxford (2003)
Barker, J.A., Henderson, D.: What is ”liquid” ? understanding the states of matter. Rev. Mod. Phys. 48(4), 587 (1976)
Born, M.: Natural Philosophy of Cause and Chance. Read Books, Vancouver (1948)
Brody, T.A., Flores, J., French, J.B., Mello, P.A., Pandey, A., Wong, S.S.M.: Random-matrix physics: spectrum and strength fluctuations. Rev. Mod. Phys. 53(3), 385 (1981)
Cecconi, F., Cencini, M., Falcioni, M., Vulpiani, A.: Predicting the future from the past: an old problem from a modern perspective. Am. J. Phys. 80, 1001 (2012)
Davies, P.: Why is the physical world so comprehensible? In: Zurek, W.H. (ed.) Complexity, Entropy and the Physics of Information, pp. 61. Santa Fe Institute (1990)
Dorato, M.: Why is the language of nature mathematical? In: Antonini, G., Altamore, A. (eds) Galileo and the Renaissance Scientific Discourse, pp. 65. Nuova Cultura (2010)
Duhem, P.: The Aim and Structure of Physical Theory. Newblock Princeton University Press, Princeton (1991)
Hansen, J.P., McDonald, I.R.: Theory of Simple Liquids. Academic Press Inc., London (1986)
Jaynes, E.T.: Foundations of probability theory and statistical mechanics. In: Bunge, M. (ed.) Delaware Seminar in the Foundations of Physics, vol. 1, p. 76. Springer-Verlag, Berlin (1967)
Jona-Lasinio. G.: La matematica come linguaggio delle scienze della natura. In Matematica cultura e societá, pp. 93. Quaderni della Scuola Normale Superiore di Pisa (2005)
Jona-Lasinio, G.: Renormalization group and probability theory. Phys. Rep. 352(4), 439 (2001)
Kac, M.: Probability and Related Topics in Physical Sciences. American Mathematical Society (1957)
Kantz, H., Schreiber, T.: Nonlinear Time Series Analysis. Cambridge University Press, Cambridge (1997)
Li, M., Vitanyi, P.: Inductive reasoning and Kolmogorov complexity. J. Comput. Syst. Sci. 44(2), 343 (1992)
Ma, S.K.: Statistical Mechanics. World Scientific, Singapore (1985)
McAllister, J.W.: Algorithmic randomness in empirical data. Stud. Hist. Philos. Sci. Part A 34(3), 633 (2003)
Miller, J., Weichman, P.B., Cross, M.C.: Statistical mechanics, Euler’s equation, and Jupiter’s red spot. Phys. Rev. A 45(4), 2328 (1992)
Newton. I: Optics. Dover Publications, New York (1957)
Onsager, L., Machlup, S.: Fluctuations and irreversible processes. Phys. Rev. 91(6), 1505 (1953)
Peres, A.: Unperformed experiments have no results. Am. J. Phys 46(7), 745–747 (1978)
Silsbee, R.H.: Focusing in collision problems in solids. J. Appl. Phys. 28(11), 1246 (1957)
Solomonoff, R.J.: A formal theory of inductive inference. part i and ii. Inf. Control 7(1), 1 (1964)
Uffink, J.: Can the maximum entropy principle be explained as a consistency requirement? Stud. Hist. Philos. Mod. Phys. 26B(3), 223 (1995)
Wheeler, J.A.: On recognizing a law without law, Öersted medal response at the joint APS-AAPT meeting, Am. J. Phys. 51, 398, 25 January 1983, New York, (1983)
Wheeler, J.A.: Recent thinking about the nature of the physical world: it from bit. Ann. N. Y. Acad. Sci. 655(1), 349 (1992)
Wigner, E.P.: Events, laws of nature, and invariance principles. Nobel Lectures in Physics, (1963)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
Copyright information
© 2014 Springer International Publishing Switzerland
About this chapter
Cite this chapter
Chibbaro, S., Rondoni, L., Vulpiani, A. (2014). Some Conclusions and Random Thoughts. In: Reductionism, Emergence and Levels of Reality. Springer, Cham. https://doi.org/10.1007/978-3-319-06361-4_7
Download citation
DOI: https://doi.org/10.1007/978-3-319-06361-4_7
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-06360-7
Online ISBN: 978-3-319-06361-4
eBook Packages: Physics and AstronomyPhysics and Astronomy (R0)