Abstract
Philip Anderson’s seminal paper `More Is Different’ is explicit in its arguments for the failure of construction methods in some areas of physics and inexplicit about the consequences of those failures. I argue that as published, Anderson’s position is obviously consistent with a reductionist position and, contrary to many casual claims, does not provide evidence for the existence of emergent phenomena. Various emergentist positions are defined and some recent undecidability results about infinite and finite Ising lattices by Barahona and by Gu et al. are examined. The former do not provide evidence for the existence of ontologically emergent states in real systems but they do provide insight into prediction-based accounts of emergence and the limits of certain theoretical representations. The latter results bear primarily on claims of weak emergence and provide support for Anderson’s claims. A pressing open problem is to articulate what counts as a novel physical property.
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- 1.
Note that this definition allows that not all fundamental laws and fundamental facts can be captured in whatever representational apparatus is used for fundamental physics, be it mathematical, computational, diagrammatic, or some other type.
- 2.
They are used in some of Anderson’s later writings, such as Anderson and Stein (1984).
- 3.
A different and perhaps more precise definition of ‘macroscopic’ is: A macroscopic system is one whose equation of state is independent of its size and an ideal macroscopic system is one containing an infinite number of particles, the density of which is finite. A real macroscopic system is one that is sufficiently large that its equation of state is empirically indistinguishable from an ideal macroscopic system (Sewell 1986, p. 4).
- 4.
Brian McLaughlin asserts that Broad is using a semantic conception of deduction here; i.e. B is deducible from A if and only if when A is true, B is true (McLaughlin 1992), but I shall use the more standard proof-theoretic idea of deducibility.
- 5.
- 6.
The Halting Problem, which is actually a theorem, tells us that there is no effective procedure for determining in advance whether a program for a partial recursive function will halt on an arbitrary input or not.
- 7.
- 8.
- 9.
The lattices have no properties beyond those specified and so nothing specifically quantum mechanical should be imputed to them.
- 10.
The argument given in Gu et al. 2009 is not in the form of a formal proof and it omits details at various points. The exact way in which limits on the infinite lattice are justified is therefore hard to assess.
- 11.
This simplification of the argument omits details that can be found in Gu et al. (2009), p. 838.
- 12.
Gu et al. are clear both that their results, proved for two-dimensional lattices, can be extended to higher dimensions and that there are other interpretations of Ising models besides magnetic systems. Nevertheless, because appeals are made to features such as ground states and Hamiltonians, those other interpretations are irrelevant unless appropriate interpretations of H, Ei, and so on can be given that correspond to legitimate features of the concrete systems. I note that some well-known features of ferromagnets such as phase transitions play no role in the results here discussed.
- 13.
It is not impossible because if our universe contained at least two infinite dimensions, not necessarily spatial or temporal, within which the required features of the Ising lattice could be embedded, a real physical system would exist with undecidable properties. Finite dimensions with periodic boundary conditions would not do, but a two dimensional unbounded system that could be extended arbitrarily far in either dimension would.
- 14.
For an early discussion of why novel values are not emergent, see Teller (1992).
- 15.
This is a real, rather than invented, example of the problem discussed in Dennett (1991).
- 16.
The appeal to autonomous principles such as localization and continuous symmetry breaking seems to be a central feature of Robert Laughlin and David Pine’s approach to emergence (Laughlin and Pine 2000).
- 17.
By ‘empirical’ I do not mean ‘empiricist’ in any traditional sense of the term but the results of measurement, instrument output, or experiment having causal origins in the subject matter, however remote those are from human observational capacities. Discussions of such liberalized approaches can be found in Shapere (1982), Humphreys (1999), Bogen (2011).
- 18.
For one such example, see Humphreys (1994).
- 19.
One exception to this neglect of the details of how axiomatic theories are applied is Suppes (1962), where a hierarchy of models is used to connect theory with data. Suppes explicitly allows that certain features of theory application do not lend themselves to formal treatment.
- 20.
‘Time’ here is to be taken as the number of computational steps required and the computation is referenced to any Turing machine-equivalent computational process.
- 21.
Further finite results can be found in Gu and Perales (2012).
- 22.
Thanks to Jeremy Butterfield, Toby Cubitt, and Ashley Montanaro for comments about some of the formal results discussed here and to Margaret Morrison for comments that led to improvements in the paper as a whole. The conclusions drawn are entirely my own.
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Humphreys, P.W. (2015). More is Different…Sometimes: Ising Models, Emergence, and Undecidability. In: Falkenburg, B., Morrison, M. (eds) Why More Is Different. The Frontiers Collection. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-43911-1_8
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