Arithmetic, Set Theory, Reduction and Explanation

Philosophers of science since Nagel have been interested in the links between intertheoretic reduction and explanation, understanding and other forms of epistemic progress. Although intertheoretic reduction is widely agreed to occur in pure mathematics as well as empirical science, the relationship between reduction and explanation in the mathematical setting has rarely been investigated in a similarly serious way. This paper examines an important particular case: the reduction of arithmetic to set theory. I claim that the reduction is unexplanatory. In defense of this claim, I offer evidence from mathematical practice, and I respond to contrary suggestions due to Steinhart, Maddy, Kitcher and Quine. I then show how, even if set-theoretic reductions are generally not explanatory, set theory can nevertheless serve as a legitimate foundation for mathematics. Finally, some implications of my thesis for philosophy of mathematics and philosophy of science are discussed. In particular, I suggest that some reductions in mathematics are probably explanatory, and I propose that differing standards of theory acceptance might account for the apparent lack of unexplanatory reductions in the empirical sciences

Viewing-as explanations and ontic dependence

According to a widespread view in metaphysics and philosophy of science, all explanations involve relations of ontic dependence between the items appearing in the explanandum and the items appearing in the explanans. I argue that a family of mathematical cases, which I call “viewing-as explanations”, are incompatible with the Dependence Thesis. These cases, I claim, feature genuine explanations that aren’t supported by ontic dependence relations. Hence the thesis isn’t true in general. The first part of the paper defends this claim and discusses its significance. The second part of the paper considers whether viewing-as explanations occur in the empirical sciences, focusing on the case of so-called fictional models. It’s sometimes suggested that fictional models can be explanatory even though they fail to represent actual worldly dependence relations. Whether or not such models explain, I suggest, depends on whether we think scientific explanations necessarily give information relevant to intervention and control. Finally, I argue that counterfactual approaches to explanation also have trouble accommodating viewing-as cases

Proving Quadratic Reciprocity: Explanation, Disagreement, Transparency and Depth

Gauss’s quadratic reciprocity theorem is among the most important results in the history of number theory. It’s also among the most mysterious: since its discovery in the late 18th century, mathematicians have regarded reciprocity as a deeply surprising fact in need of explanation. Intriguingly, though, there’s little agreement on how the theorem is best explained. Two quite different kinds of proof are most often praised as explanatory: an elementary argument that gives the theorem an intuitive geometric interpretation, due to Gauss and Eisenstein, and a sophisticated proof using algebraic number theory, due to Hilbert. Philosophers have yet to look carefully at such explanatory disagreements in mathematics. I do so here. According to the view I defend, there are two important explanatory virtues—depth and transparency—which different proofs (and other potential explanations) possess to different degrees. Although not mutually exclusive in principle, the packages of features associated with the two stand in some tension with one another, so that very deep explanations are rarely transparent, and vice versa. After developing the theory of depth and transparency and applying it to the case of quadratic reciprocity, I draw some morals about the nature of mathematical explanation

Decompositions of unitary evolutions and entanglement dynamics of bipartite quantum systems

We describe a decomposition of the Lie group of unitary evolutions for a bipartite quantum system of arbitrary dimensions. The decomposition is based on a recursive procedure which systematically uses the Cartan classification of the symmetric spaces of the Lie group SO(n). The resulting factorization of unitary evolutions clearly displays the local and entangling character of each factor.Comment: 11 pages, revtex

Functional Distribution, Land Ownership and Industrial Takeoff: The Role of Effective Demand

This paper analyses how the distribution of land property rights affects industrial takeoff and aggregate income through the demand side. We study a stylized economy composed of two sectors, agriculture and manufacturing. The former produces a single subsistence good while the latter is constituted of a continuum of markets producing distinct commodities. Following Murphy et al. [20] we model industrialization as the introduction of an increasing returns technology in place of a constant returns one. However, we depart from their framework by assuming income to be distributed according to functional groups membership (landowners, capitalists, workers). We carry out an equilibrium analysis for different levels of land ownership concentration proving that, under the specified conditions, there is a non-monotonic relation between the distribution of land property rights and both industrialization and income. We clarify that non-monotonicity arises because of the way land ownership concentration affects the level and the distribution of profits among capitalists which, in turn, shape their demand. Our results suggest that i) both a too concentrated and a too diffused distribution of land property rights can be detrimental to industrialization, ii) land ownership affects the economic performance of an industrializing country by determining the demand of manufactures of both landowners and capitalists, iii) in terms of optimal land distribution there may be a tradeoff between income and industrialization.

On the Commutative Equivalence of Context-Free Languages

The problem of the commutative equivalence of context-free and regular languages is studied. In particular conditions ensuring that a context-free language of exponential growth is commutatively equivalent with a regular language are investigated

From Turing instability to fractals

Complexity focuses on commonality across subject areas and forms a natural platform for multidisciplinary activities. Typical generic signatures of complexity include: (i) spontaneous occurrence of simple patterns (e.g. stripes, squares, hexagons) emerging as dominant nonlinear modes [1], and (ii) the formation of a highly complex pattern in the form of a fractal (with comparable levels of detail spanning decades of scale). Recently, a firm connection was established between these two signatures, and a generic mechanism was proposed for predicting the fractal generating capacity of any nonlinear system [2]. The mechanism for fractal formation is of a very general nature: any system whose Turing threshold curves exhibit a large number of comparable spatial-frequency instability minima are potentially capable of generating fractal patterns. Spontaneous spatial fractals were first reported for a very simple nonlinear system: the diffusive Kerr slice with a single feedback mirror [3]. These Kerr-slice fractals are distinct from both the transverse fractal eigenmodes of unstable-cavity lasers [4], and also from the fractals found in optical soliton-supporting systems [5,6]. On the one hand, unstable-cavity fractals may be regarded as a linear superposition of diffraction patterns with different scale lengths, each of which arises from successive round-trip magnifications of an initial diffractive seed. On the other hand, fractals formed in the Kerr slice result entirely from intrinsic nonlinear dynamics (i.e. light-matter coupling leading to harmonic generation and/or four-wave mixing cascades). These processes conspire to generate new spatial frequencies that, in turn, can produce optical structure on smaller and smaller scales, down to the order of the optical wavelength. Here we report the first predictions of spontaneous fractal patterns inside driven damped ring cavities containing a thin slice of nonlinear material. Both dispersive (i.e. diffusive-relaxing Kerr [3]) and absorptive (i.e. Maxwell- Bloch saturable absorber [7]) are considered. New linear analyses have shown that the transverse instability spectra of these two cavity systems possess the requisite comparable minima that predict the capacity for the spontaneous generation of fractal patterns. Extensive numerical simulations, in both one and two transverse dimensions, have verified that both the dispersive and absorptive cavities do indeed give rise to nonlinear optical fractals in the transverse plane. Our results confirm that the mechanism for fractal formation has independence with respect to the details of the nonlinearity. An essential ingredient for the generation of fractals is the presence of a feedback mechanism [2]. Feedback drives the cascade process that is responsible for the creation of higher spatial wavenumbers, and which ultimately leads to the “structure across decades of scale” character of the fractal pattern. Cavity geometries are therefore ideal candidates as potential optical fractal generators. The simplest dispersive nonlinearity is provided by the relaxing-diffusing Kerr effect. The threshold curves possess the qualitative features necessary for the generation of spontaneous fractal patterns: successive and comparable spatial frequency minima. Rigorous simulations have shown that the Kerr cavity is indeed capable of generating fractal patterns. In a single-K configuration, where the filter attenuates all those spatial wavenumbers outside the first instability band, it is found that simple stripe patterns emerge. Once this stationary pattern has been reached, the spatial filter is removed to allow all waves to propagate. Energy is transferred to higher spatial frequencies, and the simple strip pattern acquires successive level of fine detail at a rate that depends upon the system parameters. By analysing the power spectrum P(K) it can be seen that a fractal pattern emerges relatively rapidly. Eventually, the system enters a dynamic equilibrium (within typically less than a hundred transits) where the average power spectrum remains unchanged even though the pattern continues to evolve in real space. When this statistically invariant state is attained, the pattern is referred to as a scale-dependent fractal. An appreciable portion of the dynamic state is well described by a linear relationship ln P(K) = a + bK, where a and b are constants, and this type of behaviour is one of the characteristics of a fractal pattern [2]. We have recently found that a thin-slice Maxwell-Bloch saturable absorber [7] can also generate fractal patterns. This system can be either purely absorptive or purely dispersive. Linear analysis, together with a generalized boundary condition (which allows for attenuation), yields the threshold condition for Turing instability. One finds that the threshold spectrum comprises a series of adjacent instability islands. Simulations have revealed that the Maxwell-Bloch system can also support fractals. The single-K patterns turn out to be hexagonal arrays, familiar from conventional pattern formation [1,3]. Once this state has been reached, the spatial filter is removed and one can observe a rapid transition toward a fractal pattern. The qualitative behaviour of fractals patterns in both dispersive and absorptive systems are found to be the same, confirming the assertion of independence with respect to nonlinearity. References: [1] J. B. Geddes et al., “Hexagons and squares in a passive nonlinear optical system,” Phys. Rev. A 5, 3471-3485 (1994). [2] J. G. Huang and G. S. McDonald, “Spontaneous optical fractal pattern formation,” Phys. Rev. Lett. 94, 174101 (2005). [3] G. D’Alessandro and W. J. Firth, “Hexagonal spatial patterns for a Kerr slice with a feedback mirror,” Phys. Rev. A 46, 537-548 (1992). [4] J. G. Huang et al., “Fresnel diffraction and fractal patterns from polygonal apertures,” J. Opt. Soc. Am. A 23, 2768-2774 (2006). [5] M. Soljacic and M. Segev, “Self-similarity and fractals in soliton-supporting systems,” Phys. Rev. E 61, R1048-R1051 (2000). [6] S. Sears et al., “Cantor set fractals from solitons,” Phys. Rev. Lett. 84, 1902-1905 (2000). [7] A. S. Patrascu et al., “Multi-conical instability in the passive ring cavity: linear analysis,” Opt. Commun. 91, 433-443 (1992)

Environment induced incoherent controllability

We prove that the environment induced entanglement between two non interacting, two-dimensional quantum systems S and P can be used to control the dynamics of S by means of the initial state of P. Using a simple, exactly solvable model, we show that both accessibility and controllability of S can be achieved under suitable conditions on the interaction of S and P with the environment.Comment: revtex4, 5 page

Optimal generation of entanglement under local control

We study the optimal generation of entanglement between two qubits subject to local unitary control. With the only assumptions of linear control and unitary dynamics, by means of a numerical protocol based on the variational approach (Pontryagin's Minimum Principle), we evaluate the optimal control strategy leading to the maximal achievable entanglement in an arbitrary interaction time, taking into account the energy cost associated to the controls. In our model we can arbitrarily choose the relative weight between a large entanglement and a small energy cost.Comment: 4 page