Abstract
A significant chapter of the short history of formal philosophy is related with the notion and the theory of the so-called “Social Welfare Functions (SWFs)”, as a substantial component of the “social choice theory”. One of the main uses of SWFs is aimed, indeed, at representing coherent patterns (effectively, algebraic structures of relations) of individual and collective choices/preferences, with respect to a fixed ranking of alternative social/economical states. Indeed, the SWF theory is originally inspired by Samuelson’s pioneering work on the foundations of mathematical economic analysis. It uses explicitly Gibbs’ thermodynamics of ensembles “at equilibrium” based on statistical mechanics as the physical paradigm for the mathematical theory of economic systems. In both theories, indeed, the differences and the relationships among individuals are systematically considered as irrelevant. On the contrary, in the mathematical theory of “Social Choice Functions” (SCFs) developed by Amartya Sen, the interpersonal comparison and the real-time information exchanges among different social actors and their environments—different—ethical values and constraints, included—play an essential role. This means that the inspiring physical paradigm is no longer “gas” but “fluid thermodynamics” of interacting systems passing through different “phases” of fast “dissolution/aggregation of coherent behaviors”, and then staying persistently in far from equilibrium conditions. These processes are systematically studied by the quantum field theory (QFT) of “dissipative systems”, at the basis of the physics of condensed matter, modeled by the “algebra doubling” of coalgebras. This coalgebraic modeling is highly significant for making computationally effective Sen’s SCF theory, because both based on a dynamic and not statistical weighing of the variables for interacting systems, respectively in the physical and in the social realms.
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Notes
- 1.
Indeed, using the standard ideographic symbolism, e.g., the alphabetic letters, x, y, z, for representing relations R, the ordering relation among objects using the transitive relation is represented as follows: \(((xRy \wedge yRz)\rightarrow xRz)\). As we see, in such a way the transitive relation is linear, and then in set-theoretic ordering it supposes a total ordering among sets. That is, for all sets the ordering relation \(\ge \) holds—because admitting gaps (in our case, between x and z) among sets that can be indefinitely long. Another possible transitive relation is the so-called “Euclidean relation”: \(((xRy \wedge xRz)\rightarrow yRz\vee zRy)\) that has a tree structure in which no gap among sets is admitted, and in which therefore set total ordering is not necessary, but only partial orderings are sufficient, because in this case—and this is the essential point!—transitivity is not linear. As we see, this distinction between total and partial orderings in social choice rankings is essential for understanding the novelty of Sen’s approach to social choice theory as to the classical ones based on a liberal approach in economy.
- 2.
E.g., think at Bloomberg data streaming, updating continuously the values of shares and of currencies on the worldwide financial market, active all over the day because of the different time zones, and that are changing in average every 10 s.
- 3.
There exists evidently, and many Authors noted it, a relationship between Arrow’s impossibility theorem and the famous Condorcet’s paradox on voting stated in his Essai sur l’Application de l’Analyse à la Probabilité des Decisions Rendues à la Pluralité des Voix published in 1785 [17], and considered unanimously as the official birth-date of the modern, mathematical theory of social choices. However, Sen demonstrated that Arrow’s impossibility theorem cannot be reduced to Condorcet’s paradox, before all because it would be hard to argue that majority rule would really be a plausible way of aggregating preferences in welfare economics (see [7], p. 275). See also [7], pp. 395–419, and overall the concluding chapter of Sen’s book ([7], pp. 453–472), in which he offers a deep analysis of what a modern comprehensive, because rational theory of social choice requires in our “Communication Age”, and that is far beyond any oversimplified and oversimplifying “majority criterion”.
- 4.
Indeed, the paradox derives from the fact that voter 1, 2, and 3 are ranking the three alternatives in the orders: x, y, z; y, z, x; and z, x, y, respectively. In this way, there is a preference cycle, and each alternative is defeated in a majority vote by another alternative. Now, if we do not consider preference maximization, but maximality in terms of the transitive closures of preference relations, we have the following choices: \(C\left( \left\{ {x,y}\right\} \right) ={\left\{ x\right\} }; \,C(\left\{ {y,z}\right\} )={\left\{ y\right\} }; \,C(\left\{ {z,x}\right\} )=\left\{ {z}\right\} ;\, \text {and} \;C\left\{ ({x,y,z})\right\} =\left\{ {x,y,z}\right\} \). Of course, this choice function is not binary, but satisfies all the Arrow axioms in their choice-functional version, of which at Theorem 1 (see [7], p. 316).
- 5.
- 6.
- 7.
We recall here what we illustrated in Sect. 2.3. Namely, that a “strict” partial ordering is an ordering \(R (\le )\) among sets, in which only the asymmetric case \(\left( (xRy\ne yRx)\rightarrow x\ne y \right) \) holds of the antisymmetric relation that, on the contrary, holds for the partial ordering as such. That is, where also the symmetric case, \(\left( (xRy=yRx)\rightarrow x=y \right) \), holds.
- 8.
The usage of the \(\langle bra|ket\rangle \) notation for the matrix representation of Sen’s doubled vector states—a notation made famous by the quantum formalism, even though not limited to it—is an implicit suggestion that the more effective approach to make computationally effective the two axioms of identity of Sen’s SCF theory is the usage of the quantum state superposition principle. In the framework, however, of the doubling of the degrees of freedom formalism for calculating dynamically the statistical expectations for entangled interacting systems, typical of the “many-body physics” approach to QFT. That is the fundamental physics of condensed matter physics (see Sect. 4).
- 9.
Apart from the reference to Smiths masterpiece, Sen quotes in note another less famous book of A. Smith, The Theory of Moral Sentiments, of which Sen himself cured a re-edition with his own Preface [35].
- 10.
By machine learning computer scientists generally intend the ability of a machine of simulat-ing human brain ability of learning from its past experience without any further interven-tion of the programmer. Namely, in classical or symbolic AI computational systems such as those based on M. Minsky’s frame theory [36]—which is at the basis of contemporary “object-oriented programming” technique—, when the machine encounters a new situation it selects from memory a given data-structure or frame for representing a stereotyped situation, whose details can be adapted to fit reality. Effectively, a frame is a network of nodes and relations, whose top levels are fixed because representing what is supposed to be always true (effectively, true with the highest probability) in a given situation, and lower levels have many terminals or slots to be filled with specific instances or data, according to specified conditions its assignments must meet. Of course, “frames” are programmed data-structures. Machine learning using ANN deep learning wants precisely to avoid as much as possible this human programmer intervention in the definition of the memorized reference patterns. In this sense, it is at the basis of the so-called non-symbolic approach to AI, also because in the case of the “big data” this human programmer intervention is effectively impossible, because of the complexity of the data structures involved, characterized by millions and ever billions of parameters to be taken into account.
- 11.
The non-linear sigmoid function \(\sigma (x)=\frac{ 1}{1+e^{-x}}\), and overall its immediate relative the hyperbolic tangent function \(\tanh x = \left( \frac{e^{x}-e^{-x}}{e^{x}+e^{-x}} \right) =\frac{\sinh x}{\cosh x}\) are what characterizes the learning algorithms of the actual ANNs, instead of the linear step-wise 1/0 activation function of Rosenblatt’s early “linear perceptron” [40]. This activation function is, effectively, the Heaviside function: \(H(x)=\frac{d}{dx}\text {max}\left\{ x,0 \right\} \text {for }x\ne 0\), whose values are obviously “zero” for negative arguments and “one” for positive arguments, so to make linear—namely, a summation of the neuron weights each representing a variable of the dataset—the activation function of the “inner” or “deep” neurons between the “input” and “output” and neurons. To sum up, by using the sigmoid as the activation function, the overall neuron output is no longer 0/1, but any real value between 0 and 1, so to allow to calculate the probability that some event (output) happens, given some conditions (input), instead of the simple no/yes answer of the linear ANNs. The non-linearity is then fundamental for calculating higher-order or long-range correlations among data. Indeed, mathematically, the Taylor series expansion of \(\tanh \) includes several higher order correlations, much more than sinh and cosh alone, of which tanh is the ratio, so to answer in principle to the main criticism made by M. Minsky and S. Papert at MIT [41] to the early linear ANNs structures, such as McCulloch’s and Pitt’s neural net [42] and Rosenblatt’s linear perceptron [40], who first demonstrated that a neural net is able to calculate in a desirable parallel way any Boolean logical function like a Turing Machine. Effectively, because of the prestige of M. Minsky in US computer science community, his criticism was able to block any further research in ANNs, from 60s to 80s of the last century. In 80s a non-linear multilayer perceptron with a sigmoid as activation function for the hidden or “deep” layer neurons [39], and then characterized by the gradient descent learning algorithm [38], answered Minsky’s main criticism to early linear ANNs. However, only during the last twenty years, with the large availability of ever more powerful processors—essentially arrays of GPUs—for reckoning with the computational weight of this type of ANN architectures, we have the actual development of AI-systems based on deep machine learning algorithms (see [37] for an updated synthesis).
- 12.
For a more articulated criticism to “supervised” machine learning, see [52].
- 13.
At this point, it is significant to emphasize that in US, Todd L. Hylton and his collaborators recently launched the Thermodynamic Computer Project (see https://knowm.org/thermodynamic-computing/). For instance, the so-called “thermodynamic NNs” are trying to approach a type of unsupervised solution in machine learning using the minimum free energy (in Helmholtz formalization within the statistical mechanics approach) as a selection optimization criterion. They remain, however, in the paradigm of the statistical interpretation of thermodynamics, also for modeling neurons interacting with (their representation of the) environment, using an ingenious, but anthropomorphic logic of “ question/answer” [53]. In this way, the dynamics of such a “thermodynamic neural network” is a sort of stroboscopic approximation (i.e., limited to one phase or to a small number of close phases) of what in QFT is a continuous process of phase transitions (macroscopically, a chaotic dynamics) and then, when implemented computationally, a universal thermodynamic LTS. In this limited sense, this class of ANNs is straddling (2) and (3) types above of unsupervised machine learning algorithms.
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Basti, G., Capolupo, A., Vitiello, G. (2020). The Computational Challenge of Amartya Sen’s Social Choice Theory in Formal Philosophy. In: Giovagnoli, R., Lowe, R. (eds) The Logic of Social Practices. Studies in Applied Philosophy, Epistemology and Rational Ethics, vol 52. Springer, Cham. https://doi.org/10.1007/978-3-030-37305-4_7
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