Online research in philosophy

In his last dialogue, the Laws, Plato views citizens in the polis as players in a game. Just as contemporary game theory, Plato considers games to be states of strategic interaction. Yet the game of the Laws differs from those of game theory in one important respect. Where game theory assumes that players are rational--that they choose strategies, or rules for taking action at each instant of a game, in order to maximize payoffs--Plato explores the conditions under which rationality, as game theory defines it, is possible. Plato thus agrees with game theory that rational, maximizing behavior is a necessary constituent of civic order, but not that it may simply be assumed. He concurs that rationality can describe the behavior of citizens, but not under any circumstances. It is the task of politics in Plato's city to prepare the conditions for rationality. Only once politics has done its work is maximizing, utilitarian behavior possible. Yet political preparation of the game of the city is exactly what contemporary game theory assumes away. The Athenian, who leads the discussion, describes the political preparation for rationality as "this moderate old man's game concerning laws." Political preparation of the game of the city is itself a game, because it is never possible to escape strategic interaction. It is, however, a second-order game whose play paves the way for the first-order game of lawful strategic interaction. The second-order, political game at once completes the analysis of rationality and lays the groundwork for rationality to operate. The politics of the Laws is a game, but one whose actions and players differ from the actions and players of the first-order game. The action of the political game is lawgiving. The players are the gods and god-like men who serve as lawgivers. The second-order, political game also has its own distinct rationality, linked to a payoff that is qualitatively different from the utilitarian payoffs of the first-order game. The Athenian calls the rationality of the second-order game "intelligence." He calls the payoff associated with it "joy," as distinct from the "benefits" associated with utilitarian rationality. The players in the second-order game seek to maximize joy, not benefit. Joy is the experience of play. The payoff of players in the second-order game is the game itself, not a benefit collateral to playing it. The second-order game is a "true" game, one that the players enter in order to play, not to get utilitarian payoffs.

Determinism seems incompatible with free will. However, even indeterminism seems incompatible with free will, since it seems to make free actions random. Popper contends that free agents are not bound by physical laws, even indeterministic ones, and that undetermined actions are not random if they are influenced by abstract entities. I argue that Popper could strengthen his account by drawing upon his theories of propensities and of limited rationality; but that even then his account would not fully explain why free actions are not random. I offer a solution to this problem which draws on Hornsby’s analysis of action. I then borrow an idea of Kant about self-consciousness to distinguish free agents from sub-human animals. I make a brief evaluation of Popper’s contribution.

We explore the ability to distinguish random from non-random events. Randomness is defined in terms of radioactive decay whereas non-randomness is quantified by excess repetitions (“repeat”) or alternations (“switch”) between successive bits. In the first four experiments no mention was made of randomness, probability, or related concepts in task instructions. We found superior performance in distinguishing random stimuli from repeat stimuli compared to switch stimuli. The last three experiments explicitly evoked the concept of randomness, thus allowing comparison of perceptual and conceptual performance. The ability to identify random events from switch distracters was inferior to the ability to discriminate random from switch stimuli. In contrast, for repeat stimuli the concept of randomness appears to roughly coincide with perceptual discriminability. Finally, the ability to identify or produce stimuli as random did not co-vary with the ability to discriminate random from non-random stimuli.

We compare various notions of algorithmic randomness. First we consider relativized randomness. A set is n-random if it is Martin-Löf random relative to θ(n−1). We show that a set is 2-random if and only if there is a constant c such that infinitely many initial segments x of the set are c-incompressible: C(x) ≥ |x| − c. The 'only if' direction was obtained independently by Joseph Miller. This characterization can be extended to the case of time-bounded C-complexity. Next we prove some results on lowness. Among other things, we characterize the 2-random sets as those 1-random sets that are low for Chaitin's Ω. Also, 2-random sets form minimal pairs with 2-generic sets. The r.e. low for Ω sets coincide with the r.e. K-trivial ones. Finally we show that the notions of Martin-Löf randomness, recursive randomness, and Schnorr randomness can be separated in every high degree while the same notions coincide in every non-high degree. We make some remarks about hyperimmune-free and PA-complete degrees.

the difference, within the field of physically undetermined events, between the random and the non-random is the presence or absence of a prior mental event ...

The first paper of this series (Cogito, 1992) outlined ‘the showdown phenomenon’: a live sequence of events of two distinct kinds, ‘red’ and ‘green’, which was experienced by the would-be predictor as absolutely and irreducibly unpredictable, because the predictor invariably got his or her predictions wrong. We can clearly and distinctly imagine this happening: so a perverse-random experience of this sort is evidently ‘logically possible’. This raises the question of the relation of the new sequences to ordinary ‘random’ sequences. In this paper an account is developed of the relationship of ‘perverse-randomness’ to ‘randomness’. Such an account may assist us in achieving a new, firmer conceptualization of ordinary probability, and hence in making sense of this notoriously elusive idea.

On an influential account, chaos is explained in terms of random behaviour; and random behaviour in turn is explained in terms of having positive Kolmogorov-Sinai entropy (KSE). Though intuitively plausible, the association of the KSE with random behaviour needs justification since the definition of the KSE does not make reference to any notion that is connected to randomness. I provide this justification for the case of Hamiltonian systems by proving that the KSE is equivalent to a generalized version of Shannon's communication-theoretic entropy under certain plausible assumptions. I then discuss consequences of this equivalence for randomness in chaotic dynamical systems. Introduction Elements of dynamical systems theory Entropy in communication theory Entropy in dynamical systems theory Comparison with other accounts Product versus process randomness.

Various processes are often classified as both deterministic and random or chaotic. The main difficulty in analysing the randomness of such processes is the apparent tension between the notions of randomness and determinism: what type of randomness could exist in a deterministic process? Ergodic theory seems to offer a particularly promising theoretical tool for tackling this problem by positing a hierarchy, the so-called ‘ergodic hierarchy’ (EH), which is commonly assumed to provide a hierarchy of increasing degrees of randomness. However, that notion of randomness requires clarification. The mathematical definition of EH does not make explicit appeal to randomness; nor does the usual way of presenting EH involve a specification of the notion of randomness that is supposed to underlie the hierarchy. In this paper we argue that EH is best understood as a hierarchy of random behaviour if randomness is explicated in terms of unpredictability. We then show that, contrary to common wisdom, EH is useful in characterising the behaviour of Hamiltonian dynamical systems.

Various processes are often classified as both deterministic and random or chaotic. The main difficulty in analysing the randomness of such processes is the apparent tension between the notions of randomness and determinism: what type of randomness could exist in a deterministic process? Ergodic theory seems to offer a particularly promising theoretical tool for tackling this problem by positing a hierarchy, the so-called ‘ergodic hierarchy’ (EH), which is commonly assumed to provide a hierarchy of increasing degrees of randomness. However, that notion of randomness requires clarification. The mathematical definition of EH does not make explicit appeal to randomness; nor does the usual way of presenting EH involve a specification of the notion of randomness that is supposed to underlie the hierarchy. In this paper we argue that EH is best understood as a hierarchy of random behaviour if randomness is explicated in terms of unpredictability. We then show that, contrary to common wisdom, EH is useful in characterising the behaviour of Hamiltonian dynamical systems. r 2006 Elsevier Ltd. All rights reserved.

Various processes are often classified as both deterministic and random or chaotic. The main difficulty in analysing the randomness of such processes is the apparent tension between the notions of randomness and determinism: what type of randomness could exist in a deterministic process? Ergodic theory seems to offer a particularly promising theoretical tool for tackling this problem by positing a hierarchy, the so-called ‘ergodic hierarchy’ (EH), which is commonly assumed to provide a hierarchy of increasing degrees of randomness. However, that notion of randomness requires clarification. The mathematical definition of EH does not make explicit appeal to randomness; nor does the usual way of presenting EH involve a specification of the notion of randomness that is supposed to underlie the hierarchy. In this paper we argue that EH is best understood as a hierarchy of random behaviour if randomness is explicated in terms of unpredictability. We then show that, contrary to common wisdom, EH is useful in characterising the behaviour of Hamiltonian dynamical systems. r 2006 Elsevier Ltd. All rights reserved.