An illustrated exploration of fandom that combines academic essays with artist pages and experimental texts. Fandom as Methodology examines fandom as a set of practices for approaching and writing about art. The collection includes experimental texts, autobiography, fiction, and new academic perspectives on fandom in and as art. Key to the idea of “fandom as methodology” is a focus on the potential for fandom in art to create oppositional spaces, communities, and practices, particularly from queer perspectives, but also through transnational, (...) feminist and artist-of-color fandoms. The book provides a range of examples of artists and writers working in this vein, as well as academic essays that explore the ways in which fandom can be theorized as a methodology for art practice and art history. Fandom as Methodology proposes that many artists and art writers already draw on affective strategies found in fandom. With the current focus in many areas of art history, art writing, and performance studies around affective engagement with artworks and imaginative potentials, fandom is a key methodology that has yet to be explored. Interwoven into the academic essays are lavishly designed artist pages in which artists offer an introduction to their use of fandom as methodology. Contributors Taylor J. Acosta, Catherine Grant, Dominic Johnson, Kate Random Love, Maud Lavin, Owen G. Parry, Alice Butler, SooJin Lee, Jenny Lin, Judy Batalion, Ika Willis. Artists featured in the artist pages Jeremy Deller, Ego Ahaiwe Sowinski, Anna Bunting-Branch, Maria Fusco, Cathy Lomax, Kamau Amu Patton, Holly Pester, Dawn Mellor, Michelle Williams Gamaker, The Women of Colour Index Reading Group, Liv Wynter, Zhiyuan Yang. (shrink)

Early work on the frequency theory of probability made extensive use of the notion of randomness, conceived of as a property possessed by disorderly collections of outcomes. Growing out of this work, a rich mathematical literature on algorithmic randomness and Kolmogorov complexity developed through the twentieth century, but largely lost contact with the philosophical literature on physical probability. The present chapter begins with a clarification of the notions of randomness and probability, conceiving of the former as a (...) property of a sequence of outcomes, and the latter as a property of the process generating those outcomes. A discussion follows of the nature and limits of the relationship between the two notions, with largely negative verdicts on the prospects for any reduction of one to the other, although the existence of an apparently random sequence of outcomes is good evidence for the involvement of a genuinely chancy process. (shrink)

In this paper we consider conditional random quantities (c.r.q.’s) in the setting of coherence. Based on betting scheme, a c.r.q. X|H is not looked at as a restriction but, in a more extended way, as \({XH + \mathbb{P}(X|H)H^c}\) ; in particular (the indicator of) a conditional event E|H is looked at as EH + P(E|H)H c . This extended notion of c.r.q. allows algebraic developments among c.r.q.’s even if the conditioning events are different; then, for instance, we can give a (...) meaning to the sum X|H + Y|K and we can define the iterated c.r.q. (X|H)|K. We analyze the conjunction of two conditional events, introduced by the authors in a recent work, in the setting of coherence. We show that the conjoined conditional is a conditional random quantity, which may be a conditional event when there are logical dependencies. Moreover, we introduce the negation of the conjunction and by applying De Morgan’s Law we obtain the disjoined conditional. Finally, we give the lower and upper bounds for the conjunction and disjunction of two conditional events, by showing that the usual probabilistic properties continue to hold. (shrink)

The latter half of the twentieth century has been marked by debates in evolutionary biology over the relative significance of natural selection and random drift: the so-called “neutralist/selectionist” debates. Yet John Beatty has argued that it is difficult, if not impossible, to distinguish the concept of random drift from the concept of natural selection, a claim that has been accepted by many philosophers of biology. If this claim is correct, then the neutralist/selectionist debates seem at best futile, and at worst, (...) meaningless. I reexamine the issues that Beatty raises, and argue that random drift and natural selection, conceived as processes, can be distinguished from one another. (shrink)

A widely accepted tenet of evolutionary biology is that spontaneous mutations occur randomly with regard to their fitness effect. However, since the mutation rate varies along a genome and this variation can be subject to selection, organisms might evolve lower mutation rates at loci where mutations are most deleterious or increased rates where mutations are most needed. In fact, mechanisms of targeted hypermutation are known in organisms ranging from bacteria to humans. Here we review the main forces driving the evolution (...) of local mutation rates and identify the main limiting factors. Both targeted hyper‐ and hypomutation can evolve, although the former is restricted to loci under very frequent positive selection and the latter is severely limited by genetic drift. Nevertheless, we show how an association of repair with transcription or chromatin‐associated proteins could overcome the drift limit and lead to non‐random hypomutation along the genome in most organisms.Editor's suggested further reading in BioEssays Stress‐induced mutation via DNA breaks in Escherichia coli: A molecular mechanism with implications for evolution and medicine Abstract. (shrink)

We consider two ways one might use algorithmic randomness to characterize a probabilistic law. The first is a generative chance* law. Such laws involve a nonstandard notion of chance. The second is a probabilistic* constraining law. Such laws impose relative frequency and randomness constraints that every physically possible world must satisfy. While each notion has virtues, we argue that the latter has advantages over the former. It supports a unified governing account of non-Humean laws and provides independently motivated (...) solutions to issues in the Humean best-system account. On both notions, we have a much tighter connection between probabilistic laws and their corresponding sets of possible worlds. Certain histories permitted by traditional probabilistic laws are ruled out as physically impossible. As a result, such laws avoid one variety of empirical underdetermination, but the approach reveals other varieties of underdetermination that are typically overlooked. (shrink)

According to a traditional view, scientific laws and theories constitute algorithmic compressions of empirical data sets collected from observations and measurements. This article defends the thesis that, to the contrary, empirical data sets are algorithmically incompressible. The reason is that individual data points are determined partly by perturbations, or causal factors that cannot be reduced to any pattern. If empirical data sets are incompressible, then they exhibit maximal algorithmic complexity, maximal entropy and zero redundancy. They are therefore maximally efficient carriers (...) of information about the world. Since, on algorithmic information theory, a string is algorithmically random just if it is incompressible, the thesis entails that empirical data sets consist of algorithmically random strings of digits. Rather than constituting compressions of empirical data, scientific laws and theories pick out patterns that data sets exhibit with a certain noise.Author Keywords: Algorithmic randomness; Compression; Empirical data; Information; Law; Pattern. (shrink)

It is known that the box dimension of any Martin-Löf random closed set of ${\{0,1\}^\mathbb{N}}$ is ${\log_2(\frac{4}{3})}$ . Barmpalias et al. [J Logic Comput 17(6):1041–1062, 2007] gave one method of producing such random closed sets and then computed the box dimension, and posed several questions regarding other methods of construction. We outline a method using random recursive constructions for computing the Hausdorff dimension of almost every random closed set of ${\{0,1\}^\mathbb{N}}$ , and propose a general method for random closed sets (...) in other spaces. We further find both the appropriate dimensional Hausdorff measure and the exact Hausdorff dimension for such random closed sets. (shrink)

The concept of randomness has been unjustly neglected in recent philosophical literature, and when philosophers have thought about it, they have usually acquiesced in views about the concept that are fundamentally flawed. After indicating the ways in which these accounts are flawed, I propose that randomness is to be understood as a special case of the epistemic concept of the unpredictability of a process. This proposal arguably captures the intuitive desiderata for the concept of randomness; at least (...) it should suggest that the commonly accepted accounts cannot be the whole story and more philosophical attention needs to be paid. 1. Randomness in science1.1Random systems1.2Random behaviour1.3Random sampling1.4Caprice, arbitrariness and noise2. Concepts of randomness2.1Von Mises/church/martin-löf randomness2.2KCS-randomness3. Randomness is unpredictability: preliminaries3.1Process and product randomness3.2Randomness is indeterminism?4. Predictability4.1Epistemic constraints on prediction4.2Computational constraints on prediction4.3Pragmatic constraints on prediction4.4Prediction defined5. Unpredictability6. Randomness is unpredictability6.1Clarification of the definition of randomness6.2Randomness and probability6.3Subjectivity and context sensitivity of randomness7. Evaluating the analysis[R]andomness … is going to be a concept which is relative to our body of knowledge, which will somehow reflect what we know and what we don't know. Henry E. Kyburg, Jr ([1974], p. 217)Phenomena that we cannot predict must be judged random. Patrick Suppes ([1984], p. 32). (shrink)

This research note challenges the idea that Random Somatic Mutations are entirely random, highlighting their non-equiprobable nature and their influence on evolution, involution, or indeterminacy. It recalls the Neutrosophic Theory of Evolution, extending Darwin’s theory, and emphasizes the importance of distinguishing between different senses of ‘random mutation’ in evolutionary theory.

In 1947 Donald Cary Williams claimed in The Ground of Induction to have solved the Humean problem of induction, by means of an adaptation of reasoning first advanced by Bernoulli in 1713. Later on David Stove defended and improved upon Williams’ argument in The Rational- ity of Induction (1986). We call this proposed solution of induction the ‘Williams-Stove sampling thesis’. There has been no lack of objections raised to the sampling thesis, and it has not been widely accepted. In our (...) opinion, though, none of these objections has the slightest force, and, moreover, the sampling thesis is undoubtedly true. What we will argue in this paper is that one particular objection that has been raised on numerous occasions is misguided. This concerns the randomness of the sample on which the inductive extrapolation is based. (shrink)

A set $B\subseteq\mathbb{N}$ is called low for Martin-Löf random if every Martin-Löf random set is also Martin-Löf random relative to B . We show that a $\Delta^0_2$ set B is low for Martin-Löf random if and only if the class of oracles which compress less efficiently than B , namely, the class $\mathcal{C}^B=\{A\ |\ \forall n\ K^B(n)\leq^+ K^A(n)\}$ is countable (where K denotes the prefix-free complexity and $\leq^+$ denotes inequality modulo a constant. It follows that $\Delta^0_2$ is the largest arithmetical (...) class with this property and if $\mathcal{C}^B$ is uncountable, it contains a perfect $\Pi^0_1$ set of reals. The proof introduces a new method for constructing nontrivial reals below a $\Delta^0_2$ set which is not low for Martin-Löf random. (shrink)

We develop and knit together several theodicies in order to find a more complete picture of why certain forms of animal suffering might be permitted by a perfect being. We focus on an especially potent form of the problem of evil, which arises from considering why a perfectly good, wise, and powerful God might use evolutionary mechanisms that predictably result in so much animal suffering and loss of life. There are many existing theodicies on the market, and although they offer (...) helpful resources, we combine and further develop several proposals to produce a composite theodicy that avoids certain shortcomings of the individual theodicies. An important element of our project is locating a role for randomness in cosmic and biological evolution. In particular, we show how randomness might enhance or enable certain goods, including everlasting goods, at the risk of temporary evils. (shrink)

Are there any truly ontologically random events? This paper argues that randomness is an unavoidably epistemic concept and therefore ascription of ontological randomness to any particular event or series of events can never be justified.

It is time for a new paper about open questions in the currently very active area of randomness and computability. Ambos-Spies and Kučera presented such a paper in 1999 [1]. All the question in it have been solved, except for one: is KL-randomness different from Martin-Löf randomness? This question is discussed in Section 6.Not all the questions are necessarily hard—some simply have not been tried seriously. When we think a question is a major one, and therefore likely (...) to be hard, we indicate this by the symbol ▶, the criterion being that it is of considerable interest and has been tried by a number of researchers. Some questions are close contenders here; these are marked by ▷. With few exceptions, the questions are precise. They mostly have a yes/no answer. However, there are often more general questions of an intuitive or even philosophical nature behind. We give an outline, indicating the more general questions.All sets will be sets of natural numbers, unless otherwise stated. These sets are identified with infinite strings over {0, 1}. Other terms used in the literature are sequence and real. (shrink)

The Art of Randomness teaches readers to harness the power of randomness (and Python code) to solve real-world problems in programming, science, and art through hands-on experiments-from simulating evolution to encrypting messages to making machine-learning algorithms. Each chapter describes how randomness plays into the given topic area, then proceeds to demonstrate its problem-solving role with hands-on experiments to work through using Python code.

We report on some recent work centered on attempts to understand when one set is more random than another. We look at various methods of calibration by initial segment complexity, such as those introduced by Solovay [125], Downey, Hirschfeldt, and Nies [39], Downey, Hirschfeldt, and LaForte [36], and Downey [31]; as well as other methods such as lowness notions of Kučera and Terwijn [71], Terwijn and Zambella [133], Nies [101, 100], and Downey, Griffiths, and Reid [34]; higher level randomness (...) notions going back to the work of Kurtz [73], Kautz [61], and Solovay [125]; and other calibrations of randomness based on definitions along the lines of Schnorr [117].These notions have complex interrelationships, and connections to classical notions from computability theory such as relative computability and enumerability. Computability figures in obvious ways in definitions of effective randomness, but there are also applications of notions related to randomness in computability theory. For instance, an exciting by-product of the program we describe is a more-or-less naturalrequirement-freesolution to Post's Problem, much along the lines of the Dekker deficiency set. (shrink)

We prove that there are uncountably many sets that are low for the class of Schnorr random reals. We give a purely recursion theoretic characterization of these sets and show that they all have Turing degree incomparable to 0'. This contrasts with a result of Kučera and Terwijn [5] on sets that are low for the class of Martin-Löf random reals.

Alexander Rosenberg (1994) claims that the omniscient viewpoint of the evolutionary process would have no need for the concept of random drift. However, his argument fails to take into account all of the processes which are considered to be instances of random drift. A consideration of these processes shows that random drift is not eliminable even given a position of omniscience. Furthermore, Rosenberg must take these processes into account in order to support his claims that evolution is deterministic and that (...) evolutionary biology is an instrumental science. (shrink)

A semimeasure is a generalization of a probability measure obtained by relaxing the additivity requirement to superadditivity. We introduce and study several randomness notions for left-c.e. semimeasures, a natural class of effectively approximable semimeasures induced by Turing functionals. Among the randomness notions we consider, the generalization of weak 2-randomness to left-c.e. semimeasures is the most compelling, as it best reflects Martin-Löf randomness with respect to a computable measure. Additionally, we analyze a question of Shen, a positive (...) answer to which would also have yielded a reasonable randomness notion for left-c.e. semimeasures. Unfortunately, though, we find a negative answer, except for some special cases. (shrink)

We consider random processes more general than those considered by Erdös and Rényi for generating the countable random graph. It is proved that, in the category sense, almost all random processes we consider generate the countable random graph with probability 1. Under a weak boundedness assumption we give a criterion for the random processes which generate the countable random graph almost surely. We also consider further questions asked by Jackson regarding the outcome graphs when the process fails to produce the (...) countable random graph. (shrink)

Journal of Political Philosophy, EarlyView.

Sortition, i.e. random appointment for public duty, has been employed by societies throughout the years as a firewall designated to prevent illegitimate interference between parties in a legal case and agents of the legal system. In judicial systems of modern western countries, random procedures are mainly employed to select the jury, the court and/or the judge in charge of judging a legal case. Therefore, these random procedures play an important role in the course of a case, and should comply with (...) some principles, such as transparency and complete auditability. Nevertheless, these principles are neglected by random procedures in some judicial systems, which are performed in secrecy and are not auditable by the involved parties. The assignment of cases in the Brazilian Supreme Court is an example of such a procedure, for it is performed using procedures unknown to the parties involved in the judicial cases. This article presents a review of how sortition has been historically employed by societies and discusses how Mathematical Statistics may be applied to random procedures of the judicial system, as it has been applied for almost a century on clinical trials, for example. A statistical model for assessing randomness in case assignment is proposed and applied to the Brazilian Supreme Court. As final remarks, guidelines for the development of good randomization procedures are outlined. (shrink)

Schnorr randomness is a notion of algorithmic randomness for real numbers closely related to Martin-Löf randomness. After its initial development in the 1970s the notion received considerably less attention than Martin-Löf randomness, but recently interest has increased in a range of randomness concepts. In this article, we explore the properties of Schnorr random reals, and in particular the c.e. Schnorr random reals. We show that there are c.e. reals that are Schnorr random but not Martin-Löf (...) random, and provide a new characterization of Schnorr random real numbers in terms of prefix-free machines. We prove that unlike Martin-Löf random c.e. reals, not all Schnorr random c.e. reals are Turing complete, though all are in high Turing degrees. We use the machine characterization to define a notion of "Schnorr reducibility" which allows us to calibrate the Schnorr complexity of reals. We define the class of "Schnorr trivial" reals, which are ones whose initial segment complexity is identical with the computable reals, and demonstrate that this class has non-computable members. (shrink)

A puzzle of an unmarked clock, used by Timothy Williamson to question the KK principle, was separately adapted by David Christensen and Adam Elga to critique a principle of Rational Reflection. Both authors, we argue, flout the received relationship between ideal agency and the classical distinction between systematic and random error, namely that ideal agents are subject only to the latter. As a result, these criticisms miss their mark.

A randomly selected microcosm of the people can usefully play an official role in the lawmaking process. However, there are serious issues to be confronted if such a random sample were to take on the role of a full-scale, full-time second chamber. Some skeptical considerations are detailed. There are also advantages to short convenings of such a sample to take on some of the roles of a second chamber. This article provides a response to the skeptical considerations. Precedents from ancient (...) Athens show how such short-term convenings of a deliberating microcosm can be positioned before, during, or after other elements of the lawmaking process. The article draws on experience from Deliberative Polling to show how this is both practical and productive for the lawmaking process. (shrink)

I discuss recent work in ergodic theory and statistical mechanics, regarding the compatibility and origin of random and chaotic behavior in deterministic dynamical systems. A detailed critique of some quite radical proposals of the Prigogine school is given. I argue that their conclusion regarding the conceptual bankruptcy of the classical conceptions of an exact microstate and unique phase space trajectory is not completely justified. The analogy they want to draw with quantum mechanics is not sufficiently close to support their most (...) radical conclusion. (shrink)

This is a review of the issue of randomness in quantum mechanics, with special emphasis on its ambiguity; for example, randomness has different antipodal relationships to determinism, computability, and compressibility. Following a philosophical discussion of randomness in general, I argue that deterministic interpretations of quantum mechanics are strictly speaking incompatible with the Born rule. I also stress the role of outliers, i.e. measurement outcomes that are not 1-random. Although these occur with low probability, their very existence implies (...) that the no-signaling principle used in proofs of randomness of outcomes of quantum-mechanical measurements should be reinterpreted statistically, like the second law of thermodynamics. In three appendices I discuss the Born rule and its status in both single and repeated experiments, review the notion of 1-randomness that in various guises was investigated by Kolmogorov and others and treat Bell’s Theorem and the Free Will Theorem with their implications for randomness. (shrink)

Demuth tests generalize Martin-Löf tests in that one can exchange the m-th component a computably bounded number of times. A set fails a Demuth test if Z is in infinitely many final versions of the Gm. If we only allow Demuth tests such that GmGm+1 for each m, we have weak Demuth randomness.We show that a weakly Demuth random set can be high and , yet not superhigh. Next, any c.e. set Turing below a Demuth random set is strongly (...) jump-traceable.We also prove a basis theorem for non-empty classes P. It extends the Jockusch–Soare basis theorem that some member of P is computably dominated. We use the result to show that some weakly 2-random set does not compute a 2-fixed point free function. (shrink)

We consider the nature of quantum randomness and how one might have empirical evidence for it. We will see why, depending on one’s computational resources, it may be impossible to determine whether...

Review of "Exploring Randomness" (200) and "The Unknowable" (1999) by Gregory Chaitin.

We say that A ≤LR B if every B-random number is A-random. Intuitively this means that if oracle A can identify some patterns on some real γ. In other words. B is at least as good as A for this purpose. We study the structure of the LR degrees globally and locally (i.e., restricted to the computably enumberable degrees) and their relationship with the Turing degrees. Among other results we show that whenever α in not GL₂ the LR degree of (...) α bounds $2^{\aleph _{0}}$ degrees (so that, in particular, there exist LR degrees with uncountably many predecessors) and we give sample results which demonstrate how various techniques from the theory of the c.e. degrees can be used to prove results about the c.e. LR degrees. (shrink)

We investigate the question “To what extent can random reals be used as a tool to establish number theoretic facts?” Let $\text{2-\textit{RAN\/}}$ be the principle that for every real $X$ there is a real $R$ which is 2-random relative to $X$. In Section 2, we observe that the arguments of Csima and Mileti [3] can be implemented in the base theory $\text{\textit{RCA}}_0$ and so $\text{\textit{RCA}}_0+\text{2-\textit{RAN\/}}$ implies the Rainbow Ramsey Theorem. In Section 3, we show that the Rainbow Ramsey Theorem is (...) not conservative over $\text{\textit{RCA}}_0$ for arithmetic sentences. Thus, from the Csima—Mileti fact that the existence of random reals has infinitary-combinatorial consequences we can conclude that $\text{2-\textit{RAN\/}}$ has non-trivial arithmetic consequences. In Section 4, we show that $\text{2-\textit{RAN\/}}$ is conservative over $\text{\textit{RCA}}_0+\text{\textit{B\/}$\,\Sigma$}_2$ for $\Pi^1_1$-sentences. Thus, the set of first-order consequences of $\text{2-\textit{RAN\/}}$ is strictly stronger than $P^-+I\Sigma_1$ and no stronger than $P^-+\text{\textit{B\/}$\,\Sigma$}_2$. (shrink)

n this text, we revisit part of the analysis of anti-entropy in Bailly and Longo (2009} and develop further theoretical reflections. In particular, we analyze how randomness, an essential component of biological variability, is associated to the growth of biological organization, both in ontogenesis and in evolution. This approach, in particular, focuses on the role of global entropy production and provides a tool for a mathematical understanding of some fundamental observations by Gould on the increasing phenotypic complexity along evolution. (...) Lastly, we analyze the situation in terms of theoretical symmetries, in order to further specify the biological meaning of anti-entropy as well as its strong link with randomness. (shrink)

John organized a state lottery and his wife won the main prize. You may feel that the event of her winning wasn’t particularly random, but how would you argue that in a fair court of law? Traditional probability theory does not even have the notion of random events. Algorithmic information theory does, but it is not applicable to real-world scenarios like the lottery one. We attempt to rectify that.

The Copenhagen interpretation of quantum mechanics assumes the existence of the classical deterministic Newtonian world. We argue that in fact the Newton determinism in classical world does not hold and in the classical mechanics there is fundamental and irreducible randomness. The classical Newtonian trajectory does not have a direct physical meaning since arbitrary real numbers are not observable. There are classical uncertainty relations: Δq>0 and Δp>0, i.e. the uncertainty (errors of observation) in the determination of coordinate and momentum is (...) always positive (non zero).A “functional” formulation of classical mechanics was suggested. The fundamental equation of the microscopic dynamics in the functional approach is not the Newton equation but the Liouville equation for the distribution function of the single particle. Solutions of the Liouville equation have the property of delocalization which accounts for irreversibility. The Newton equation in this approach appears as an approximate equation describing the dynamics of the average values of the position and momenta for not too long time intervals. Corrections to the Newton trajectories are computed. An interpretation of quantum mechanics is attempted in which both classical and quantum mechanics contain fundamental randomness. Instead of an ensemble of events one introduces an ensemble of observers. (shrink)

Belief in free will is widespread. The present research considered one reason why people may believe that actions are freely chosen rather than determined: they attribute randomness in behavior to free will. Experiment 1 found that participants who were prompted to perform a random sequence of actions experienced their behavior as more freely chosen than those who were prompted to perform a deterministic sequence. Likewise, Experiment 2 found that, all else equal, the behavior of animated agents was perceived to (...) be more freely chosen if it consisted of a random sequence of actions than if it consisted of a deterministic sequence; this was true even when the degree of randomness in agents’ behavior was largely a product of their environments. Together, these findings suggest that randomness in behavior—one’s own or another’s—can be mistaken for free will. (shrink)

In 1866, the young John Venn published The Logic of Chance, motivated largely by the desire to correct what he saw as deep fallacies in the reasoning of historical determinists such as Henry Buckle and in the optimistic heralding of a true social science by Adolphe Quetelet. Venn accepted the inevitable determinism implied by the physical sciences, but denied that the stable social statistics cited by Buckle and Quetelet implied a similar determinism in human actions. Venn maintained that probability statements (...) were intelligible solely as projections of the ultimate frequency of an outcome in an unlimited series of trials and therefore led to no inferences about free will. An issue here was Venn's claim that although the aggregate probability may be stable, the individual events were completely irregular and, in fact, displayed randomness. Venn's concept of randomness was really apparent, or pseudo-, randomness, which could be totally determined. Ironically, Venn's frequentist position was very useful to the growing field of statistics, which could develop correlations and predictive models empirically without having to address implications for personal freedom or fatalism. (shrink)

We investigate notions of randomness in the space ${{\mathcal C}(2^{\mathbb N})}$ of continuous functions on ${2^{\mathbb N}}$ . A probability measure is given and a version of the Martin-Löf test for randomness is defined. Random ${\Delta^0_2}$ continuous functions exist, but no computable function can be random and no random function can map a computable real to a computable real. The image of a random continuous function is always a perfect set and hence uncountable. For any ${y \in 2^{\mathbb (...) N}}$ , there exists a random continuous function F with y in the image of F. Thus the image of a random continuous function need not be a random closed set. The set of zeroes of a random continuous function is always a random closed set. (shrink)

Recent years have seen increased interest in the question of whether it is possible to provide an evolutionary game-theoretic explanation for certain kinds of social norms. I sketch a proof of a general representation theorem for a large class of evolutionary game-theoretic models played on a social network, in hope that this will contribute to a greater understanding of the long-term evolutionary dynamics of such models, and hence the evolution of social norms.

We investigate various ways of introducing axioms for randomness in set theory. The results show that these axioms, when added to ZF, imply the failure of AC. But the axiom of extensionality plays an essential role in the derivation, and a deeper analysis may ultimately show that randomness is incompatible with extensionality.

How realistic is it to adopt a quantum random walk model to account for decisions involving two choices? Here, we discuss the neural plausibility and the effect of initial state and boundary thresholds on such a model and contrast it with various features of the classical random walk model of decision making.