Abstract:
The notion of the quantum angle is introduced. The quantum angle turns out to be a metric on the set of physical states of a quantum system. Its kinematics and dynamics is studied. The certainty principle for quantum systems is formulated and proved. It turns out that the certainty principle is closely connected with the Heisenberg uncertainty principle (it presents, in some sense, an opposite point of view). But at the same time the certainty principle allows to give rigorous formulations for wider class of problems (it allows to rigorously interpret and ground the analogous inequalities for the pairs of quantities like time - energy, angle - angular momentum etc.)

Abstract:
The certainty principle (2005) allowed to conceptualize from the more fundamental grounds both the Heisenberg uncertainty principle (1927) and the Mandelshtam-Tamm relation (1945). In this review I give detailed explanation and discussion of the certainty principle, oriented to all physicists, both theorists and experimenters.

Abstract:
Infinite games where several players seek to coordinate under imperfect information are known to be intractable, unless the information flow is severely restricted. Examples of undecidable cases typically feature a situation where players become uncertain about the current state of the game, and this uncertainty lasts forever. Here we consider games where the players attain certainty about the current state over and over again along any play. For finite-state games, we note that this kind of recurring certainty implies a stronger condition of periodic certainty, that is, the events of state certainty ultimately occur at uniform, regular intervals. We show that it is decidable whether a given game presents recurring certainty, and that, if so, the problem of synthesising coordination strategies under w-regular winning conditions is solvable.

Abstract:
The certainty of judgment (or self-confidence) has been traditionally studied in relation with the accuracy. However, from an observer's viewpoint, certainty may be more closely related to the consistency of judgment than to its accuracy: consistent judgments are objectively certain in the sense that any external observer can rely on these judgments to happen. The regions of certain vs. uncertain judgment were determined in a categorical rating experiment. The participants rated the size of visual objects on a 5-point scale. There was no feedback so that there were no constraints of accuracy. Individual data was examined, and the ratings were characterized by their frequency distributions (or categories). The main result was that the individual categories always presented a core of certainty where judgment was totally consistent, and large peripheries where judgment was inconsistent. In addition, the geometry of cores and boundaries exhibited several phenomena compatible with the literature on visual categorical judgment. The ubiquitous presence of cores in absence of accuracy constraints provided insights about objective certainty that may complement the literature on subjective certainty (self-confidence) and the accuracy of judgment.

Abstract:
In the Bayesian approach to quantum mechanics, probabilities--and thus quantum states--represent an agent's degrees of belief, rather than corresponding to objective properties of physical systems. In this paper we investigate the concept of certainty in quantum mechanics. Particularly, we show how the probability-1 predictions derived from pure quantum states highlight a fundamental difference between our Bayesian approach, on the one hand, and Copenhagen and similar interpretations on the other. We first review the main arguments for the general claim that probabilities always represent degrees of belief. We then argue that a quantum state prepared by some physical device always depends on an agent's prior beliefs, implying that the probability-1 predictions derived from that state also depend on the agent's prior beliefs. Quantum certainty is therefore always some agent's certainty. Conversely, if facts about an experimental setup could imply agent-independent certainty for a measurement outcome, as in many Copenhagen-like interpretations, that outcome would effectively correspond to a preexisting system property. The idea that measurement outcomes occurring with certainty correspond to preexisting system properties is, however, in conflict with locality. We emphasize this by giving a version of an argument of Stairs [A. Stairs, Phil. Sci. 50, 578 (1983)], which applies the Kochen-Specker theorem to an entangled bipartite system.

Abstract:
Persistent activity in neuronal populations has been shown to represent the spatial position of remembered stimuli. Networks that support bump attractors are often used to model such persistent activity. Such models usually exhibit translational symmetry. Thus activity bumps are neutrally stable, and perturbations in position do not decay away. We extend previous work on bump attractors by constructing model networks capable of encoding the certainty or salience of a stimulus stored in memory. Such networks support bumps that are not only neutrally stable to perturbations in position, but also perturbations in amplitude. Possible bump solutions then lie on a two-dimensional attractor, determined by a continuum of positions and amplitudes. Such an attractor requires precisely balancing the strength of recurrent synaptic connections. The amplitude of activity bumps represents certainty, and is determined by the initial input to the system. Moreover, bumps with larger amplitudes are more robust to noise, and over time provide a more faithful representation of the stored stimulus. In networks with separate excitatory and inhibitory populations, generating bumps with a continuum of possible amplitudes, requires tuning the strength of inhibition to precisely cancel background excitation.

Abstract:
In his Discourse on the Method of Rightly Conducting the Reason, and Seeking Truth in the Sciences, Rene Descartes sought ``clear and certain knowledge of all that is useful in life.'' Almost three centuries later, in ``The foundations of mathematics,'' David Hilbert tried to ``recast mathematical definitions and inferences in such a way that they are unshakable.'' Hilbert's program relied explicitly on formal systems (equivalently, computational systems) to provide certainty in mathematics. The concepts of computation and formal system were not defined in his time, but Descartes' method may be understood as seeking certainty in essentially the same way. In this article, I explain formal systems as concrete artifacts, and investigate the way in which they provide a high level of certainty---arguably the highest level achievable by rational discourse. The rich understanding of formal systems achieved by mathematical logic and computer science in this century illuminates the nature of programs, such as Descartes' and Hilbert's, that seek certainty through rigorous analysis.

Abstract:
The marker surely possesses at least three different functions: as an intrapredicative adverb of manner, an extrapredicative epistemic adverb and a discourse adverb. In this paper I will propose a single characterisation of surely within the framework of the Theory of Enunciative Operations (cf. Culioli, 1990, 1999a and 1999b). We will see that, depending on variable contextual parameters, surely assumes one of its three possible values. More precisely, surely marks a correspondence between, on the one hand, a preconstructed trajectory leading from a start point to an end point (from (p, non-p) to p), and on the other, the same trajectory constructed in the speech situation by the speaker. When the trajectory bears upon the mode of realisation of a process, surely assumes its intrapredicative value, when it bears upon the passage from a source situation to a projected situation, surely functions extrapredicatively, as an epistemic adverb. The discourse adverb function of surely poses some problems for analysis. When epistemic, surely appears to mark certainty, but when discursive, it appears rather to mark doubt, disbelief or incomprehension, according to context. I propose that when surely is used in this way, it marks the speaker’s endorsement of a preconstructed trajectory while also acknowledging the presence of a preconstructed, counteroriented trajectory, endorsed by another speaker. The shift from certainty to doubt might be explained as the consequence of the speaker’s recognition of a wider discursive context. The various contextual values of the discourse adverb surely (cf. Downing, 2001, for example) depend, among other factors, on how the speaker positions his or her discourse relative to other enunciative instances. Le marqueur anglais surely possède au moins trois fonctions différentes : adverbe de manière intraprédicatif, adverbe épistémique extraprédicatif et adverbe de discours. Dans cet article je proposerai une caractérisation unique de surely dans le cadre de la Théorie des Opérations énonciatives (cf. Culioli, 1990, 1999a et 1999b). Nous verrons que, selon des paramètres contextuels variables, surely prend l’une de ses trois valeurs possibles. Plus précisément, surely marque une correspondance entre, d’un c té, une trajectoire préconstruite menant d’un point de départ vers un point d’arrivée (de (p, non-p) vers p), et, de l’autre, la même trajectoire construite dans la situation d’énonciation par l’énonciateur. Lorsque la trajectoire porte sur le mode de réalisation d’un procès, surely prend sa valeur intraprédicative ; lorsque la tr