Supra-Bayesianism is the Bayesian response to learning the opinions of others. Probability pooling constitutes an alternative response. One natural question is whether there are cases where probability pooling gives the supra-Bayesian result. This has been called the problem of Bayes-compatibility for pooling functions. It is known that in a common prior setting, under standard assumptions, linear pooling cannot be nontrivially Bayes-compatible. We show by contrast that geometric pooling can be nontrivially Bayes-compatible. Indeed, we show (...) that, under certain assumptions, geometric and Bayes-compatible pooling are equivalent. Granting supra-Bayesianism its usual normative status, one upshot of our study is thus that, in a certain class of epistemic contexts, geometric pooling enjoys a normative advantage over linear pooling as a social learning mechanism. We discuss the philosophical ramifications of this advantage, which we show to be robust to variations in our statement of the Bayes-compatibility problem. (shrink)

Much of our information comes to us indirectly, in the form of conclusions others have drawn from evidence they gathered. When we hear these conclusions, how can we modify our own opinions so as to gain the benefit of their evidence? In this paper we study the method known as geometric pooling. We consider two arguments in its favour, raising several objections to one, and proposing an amendment to the other.

Suppose several individuals (e.g., experts on a panel) each assign probabilities to some events. How can these individual probability assignments be aggregated into a single collective probability assignment? This article reviews several proposed solutions to this problem. We focus on three salient proposals: linear pooling (the weighted or unweighted linear averaging of probabilities), geometric pooling (the weighted or unweighted geometric averaging of probabilities), and multiplicative pooling (where probabilities are multiplied rather than averaged). We present axiomatic (...) characterisations of each class of pooling functions (most of them classic, but one new) and argue that linear pooling can be justified procedurally, but not epistemically, while the other two pooling methods can be justified epistemically. The choice between them, in turn, depends on whether the individuals' probability assignments are based on shared information or on private information. We conclude by mentioning a number of other pooling methods. (shrink)

We often learn the opinions of others without hearing the evidence on which they're based. The orthodox Bayesian response is to treat the reported opinion as evidence itself and update on it by conditionalizing. But sometimes this isn't feasible. In these situations, a simpler way of combining one's existing opinion with opinions reported by others would be useful, especially if it yields the same results as conditionalization. We will show that one method---upco, also known as multiplicative pooling---is specially suited (...) to this role when the opinions you wish to pool concern hypotheses about chances. The result has interesting consequences: it addresses the problem of disagreement between experts; and it sheds light on the social argument for the uniqueness thesis. (shrink)

How should your opinion change in light of an epistemic peer's? We show that the pooling rule known as "upco" is the unique answer satisfying some natural desiderata. If your revised opinion will impact your other views by Jeffrey conditionalization, then upco is the only standard pooling rule that ensures the order in which peers are consulted makes no difference. Popular alternatives like linear pooling, geometric pooling, and harmonic pooling cannot boast the same. In (...) fact, no alternative can that possesses four minimal properties these proposals all share. (shrink)

A group is often construed as one agent with its own probabilistic beliefs (credences), which are obtained by aggregating those of the individuals, for instance through averaging. In their celebrated “Groupthink”, Russell et al. (2015) require group credences to undergo Bayesian revision whenever new information is learnt, i.e., whenever individual credences undergo Bayesian revision based on this information. To obtain a fully Bayesian group, one should often extend this requirement to non-public or even private information (learnt by not all or (...) just one individual), or to non-representable information (not representable by any event in the domain where credences are held). I pro- pose a taxonomy of six types of ‘group Bayesianism’. They differ in the information for which Bayesian revision of group credences is required: public representable information, private representable information, public non-representable information, etc. Six corre- sponding theorems establish how individual credences must (not) be aggregated to ensure group Bayesianism of any type, respectively. Aggregating through standard averaging is never permitted; instead, different forms of geometric averaging must be used. One theorem—that for public representable information—is essentially Russell et al.’s central result (with minor corrections). Another theorem—that for public non-representable information—fills a gap in the theory of externally Bayesian opinion pooling. (shrink)

There are many reasons we might want to take the opinions of various individuals and pool them to give the opinions of the group they constitute. If all the individuals in the group have probabilistic opinions about the same propositions, there is a host of pooling functions we might deploy, such as linear or geometric pooling. However, there are also cases where different members of the group assign probabilities to different sets of propositions, which might overlap a (...) lot, a little, or not at all. There are far fewer proposals for how to proceed in these cases, and those there are have undesirable features. I begin by considering four proposals and arguing that they don't work. Then I'll describe my own proposal, which is intended to cover the situation in which we want to pool the individual opinions in order to ascribe an opinion to the group considered as an agent in its own right. (shrink)

Can a group be an orthodox rational agent? This requires the group's aggregate preferences to follow expected utility (static rationality) and to evolve by Bayesian updating (dynamic rationality). Group rationality is possible, but the only preference aggregation rules which achieve it (and are minimally Paretian and continuous) are the linear-geometric rules, which combine individual values linearly and combine individual beliefs geometrically. Linear-geometric preference aggregation contrasts with classic linear-linear preference aggregation, which combines both values and beliefs linearly, but achieves (...) only static rationality. Our characterisation of linear-geometric preference aggregation has two corollaries: a characterisation of linear aggregation of values (Harsanyi's Theorem) and a characterisation of geometric aggregation of beliefs. (shrink)

If a group is modelled as a single Bayesian agent, what should its beliefs be? I propose an axiomatic model that connects group beliefs to beliefs of group members, who are themselves modelled as Bayesian agents, possibly with different priors and different information. Group beliefs are proven to take a simple multiplicative form if people’s information is independent, and a more complex form if information overlaps arbitrarily. This shows that group beliefs can incorporate all information spread over the individuals without (...) the individuals having to communicate their (possibly complex and hard-to-describe) private information; communicating prior and posterior beliefs sufices. JEL classification: D70, D71.. (shrink)

How can different individuals' probability assignments to some events be aggregated into a collective probability assignment? Classic results on this problem assume that the set of relevant events -- the agenda -- is a sigma-algebra and is thus closed under disjunction (union) and conjunction (intersection). We drop this demanding assumption and explore probabilistic opinion pooling on general agendas. One might be interested in the probability of rain and that of an interest-rate increase, but not in the probability of rain (...) or an interest-rate increase. We characterize linear pooling and neutral pooling for general agendas, with classic results as special cases for agendas that are sigma-algebras. As an illustrative application, we also consider probabilistic preference aggregation. Finally, we compare our results with existing results on binary judgment aggregation and Arrovian preference aggregation. This paper is the first of two self-contained, but technically related companion papers inspired by binary judgment-aggregation theory. (shrink)

When faced with uncertainty, consequentialists often advocate choosing the option with the largest expected utility, as calculated using the arithmetic average. I provide some arguments to suggest that instead, one should consider choosing the option with the largest geometric average of utility. I explore the difference between these two approaches in a variety of ethical dilemmas and argue that geometric averaging has some appealing properties as a normative decision-making tool.

At some point in the Incessu Animalium, Aristotle appeals to some geometrical claims in order to explain why animal progression necessarily involves the bending (of the limbs), and this appeal to geometrical claims might be taking as violating the recommendation to avoid “kind-crossing” (as found in the Posterior Analytic). But a very unclear notion of kind-crossing has been assumed in most debates. I will argue that kind-crossing in the Posterior Analytics does not mean any employment of premises from a discipline (...) other than that to which the explanandum belongs. Kind-crossing was meant to cover a specific sort of employment of premises from a different discipline, namely, the case in which premises from a discipline X are taken as the most important explanatory factor that delivers the fullest appropriate explanation of an explanandum within discipline Y. If this is so, the employment of geometrical premises in the Incessu Animalium is not an instance of the prohibited kind-crossing, but something that is in line with the theory of the Posterior Analytics. (shrink)

There can be situations in which, if I contribute to a pool of resources for helping a large number of people, the difference that my contribution makes to any of the people helped from the pool will be imperceptible at best, and maybe even non-existent. And this can be the case where it is also true that giving the same amount directly to one of the intended beneficiaries of the pool would have made a very large difference to her. Can (...) non-contribution to the pool be morally justified on this ground? I argue that it cannot. For, first, this line of thought leaves unaffected any reasons for holding that failing to perform the direct action of benefiting someone greatly would be wrong. But the pooling system of helping people is often better than a system separating the help which is given — better because of the perceptible difference it makes to its beneficiaries. If so, failing to contribute to the pool will be at least as wrong as failing to have helped directly would have been. The paper clarifies and defends an argument of this form, showing how it can be formulated in a way that avoids apparent counterexamples, and identifying the assumptions on which it rests. (shrink)

How can different individuals' probability functions on a given sigma-algebra of events be aggregated into a collective probability function? Classic approaches to this problem often require 'event-wise independence': the collective probability for each event should depend only on the individuals' probabilities for that event. In practice, however, some events may be 'basic' and others 'derivative', so that it makes sense first to aggregate the probabilities for the former and then to let these constrain the probabilities for the latter. We formalize (...) this idea by introducing a 'premise-based' approach to probabilistic opinion pooling, and show that, under a variety of assumptions, it leads to linear or neutral opinion pooling on the 'premises'. This paper is the second of two self-contained, but technically related companion papers inspired by binary judgment-aggregation theory. (shrink)

The paper is an introduction to geometric algebra and geometric calculus for those with a knowledge of undergraduate mathematics. No knowledge of physics is required. The section Further Study lists many papers available on the web.

The purpose of this work is to address what notion of geometrical object and geometrical figure we have in different kinds of geometry: practical, pure, and applied. Also, we address the relation between geometrical objects and figures when this is possible, which is the case of pure and applied geometry. In practical geometry it turns out that there is no conception of geometrical object.

How is knowledge of geometry developed and acquired? This central question in the philosophy of mathematics has received very different answers. Spelke and colleagues argue for a “core cognitivist”, nativist, view according to which geometric cognition is in an important way shaped by genetically determined abilities for shape recognition and orientation. Against the nativist position, Ferreirós and García-Pérez have argued for a “culturalist” account that takes geometric cognition to be fundamentally a culturally developed phenomenon. In this paper, I (...) argue that when understood as moderate versions supported by the state-of-the-art research, the nativist and culturalist views are in fact possible to reconcile. While Ferreirós and García-Pérez present the work of Spelke and colleagues as implying that geometric cognition is genetically determined, I argue that they fail to appreciate the role that Spelke and colleagues see for cultural factors. On this basis, I provide theoretical and terminological clarifications and show that moderate versions of the nativist and culturalist view are in fact consistent with each other. I then propose a unifying theoretical framework for future study that can integrate the two accounts in ontogeny by moving beyond the crude nature (nativism) vs. nurture (culturalism) dichotomy. (shrink)

This paper examines Aristotle's discussion of the priority of actuality to potentiality in geometry at Metaphysics Θ9, 1051a21–33. Many scholars have assumed what I call the "geometrical construction" interpretation, according to which his point here concerns the relation between an inquirer's thinking and a geometrical figure. In contrast, I defend what I call the "geometrical analysis" interpretation, according to which it concerns the asymmetrical relation between geometrical propositions in which one is proved by means of the other. His argument as (...) so construed is ultimately based on the asymmetrical relation between the corresponding geometrical facts. Then I explore this ontological priority in geometry by drawing attention to a parallel passage, Posterior Analytics II.11, 94a24–35, where Aristotle explains the relation between the same geometrical propositions in connection to material causation. (shrink)

All acknowledged proofs of the Four Colour Theorem (4CT) are computerdependent. They appeal to the existence, and manual identification, of an ‘unavoidable’ set containing a sufficient number of explicitly defined configurations—each evidenced only by a computer as ‘reducible’—such that at least one of the configurations must occur in any chromatically distinguished, putatively minimal, planar map. For instance, Appel and Haken ‘identified’ 1,482 such configurations in their 1977, computer-dependent, proof of 4CT; whilst Neil Robertson et al ‘identified’ 633 configurations as sufficient (...) in their 1997, also computer-dependent, proof of 4CT. However, treating any specific number of ‘reducible’ configurations in an ‘unavoidable’ set as sufficient entails a minimum number as necessary and sufficient. We now show that the minimum number of such configurations can only be the one corresponding to the ‘unavoidable’ set of the single, ‘reducible’, 4-sided configuration identified by Alfred Kempe in his, seemingly fatally flawed, 1879 ‘proof’ of 4CT. We shall further show that although Kempe fallaciously concluded that a 5-sided configuration was also in the ‘unavoidable’ set, and appealed to unproven properties of ‘Kempe’ chains in a graphical representation to then argue for its ‘reducibility’, neither flaw in his ‘proof’ is fatal when the argument is expressed geometrically; and that, essentially, Kempe correctly argued that any planar map which admits a chromatic differentiation with a five-sided area C that shares non-zero boundaries with four, all differently coloured, neighbours can be 4-coloured. (shrink)

General relativity (GR) describes gravity through the curvature of spacetime. However, there are two equivalents of GR that describe flat spacetimes with gravitational effects attributed to torison or non-metricity. These theories, together with GR, are known as the geometrical trinity of gravity and are said to present a case of underdetermination by Wolf et al. (2024). In this article, I argue against this stance by examining the empirical equivalence and possible interpretations of the trinity. I propose a unifying framework where (...) the trinity emerge as different facets of the same underlying ontology, thereby breaking down the underdetermination. (shrink)

A description of a geometrical model of the structure of the volume of the universe has to clarify the existence of universal conservation laws, universal constants and universal principles (like the principle of non-locality). But the model envelopes observable and detectable phenomena too, so it is reasonable to expect calculated relations too.

This study analyses the predictions of the General Theory of Relativity (GTR) against a slightly modified version of the standard central mass solution (Schwarzschild solution). It is applied to central gravity in the solar system, the Pioneer spacecraft anomalies (which GTR fails to predict correctly), and planetary orbit distances and times, etc (where GTR is thought consistent.) -/- The modified gravity equation was motivated by a theory originally called ‘TFP’ (Time Flow Physics, 2004). This is now replaced by the ‘ (...) class='Hi'>Geometric Model’, 2014 [20], which retains the same theory of gravity. This analysis is offered partially as supporting detail for the claim in [20] that the theory is realistic in the solar system and explains the Pioneer anomalies. The overall conclusion is that the model can claim to explain the Pioneer anomalies, contingent on the analysis being independently verified and duplicated of course. -/- However the interest lies beyond testing this theory. To start with, it gives us a realistic scale on which gravity might vary from the accepted theory, remain consistent with most solar-scale astronomical observations. It is found here that the modified gravity equation would appear consistent with GTR for most phenomena, but it would retard the Pioneer spacecraft by about the observed amount (15 seconds or so at time). Hence it is a possible explanation of this anomaly, which as far as I know remains unexplained now for 20 years. -/- It also shows what many philosophers of science have emphasized: the pivotal role of counterfactual reasoning. By putting forward an exact alternative solution, and working through the full explanation, we discover a surprising ‘counterfactual paradox’: the modified theory slightly weakens GTR gravity – and yet the effect is to slow down the Pioneer trajectory, making it appear as if gravity is stronger than GTR. The inference that “there must be some tiny extra force…” (Musser, 1998 [1]) is wrong: there is a second option: “…or there may be a slightly weaker form of gravity than GTR.” . (shrink)

This study presents a new type of foundational model unifying quantum theory, relativity theory and gravitational physics, with a novel cosmology. It proposes a six-dimensional geometric manifold as the foundational ontology for our universe. The theoretical unification is simple and powerful, and there are a number of novel empirical predictions and theoretical reductions that are strikingly accurate. It subsequently addresses a variety of current anomalies in physics. It shows how incomplete modern physics is by giving an example of a (...) theory that is genuinely unified. In doing this, it radically alters the metaphysical interpretation of the nature of time, space and matter currently interpreted from modern physics. It also profoundly challenges materialist expectations about a naturalistic account our own existence. I contend here that there is sufficient evidence to support this theory as a leading paradigm for a unified foundational theory. (shrink)

This essay offers an in-depth reading of the geometrical illustration of Ethics IIP8S and shows how it can be used to explicate the whole architecture of Spinoza’s system by specifying the way in which all the key structural features of his basic ontology find their analogies in the example. The illustration can also throw light on Spinoza’s ontology of finite things and inform us about what is at stake when we form universal ideas. In general, my reading of IIP8S thus (...) elucidates what it means, for Spinoza, to think geometrically or to consider geometry as a model: fundamentally, geometricity is not a form of exposition but the way in which reality itself is structured. (shrink)

We study shared intentions in what we call “loose groups”. These are groups that lack a codified organizational structure, and where the communication channels between group members are either unreliable or not completely open. We start by formulating two desiderata for shared intentions in such groups. We then argue that no existing account meets these two desiderata, because they assume either too strong or too weak an epistemic condition, that is, a condition on what the group members know and believe (...) about what the others intend, know, and believe. We propose an alternative, pooled knowledge, and argue that it allows formulating conditions on shared intentions that meet the two desiderata. (shrink)

Luminance and color are strong and self-sufficient cues to pictorial depth in visual scenes and images. The present study investigates the conditions Under which luminance or color either strengthens or overrides geometric depth cues. We investigated how luminance contrasts associated with color contrast interact with relative height in the visual field, partial occlusion, and interposition in determining the probability that a given figure is perceived as ‘‘nearer’’ than another. Latencies of ‘‘near’’ responses were analyzed to test for effects of (...) attentional selection. Figures in a pair were supported by luminance contrast or isoluminant color contrast and combined with one of the three geometric cues. The results of Experiment 1 show that luminance contrasts associated with hue, when it does not interact with other hues, produces the same effects as achromatic luminance contrasts: The probability of‘‘near’’ increases with luminance contrast while the latencies for ‘‘near’’ responses decrease. Partial occlusion is found to be a strong enough pictorial cue to support a weaker red luminance contrast. Interposition cues lose out against cues of spatial position and partial occlusion. The results of Experiment 2, with isoluminant displays of varying color contrast, reveal that red color contrast on a light background supported by any of the three geometric cues wins over green or white supported by any of the three geometric cues. On a dark background, red color contrast supported by the interposition cue loses out against green or white color contrast supported by partial occlusion. These findings reveal that color is not an independent depth cue, but is strongly influenced by luminance contrast and stimulus geometry. Systematically shorter response latencies for stronger ‘‘near’’ percepts demonstrate that selective visual attention reliably detects the most likely depth cue combination in a given configuration. (shrink)

The cognitive basis of geometry is still poorly understood, even the ‘simpler’ issue of what kind of representation of geometric objects we have. In this work, we set forward a tentative model of the neural representation of geometric objects for the case of the pure geometry of Euclid. To arrive at a coherent model, we found it necessary to consider earlier forms of geometry. We start by developing models of the neural representation of the geometric figures of (...) ancient Greek practical geometry. Then, we propose a related model for the earliest form of pure geometry – that of Hippocrates of Chios. Finally, we develop the model of the neural representation of the geometric objects of Euclidean geometry. The models are based on the hub-and-spoke theory. In our view, the existence of specific models opens the possibility of addressing the relationship between geometric figures and geometric objects, in a novel way, in terms of their neural representation. (shrink)

This paper examines the second geometrical problem in the Meno. Its purpose is to explore the implication of Cook Wilson’s interpretation, which has been most widely accepted by scholars, in relation to the nature of hypothesis. I argue that (a) the geometrical hypothesis in question is a tentative answer to a more basic problem, which could not be solved by available methods at that time, and that (b) despite the temporary nature of a hypothesis, there is a rational process for (...) formulating it. The paper also contains discussion of the method of analysis, problem reduction and a diorism, which have often been ambiguously explained in relation to the geometrical problem in question. (shrink)

In this article, we address an existing lacuna in the sociology of the senses, by employing sociological phenomenology to illuminate the under-researched sense of temperature, as lived by a social group for whom water temperature is particularly salient: competitive pool swimmers. The research contributes to a developing ‘sensory sociology’ that highlights the importance of the socio-cultural framing of the senses and ‘sensory work’, but where there remains a dearth of sociological exploration into senses extending beyond the ‘classic five’ sensorium. Drawing (...) on data from a three-year ethnographic study of competitive swimmers in the UK, our analysis explores the rich sensuousities of swimming, and highlights the role of temperature as fundamentally affecting the affordances offered by the aquatic environment. The article contributes original theoretical perspectives to the sociology of the senses and of sport in addressing the ways in which social actors in the aquatic environment interact, both intersubjectively and intercorporeally, as thermal beings. (shrink)

In this paper, we will make explicit the relationship that exists between geometric objects and geometric figures in planar Euclidean geometry. That will enable us to determine basic features regarding the role of geometric figures and diagrams when used in the context of pure and applied planar Euclidean geometry, arising due to this relationship. By taking into account pure geometry, as developed in Euclid’s Elements, and practical geometry, we will establish a relation between geometric objects and (...) figures. Geometric objects are defined in terms of idealizations of the corresponding figures of practical geometry. We name the relationship between them as a relation of idealization. This relation, existing between objects and figures, is what enables figures to have a role in pure and applied geometry. That is, we can use a figure or diagram as a representation of geometric objects or composite geometric objects because the relation of idealization corresponds to a resemblance-like relationship between objects and figures. Moving beyond pure geometry, we will defend that there are two other ‘layers’ of representation at play in applied geometry. To show that, we will consider Euclid’s Optics. (shrink)

Focused attention is needed to perceive change (Rensink et al., 1997; Psychological Science, 8: 368-373) . But how much attentional processing is given to an item? And does this depend on the nature of the task? To answer these questions, "flicker" displays were created, where an original and a modified image continually alternated, with brief blanks between them. Each image was an array of simple figures, half being horizontal and the other half vertical. In half the trials, one of the (...) items changed orientation; in the remainder, all items remained the same. Observers were asked to detect whether a change was occurring in each trial. -/- Within-observer comparisons for different shapes of items yielded two classes of speed: relatively fast search (50 ms/item) for simple lines, and slower search (90 ms/item) for compound figures made of two or more lines. Evidently, more processing was given to compound figures, even though shape was irrelevant to the task. A similar set of experiments asked observers to detect changes in contrast polarity. Here, no differences were found for different shapes, indicating that geometric information is not processed as much when the task involves only nongeometric properties. (shrink)

All accepted proofs of the Four Colour Theorem (4CT) are computer-dependent; and appeal to the existence, and manual identification, of an ‘unavoidable’ set containing a sufficient number of explicitly defined configurations—each evidenced only by a computer as ‘reducible’—such that at least one of the configurations must occur in any chromatically distinguished, minimal, planar map. For instance, Appel and Haken ‘identified’ 1,482 such configurations in their 1977, computer-dependent, proof of 4CT; whilst Neil Robertson et al ‘identified’ 633 configurations as sufficient in (...) their 1997, also computer-dependent, proof of 4CT. However, treating any specific number of ‘reducible’ configurations in an ‘unavoidable’ set as sufficient entails a minimum number as necessary and sufficient. We now show that the minimum number of such configurations can only be the one corresponding to the ‘unavoidable’ set of a ‘reducible’, 4-sided configuration identified by Alfred Kempe in his, seemingly fatally flawed, 1879 ‘proof’ of 4CT. Although Kempe appealed to putative properties of ‘Kempe’ chains in a graphical representation to fallaciously argue that a 5-sided configuration was also in the ‘unavoidable’ set, and ‘reducible’, we shall show why the flaw in his ‘proof’ is not fatal when the argument is expressed geometrically; and that, essentially, Kempe correctly argued that any planar map which admits a chromatic differentiation with a five-sided area C that shares non-zero boundaries with four, all differently coloured, neighbours can be 4-coloured. (shrink)

In the Critique of Pure Reason, Kant defends the mathematically deterministic world of physics by arguing that its essential features arise necessarily from innate forms of intuition and rules of understanding through combinatory acts of imagination. Knowing is active: it constructs the unity of nature by combining appearances in certain mandatory ways. What is mandated is that sensible awareness provide objects that conform to the structure of ostensive judgment: “This (S) is P.” -/- Sensibility alone provides no such objects, so (...) the imagination compensates by combining passing point-data into “pure” referents for the subject-position, predicate-position, and copula. The result is a cognitive encounter with a generic physical object whose characteristics—magnitude, substance, property, quality, and causality—are abstracted as the Kantian categories. Each characteristic is a product of “sensible synthesis” that has been “determined” by a “function of unity” in judgment. -/- Understanding the possibility of such determination by judgment is the chief difficulty for any rehabilitative reconstruction of Kant’s theory. I will show that Kant conceives of figurative synthesis as an act of line-drawing, and of the functions of unity as rules for attending to this act. The subject-position refers to substance, identified as the objective time-continuum; the predicate-position, to quality, identified as the continuum of property values (constituting the second-order type named by the predicate concept). The upshot is that both positions refer to continuous magnitudes, related so that one (time-value) is the condition of the other (property-value). -/- Kant’s theory of physically constructive grammar is thus equivalent to the analytic-geometric formalism at work in the practice of mathematical physics, which schematizes time and state as lines related by an algebraic formula. Kant theorizes the subject–predicate relation in ostensive judgment as an algebraic time–state function. When aimed towards sensibility, “S is P” functions as the algebraic relation “t → ƒ(t).”. (shrink)

We give a brief overview of the evolution of mathematics, starting from antiquity, through Renaissance, to the 19th century, and the culmination of the train of thought of history’s greatest thinkers that lead to the grand unification of geometry and algebra. The goal of this paper is not a complete formal description of any particular theoretical framework, but to show how extremisation of mathematical rigor in requiring everything be drivable directly from first principles without any arbitrary assumptions actually leads to (...) relaxing the computational difficulty along with maximal conceptual clarity. With this, we consider a revision of the foundations of elementary geometry and algebra based on the work of Grassmann and Clifford and apply it to conceptual and practical problems of past and present modern mathematics and mathematical physics. (shrink)

Review of Geometric Possibility (2011), by Gordon Belot. Oxford and New York: Oxford University Press. x + 219 pp.

During 17th century a scientific controversy existed on the derivation of Snell’s laws of reflection and refraction. Descartes gave a derivation of the laws, independent of the minimality of travel time of a ray of light between two given points. Fermat and Leibniz gave a derivation of the laws, based on the minimality of travel time of a ray of light between two given points. Leibniz’s calculus method became the standard method of derivation of the two laws. We demonstrate in (...) this article that Snell’s law of reflection follows from simple results of geometry. We do not use the concept of motion or the time of travel in our demonstration. That is, time plays no role in our demonstration. (shrink)

This article analyzes the value of geometric models to understand matter with the examples of the Platonic model for the primary four elements (fire, air, water, and earth) and the models of carbon atomic structures in the new science of crystallography. How the geometry of these models is built in order to discover the properties of matter is explained: movement and stability for the primary elements, and hardness, softness and elasticity for the carbon atoms. These geometric models appear (...) to have a double quality: firstly, they exhibit visually the scientific properties of matter, and secondly they give us the possibility to visualize its whole nature. Geometrical models appear to be the expression of the mind in the understanding of physical matter. (shrink)

Here we propose a discussion about the "mandorla" or "vesica piscis". It is a type of 2-dimensional lens, that is, a geometric shape formed by the intersection of two circles with the same radius, intersecting in such a way that the centre of each circle lies on the perimeter of the other. The aim of the discussion is that of understanding when such a geometric shape became a symbol and when this symbol received a specific name. We will (...) find that the name "mandorla" was used long before the term "vesica piscis", which is the Latin translation of the German "fischblosen" used by Albrecht Dürer in his book on geometry. Therefore, the name invented by Dürer was not used by the painter for a sacred form. However, after the middle of the nineteenth century the term "vesica piscis" exploded in literature. Its use was criticized and, at the same time, it was stressed that the proper term for the symbol is "mandorla". Nonetheless, the "vesica piscis" continues to be largely used in the sacred geometry, which ascribes symbolic and sacred meanings to certain geometric shapes and proportions. In the proposed discussion, we will also show that, recently, the 2-dimensional lenticular symbol has been related to the Pythagorean philosophy. It is told that the followers of this philosophy had the habit of using an apple for symbolic communications. Sliced across, the core of the apple is displaying a pentagram, but sliced lengthwise it forms two intersecting circles, that is a "mandorla". Then, the investigation about terms and related uses of this specific geometric shape is the core of the matter of our discussion or, let us tell, the "core of the apple", in a Pythagorean-like approach to find how a German "fischblosen" evolved into a sacred symbol, thanks to a translation into Latin. (shrink)

Many species rely on the three-dimensional surface layout of an environment to find a desired goal following disorientation. They generally do so to the exclusion of other important spatial cues. Two influential frameworks for explaining that phenomenon are provided by geometric-module theories and view-matching theories of reorientation respectively. The former posit a module that operates only on representations of the global geo- metry of three-dimensional surfaces to guide behavior. The latter place snapshots, stored representations of the subject’s two-dimensional retinal (...) stimulation at specific locations, at the heart of their accounts. In this paper, I take a fresh look at the debate between them. I begin by making a case that the empirical evidence we currently have does not clearly favor one framework over the other, and that the debate has reached something of an impasse. Then, I present a new explanatory problem—the representation selection problem—that offers the pro- spect of breaking the impasse by introducing a new type of explanatory consideration that both frameworks must address. The representation selection problem requires explaining how subjects can reliably select the relevant representation with which they initiate the reorientation process. I argue that the view-matching framework does not have the resources to address this problem, while a certain type of theory within the geometric-module framework can provide a natural response to it. In showing this, I develop a new geometric-module theory. (shrink)

Back to the days on childhood, to a regular café on a small village where time goes by.

Newton frequently characterized his methodology as distinctive and capable of achieving greater evidential support than that of his contemporaries, due to its mathematical character. Newton's pronouncements reflect a striking position regarding the role of mathematics in natural philosophy. We can give an initial characterization of his position by considering two questions central to seventeenth century debates about the applicability of mathematics. First, how are we to understand the distinctive universality and necessity of mathematical reasoning? One common way to preserve the (...) demonstrative character of mathematics was to restrict its domain, as far as possible, to pure abstractions. The subject matter of mathematics is then taken to be abstracted from the changeable natural world, consisting of quantity and magnitude themselves rather than the objects bearing quantifiable properties. Yet restricting the domain in this way leaves it difficult to see how mathematics relates to natural phenomena. How could the book of nature be written in a language of pure abstractions? Second, what is the proper role of mathematical reasoning in natural philosophy? Many followed Aristotle in consigning mathematics to a subordinate role. Mathematics could not fulfill the main aim of natural philosophy, namely to provide demonstrations reflecting the essences of things and nature's causal order. A related concern was also pressing for the mechanical philosophers: a merely mathematical demonstration fails to provide an intelligible mechanical explanation. The very title of Newton's masterpiece, Philosophiae Naturalis Principia Mathematica, would have been extremely perplexing: how could natural philosophy be based on mathematical principles? My aim is to articulate Newton's position regarding the mathematization of nature in response to these two concerns. (shrink)

The first article in this issue attempts to refute the concept of Altruism and calls it akin to Selfishness. The arguments are logically set in the way like that of Spinoza’s method of demonstration, with Axioms, Definitions, Propositions and Notes: so as to make them exact and precise. Interestingly, the writer introduces a new concept of Credit and through various other original propositions and examples rebuts the altruistic nature which is generally ascribed to humans.

Scientists use models to know the world. It i susually assumed that mathematicians doing pure mathematics do not. Mathematicians doing pure mathematics prove theorems about mathematical entities like sets, numbers, geometric figures, spaces, etc., they compute various functions and solve equations. In this paper, I want to exhibit models build by mathematicians to study the fundamental components of spaces and, more generally, of mathematical forms. I focus on one area of mathematics where models occupy a central role, namely homotopy (...) theory. I argue that mathematicians introduce genuine models and I offer a rough classification of these models. (shrink)

In this contribution we will present a generalization of de Finetti's betting game in which a gambler is allowed to buy and sell unknown events' betting odds from more than one bookmaker. In such a framework, the sole coherence of the books the gambler can play with is not sucient, as in the original de Finetti's frame, to bar the gambler from a sure-win opportunity. The notion of joint coherence which we will introduce in this paper characterizes those coherent books (...) on which sure- win is impossible. Our main results provide geometric characterizations of the space of all books which are jointly coherent with a xed one. As a consequence we will also show that joint coherence is decidable. (shrink)

Galileo’s dictum that the book of nature “is written in the language of mathematics” is emblematic of the accepted view that the scientific revolution hinged on the conceptual and methodological integration of mathematics and natural philosophy. Although the mathematization of nature is a distinctive and crucial feature of the emergence of modern science in the seventeenth century, this volume shows that it was a far more complex, contested, and context-dependent phenomenon than the received historiography has indicated, and that philosophical controversies (...) about the implications of mathematization cannot be understood in isolation from broader social developments related to the status and practice of mathematics in various commercial, political, and academic institutions. (shrink)

As has recently been highlighted, despite the prevalence of methodological “confessional tales” in ethnography generally, the challenges of undertaking ethnographic research specifically in institutional sports settings remain underexplored. Drawing on data from a 3-year ethnographic study of competitive swimming in the United Kingdom (UK), here we explore some of the practical challenges of balancing different elements of the researcher’s role when undertaking ethnographic “insider” research in familiar settings. In particular, we consider the difficulties of balancing the role of a doctoral (...) researcher and the chosen research role of a volunteer coach with a competitive swimming program. Employing the anthropological concept of liminality, we also analyze the lived challenges of leaving a highly familiar field and entering a state of liminality, where the researcher was caught on the threshold betwixt and between a return to full-time employment in the former “known” role of coach, and a move forward to embrace a new “unknown” role as a full-time member of academic staff. (shrink)

Models of collective deliberation often assume that the chief aim of a deliberative exchange is the sharing of information. In this paper, we argue that an equally important role of deliberation is to draw participants’ attention to pertinent questions, which can aid the assembly and processing of distributed information by drawing deliberators’ attention to new issues. The assumption of logical omniscience renders classical models of agents’ informational states unsuitable for modelling this role of deliberation. Building on recent insights from psychology, (...) linguistics and philosophy about the role of questions in speech and thought, we propose a different model in which beliefs are treated as answers directed at specific questions. Here, questions are formally represented as partitions of the space of possibilities and individuals’ information states as sets of questions and corresponding partial answers to them. The state of conversation is then characterised by individuals’ information together with the questions under discussion, which can be steered by various deliberative inputs. Using this model, deliberation is then shown to shape collective decisions in ways that classical models cannot capture, allowing for novel explanations of how group consensus is achieved. (shrink)

Article Authors Metrics Comments Media Coverage Abstract Author Summary Introduction Results Discussion Supporting information Acknowledgments Author Contributions References Reader Comments (0) Media Coverage (0) Figures Abstract During language processing, humans form complex embedded representations from sequential inputs. Here, we ask whether a “geometrical language” with recursive embedding also underlies the human ability to encode sequences of spatial locations. We introduce a novel paradigm in which subjects are exposed to a sequence of spatial locations on an octagon, and are asked to (...) predict future locations. The sequences vary in complexity according to a well-defined language comprising elementary primitives and recursive rules. A detailed analysis of error patterns indicates that primitives of symmetry and rotation are spontaneously detected and used by adults, preschoolers, and adult members of an indigene group in the Amazon, the Munduruku, who have a restricted numerical and geometrical lexicon and limited access to schooling. Furthermore, subjects readily combine these geometrical primitives into hierarchically organized expressions. By evaluating a large set of such combinations, we obtained a first view of the language needed to account for the representation of visuospatial sequences in humans, and conclude that they encode visuospatial sequences by minimizing the complexity of the structured expressions that capture them. (shrink)