We present an approach in which ancient Greek mathematical proofs by Hippocrates of Chios and Euclid are addressed as a form of (guided) intentional reasoning. Schematically, in a proof, we start with a sentence that works as a premise; this sentence is followed by another, the conclusion of what we might take to be an inferential step. That goes on until the last conclusion is reached. Guided by the text, we go through small inferential steps; in each one, we go (...) through an autonomous reasoning process linking the premise to the conclusion. The reasoning process is accompanied by a metareasoning process. Metareasoning gives rise to a feeling-knowing of correctness. In each step/cycle of the proof, we have a feeling-knowing of correctness. Overall, we reach a feeling of correctness for the whole proof. We suggest that this approach allows us to address the issues of how a proof functions, for us, as an enabler to ascertain the correctness of its argument and how we ascertain this correctness. (shrink)

Une géodésie spatio-temporelle pervertie résulte des notions de règles et d’horloges variables, qui sont prises pour avoir leur longueur et leur rythme affectés par le champ gravitationnel. D’autre part ce que nous pourrions appeler une géodésie concrète repose sur les notions de règles et d’horloges invariables de mesure d’unité. En fait, il s’agit d’une hypothèse de base de la relativité générale. Les règles et les horloges variables conduisent à une géodésie pervertie dans le sens où un espace-temps courbe pourrait être (...) vu comme provenant du départ de l’espace-temps Minkowskien comme un effet du champ gravitationnel sur le rythme d’horloges et la longueur des règles. Dans le cas d’une géodésie concrète, nous avons « directement » un espace-temps courbe dont la courbure peut être déterminée en utilisant des règles et des horloges de mesure d’unité (invariables). Dans cet article, nous défendrons la plausibilité que le point de vue d’Einstein sur la géométrie par rapport à la relativité générale est imprégné par une géodésie pervertie. (shrink)

in this paper we return to Marshall Clagett’s view about the existence of an ancient Greek geometry of motion. It can be read in two ways. As a basic presentation of ancient Greek geometry of motion, followed by some aspects of its further development in landmark works by Galileo and Newton. Conversely, it can be read as a basic presentation of aspects of Galileo’s and Newton’s mathematics that can be considered as developments of a geometry of motion that was first (...) conceived by ancient Greek mathematicians. (shrink)

The purpose of this work is to address the relation existing between ancient Greek practical geometry and ancient Greek pure geometry. In the first part of the work, we will consider practical and pure geometry and how pure geometry can be seen, in some respects, as arising from an idealization of practical geometry. From an analysis of relevant extant texts, we will make explicit the idealizations at play in pure geometry in relation to practical geometry, some of which are basically (...) explicit in definitions, like that of segments in Euclid’s Elements. Then, we will address how in pure geometry we, so to speak, “refer back” to practical geometry. This occurs in two ways. One, in the propositions of pure geometry. The other, when applying pure geometry. In this case, geometrical objects can represent practical figures like, e.g., a practical segment. (shrink)

It has been argued in relation to Old Babylonian mathematical procedure texts that their validity or correctness is self-evident. One “sees” that the procedure is correct without it having, or being accompanied by, any explicit arguments for the correctness of the procedure. Even when agreeing with this view, one might still ask about how is the correctness of a procedure articulated? In this work, we present an articulation of the correctness of ancient Egyptian and Old Babylonian mathematical procedure texts – (...) mathematical texts presenting the solution of problems. We endeavor to make explicit and explain how and why the procedures are reliable over and above the fact that their correctness is intuitive. (shrink)

Einstein's gravitational redshift derivation in his famous 1916 paper on general relativity seems to be problematic, being mired in what looks like conceptual difficulties or at least contradictions or gaps in his exposition. Was this derivation a blunder? To answer this question, we will consider Einstein’s redshift derivations from his first one in 1907 to the 1921 derivation made in his Princeton lectures on relativity. This will enable to see the unfolding of an interdependent network of concepts and heuristic derivations (...) in which previous ideas inform and condition later developments. The resulting derivations and views on coordinates and clocks are in fact not without inconsistencies. However, we can see these difficulties as an aspect of an evolving network understood as a “work in progress”. (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)

When adopting a sound logical system, reasonings made within this system are correct. The situation with reasonings expressed, at least in part, with natural language is much more ambiguous. One way to be certain of the correctness of these reasonings is to provide a logical model of them. To conclude that a reasoning process is correct we need the logical model to be faithful to the reasoning. In this case, the reasoning inherits, so to speak, the correctness of the logical (...) model. There is a weak link in this procedure, which we call the faithfulness problem: how do we decide that the logical model is faithful to the reasoning that it is supposed to model? That is an issue external to logic, and we do not have rigorous formal methods to make the decision. The purpose of this paper is to expose the faithfulness problem (not to solve it). For that purpose, we will consider two examples, one from the geometrical reasoning in Euclid’s Elements and the other from a study on deductive reasoning in the psychology of reasoning. (shrink)

A re-evaluation of the notion of vacuum in quantum electrodynamics is presented, focusing on the vacuum of the quantized electromagnetic field. In contrast to the ‘nothingness’ associated to the idea of classical vacuum, subtle aspects are found in relation to the vacuum of the quantized electromagnetic field both at theoretical and experimental levels. These are not the usually called vacuum effects. The view defended here is that the so-called vacuum effects are not due to the ground state of the quantized (...) electromagnetic field. Nevertheless it is possible to maintain an empirically demonstrable notion of vacuum state that is consistent with the interpretation of the formalism of the theory. (shrink)

Philosophy of Physics has emerged recently as a scholarly important subfield of philosophy of science. However outside the small community of experts it is not a well-known field. It is not clear even to experts the exact nature of the field: how much philosophical is it? What is its relation to physics? In this work it is presented an overview of philosophy of physics that tries to answer these and other questions.

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.

In this work, we consider the views of three exponents of major areas of linguistics – Levelt (psycholinguistics), Jackendoff (theoretical linguistics), and Gil (field linguistics) – regarding the issue of the universality or not of the conceptual structure of languages. In Levelt’s view, during language production, the conceptual structure of the preverbal message is language-specific. In Jackendoff’s theoretical approach to language – his parallel architecture –, there is a universal conceptual structure shared by all languages, in contradiction to Levelt’s view. (...) In Gil’s work on Riau Indonesian, he proposes a conceptual structure that is quite different from that of English, adopted by Jackendoff as universal. We find no reason to disagree with Gil’s view. In this way, we take Gil’s work as vindicating Levelt’s view that during language production preverbal messages are encoded with different conceptual structures for different languages. (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)

This essay exemplifies a cognitive approach to literary and film studies, with particular emphasis on fictional reimagining of legal institutions. It draws on research of cognitive scientists who study metacognition—specifically, the difference between reflective and intuitive beliefs—to suggest that courtroom dramas, such as Billy Wilder’s Witness for the Prosecution (1957), can manipulate their viewers into believing something that they, on some level, know cannot be true. In this case, viewers accept the not guilty verdict by the jury even though “the (...) facts in the case” are “simple” and point to the guilt of the defendant. I show how different versions of Witness for the Prosecution—from Agatha Christie’s original short story (1925) and her subsequent play based on that story (1953), to the Hollywood film (1957), and the BBC mini-series (2016)—trigger our reflective beliefs about the defendant’s innocence, so that we find the verdict satisfying (at least for a short while). I conclude by considering literary payoffs of this manipulation, especially in the context of a culture that subscribes to a view of the mind as, in principle, knowable, and thus readable by skillful others. (shrink)

I hold profound respect for all individuals, regardless of gender, race, sexual orientation, or any other characteristic, advocating strongly for equality and equity worldwide. Throughout my career as a scientist, I have demonstrated this respect, including for female students and colleagues, supporting them with recommendations and accolades. I have always cherished diversity in my academic and professional circles and I hugely value the contributions of women like Marie Curie and Maryam Mirzakhani, as well as authors such as Agatha Christie (...) and J.K. Rowling. In my personal life, I express my love and admiration for my girlfriend through poetry and gestures, while also honoring my parents, especially my mother, who instilled in me a deep respect for all individuals. Raised in a family that values and respects women, I have been positively impacted by my mother, a retired teacher, and my father, a novelist who often portrays the life of Turkmen women in his works. Overall, my commitment to respecting and valuing all individuals, regardless of their background or identity, remains unwavering. (shrink)

Le Père Ignace Carbonnelle, l'un des principaux fondateurs de la Société scientifique de Bruxelles en 1875 et son secrétaire général depuis cette époque, décède inopinément en 1889 après une quin­zaine d'années durant lesquelles il fut «l'homme fort» de ladite Société. Aussitôt, la Revue des questions scienti­fiques annonce la triste nouvelle, promettant, pour un prochain numéro, une étude détaillée de sa vie et de son œuvre. Elle ne paraîtra jamais, de sorte que sa mort ne fut pas saluée avec l'ampleur qu'on (...) était en droit d'attendre. Et pour cause ! Au ter­me d'une enquête digne d'un roman policier, cette étude révèle que Rome, agacée par l'atomisme de Carbonnelle, profita de sa mort pour rappeler à l'ordre la Société en l'invitant à marcher dans les pas de l'Aquinate. En réponse à cette pressante invi­tation et par un excès de zèle non requis, la Société élut comme président le célèbre thomiste français Edmond Domet de Vor­ges, cependant que le mathématicien Paul Mansion s'attacha d'établir, à partir des publications de Pierre Duhem prônant un retour à une physique des qualités, que la Société se conformait bien, mais à sa manière, aux injonctions romaines. ––– Fr. Ignace Carbonnelle, who founded the Brussels Scientific Society in 1875 and was from this date onwards her Secretary General, passed away suddenly in 1889, after fifteen years during which he was the leading figure of the aforementioned Society. “La Revue des Question scientifique” announced the sad news but promised that their next publication would include a detailed article on the great man and his works. Sadly this article never appeared, meaning that the passing of Fr. Carbonnelle was not marked with the importance which we might have otherwise expected. For what reason, you may well ask? After a detailed investigation, worthy of Agatha Christie herself, it would appear that Rome had been rather alarmed by the “atomism” present in Fr. Carbonnelle's reflections, and had taken the opportunity of his death to invite the Scientific Society to opinions more in line with those of St Thomas. In response to this firmly worded invitation, and with overzealous spontaneity, the Society elected the celebrated French Thomiste Edmond Domet de Vorges as their new President. At the same time the mathematician Paul Mansion relied on the publications of Pierre Duhem, which advocated a stronger adhesion to a physics of an object's qualities, to establish that the Society was, in fact, well aligned with the desires of Rome. (shrink)