We show that plane hyperbolic geometry, expressed in terms of points and the ternary relation of collinearity alone, cannot be expressed by means of axioms of complexity at most ∀∃∀, but that there is an axiom system, all of whose axioms are ∀∃∀∃ sentences. This remains true for Klingenberg's generalized hyperbolic planes, with arbitrary ordered fields as coordinate fields.

Three things are presented: How Hilbert changed the original construction postulates of his geometry into existential axioms; In what sense he formalized geometry; How elementary geometry is formalized to present day's standards.

Three things are presented: How Hilbert changed the original construction postulates of his geometry into existential axioms; In what sense he formalized geometry; How elementary geometry is formalized to present day's standards.

Three things are presented: How Hilbert changed the original construction postulates of his geometry into existential axioms; In what sense he formalized geometry; How elementary geometry is formalized to present day's standards.

Elementary geometry can be axiomatized constructively by taking as primitive the concepts of the apartness of a point from a line and the convergence of two lines, instead of incidence and parallelism as in the classical axiomatizations. I first give the axioms of a general plane geometry of apartness and convergence. Constructive projective geometry is obtained by adding the principle that any two distinct lines converge, and affine geometry by adding a parallel line construction, etc. (...) Constructive axiomatization allows solutions to geometric problems to be effected as computer programs. I present a formalization of the axiomatization in type theory. This formalization works directly as a computer implementation of geometry. (shrink)

ABSTRACT A spatial logic is a modal logic of which the models are the mathematical models of space. Successively considering the mathematical models of space that are the incidence geometry and the projective geometry, we will successively establish the language, the semantical basis, the axiomatical presentation, the proof of the decidability and the proof of the completeness of INC, the modal multilogic of incidence geometry, and PRO, the modal multilogic of projective geometry.

The uniqueness of the parallel lines is independent from the analogous statement on parallel planes and the usual further axioms of three-dimensional affine geometry. MSC: 51A15, 03F65.

Hyperbolic geometry can be axiomatized using the notions of order andcongruence (as in Euclidean geometry) or using the notion of incidencealone (as in projective geometry). Although the incidence-based axiomatizationmay be considered simpler because it uses the single binary point-linerelation of incidence as a primitive notion, we show that it issyntactically more complex. The incidence-based formulation requires some axioms of the quantifier-type forallexistsforall, while the axiom system based on congruence and order can beformulated using only (...) forallexists-axioms. (shrink)

Hyperbolic geometry can be axiomatized using the notions of order andcongruence (as in Euclidean geometry) or using the notion of incidencealone (as in projective geometry). Although the incidence-based axiomatizationmay be considered simpler because it uses the single binary point-linerelation of incidence as a primitive notion, we show that it issyntactically more complex. The incidence-based formulation requires some axioms of the quantifier-type \forall\exists\forall, while the axiom system based on congruence and order can beformulated using only (...) \forall\exists-axioms. (shrink)

Thomas Reid (1710–1796) presented a two-dimensional geometry of the visual field in his Inquiry into the human mind (1764), whose axioms are different from those of Euclidean plane geometry. Reid’s ‘geometry of visibles’ is the same as the geometry of the surface of the sphere, described without reference to points and lines outside the surface itself. Interpreters of Reid seem to be divided in evaluating the significance of his geometry of visibles in the history of (...) the discovery of non-Euclidean geometries. The question will be reexamined with particular attention given to his unpublished manuscripts. These include comments on Saccheri’s work and Reid’s repeated attempts to derive Euclid’s parallel postulate from the axioms of incidence. (shrink)

This is a reprinting of Ken Pledger’s PhD thesis, submitted to the University of Warsaw in 1980 with the degree awarded in 1981. It develops a one-sorted approach to the theory of plane geometry, based on the idea that the usually two-sorted theory “can be made one-sorted by keeping careful account of whether the incidence relation is iterated an even or odd number of times”.The one-sorted structures can also serve as Kripke frames for modal logics, and the thesis (...) defines and studies two such logics that are validated by projective planes and elliptic planes respectively. It raises questions of logical completeness for these systems that are addressed in the first article of this journal issue. (shrink)

Generalizedn-gons are certain geometric structures (incidence geometries) that generalize the concept of projective planes (the nontrivial generalized 3-gons are exactly the projective planes).In a simplified world, every generalizedn-gon of finite Morley rank would be an algebraic one, i.e., one of the three families described in [9] for example. To our horror, John Baldwin [2], using methods discovered by Hrushovski [7], constructed ℵ1-categorical projective planes which are not algebraic. The projective planes that Baldwin constructed fail to be algebraic in a (...) dramatic way.Indeed, every algebraic projective plane over an algebraically closed field is Desarguesian [12]. In particular, an algebraically closed field (isomorphic to the base field) can be interpreted in every one of them. However, in the projective planes that Baldwin constructed, one cannot even interpret an infinite group.In this article we show that the same phenomenon occurs for the generalizedn-gons ifn≥ 3 is an odd integer. For each suchnwe constructmany nonisomorphic generalizedn-gons of finite Morley rank that do not interpret an infinite group. As one may expect, our method is inspired by Hrushovski and Baldwin, and we follow Baldwin's line of approach. Quite often our proofs are a verification of the fact that the proofs of Baldwin [2] forn= 3 carry over to an arbitrary positive odd integern(which is sometimes far from being obvious). As in [2], we begin by defining a certain collection of finite graphsK* and a binary relation ≤ on these graphs. We show that (K*, ≤) satisfies the amalgamation property. (shrink)

The present article provides a mereological analysis of Euclid’s planar geometry as presented in the first two books of his Elements. As a standard of comparison, a brief survey of the basic concepts of planar geometry formulated in a set-theoretic framework is given in Section 2. Section 3.2, then, develops the theories of incidence and order using a blend of mereology and convex geometry. Section 3.3 explains Euclid’s “megethology”, i.e., his theory of magnitudes. In Euclid’s system (...) of geometry, megethology takes over the role played by the theory of congruence in modern accounts of geometry. Mereology and megethology are connected by Euclid’s Axiom 5: “The whole is greater than the part.” Section 4 compares Euclid’s theory of polygonal area, based on his “Whole-Greater-Than-Part” principle, to the account provided by Hilbert in his Grundlagen der Geometrie. An hypothesis is set forth why modern treatments of geometry abandon Euclid’s Axiom 5. Finally, in Section 5, the adequacy of atomistic mereology as a framework for a formal reconstruction of Euclid’s system of geometry is discussed. (shrink)

ZusammenfassungDas Axiomensystem der ebenen projektiven Geometrie ist dual in dem Sinne, dass es bei Vertauschung der Begriffe « Punkt » und « Gerade » in sich übergeht. Daraus folgt, dass mit jedem Satz auch der duale Satz aus den Axiomen beweisbar ist. Dagegen kann aus der Dualität des Axiomensystems nicht geschlossen werden, dass in einem Modell mit jedem Satz auch der duale Satz gilt; noch weniger folgt, dass ein Modell eine eineindeutige Abbildung zulässt, welche Punkte and Geraden unter Erhaltung der (...) Inzidenz vertauscht. Im Falle der projektiven Geometrie sind solche Modelle bekannt; für die einfache Typentheorie wird gezeigt, dassa die Existenz solcher Modelle äquivalent ist mit der Widerspruchsfreiheit des Systems « New Foundations ».Zusatz. Der folgende Satz beantwortet die beiden in der Arbeit aufgeworfenen Fragen: Eine vollständige Theorie mit einem Automorphismus hat ein Modell, welches einen entsprechenden Automorphismus zulässt. Insbesondere ist somit NF widerspruchsfrei, wenn die einfache Typentheorie mit den zusätzlichen Axiomen S ≡ S* widerspruchsfrei ist.RésuméLe système d'axiomes de la géométrie projective plane est dualistique dans le sens qu'il se transforme en lui‐même si I'on y échange les notions « points » et « droite ». Cette dualité entraîne un parallélisme des théorémes et de leurs démonstrations. Par contre, la dualité d'un système d'axiomes ne nous permet pas de conclure au parallélisme des énoncés qui soient vrais dans un modéle donné du systéme; moins encore peut‐on insérer I'existence d'une transformation biunivoque échangeant points et droites et conservant la relation d'incidence. Pour la géométric projective, de tels modèles sont bien connus; pour la théorie simple des types, I'existence d'un tel modèle est équivalent a la cohérence du système « New Foundations ».Remarque. Le théoréme suivant résout les deux problèmes proposés dans le travail: Toute théorie complète admettant un automorphisme possède un modèle admettant un automorphisme correspondant. NF est par conséquent cohérent si la théorie simple des types avec les axiomes supplémentaires S ≡ S * est cohérente.The axiom system of plane projective geometry is dual in the sense that it is transformed into itself by exchange of the notions « point » and « line.» It follows that for every theorem the dual sentence is also a theorem. However, from the duality of the axiom system one cannot conclude that in a model the truth of a sentence implies that of the dual sentence; even less can one conclude that each model admits a 1‐1‐transformation interchanging points and lines and preserving the incidence relation. For projective geometry, models of this kind are well known. For the simple theory of types it is shown that the existence of such models is equivalent to the consistency of « New Foundations. »Additional remark. The following theorem answers both of the questions proposed in the paper: If it is complete, then a theory with an automorphism has a model with a corresponding automorphism. NF is therefore consistent if simple theory of type with the additional axioms S ≡ S* is consistent. (shrink)

In lieu of an abstract, here is a brief excerpt of the content:Journal of the History of Philosophy 42.2 (2004) 234-235 [Access article in PDF] Leonard Lawlor. Derrida and Husserl: The Basic Problem of Phenomenology. Bloomington: Indiana University Press, 2002. Pp. xii + 286. Paper, $19.95. Ever since Derrida began producing his interpretive critical studies on the giant figures of Husserl and Heidegger, a book of the kind Lawlor offers has been needed. Framing his study by drawing directly from Derrida's (...) texts themselves, Lawlor takes up in detail Derrida's interpretation and critique of Husserl's phenomenology to show how he moves beyond the framework he identifies as Husserl's in developing his "deconstruction."In the course of a study thus primarily internal to Derrida's texts, Lawlor provides one of the greatest strengths of the book, his fine treatment of the way the work of French thinkers subsequent to Husserl helped Derrida develop the distinctive enterprise of "deconstruction": specifically, Jean Cavaillès, Tran-Duc-Thao (ch. 4, parts 1-2), Jean Hyppolyte (ch. 5, part 1), and Emmanuel Levinas (chs. 6-8). Determinative prior to that, however, is the influence Lawlor shows stemming from Eugen Fink's published writings (chapter 1). Impressive as well throughout (though his "internalist" restriction also shows serious faults—mentioned later) is the exegetical subtlety Lawlor exercises upon Derrida's writings, those concentrating on Husserl from 1953 to 1967, together with one on Lévinas ("Violence et métaphysique," 1964/1967) and three later ones no longer dealing with Husserl (principally "Les fins de l'homme," 1972, De l'esprit, 1987, on Heidegger, and "Spectres de Marx," 1993). It is a choice of texts that is eminently justifiable, even if the necessary inclusion of the latter introduces a dimension that needs to be contrasted more incisively with the way the Husserl-centered material is focused.Lawlor is excellent in showing a shift in interpretive focus and character taking place as Derrida moves from his treatment of Husserl in Le problème de la genèse dans la philosophie de Husserl (1953-54) through his "Introduction" to Husserl's Origine de la géométrie (1962) to La voix et le phénomène (1967) (chs. 4-6). Derrida's interpretive critique of Husserl thus begins with the primacy of the problem of "genesis," of "origin," then moves in transition through the second book (via the notable interlude on Lévinas, ch. 6) to the decisive break from Husserl in the third (ch. 7). With this last, La voix et le phénomène, Lawlor writes, "[t]he problem of the sign has come to replace, for Derrida, the problem of genesis" (166). Here instead of the transcendental power of origination being the intuition-effecting subjectivity of Husserl's Ideas I, a double replacement occurs by which a) "the voice" becomes for Derrida the "ultra-transcendental," and b) its constitutive dynamic, "temporality," is cast as a movement of pure "auto-affection" in a process called "archi-writing," unceasingly generating a supply of element upon element in a "supplement"-driven articulation of meaning (192-97)—in sum, différance as co-incident "differentiation" and "deferral" (196-205). This is the burden of the section (chs. 6 and 7) entitled "The End of Phenomenology and Ontology," closing a debate between Husserl—"phenomenology"—and Heidegger—"ontology." With this the new, post-phenomenological, post-ontological effort begins, the era in which Derrida's philosophic work, now after the definitive "deconstruction of metaphysics," gradually moves ethico-political issues into the foreground, though "[d]ifférance and supplementarity" continue as basic concepts (211-12). Here no foundational grounding underlies clarifications in the region of meaning and value, or rather a "non-foundation" subtends any gaining of insight, radically qualifying its claims. One's efforts deploy in ever-renewed consideration of issues guided by something "deeper than being," by a "non-foundational," "non-ontological" "aporia" "experienced" in deconstructive ultimacy as "the unheard-of question" (234).About all this, some reservations have to be raised. First of all, it is quite clear that in fact the question of "genesis" remains; the whole... (shrink)

El ensayo trata de presentar la fenomenología de los actos racionales, actos que son fundamentales, en todo caso, para la ciencia. En la primera parte se expone la fenomenología de la razón tal como aparece en la cuarta sección del libro de Husserl Ideas I. En la segunda parte, teniendo a la vista El origen de la geometría, se expone esa fenomenología de la razón aplicada a la explicación de la ciencia histórica, superando el relativismo en que se ve envuelta, (...) superación que también vale para la antropología cultural. Si en la primera parte se desvela la razón como legitimidad de lo que se da en sí mismo, en la segunda se expone la importante elucidación del apriori de la historia con el que cuenta el historiador y frente al cual nada puede el relativismo. Como conclusión se explica de qué manera incide la fenomenología de la razón de la primera parte en las ciencias humanas.The essay tries to present the phenomenology of rational acts, acts that are fundamental, in any case, for science. In the first part the phenomenology of reason is exposed as it appears in the fourth section of Husserl's book Ideas I. In the second part, considering The Origin of Geometry, this phenomenology of reason applied to explanation of historical science, overcoming the relativism in which it is involved, an overcoming that also applies to cultural anthropology. If in the first part reason is revealed as the legitimacy of what is given in itself, in the second part the important elucidation of the apriori of history is exposed with which the historian must count and against which relativism can do nothing. As a conclusion, it is explained how the phenomenology of reason of the first part affects the human sciences. (shrink)

Frequency selective surface -based intelligent spatial filters are capturing the eyes of the researchers by offering a dynamic behavior when exposed to the electromagnetic radiations. In this manuscript, a concept of creating complementary structures which stems from Babinet’s principle is illustrated. A hybrid complementary pair of FSS comprising double square loop FSS and double square slot FSS on either side of the dielectric substrate is proposed. DSLFSS offers band-pass behavior and can be placed as a superstrate, whereas DSSFSS behaves as (...) a band-stop intelligent spatial filter that blocks the radiations falling on it, thus making them applicable for use as a substrate. The technique utilized for analyzing DSLFSS and DSSFSS structures is based on the equivalent circuit modeling and transmission line methodology. The CPFSS structure offers the design simplicity, hence, suitable for placing them with the printed patch antenna radiators in wireless networking devices operating in sub-6 GHz 5G spectrum. DSLFSS offers band-pass behavior ranging from 2.99 to 5.56 GHz, whereas DSSFSS offers band-stop behavior ranging from 2.85 to 5.42 GHz covering all n77, n78, and n79 bands of FR1 spectrum of sub-6 GHz 5G range. The passband and the stopband offered by the two structures of CPFSS geometry are stable to oblique angles of incidence and the proposed design also offers polarization-independent behavior. The thickness of the dielectric region existing within the pair of designed structures is critical for the location of the passbands and the stopbands. The impact of the overall thickness of the dielectric substrate on the passbands and stopbands is also reported in this article. (shrink)

Ken Pledger devised a one-sorted approach to the incidence relation of plane geometries, using structures that also support models of propositional modal logic. He introduced a modal system 12g that is valid in one-sorted projective planes, proved that it has finitely many non-equivalent modalities, and identified all possible modality patterns of its extensions. One of these extensions 8f is valid in elliptic planes. These results were presented in his 1980 doctoral dissertation, which is reprinted in this issue of the (...) Australasian Journal of Logic. Here we show that 12g and 8f are strongly complete for validity in their intended one-sorted geometrical interpretations, and have the finite model property. The proofs apply standard technology of modal logic together with a step-by-step procedure introduced by Yde Venema for constructing two-sorted projective planes. (shrink)

L’article vise à établir un lien entre les travaux de Russell sur les fondements de la géométrie et ses célèbres textes des années 1911-1914 sur la perception, la matière et les sense-data. Nous insistons d’abord sur le fait que la notion russellienne de donnée sensorielle n’est pas phénoménaliste : les sense-data des Problèmes de Philosophie sont des objets extérieurs aussi peu mentaux que les corps matériels. L’accent est ensuite mis sur l’article On Matter : Russell y introduit pour la première (...) fois l’idée que les sense-data n’ont pas une, mais deux positions — une position où ils sont vus et une position d’où ils sont vus. Cette manœuvre permet à Russell de considérer la donnée sensorielle comme le point d’intersection neutre de deux ensembles fondamentalement différents : une série « physique », matérielle, et une série de « perspectives », mentales. L’idée qu’une même chose peut être classée sous différentes catégories, donc que le mode de classification aristotélicien en espèce et en genre n’est pas le seul concevable, est à la fois la clé de voûte de la théorie des relations entre « sense-data » et « physical things » de 1914 et le nerf du ralliement ultérieur au « monisme neutre ». Nous défendons, dans une troisième partie, que ce canevas n’est lui-même qu’un lointain écho de la conception de la géométrie projective définie comme théorie générale des relations d’incidence développée par Russell dans The Principles of Mathematics (1903). (shrink)

L’article vise à établir un lien entre les travaux de Russell sur les fondements de la géométrie et ses célèbres textes des années 1911-1914 sur la perception, la matière et les sense-data. Nous insistons d’abord sur le fait que la notion russellienne de donnée sensorielle n’est pas phénoménaliste : les sense-data des Problèmes de Philosophie sont des objets extérieurs aussi peu mentaux que les corps matériels. L’accent est ensuite mis sur l’article On Matter : Russell y introduit pour la première (...) fois l’idée que les sense-data n’ont pas une, mais deux positions — une position où ils sont vus et une position d’où ils sont vus. Cette manœuvre permet à Russell de considérer la donnée sensorielle comme le point d’intersection neutre de deux ensembles fondamentalement différents : une série « physique », matérielle, et une série de « perspectives », mentales. L’idée qu’une même chose peut être classée sous différentes catégories, donc que le mode de classification aristotélicien en espèce et en genre n’est pas le seul concevable, est à la fois la clé de voûte de la théorie des relations entre « sense-data » et « physical things » de 1914 et le nerf du ralliement ultérieur au « monisme neutre ». Nous défendons, dans une troisième partie, que ce canevas n’est lui-même qu’un lointain écho de la conception de la géométrie projective définie comme théorie générale des relations d’incidence développée par Russell dans The Principles of Mathematics (1903). (shrink)

Next SectionBackground Scientific papers are retracted for many reasons including fraud (data fabrication or falsification) or error (plagiarism, scientific mistake, ethical problems). Growing attention to fraud in the lay press suggests that the incidence of fraud is increasing. Methods The reasons for retracting 742 English language research papers retracted from the PubMed database between 2000 and 2010 were evaluated. Reasons for retraction were initially dichotomised as fraud or error and then analysed to determine specific reasons for retraction. Results Error (...) was more common than fraud (73.5% of papers were retracted for error (or an undisclosed reason) vs 26.6% retracted for fraud). Eight reasons for retraction were identified; the most common reason was scientific mistake in 234 papers (31.5%), but 134 papers (18.1%) were retracted for ambiguous reasons. Fabrication (including data plagiarism) was more common than text plagiarism. Total papers retracted per year have increased sharply over the decade (r=0.96; p<0.001), as have retractions specifically for fraud (r=0.89; p<0.001). Journals now reach farther back in time to retract, both for fraud (r=0.87; p<0.001) and for scientific mistakes (r=0.95; p<0.001). Journals often fail to alert the naïve reader; 31.8% of retracted papers were not noted as retracted in any way. Conclusions Levels of misconduct appear to be higher than in the past. This may reflect either a real increase in the incidence of fraud or a greater effort on the part of journals to police the literature. However, research bias is rarely cited as a reason for retraction. (shrink)

Traditional geometry concerns itself with planimetric and stereometric considerations, which are at the root of the division between plane and solid geometry. To raise the issue of the relation between these two areas brings with it a host of different problems that pertain to mathematical practice, epistemology, semantics, ontology, methodology, and logic. In addition, issues of psychology and pedagogy are also important here. To our knowledge there is no single contribution that studies in detail even one of the (...) aforementioned areas. (shrink)

We investigate two under-explored factors in mitigating the risk of corporate fraud and regulatory enforcement against fraud, namely institutional investors and political connections. The role of institutional investors in the effective monitoring of a firm’s management is well established in the literature. We further observe that firms that have a large proportion of their shares held by institutional investors have a lower incidence of enforcement actions against corporate fraud. The importance of political connections for enterprises, whether in a developed (...) market such as the United States or an emerging market such as China, has been established by previous studies. However, we find evidence of another positive effect of political connections: they may reduce the incidence of enforcement action against corporate fraud. We also find that political connections play a more significant role in reducing regulatory enforcement incidents against non-state-owned enterprises and firms in weaker legal environments, whereas institutional ownership plays a more important role in reducing regulatory enforcement incidents against state-owned enterprises. (shrink)

Derrida's introduction to his French translation of Husserl's essay "The Origin of Geometry," arguing that although Husserl privileges speech over writing in an account of meaning and the development of scientific knowledge, this privilege is in fact unstable.

§30. Significance of Desargues's theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64 CHAPTER VI. PASCAL'S THEOREM. §31. ...

If there is an area of discourse in which disagreement is virtually absent, it is mathematics. After all, mathematicians justify their claims with deductive proofs: arguments that entail their conclusions. But is mathematics really exceptional in this respect? Looking at the history and practice of mathematics, we soon realize that it is not. First, deductive arguments must start somewhere. How should we choose the starting points (i.e., the axioms)? Second, mathematicians, like the rest of us, are fallible. Their ability to (...) recognize whether a putative proof is correct is not infallible. In most cases, disagreement over the correctness of a putative proof is, however, evanescent. Once an error is spotted and communicated, the disagreement disappears. But this is not always the case. Sometimes it is recalcitrant; that is, it persists over time. In order to zoom in on this type of disagreement and explain its very possibility, we focus on a single case study: a decades-long (1921-1949) controversy between Federigo Enriques and Francesco Severi, two prominent exponents of the Italian school of algebraic geometry. We suggest that the instability of the mathematical community to which they belonged can be explained by the gap between an abstract criterion of rigor and local criteria of acceptability. It is this instability that made the existence of recalcitrant disagreement over putative proofs possible. We do not condemn speculative mathematics but rather its pretense of being rigorous mathematics. In this respect, we show that the overly self-confident Severi and the more intuitive, visionary Enriques had a completely different attitude. (shrink)

Graduate-level textbook in general relativity.

Edmund Husserl has argued that we can intuit essences and, moreover, that it is possible to formulate a method for intuiting essences. Husserl calls this method ‘ideation’. In this paper I bring a fresh perspective to bear on these claims by illustrating them in connection with some examples from modern pure geometry. I follow Husserl in describing geometric essences as invariants through different types of free variations and I then link this to the mapping out of geometric invariants in (...) modern mathematics. This view leads naturally to different types of spatial ontologies and it can be used to shed light on Husserl's general claim that there are different ontologies in the eidetic sciences that can be systematically related to one another. The paper is rounded out with a consideration of the role of ideation in the origins of modern geometry, and with a brief discussion of the use of ideation outside of pure geometry. What would be the study that would draw the soul away from the world of becoming to the world of being?… Geometry and arithmetic would be among the studies we are seeking … a philosopher must learn them because he must arise out of the region of generation and lay hold on essence or he can never become a true reckoner … they facilitate the conversion of the soul itself from the world of generation to essence and truth… they are knowledge of that which always is and not of something which at some time comes into being and passes away. (shrink)

This paper examines with the use of aggregate data from the U.S. Department of Justicethe extent of contemporary white-collar crime as a consequence of multiple normative standards existing within corporations. Given the implications of globalization, the desire for increased profits, and the declining role of regulatory agencies across much of the world (save for Europe, Japan, Mexico and India), paper suggests that the incidence of corporate deviance is likely to increase in the foreseeable future.

This paper assumes the success of arguments against the view that informal mathematical proofs secure rational conviction in virtue of their relations with corresponding formal derivations. This assumption entails a need for an alternative account of the logic of informal mathematical proofs. Following examination of case studies by Manders, De Toffoli and Giardino, Leitgeb, Feferman and others, this paper proposes a framework for analysing those informal proofs that appeal to the perception or modification of diagrams or to the inspection or (...) imaginative manipulation of mental models of mathematical phenomena. Proofs relying on diagrams can be rigorous if it is easy to draw a diagram that shares or otherwise indicates the structure of the mathematical object, the information thus displayed is not metrical and it is possible to put the inferences into systematic mathematical relation with other mathematical inferential practices. Proofs that appeal to mental models can be rigorous if the mental models can be externalised as diagrammatic practice that satisfies these three conditions. (shrink)

The “origins” of (geometric) space is examined from the perspective of the so-called “conceptual space” or “semantic space”. Semantic space is characterized by its fundamental “locality” that generates an “implicit” mode of geometrizing. This view is examined from within three perspectives. First, the role that various diagrammatic entities play in the everyday life and pragmatic activities of selected ethnic groups is illustrated. Secondly, it is shown how conceptual spaces are fundamentally linked to the meaning effects of particular natural languages and (...) these are very different from the global and universal aspects of Euclidean spaces. Thirdly, it is contended that these modes of creating body and culture-based spatial frameworks and related cosmogonies and cosmologies can be described as forms of “latent geometry” that initially appear unexplainable in any rational way. Nonetheless, and thanks to the deep mathematical reflections provided by René Thom, it is illustrated how the various ways of generating space can be further analyzed as distortions of mainstream spatialization furnished by Euclidean geometry that established the dominant universality of the ideas of space (and time). (shrink)

The paper describes an algorithm for semantic representation of behavioral contexts relative to a dichotomic decision alternative. The contexts are represented as quantum qubit states in two-dimensional Hilbert space visualized as points on the Bloch sphere. The azimuthal coordinate of this sphere functions as a one-dimensional semantic space in which the contexts are accommodated according to their subjective relevance to the considered uncertainty. The contexts are processed in triples defined by knowledge of a subject about a binary situational factor. The (...) obtained triads of context representations function as stable cognitive structure at the same time allowing a subject to model probabilistically-variative behavior. The developed algorithm illustrates an approach for quantitative subjectively-semantic modeling of behavior based on conceptual and mathematical apparatus of quantum theory. (shrink)

Cognitive biases play an important role in creating and perpetuating problems that lead to foodborne illness outbreaks. By using insights from behavioral ethics, we argue that sometimes people engage in unethical behavior that increases the likelihood of foodborne illness outbreaks without necessarily intending to or being consciously aware of it. We demonstrate these insights in an analysis of the 2011 Listeriosis outbreak in the U.S. from the consumption of contaminated cantaloupes. We then provide policy implications that can improve our understanding (...) of other kinds of disease outbreaks and epidemics. (shrink)

The standard coordinate-based formulation of the space-time theory of special relativity (Minkowski geometry) is philosophically unsatisfactory for various reasons. We here present an explicit axiomatic formulation of that theory in terms of primitives with a definitive physical interpretation, prove its equivalence to the standard coordinate formulation, and draw various philosophical conclusions concerning the physical content and assumptions of the space-time theory. The prevalent causal interpretation of physical Minkowski geometry deriving from Reichenbach is criticised on the basis of the (...) present formulation. (shrink)

Brown and Pooley’s ‘dynamical approach’ to physical theories asserts, in opposition to the orthodox position on physical geometry, that facts about physical geometry are grounded in, or explained by, facts about dynamical fields, not the other way round. John Norton has claimed that the proponent of the dynamical approach is illicitly committed to spatiotemporal presumptions in ‘constructing’ space-time from facts about dynamical symmetries. In this article, I present an abstract, algebraic formulation of field theories and demonstrate that the (...) proponent of the dynamical approach is not committed, in special relativity, to the illicit presumptions to which Norton refers. (shrink)

Minkowski geometry is axiomatized in terms of the asymmetric binary relation of optical connectibility, using ten first-order axioms and the second-order continuity axiom. An axiom system in terms of the symmetric binary optical connection relation is also presented. The present development is much simpler than the corresponding work of Robb, upon which it is modeled.

In the traditional formalism of quantum mechanics, a simple direct proof of the Spin Geometry Theorem of Penrose is given; and the structure of a model of the ‘space of the quantum directions’, defined in terms of elementary SU-invariant observables of the quantum mechanical systems, is sketched.