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.

H. Tietze has proved algebraically that the geometry of uniquely determined ruler and compass constructions coincides with the geometry of ruler and set square constructions. We provide a new proof of this result via new universal axiom systems for Euclidean planes of characteristic ≠ 2 in languages containing only operation symbols.

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.

We provide a quantifier-free axiom system for plane hyperbolic geometry in a language containing only absolute geometrically meaningful ternary operations (in the sense that they have the same interpretation in Euclidean geometry as well). Each axiom contains at most 4 variables. It is known that there is no axiom system for plane hyperbolic consisting of only prenex 3-variable axioms. Changing one of the axioms, one obtains an axiom system for plane Euclidean geometry, expressed in the same language, all of whose (...) axioms are also at most 4-variable universal sentences. We also provide an axiom system for plane hyperbolic geometry in Tarski's language L B which might be the simplest possible one in that language. (shrink)

Problem 2 at the 56th International Mathematical Olympiad asks for all triples of positive integers for which ab−c, bc−a, and ca−b are all powers of 2. We show that this problem requires only a primitive form of arithmetic, going back to the Pythagoreans, which is the arithmetic of the even and the odd.

Reverse analyses of three proofs of the Sylvester-Gallai theorem lead to three different and incompatible axiom systems. In particular, we show that proofs respecting the purity of the method, using only notions considered to be part of the statement of the theorem to be proved, are not always the simplest, as they may require axioms which proofs using extraneous predicates do not rely upon.

We show that the first-order theory of a large class of plane geometries and the first-order theory of their groups of motions, understood both as groups with a unary predicate singling out line-reflections, and as groups acting on sets, are mutually inter-pretable.

Using the axiom system provided by Carsten Augat in [1], it is shown that the only 6-variable statement among the axioms of the axiom system for plane hyperbolic geometry (in Tarski's language L B =), we had provided in [3], is superfluous. The resulting axiom system is the simplest possible one, in the sense that each axiom is a statement in prenex form about at most 5 points, and there is no axiom system consisting entirely of at most 4-variable statements.

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.

In this paper we provide quantifier-free, constructive axiomatizations for 2-dimensional absolute, Euclidean, and hyperbolic geometry. The main novelty consists in the first-order languages in which the axiom systems are formulated.

In this paper we provide a quantifier-free, constructive axiomatization of metric-Euclidean and of rectangular planes . The languages in which the axiom systems are expressed contain three individual constants and two ternary operations. We also provide an axiom system in algorithmic logic for finite Euclidean planes, and for several minimal metric-Euclidean planes. The axiom systems proposed will be used in a sequel to this paper to provide ‘the simplest possible’ axiom systems for several fragments of plane Euclidean geometry.

We prove that, in the framework of ordered geometry, the inner form of the Pasch axiom does not imply its outer form . We also show that OP can be properly split into IP and the weak Pasch axiom.

This paper provides a definitive answer, based on considerations derived from first-order logic, to the question regarding the status of elementary geometry, whether elementary geometry can be reduced to algebra. The answer we arrive at is negative, and is based on a series of structural questions that can be asked only inside the geometric formal theory, as well as the consideration of reverse geometry, which is the art of finding minimal axiom systems strong enough to prove certain geometrical theorems, given (...) that there are no algebraic structures that one could associate with those minimal axiom systems. (shrink)

As with the development of several logical notions, it is shown that the concept of resource-consciousness, i. e. the concern over the number of times that a given sentence is used in the proof of another sentence, has its origin in the foundations of geometry, pre-dating its appearence in logical circles as BCK-logic or affine logic.

In this paper we provide quantifier-free, constructive axiomatizations for several fragments of plane Euclidean geometry over Euclidean fields, such that each axiom contains at most 4 variables. The languages in which they are expressed contain only at most ternary operations. In some precisely defined sense these axiomatizations are the simplest possible.

In this paper we provide a quantifier-free constructive axiomatization for Euclidean planes in a first-order language with only ternary operation symbols and three constant symbols . We also determine the algorithmic theories of some ‘naturally occurring’ plane geometries.

It is argued that Brouwer’s philosophy of mathematics makes perfect sense if viewed from an Eastern philosophical perspective, as a mathematics in what Erich Fromm called “the being mode of existence.” The difficulty Western philosophers have accepting its validity under Brouwer’s own justifications is that mathematics is one of the highest prized treasures of Western philosophy (those footnotes to Plato’s dialogues).

We provide a universal axiom system for plane hyperbolic geometry in a firstorder language with two sorts of individual variables, ‘points’ and ‘lines’ , containing three individual constants, A0, A1, A2, standing for three non-collinear points, two binary operation symbols, φ and ι, with φ = l to be interpreted as ‘[MATHEMATICAL SCRIPT SMALL L] is the line joining A and B’ , and ι = P to be interpreted as [MATHEMATICAL SCRIPT SMALL L]P is the point of intersection of (...) g and h , and two binary operation symbols, π1 and —2, with πi = g to be interpreted as ‘g is one of the two limiting paralle lines from P to [MATHEMATICAL SCRIPT SMALL L]. (shrink)

By rephrasing quantifier-free axioms as rules of derivation in sequent calculus, we show that the generalized Steiner–Lehmus theorem admits a direct proof in classical logic. This provides a partial answer to a question raised by Sylvester in 1852. We also present some comments on possible intuitionistic approaches.

We provide the syntactic equivalent for the theorem stating that all epimorphisms of finite projective planes are isomorphisms. The definition of the inequality relation that we provide adds little to our understanding of the theorem, since its very validity can be discerned only from the validity of the model-theoretic theorem regarding epimorphisms.

In 1893, Schatunowsky showed that 30 is the largest number all of whose totatives are primes; we show that this result cannot be proved, in any form, in Chebyshev's theorem (Bertrand's postulate), even if all irreducibles are primes. Bonse's inequality is shown to be indeed weaker than Chebyshev's theorem. Schatunowsky's theorem holds in together with Bonse's inequality, the existence of the greatest prime dividing certain types of numbers, and the condition that all irreducibles be prime.

We proved in the first part [1] that plane geometry over Pythagorean fields is axiomatizable by quantifier-free axioms in a language with three individual constants, one binary and three ternary operation symbols. In this paper we prove that two of these operation symbols are superfluous.

This paper continues the investigations begun in [6] and continued in [7] about quantifier-free axiomatizations of plane Euclidean geometry using ternary operations. We show that plane Euclidean geometry over Archimedean ordered Euclidean fields can be axiomatized using only two ternary operations if one allows axioms that are not first-order but universal Lw1,w sentences. The operations are: the transport of a segment on a halfline that starts at one of the endpoints of the given segment, and the operation which produces one (...) of the intersection points of a perpendicular on a diameter of a circle with that circle. MSC: 03F65, 51M05, 51M15. (shrink)

Two results of elementary number theory, going back to Kürschák and Nagell, stating that the sums $\sum_{i=1}^k \frac{m_i}{n+i}$ (with $k\geq 1$, $(m_i, n+i)=1$, $m_i\lessthan n+i$) and $\sum_{i=0}^k \frac{1}{m+in}$ (with $n, m, k$ positive integers) are never integers, are shown to hold in $\mathrm{PA}^{-}$, a very weak arithmetic, whose axiom system has no induction axiom.

We study the class of Sperner spaces, a generalized version of affine spaces, as defined in the language of pointline incidence and line parallelity. We show that, although the class of Sperner spaces is a pseudo-elementary class, it is not elementary nor even ℒ∞ω-axiomatizable. We also axiomatize the first-order theory of this class.

Parsa-Parsi et al systematically discuss and elucidate contentious and non-controversial ethical issues that emerged during the ICoME (International Code of Medical Ethics) revision process and the consensus they achieved. The ethical issues discussed include the physician’s duty to act in the best interests of patients and to ensure they are protected from the unjustifiable risk of harm, respect for patient autonomy and the duties of physicians during emergencies, among others. This paper examines paragraph 26, which requires doctors to provide only (...) medically justified remote treatment, ensure patients offer informed consent and adequately protect their confidentiality.1 At the peak of the COVID-19 pandemic, many countries in Europe and America implemented remote-by-default healthcare policies. At that time, the policy aimed to ensure that patients enjoyed continued access to care while protecting healthcare professionals’ health and well-being. After the COVID-19 pandemic, this policy remained in place. Regardless of the nature of their ailment, patients have consistently preferred in-person care out of concern that their health issues might not be accurately diagnosed and that they may not receive quality care …. (shrink)

This book examines leadership strategies that allow administrators to be proactive, visionary, and flexible while increasing collaboration, open communication, and closely integrating theory and practice to ensure successful administration in higher education settings.

Humans are moral creatures. Among all life on Earth, we alone experience rich moral emotions, follow complex rules governing how we treat one another, and engage in moral dialogue. But how did human morality evolve? And can humans become morally evolved? -/- In A Better Ape, Victor Kumar and Richmond Campbell draw on the latest research in the biological and social sciences to explain the key role that morality has played in human evolution. They explore the moral traits that (...) humans share with chimpanzees; how a more complex moral mind enabled Homo sapiens to arise and out-compete other human species; and the place of morality alongside historic revolutions in technology and social organization. Throughout the book, Kumar and Campbell argue that morality co-evolved with intelligence and complex sociality. Morality prevents societal collapse and enables complex knowledge. -/- After unearthing the ancient origins of human morality, Kumar and Campbell use evolutionary theory to deliver profound insights about how to advance moral progress and resist moral regress, such as reducing animal suffering on industrial farms; capitalizing on the recent revolution in gay rights to foster a nascent revolution in transgender rights; opposing intersectional inequality that impacts women and people of color in lower socioeconomic classes; and addressing major problems of global inequality, especially impending crises of injustice caused by anthropogenic climate change. Understanding how we evolved--and how we continue to evolve--can help us become a better ape. (shrink)

Many philosophers endorse the Transparency Thesis, the claim that by introspection one cannot become aware of one's experience. Recently, some authors have suggested that the Transparency Thesis is challenged by introspective states reached under mindfulness. We label this the Mindfulness Opacity Hypothesis. The present paper develops the hypothesis in important new ways. First, we motivate the hypothesis by drawing on recent clinical psychology and cognitive science of mindfulness. Secondly, we develop the hypothesis by describing the implied shift in experiential perspective, (...) the scope of introspectable qualities, and the level of skill. Thirdly, we defend the hypothesis against various philosophical arguments. We conclude that the Mindfulness Opacity Hypothesis is empirically and theoretically well motivated and supported. (shrink)

The book outlines the contribution of passivity to the constitution of phenomena as diverse as temporal syntheses, perceptual associations, memory fulfillment and cross-cultural communication.

This article is an update of our previous article in this SGJ journal, titled: On Gödel's Incompleteness Theorem, Artificial Intelligence & Human Mind. We provide some commentary on the latest developments around AI, humanoid robotics, and future scenario. Basically, we argue that a more thoughtful approach to the future is "techno-realism.".

Examines to what extent Heidegger accepted the Nazi philosophy, assesses his anti-Semitism, and looks at the links between philosophy and politics.