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.

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.

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)

I INTRODUCTION: RESPECT, EQUALITY AND THE AUTONOMY OF MORALITY We often invoke a notion of respect to express our sense of human equality. ...

Everyone, even when asserting unchallengeable authority from God or Science, thinks in language, in words and phrases, in expressions of moral, social and political impact, fighting words and words with and over which we fight. However, debates among the educated can be irrelevant elsewhere, ineffective against the highly motivated whose dogma instructs and guides them, their voting and their arming. The degeneration of “democracy” to “tyranny” such as Plato’s Republic postulated threatens in some lands “of the free,” while in others (...) it seems inexorably in progress. Today “democracy,” republic,” “liberty/freedom,” “justice,” “law and order” are adduced by their actual enemies to complement “nationalism,” “patriotism,” and frightening “empire.” Extremists on both sides decry “Tyranny!” each in fact seeking authoritarian power. All of which have instructive ancient precedents: in the History of the Peloponnesian War Thucydides offers horrific accounts of ancient nation-states “long ago and far away,” more or less democratic, that abandoned common sense, moral sense, and with them regard for value words and truth itself. In the early 2020s CE many persons and governments have done little better during the still lingering pandemic. A second horror is prompted by Thucydides’ report of civil war on Corcyra, telling how faction spawned riots, coups d’état, and outright civil war. The twenty-first-century media are uniquely potent, yet mechanics of seduction are timeless. Likewise techniques for winning true believers and votes, as illustrated by passages in Mein Kampf on the language of propaganda, its production and reception. Our hope is therefore Education for a New Era to help students to discover pitfalls of media(ted) “information,” to achieve independent critical thinking at levels of basic language use and foundations of argument. Besides recognizing mis- and disinformation as harmful, they must learn themselves to judge thoughtless use, cynical abuse, or, worse yet, brazen disdain of vocabulary indicating moral/ethical value. Scientists have key roles in this drama; humanists larger ones yet. (shrink)

In lieu of an abstract, here is a brief excerpt of the content:The Rhetoric of Philosophical Politics in Plato's Seventh LetterV. Bradley LewisThe name Syracuse has come to stand as an emblem of the problematic relationship between philosophy and politics. While the sources1 differ on specifics, we can be confident that Plato visited there at least three times between 387 and 362 B.C. On his first trip, during the reign of Dionysius I, he became acquainted with Dion, the tyrant's brother-in-law. (...) When the elder Dionysius died, Dion, a devotee of philosophy, apparently contrived for the old man's son, namesake, and successor to invite Plato back, believing that the young tyrant could be philosophically educated and convinced to lay down tyranny in favor of a new regime based on the rule of law. The plan was defeated, however, by a combination of Dionysius's dissolute life and opposition by Dion's enemies at court. Dion was exiled, and Plato returned to the Academy. Dion implored Plato to return to Syracuse on the basis of reports of Dionysius II's renewed resolution to take up philosophy and by the tyrant himself (who seems to have made some thinly veiled threats against the still-exiled Dion). Plato's second visit to Dionysius was no more successful than the first, however, and he considered himself fortunate to have escaped with his life. After a meeting at which Plato recounted the events of his final trip, Dion marshaled an expeditionary force, which included volunteers from the Academy, and set out to expel Dionysius II by violence. The effort succeeded, but the new regime was itself riven by faction, and Dion was assassinated in 353.2The principal legacy of this ill-starred episode is contained in the letters that have come down to us under the name of Plato, particularly those numbered seven and eight. Scholarly discussion of Plato's Seventh Letter has usually focused on three questions: the doctrinal content of the letter and its relationship to the doctrinal content of the dialogues; the character of the letter as an apologia for Plato's Syracusan activities; and the general authenticity of the letter. While the second of these questions focuses correctly [End Page 23] on the rhetorical character of the letter, it ignores the most obvious aspect, namely, the letter's explicit addressees: the friends and comrades of Dion, and their request for political advice. If we take this rhetorical element with sufficient seriousness, we will see that the content of the letter is fully consistent with the teachings of the political dialogues, a point that tells in favor of the authenticity of the letter. Specifically, a focus on the rhetorical character of the letter reveals that its content is determined mostly by what emerges as an effort by Plato to influence his correspondents by moderating their understandable desire for vengeance and thus to forestall further bloodshed and disorder in Syracuse. What is most important about the letter, then, is not so much its account of Plato's actions in Syracuse, but its unveiling of his intentions with respect to Dion's followers. It is not the philosophical-political blueprint, sketched vaguely in the Seventh Letter, that is most important,3 but the concrete effort to mitigate the passions of his correspondents. Read in this light, the letter becomes even more important, since it contradicts the popular image of Plato as a proponent of dangerous political utopianism and portrays a more subtle relationship between philosophy and politics.1. Dion's intentions, Plato's apologyTo the extent that the rhetoric of the letter has been treated, commentators have usually focused on Plato's (or whoever the author) defense of his own actions in Sicily.4 While there is certainly warrant for this approach, the most important fact about the letter concerns its intended audience, the "relatives and comrades of Dion."5 As Plato explains in the opening of the letter, these are the men who have sought not only his advice, but his cooperation in completing the political project halted by Dion's death at the hands of several of his other purported "allies."The most striking feature of the rhetorical structure is that, while the letter... (shrink)

In lieu of an abstract, here is a brief excerpt of the content:The Rhetoric of Philosophical Politics in Plato's Seventh LetterV. Bradley LewisThe name Syracuse has come to stand as an emblem of the problematic relationship between philosophy and politics. While the sources1 differ on specifics, we can be confident that Plato visited there at least three times between 387 and 362 B.C. On his first trip, during the reign of Dionysius I, he became acquainted with Dion, the tyrant's brother-in-law. (...) When the elder Dionysius died, Dion, a devotee of philosophy, apparently contrived for the old man's son, namesake, and successor to invite Plato back, believing that the young tyrant could be philosophically educated and convinced to lay down tyranny in favor of a new regime based on the rule of law. The plan was defeated, however, by a combination of Dionysius's dissolute life and opposition by Dion's enemies at court. Dion was exiled, and Plato returned to the Academy. Dion implored Plato to return to Syracuse on the basis of reports of Dionysius II's renewed resolution to take up philosophy and by the tyrant himself (who seems to have made some thinly veiled threats against the still-exiled Dion). Plato's second visit to Dionysius was no more successful than the first, however, and he considered himself fortunate to have escaped with his life. After a meeting at which Plato recounted the events of his final trip, Dion marshaled an expeditionary force, which included volunteers from the Academy, and set out to expel Dionysius II by violence. The effort succeeded, but the new regime was itself riven by faction, and Dion was assassinated in 353.2The principal legacy of this ill-starred episode is contained in the letters that have come down to us under the name of Plato, particularly those numbered seven and eight. Scholarly discussion of Plato's Seventh Letter has usually focused on three questions: the doctrinal content of the letter and its relationship to the doctrinal content of the dialogues; the character of the letter as an apologia for Plato's Syracusan activities; and the general authenticity of the letter. While the second of these questions focuses correctly [End Page 23] on the rhetorical character of the letter, it ignores the most obvious aspect, namely, the letter's explicit addressees: the friends and comrades of Dion, and their request for political advice. If we take this rhetorical element with sufficient seriousness, we will see that the content of the letter is fully consistent with the teachings of the political dialogues, a point that tells in favor of the authenticity of the letter. Specifically, a focus on the rhetorical character of the letter reveals that its content is determined mostly by what emerges as an effort by Plato to influence his correspondents by moderating their understandable desire for vengeance and thus to forestall further bloodshed and disorder in Syracuse. What is most important about the letter, then, is not so much its account of Plato's actions in Syracuse, but its unveiling of his intentions with respect to Dion's followers. It is not the philosophical-political blueprint, sketched vaguely in the Seventh Letter, that is most important,3 but the concrete effort to mitigate the passions of his correspondents. Read in this light, the letter becomes even more important, since it contradicts the popular image of Plato as a proponent of dangerous political utopianism and portrays a more subtle relationship between philosophy and politics.1. Dion's intentions, Plato's apologyTo the extent that the rhetoric of the letter has been treated, commentators have usually focused on Plato's (or whoever the author) defense of his own actions in Sicily.4 While there is certainly warrant for this approach, the most important fact about the letter concerns its intended audience, the "relatives and comrades of Dion."5 As Plato explains in the opening of the letter, these are the men who have sought not only his advice, but his cooperation in completing the political project halted by Dion's death at the hands of several of his other purported "allies."The most striking feature of the rhetorical structure is that, while the letter... (shrink)

This debate, I shall argue, has everything to do with Aristotle. Aristotle raises the charge of epiphenomenalism himself against a theory that seems to have close affinities to his own, and he offers what has the makings of an emergentist response. This leads to controversy within his own school. We find opponents ranged on both sides, starting with his own pupils, several of whom are stout defenders of epiphenomenalism, and culminating in the developed emergentism of later commentators. Aristotle’s theory and (...) the debate that ensued are thus quite relevant to contemporary discussions. But first we need to get clear on terms. (shrink)

Aristotle not only formulates the problem of intentionality explicitly, he makes a solution to it a requirement for any adequate theory of mind. His own solution, however, is not to be found in his theory of sensation, as Brentano and others have thought. In fact, it is precisely because Aristotle regards this theory as inadequate that he goes on to argue for a distinct new ability he calls "phantasia." The theory of content he develops on this basis (unlike Brentano's) is (...) profoundly naturalistic: it is a representational theory, formulated in terms of the causal powers and physical magnitudes of the body. (shrink)

Aristotle not only fonnulates the problem of intentionality explicitly, he makes a solution to it a requirement for any adequate theory of mind. His own solution, however, is not to be found in his theory of sensation, as Brentano and others have thought. In fact, it is precisely because Aristotle regards this theory as inadequate that he goes on to argue for a distinct new ability he calls “phantasia.” The theory of content he develops on this basis (unlike Brentano’s) is (...) profoundly naturalistic: it is a representational theory, formulated in tenns of the causal powers and physical magnitudes of the body. (shrink)

This debate, I shall argue, has everything to do with Aristotle. Aristotle raises the charge of epiphenomenalism himself against a theory that seems to have close affinities to his own, and he offers what has the makings of an emergentist response. This leads to controversy within his own school. We find opponents ranged on both sides, starting with his own pupils, several of whom are stout defenders of epiphenomenalism, and culminating in the developed emergentism of later commentators. Aristotle’s theory and (...) the debate that ensued are thus quite relevant to contemporary discussions. But first we need to get clear on terms. (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.