Metaphysics, geometry, and the problems with diagrams -- The Pythagorean theorem: Euclid I.47 and VI.31 -- Thales and geometry: Egypt, Miletus, and beyond -- Pythagoras and the famous theorems -- From the Pythagorean theorem to the construction of the cosmos out of right triangles.

Previously we saw that racist prejudice is supported by false history. The false history of the Greek origins of mathematics is reinforced by a bad philosophy of mathematics. There is no evidence for the existence of Euclid. The “Euclid” book does not contain a single axiomatic proof, as was exposed over a century ago. Such was never the intention of the actual author. The book was brazenly reinterpreted, since axiomatic proof was a church political requirement, and used in church rational (...) theology adopted during the Crusades, as a counter to Islamic rational theology. Deductive proofs are MORE fallible than inductive or empirical proofs. Even a validly proved mathematical theorem, such as the “Pythagorean” theorem, is invalid knowledge in the real world. There is no concept of approximate truth in formal mathematics. Nevertheless, the myth of “superior” axiomatic proofs in the “Euclid” book continues to be reiterated by Western historians, and colonial education teaches axiomatic mathematics. Actually, superior practical value comes from the two “Pythagorean” calculations well known toIndian/Egyptian tradition, but unknown to Greeks. The advantage of related decolonized courses in mathematics has been pedagogically demonstrated. But understanding and political will are needed to change colonial/church education. (shrink)

To eliminate racist prejudices, it is necessary to identify the root cause of racism. American slavery preceded racism, and it was closely associated with genocide. Accordingly, we seek the unique cause of the unique event of genocide + slavery. This was initially justified by religious prejudice, rather than colour prejudice. This religious justification was weakened when many Blacks converted to Christianity, after the trans-Atlantic slave trade. The curse of Kam, using quick visual cues to characterize Blacks as inferior Christians, was (...) inadequate. Hence, the church fell back on an ancient trick of using false history as secular justification for Christian superiority. This trick had resulted in a false history of science during the Crusades when scientific knowledge in translated Arabic texts was indiscriminately attributed to the early Greeks, without evidence. This false history enabled belief in religious superiority to mutate into a secular belief in White superiority. After colonialism, and the Aryan race conjecture, the belief in White superiority further mutated into a belief in Western civilizational superiority, openly propagated today by colonial education. Hence, to eliminate racist prejudice, it is necessary to engage simultaneously with the allied prejudices about Christian/White/Western superiority, based on the same false history of science. (shrink)

A new form of the Hyperbolic Pythagorean Theorem, which has a striking intuitive appeal and offers a strong contrast to its standard form, is presented. It expresses the square of the hyperbolic length of the hypotenuse of a hyperbolic right-angled triangle as the “Einstein sum” of the squares of the hyperbolic lengths of the other two sides, Fig. 1, thus completing the long path from Pythagoras to Einstein. Following the pioneering work of Varičak it is well known that (...) relativistic velocities are governed by hyperbolic geometry in the same way that prerelativistic velocities are governed by Euclidean geometry. Unlike prerelativistic velocity composition, given by the ordinary vector addition, the composition of relativistic velocities, given by the Einstein addition, is neither commutative nor associative due to the presence of Thomas precession. Following the discovery of the mathematical regularity that Thomas precession stores, it is now possible to extend Thomas precession by abstraction, (i) allowing hyperbolic geometry to be studied by means of analogies that it shares with Euclidean geometry; and, similarly (ii) allowing velocities and accelerations in relativistic mechanics to be studied by means of analogies that they share with velocities and accelerations in classical mechanics. The abstract Thomas precession, called the Thomas gyration, gives rise to gyrovector space theory in which the prefix gyro is used extensively in terms like gyrogroups and gyrovector spaces, gyroassociative and gyrocommutative laws, gyroautomorphisms, gyrotranslations, etc. We demonstrate the superiority of our gyrovector space formalism in capturing analogies by deriving the Hyperbolic Pythagorean Theorem in a form fully analogous to its Euclidean counterpart, thus contrasting it with the standard form in which the Hyperbolic Pythagorean Theorem is known in the literature. The hyperbolic metric, which supports the Hyperbolic Pythagorean Theorem, has a dual metric. We show that the dual metric does not support a Pythagorean theorem but, instead, it supports the π-Theorem according to which the sum of the three dual angles of a hyperbolic triangle is π. (shrink)

Archytas of Tarentum is one of the three most important philosophers in the Pythagorean tradition, a prominent mathematician, who gave the first solution to the famous problem of doubling the cube, an important music theorist, and the leader of a powerful Greek city-state. He is famous for sending a trireme to rescue Plato from the clutches of the tyrant of Syracuse, Dionysius II, in 361 BC. This 2005 study was the first extensive enquiry into Archytas' work in any language. (...) It contains original texts, English translations and a commentary for all the fragments of his writings and for all testimonia concerning his life and work. In addition there are introductory essays on Archytas' life and writings, his philosophy, and the question of authenticity. Carl A. Huffman presents an interpretation of Archytas' significance both for the Pythagorean tradition and also for fourth-century Greek thought, including the philosophies of Plato and Aristotle. (shrink)

A guiding thread in Western thought is that the world has a mathematical structure. This essay articulates this thread by making use of a cardboard teaching aid that illustrates the Pythagorean Theorem and uses this teaching aid as a starting point for discussion about a variety of philosophical and historical topics. To name just a few, the aid can be used to segue into a discussion of the Pythagorean association of shapes with numbers, the nature of deductive (...) argumentation, the demonstration of part of the Theorem in the Meno, the possible origin of the Theorem in Egypt, the influence of Pythagoreans upon Plato, or even the relation of the Pythagorean Theorem to Fermat’s Last Theorem. (shrink)

The text is a continuation of the article of the same name published in the previous issue of Philosophical Alternatives. The philosophical interpretations of the Kochen- Specker theorem (1967) are considered. Einstein's principle regarding the,consubstantiality of inertia and gravity" (1918) allows of a parallel between descriptions of a physical micro-entity in relation to the macro-apparatus on the one hand, and of physical macro-entities in relation to the astronomical mega-entities on the other. The Bohmian interpretation ( 1952) of quantum mechanics (...) proposes that all quantum systems be interpreted as dissipative ones and that the theorem be thus derstood. The conclusion is that the continual representation, by force or (gravitational) field between parts interacting by means of it, of a system is equivalent to their mutual entanglement if representation is discrete. Gravity (force field) and entanglement are two different, correspondingly continual and discrete, images of a single common essence. General relativity can be interpreted as a superluminal generalization of special relativity. The postulate exists of an alleged obligatory difference between a model and reality in science and philosophy. It can also be deduced by interpreting a corollary of the heorem. On the other hand, quantum mechanics, on the basis of this theorem and of V on Neumann's (1932), introduces the option that a model be entirely identified as the modeled reality and, therefore, that absolutely reality be recognized: this is a non-standard hypothesis in the epistemology of science. Thus, the true reality begins to be understood mathematically, i.e. in a Pythagorean manner, for its identification with its mathematical model. A few linked problems are highlighted: the role of the axiom of choice forcorrectly interpreting the theorem; whether the theorem can be considered an axiom; whether the theorem can be considered equivalent to the negation of the axiom. (shrink)

In lieu of an abstract, here is a brief excerpt of the content:Reviewed by:Logos and Alogon: Thinkable and Unthinkable in Mathematics, from the Pythagoreans to the Moderns by Arkady PlotnitskyNoam CohenPLOTNITSKY, Arkady. Logos and Alogon: Thinkable and Unthinkable in Mathematics, from the Pythagoreans to the Moderns. Cham: Springer, 2023. xvi + 294 pp. Cloth, $109.99The limits of thought in its relations to reality have defined Western philosophical inquiry from its very beginnings. The shocking discovery of the incommensurables in Greek mathematics (...) not only reintroduced these limits with greater force but also instigated two different forms of coping with the unthinkable in Western philosophy, mathematics, and science. As Arkady Plotnitsky sets forth, in response to the incommensurability crisis in the ancient Greek intellectual world, there appeared two different attitudes toward the alogon, that is, the unthinkable or irrational. Plato and his followers sought to overcome the unthinkable, of which the incommensurables in mathematics are only an example, through philosophy. In contrast, the Pythagoreans continued to hold on to mathematics as the primary means of knowing the world, but put a larger emphasis on geometry. Plotnitsky [End Page 359] claims that by not putting aside the primacy of mathematics, the Pythagorean shift from arithmetic to geometry established in Western thought a certain tolerance toward the unthinkable within the realm of the thinkable, “a possibility of the alogon in the logos.” Until modern times this remained no more than a possibility. However, in modern mathematics and mathematical physics Plotnitsky identifies a certain fulfillment of this potential, a “radical Pythagorean mathematics.” In his comprehensive and thought-provoking study, Plotnitsky recounts the development of such radical modern Pythagoreanism, not only to tell its history but also to suggest that the unthinkable is a condition of possibility for thought, and that a proper recognition of this role it plays is crucial for the advancement of mathematical thinking and thinking in general.Plotnitsky begins by portraying in broad strokes the two defining characteristics of radical Pythagorean mathematics: the presence of the unthinkable within mathematical thought itself, and the interplay of algebra and geometry. Relying on an extensive knowledge of the history of modern mathematics and physics, in the first two chapters of the book Plotnitsky gradually demonstrates the key role that the alogon plays in some theories of modern mathematical thought, such as Gödel’s incompleteness theorems and quantum mechanics. He also presents and clarifies his own conception of the history of Western thought as a creative process of concept-forming. By creating rather than discovering concepts, great mathematicians break new ground for thinking. The book as a whole argues in length for these two main theses.In chapter 3, the author addresses the thought of Bernhard Riemann to make the case that Riemann’s mathematical thought puts an emphasis on creative conceptual thinking and is not defined merely by calculations or logic. Such thinking is problem-posing rather than axiomatic. Riemann’s central innovation in this respect is the concept of manifold (Mannigfaltigkeit), which enabled him to think anew space and geometry in a radical non-Euclidean fashion, implying an infinite number of possible geometries. Plotnitsky argues—borrowing Deleuze and Guattari’s terminology—that this opened up a “plane of immanence,” which is to say that Riemann created a concept-generating movement of thought. However, we cannot grasp the generation of concepts in modern mathematics without giving proper attention to the interplay of algebra and geometry that defines it. Chapter 4 demonstrates this interweaving of the two fields by recounting the history of the idea of the curve in modernity, from the analytic geometry and calculus of Fermat and Descartes to its modification in the concept of a Riemann surface, up to its more recent developments in the algebraic geometry of Weil and Grothendieck. It gives a historical account of the algebraization of the curve that at the same time brings into relief the irreducible geometrical aspect, continuity, of all modern mathematical conceptualizations of the curve.The next two chapters provide further case studies. Chapter 5 expands and solidifies the argument of chapter 2. It discusses three examples of mathematical practice, each displayed by a pair of important [End Page 360] mathematicians: Abel and Galois, Lobachevsky and Riemann, Weil and Grothendieck. Each case... (shrink)

One of the foremost scientists and thinkers of our time, David Bohm worked alongside Oppenheimer and Einstein. In Science, Order and Creativity he and physicist F. David Peat propose a return to greater creativity and communication in the sciences. They ask for a renewed emphasis on ideas rather than formulae, on the whole rather than fragments, and on meaning rather than mere mechanics. Tracing the history of science from Aristotle to Einstein, from the Pythagorean theorem to quantum mechanics, (...) the authors offer intriguing new insights into how scientific theories come into being, how to eliminate blocks to creativity and how science can lead to a deeper understanding of society, the human condition and the human mind itself. Science, Order and Creativity looks to the future of science with elegance, hope and enthusiasm. (shrink)

One of the foremost scientists and thinkers of our time, David Bohm worked alongside Oppenheimer and Einstein. In _Science, Order and Creativity_ he and physicist F. David Peat propose a return to greater creativity and communication in the sciences. They ask for a renewed emphasis on ideas rather than formulae, on the whole rather than fragments, and on meaning rather than mere mechanics. Tracing the history of science from Aristotle to Einstein, from the Pythagorean theorem to quantum mechanics, (...) the authors offer intriguing new insights into how scientific theories come into being, how to eliminate blocks to creativity and how science can lead to a deeper understanding of society, the human condition and the human mind itself. _Science, Order and Creativity_ looks to the future of science with elegance, hope and enthusiasm. (shrink)

Given Lewis’s views about recombination and spatial relations, there are possible worlds in which space is discrete and yet the Pythagorean theorem is true – contrary to the so-called Weyl-Tile argument that concluded that the Pythagorean theorem must fail if space is discrete.

Down through the ages, logic has adopted many strange and awkward technical terms: assertoric, prove, proof, model, constant, variable, particular, major, minor, and so on. But truth-value is a not a typical example. Every proposition, even if false, no matter how worthless, has a truth-value:even “one plus two equals four” and “one is not one”. In fact, every two false propositions have the same truth-value—no matter how different they might be, even if one is self-contradictory and one is consistent. It (...) is not such a big surprise that every true proposition, no matter how worthless, has a truth-value. Every proposition, whether valuable or worthless, has a truth-value.But it is a little strange that every two true propositions, no matter how different they might be, have the same truth-value. The Pythagorean Theorem has the same truth-value as the proposition that one is one. It is a stretch to think of truth-values as values in any of the normal senses of the word ‘value’. (shrink)

“The problem of universals” in general is a historically variable bundle of several closely related, yet in different conceptual frameworks rather differently articulated metaphysical, logical, and epistemological questions, ultimately all connected to the issue of how universal cognition of singular things is possible. How do we know, for example, that the Pythagorean theorem holds universally, for all possible right triangles? Indeed, how can we have any awareness of a potential infinity of all possible right triangles, given that we (...) could only see a finite number of actual ones? How can we universally indicate all possible right triangles with the phrase ‘right triangle’? Is there something common to them all signified by this phrase? If so, what is it, and how is it related to the particular right triangles? The medieval problem of universals is a logical, and historical, continuation of the ancient problem generated by Plato's (428-348 B.C.) theory answering such a bundle of questions, namely, his theory of Ideas or Forms. (shrink)

A solution of the zeno paradoxes in terms of a discrete space is usually rejected on the basis of an argument formulated by hermann weyl, The so-Called tile argument. This note shows that, Given a set of reasonable assumptions for a discrete geometry, The weyl argument does not apply. The crucial step is to stress the importance of the nonzero width of a line. The pythagorean theorem is shown to hold for arbitrary right triangles.

The purpose of the conference "On Pythagoreanism", held in Brasilia in 2011, was to bring together leading scholars from all over the world to define the status quaestionis for the ever-increasing interest and research on Pythagoreanism in the 21st century. The papers included in this volume exemplify the variety of topics and approaches now being used to understand the polyhedral image of one of the most fascinating and long-lasting intellectual phenomena in Western history. Cornelli's paper opens the volume by charting (...) the course of Pythagorean studies over the past two centuries. The remaining contributions range chronologically from Pythagoras and the early Pythagoreans of the archaic period (6th-5th centuries BCE) through the classical, hellenistic and late antique periods, to the eighteenth century. Thematically they treat the connections of Pythagoreanism with Orphism and religion, with mathematics, metaphysics and epistemology and with politics and the Pythagorean way of life. (shrink)

A solution of the zeno paradoxes in terms of a discrete space is usually rejected on the basis of an argument formulated by hermann weyl, The so-Called tile argument. This note shows that, Given a set of reasonable assumptions for a discrete geometry, The weyl argument does not apply. The crucial step is to stress the importance of the nonzero width of a line. The pythagorean theorem is shown to hold for arbitrary right triangles.

In lieu of an abstract, here is a brief excerpt of the content:Reviewed by:Natural Philosophy: On Retrieving a Lost Disciplinary Imaginary by Alister E. McGRATHJack ZupkoMcGRATH, Alister E. Natural Philosophy: On Retrieving a Lost Disciplinary Imaginary. Oxford: Oxford University Press, 2023. viii + 248 pp. Cloth, $39.95This book attempts to retrieve and reimagine the tradition of natural philosophy as an antidote for what the author sees as the fragmented, instrumentalized, and ethically disengaged understanding of the natural world most of us (...) have today. The idea is not to reinstate the older vision of Aristotle and Thomas Aquinas, with its final causes and animated spheres, or even the early modern culmination of natural philosophy in Newton’s universe of mathematically fixed natural laws; rather, the author hopes to engender a newer and deeper understanding of nature that is at once scientific, ethical, and poetic, which he explains in terms of the conceptual framework of Popper’s “Three Worlds”: objective, subjective, and theoretical. What is recovered thereby is said to be a “lost conceptual space” wherein human beings learn both about and from nature, modes of knowing rendered obsolete by disciplinary specialization in the sciences as well as our commitment to objective methods that leave no room for ethical or spiritual approaches to understanding the natural world and our place in it.The author covers quite a bit of ground on the historical side, from the ancient Greeks through the Middle Ages, moving on to the idea of natural philosophy as it was gradually transformed during the early modern period by Copernicus, Kepler, Galileo, Bacon, Boyle, and Newton—a picture finally eclipsed by the objective, empirical concept of natural science familiar to us today. Curiously, the book says very little about the Platonic tradition of natural philosophy that grew up around the Timaeus. Because it was available in Latin (or the first half of it, anyway, in a Latin translation by the Roman Neoplatonist Calcidius, who also wrote an influential commentary on the work), the Timaeus was, for almost a millennium, the authoritative text in natural philosophy in the West, at least until Aristotle’s nonlogical and scientific writings were recovered in the twelfth century. The Timaeus was the subject of many commentaries and philosophical discussions, from Calcidius in the fourth century to Bernardus Silvestris, William of Conches, and Alan of Lille in the twelfth. It fell out of favor among philosophers with the rise of Aristotle’s Physics and Aristotelian natural philosophy, though it continued to shape the popular imagination of nature in other genres, such as art and literature—recall, for example, the well-known image of “God the Geometer” (c. 1230), measuring the cosmos with a compass. If the author’s project is about “retrieving” natural philosophy “as a lost disciplinary imaginary” (the book’s subtitle), then surely the Platonic vision of the universe deserves equal billing with the Aristotelian. [End Page 158]That said, there is much to like in the project sketched here. The author’s diagnosis of the epistemic malaise of modern-day science sounds right to anyone familiar with the history of philosophy and natural science. So, even though we can now correctly describe the velocity constant of earth’s gravity as 9.8 m/s2, among other scientific achievements, we take ourselves to be wearing a different hat when we wax poetic at the beauty of the sky and the stars. Early-and pre-modern authors were more adept at bringing together science and poetry, but that might be because their epistemic position was radically different from our own. Calcidius, for example, finds it easy to draw moral and aesthetic conclusions from the motions of the cosmos because he really believed heavenly bodies are, like us, animated by souls. Even a materialist like Lucretius accepts that the gods exist; his (still) very radical thesis is that they don’t care about us. (And why should they? The Pythagorean theorem is much more beautiful and edifying to consider than the troubles of ignorant and weak-willed mortals.) Enlightened moderns that we are, we no longer believe the heavens are populated by super-intelligent beings; yet we continue to marvel at Van Gogh’s Starry Night and feel it is wrong to cause gratuitous... (shrink)

Like others who work in philosophy, I asked myself from time to time what I was trying to do in my philosophizing. The natural answer seemed to be that I was trying to understand the world, and to do so by taking any thing or event that puzzled me and pressing the question Why? till I arrived at the understanding I sought. And what does understanding anything mean? It means to explain it or to render it intelligible. And when does (...) it become intelligible? Only when it is seen in context, and seen as required by that context. This requirement is of various kinds. Sometimes it is causal, as when an attack of malaria is explained by the bite of the anopheles mosquito. Sometimes it is a means-end relation, as when the presence of a rudder is explained as a means of guiding a ship. Sometimes it is logical, as when the Pythagorean theorem is explained by showing that it follows necessarily from other accepted propositions of geometry. (shrink)

How the concept of proof has enabled the creation of mathematical knowledge. The Story of Proof investigates the evolution of the concept of proof--one of the most significant and defining features of mathematical thought--through critical episodes in its history. From the Pythagorean theorem to modern times, and across all major mathematical disciplines, John Stillwell demonstrates that proof is a mathematically vital concept, inspiring innovation and playing a critical role in generating knowledge. Stillwell begins with Euclid and his influence (...) on the development of geometry and its methods of proof, followed by algebra, which began as a self-contained discipline but later came to rival geometry in its mathematical impact. In particular, the infinite processes of calculus were at first viewed as "infinitesimal algebra," and calculus became an arena for algebraic, computational proofs rather than axiomatic proofs in the style of Euclid. Stillwell proceeds to the areas of number theory, non-Euclidean geometry, topology, and logic, and peers into the deep chasm between natural number arithmetic and the real numbers. In its depths, Cantor, Gödel, Turing, and others found that the concept of proof is ultimately part of arithmetic. This startling fact imposes fundamental limits on what theorems can be proved and what problems can be solved. Shedding light on the workings of mathematics at its most fundamental levels, The Story of Proof offers a compelling new perspective on the field's power and progress. (shrink)

The richness of humanity is the diversity of its cultures, but now as never before the destructive power of modern technology and threatening ecological disasters make it necessary that we all recognize we are many peoples of one world. Complementing the diversity of our different cultures, the growth of a common, scientific knowledge inspires the hope that we may achieve and share a secondary culture of ideas. Computers, which can help represent explicitly the best ideas of modern science, can aid (...) in the diffusion of such powerful ideas to create a popular, secondary, scientific culture.We propose that a primary objective of learning environment design should be the development of thinkable models in the minds of students; further, since thinkable models represent a level of knowledge deeper than superficial terminology and specific graphical forms, such may be an objective suitable for various people in different cultures. To pursue these notions in some detail, a taxonomy of models is developed and the issue of how representations relate to human modes of perception and action is raised. The notions are explored first through the contrasting of a half-dozen approaches to the Pythagorean Theorem; then through describing polylingual word worlds and the Rosetta disk project.Common experiences produce common models —not only public models but the cognitive structures built from those shared experiences; and shared models will permit enhanced communication and understanding. If sharing experiences through play with computer based learning environments in different languages permits children to develop common models of the world, this will ultimately enhance mutual understanding between people in different places. This objective is now within reach. (shrink)

While collaboration has always played an important role in many cases of discovery and creation, recent developments such as the web facilitate and encourage collaboration at scales never seen before, even in areas such as mathematics, where contributions by single individuals have historically been the norm. This new scenario poses a challenge at the theoretical level, as it brings out the importance of various issues which, as of yet, have not been sufficiently central to the study of problem-solving, discovery, and (...) creativity. We analyze the case of collective and web-based proof events in mathematics, which share their temporal and social nature with every case of collective problem-solving. We propose that some ideas from cognitive architectures, in particular, the notion of codelet—understood as an agent engaged in one of a multitude of available tasks—can illuminate our understanding of collective problem-solving and act as a natural bridge from some of the theoretical aspects of collective, web-based discovery to the practical concern of designing cognitively inspired systems to support collective problem-solving. We use the Pythagorean Theorem and its many proofs as a case study to illustrate our approach. (shrink)

continent. 1.3 (2011): 201-207. “Poetry is experience, linked to a vital approach, to a movement which is accomplished in the serious, purposeful course of life. In order to write a single line, one must have exhausted life.” —Maurice Blanchot (1982, 89) Nikos Karouzos had a communist teacher for a father and an orthodox priest for a grandfather. From his four years up to his high school graduation he was incessantly educated, reading the entire private library of his granddad, comprising mainly (...) the Orthodox Church Fathers and the ancient classics. Later on in his life he sold the library for money, only to buy a little more time before he went broke again: “I carry sorrow like a cage/ I got no birds/ as I come back from her hair/ I see the profit emptied/ cool from terror.” (1993, Ι, 101) Nowadays—and ridiculously recently—we are more than apt to speak of a certain insouciance pertaining to the Greek form of expenditure: expenditure without any type of investment, sometimes not (even) symbolic. In the vaults of the European unconscious, this imprudent stance still conjures a “capital punishment” in the form of de-capitation, reflecting back the fear of excess and the terror of its consequences.1 Thus the Greek way of living is very close to what a stranger could have termed as “the Greek way of dying.” Yet, what is not often understood is that the Greek esprit , the same one that invented the maxim μηδ?ν ?γαν (“nothing in excess”), does not experience death solely as an imminent fear of bankruptcy. For it miraculously combines its unconditional acceptance with the promise of a resurrection that is always there in the scents of Spring, from Dionysus to the orthodox Christ: “Everyone resurrects himself through dying [….] Resurrection is the switching of mortalities.” (2002, 94) Hence, if an entire library was imprudently traded for a plate with seven olives, a sliced tomato and some fresh goat cheese, let us not be fooled: this is not the meal of the pauper, but that of “an aristocrat from God” (1993, II, 491) feeding his eloquent loudmouth, ridiculously cut open in an otherwise rock-solid caput . *** To speak of the Greek experience is to speak about a half-dead language that still utters in life what is seemingly excluded from it and thus forbidden to be talked about: death. Death as anything that is out of this world, as something that will never return . Still, along with an experience of death that recently waned as a worn-out academic fashion, the Greek experience is dead too. Nearly 2300 years of written words separating Homer from the fall of Byzantium equal 500 books on a library traded for money: “I am with the killed. Hence my deepest solitude. I do not feel this tremendous macho society, beyond from the fact that it is a ruthlessly consuming one. It is me who pays all the time .” (2002, 57) Let us do the math and see how much this dealing with the Greek experience is costing the poet and how much dealing with the Greek experience might cost us today. *** “I do not guarantee a single word.” (Karouzos, 1993, II, 454) Through its various dialects and forms, the Greek language speaks through an incessant historical dissemination. Anyone who is aware of this terrifying polymorphy and still calls himself a poet, must stand against a white sheet of paper, pen-in-hand, with a very particular duty: to be as fully inconsistent as possible. In the formally ironic uniqueness of the poem, the poet functions as “a band-aid for lesser and greater antinomies.” (1993, I, 251) Of course these antinomies are not exhausted in language—they are historical, political and, alas, existential. Yet, beyond appearances , it is language that ruthlessly encodes them through history, submitting the poet to the temptation of placing them one next to the other on a single white sheet. The closer their neighboring, the greater the scattering of the writer in the ironic uniqueness of the paper. In a work where poetic license is described as a “freedom-impasse” (2002, 51), this task is undertaken in full conscience of its personal consequences: “what I am interested in is to escape my individuality (envisioning the non-ego) […] Nevertheless, my dissociations never achieve duration.” (2002, 74-75) Hence, though poetry is the “deserted direction of will,” (1998, 62) the stance of the poet is not that of a Nietzschean “great man.”2 Whereas drunkenness provides a thread between the poem and the excess it both presupposes and infuses (“Poetry always enlarges. ‘Drunkenness’ is nothing more than that,” (2002, 85)), whereas language flashes in loudmouth spurts of déraison (“When I am alone I do something else. I utter words. For example, while having my ouzo and listening to music I am most likely capable of randomly shouting ‘Electricity!’” (2002, 138)), the poet never gives in to the double affirmation that would eventually risk the “element of pleasure in discourse.” (2002, 80)” It is exactly because the gap of the antinomies is so vast that poetry is not meant to be written with a hammer. Neither does writing consist in a reevaluation of those elements. In short, poetry is not an affirmation of difference, but the surprising beauty of chiming antinomies. We might never transcend them, but it is up to us to put them together. Yet again, this voisinage is not to be identified with the necessity of chance as an eternal return.4 On the contrary it is a return from that very return : Let us treat Yes as a No to No and No as a Yes […] And let us not forget that this pissed affirmation crumbled down Nietzsche’s intellect in the dark paths of this world. (2002, 88) This return from the “vicious circle” should in no way be taken as a form of artistic prudence. Rather it can be seen as a dribble of demonic inconsistency as Dionysus transforms himself into a Christ that is in turn de-theologized: “Who can forbid that? Every man is capable of his own theology, nothing can stop him.” (2002, 73-74) This is a turn towards an existential vision of the world, historically coinciding with a particular political defeat of the Left, to which the poet devoted all his life: after the defeat of the popular front I raised the question “why do we exist?” while others were asking “why we failed.” (2002, 57)5 But most of all it is a return to poetry that exceeds the existential question itself, going beyond the issue of faith. Poetry comes in as a question of return after a defeat that is confessed in full profanity. Though it accepts the necessity of the defeat, it does not affirm it. Though it negates it, the negation is not in the name of a promise to be delivered in the kingdom of heavens. Following a historical and existential defeat, poetry is born post mortem . It signals a return to Christ as “groundless religiousness in the surprise of the real as such” (2002, 72) which is at once a return to the refuge of childhood. It is not a question of endurance towards an eternal return. Rather it is a question on the possibility of an existential return. Returning in a world as someone who cannot enjoy any returns exactly because he is averse to guarantees. A return without returns. *** PHOTOCOPY OF HAPPINESS When I was young I used to pin down cicadas and step on ant nests. I used to stand there silent for hours. With threads I decapitated bees. Now I am a dead man breathing. (Karouzos, 1993, II, 336.) The return to the “paradise of childhood” (2002, 68) constitutes the devouring refuge of its own memory when defeat becomes the synonym of adulthood. The political struggles of a young communist in a country torn up by a civil war right after World War II fraught with incarcerations and exiles. Historically they resulted in the defeat of the communist movement in Greece and opened up a long turbulent road that would peak in the dictatorship of 1967. Existentially they led to a series of disillusionments: mental breakdown, divorce, abandonment of studies in law school. To the question “why do we exist” the answer was poetry. Still this exposé should not be read as a sweeping chronography of a man. For it actually happens to coincide with the historical fate of a nation that after its modern constitution never stopped dreaming of the glories of its past youth, in a present that was (and is) sweepingly disappointing. But isn’t this return to youth finally a way of compensating for a loss of youth that necessarily results in a losing adulthood? Will Greeks, the eternal children of Plato and Nietzsche, ever learn? How to return without dying, how to remember without wasting time? WE ARE IN THE OPEN MEANING Living the coldest mauve night right across Parthenon I went to take a piss on some foliage and I was enjoying the foliage as it was steaming. (Karouzos, 1993, II, 340) Beyond the historical tragedies of modern Greece and away from any personal disappointments, the relation that this land holds with language and history is mediated through the Greek light—whose omnipresence is the very condition of its transcendence. All historical contradictions from Dionysus to Christ took place under this light; all those disseminated dialects were spoken beneath its warmth. To paraphrase Lévinas,6 light is both the condition of the world and of our withdrawal from it—a withdrawal towards the invisibility of God, of the dead, of meta-physics, resulting from the temptation that all is still here, behind this light the visibility of which they once evaded: “birds, the allurement of God.” (1993, I, 17) Under that luminous sky, if Greeks can do anything at all, it is to envision a return that will never come. All they can do is write poetry—which is doing nothing; other than lending an ear to a disseminated language whispering a unity that cannot be promised, as an adulthood in defeat is ready to recognize. Trying “to trap the invisible in visibility” (1993, II, 483) they forget that they have grown up and one day they die—with the promise of return. Hence an additional meaning must be given to this return to childhood. It does not signal salvation but rather its promise. To the ears of an aged continent it means that the return to/of poetry is a losing game, a return without returns. “Europe, Europe… you are nothing more than the continuation of Barabbas.” (1993, I, 295) *** “Life is not there to verify theories.”7 The records show that Karouzos was finally given a second-class pension from the Greek government, at the time when he was being recognized by the literary press as one of Greece’s major contemporary poets. Being neither a bourgeois nor a nobelist, he proclaimed himself to be an anarcho-communist, unconditionally faithful to the utopia of a classless society. He also drank, heavily. “Capitalism made an animal out of man/ Marxism made an animal out of truth/ Shut up.” (1993, II, 369) Perhaps one of the most scandalous divides of our times has been the one between the living and the dead, the latent prohibition that the living should not be concerned with the dead based on the mere impossibility of the dead to be concerned with the living in the first place. What adds to this scandal is that this divide abuses anything that cannot return to us by subsuming it under the same futility. Hence death is no longer “loss” in the usual sense. It no more refers to the things we lost but to our “loss” (of time, money and well-being) as we insist to dwell on them. Death is a waste —of time. It is this waste that we find in the insouciance of Greek expenditure; the waste in dealing with a language that most of its historical part is no longer spoken (a dead language); the waste in translating a poet who is ex definitio untranslatable; the waste of his vision, his money, his life. The waste of dealing with anything that cannot return and that cannot bring in any returns . But it is also the waste of life that poetry itself presupposes, the waste of dealing with invisibility, with anything that is out of this world and thus invokes the fear of death that is in turn—and surprisingly— nowhere to be found. Instead of death, what is there, beyond the light, is the being without us (to recall the Lévinasian il y a ), the mumble of our own nothingness which constitutes the price to be paid for writing poetry under an evergreek light. To understand this to-and-fro, is to realize that poetry is something out of this world that nevertheless takes place in this world, by virtue of this world. But to ridiculously equate this to-and-fro with death as non-existence, is to exile poetry along with its own possibility from this same old world: I do not believe that poetry will ever disappear from this world. […] But I am also sure that it does not have many chances of playing, as you say, a redemptive role in our vertiginous technological future. Without being endangered as a creative need, it will be placed on the side of history. (2002, 32) It is mainly there that we would like to locate the meaning of Nikos Karouzos’s poetry today. If we are willing to include the Greek original it is because we consider that it will be both a waste of time for us to do so (since most of you cannot read Greek) and because it might induce you to the even larger waste of learning it. We would additionally be glad if this small introduction served as an equally wasteful, academically useless piece of reading, gesturing towards a taboo of investing in anything Greek—that is in anything dead among the living, in anything that will never come back and maybe was never here in the first place. This is the only way to reserve for ourselves the possibility of poetry and preserve the light of its promise. “To return: that is the miracle.” (1993, I, 17) Athens, Greece July 10, 2011 NOTES 1 “ Capitale (a Late Latin word based on caput “head”) emerged in the twelfth to thirteenth centuries in the sense of funds, stock of merchandise, sum of money or money carrying interest.[…] The word and the reality it stood for appear in the sermons of St. Bernardino of Siena (1380-1444) ‘… quamdam seminalem rationem lucrosi quam communiter capitale vocamus ’, ‘that prolific cause of wealth that we commonly call capital’” (Braudel, 1992, 232-233). 2 “A great man – a man whom nature has constructed and invented in the grand style – what is he? First : there is a long logic in all of his activity, hard to survey because of its length, and consequently misleading; he has the ability to extend his will across great stretches of his life and to despise and reject everything petty about him, including even the fairest, ‘divinest’ things in the world.” (Nietzsche, 1968, 505, §962) 3 Cf. Deleuze, 1986, 186-9. 4 “Return is the being of becoming, the unity of multiplicity, the necessity of chance: the being of difference as such or the eternal return.” (Deleuze, 189)5 In this 1982 interview Karouzos refers to the defeat of the communist movement in Greece after the civil war of 1946-1949 between the Governmental Army and the Democratic Army of Greece, the military division of the Greek communist party. Karouzos’s father was a member of the communist National Liberation front (EAM) members of which formed the mainl resistance movement (ELAS) in Greece during WWII. The poet was a member of the United Panhellenic Organization of Youth (EPON), which was a youth division of EAM. After the end of the war, ELAS was called to disarmament in view of the formation of a National Army. The members of EAM resigned from the government of national unity and a series of protests led to a 3 year civil war between ELAS and the Government Army. After the defeat of ELAS the Communist party was outlawed and many communists were exiled in deserted Greek islands. Karouzos, who took action in the Greek resistance and was active during the Greek civil war, was exiled in Icaria on 1947 and in Makronisos on 1953 where he was called two years earlier to do his military service. 6 “Existence in the world qua light, which makes desire possible, is then, in the midst of being, the possibility of detaching oneself from being. To enter into being is to link up with objects; it is in effect a bond that is already tainted with nullity. It is already to escape anonymity. In this world where everything seems to affirm our solidarity with the totality of existence, where we are caught up in the gears of a universal mechanism, our first feeling, our ineradicable illusion, is a feeling or illusion of freedom. To be in the world is this hesitation, this interval in existing, which we have seen in the analyses of fatigue and the present.” (Lévinas, 1988,43-4) 7from a TV interview. THE POETRY OF NIKOS KAROUZOS From Redwriter [ Ερυθρογρ?φος ], in Collected Works, Vol . II, Athens: Ikaros. 1993, 463-4. [Athens: Apopeira, 1990, 9-10.] JESUS ANTI-OEDIPUS* Why were we once saving till the fifth day the Paschal lamb? The scriptures say: with this victim’s meat we ought to cleanse all of our senses. To give life to the whitest wave miraculous odors in the extension of the chest. Ideals to death. For us to witness the illustrious dawns of Absinthe and for the soul to be a deeply carved ornament on the fore we named will. Then many deer running amidst the evergreen growth with watery hymns, then, the unintended angels descending with heights spiraling in their momentum offer themselves to the luminous Adventure. Whence the need for speech and our sufferings covered with flags like the glorified dead. An incorporeal finger pointing at the flamboyant and fragrant holocausts within the tired horizons and the exhausted breadths with joys of the mistletoe on fire and shivering all of the green-leaved love. Something would have chirped again had we not driven it away— maybe the ewe’s grass. Something would have called us to the eternal resurrection— maybe the grace of spring. But now our heart is fiercely blinded, withdrawn to the appearances of Hades. Silence and ice incessantly cover the Infant Spirit in thatched times. While the cherry trees are dreaming faintly glowing in absolute darkness there’s nothing the Babylonians can contemplate in labs with automatic colored lights. How to rejoice now in the fifth day of the lamb? We have forked the stars. We’ve gradually become supposedly sublime with plucked chimeras in our hands. Avenged relentlessly by science. *Though the poem appears in a 1990 collection, it was actually written in 1968. Hence the reader must be aware in not associating Karouzos’ Anti-Oedipus with the famous 1972 work of Deleuze and Guattari, given that the former precedes the latter. ΙΗΣΟΥΣ ΑΝΤΙ-ΟΙΔΙΠΟΥΣ Γιατ? κρατο?σαμε κ?ποτε ?ς την π?μπτη μ?ρα το πασχαλιν? πρ?βατο; Τα κε?μενα λ?νε: μ’ αυτο? του θ?ματος το κρ?ας ?πρεπε να καθαρ?σουμε ?λες τις αισθ?σεις. Για να δ?σουμε κ?μα στη ζω? λευκ?τατο θαυμ?σιες ευωδι?ς στην ?κταση του στ?θους. Ιδανικ? στο χ?ρο. Για να βλ?πουμε τα λαμπρ? χαρ?ματα του ?ψινθου και να ’ναι η ψυχ? στ?λισμα βαθυχ?ρακτο της πλ?ρης που την ε?παμε θ?ληση. Τ?τε πολλ?ς δορκ?δες τρ?χοντας αν?μεσα στην καταπρ?σινη φ?ση με τους υδ?τινους ?μνους, τ?τε, κατεβα?νοντας οι αθ?λητοι ?γγελοι με κυλινδο?μενο το ?ψος στην ορμ? τους χαρ?ζονται της αστραφτερ?ς Περιπ?τειας. ?θεν η μιλι? γι’ αυτ? χρει?ζεται και τα δειν? σκεπ?ζονται ωσ?ν τους τιμημ?νους νεκρο?ς με σημα?ες. Ασ?ματο δ?χτυλο δε?χνει τα φλογ?δη και μοσχοβ?λα ολοκαυτ?ματα στους κουρασμ?νους ορ?ζοντες στα εξουθενωμ?να πλ?τη καθ?ς αν?βουν οι χαρ?ς του νερα?δ?ξυλου και τρ?μει ολ?κληρη η πρασιν?φυλλη αγ?πη. Κ?τι θα κελαηδο?σε π?λι αν δεν το δι?χναμε— μπορε? της προβατ?νας το χορτ?ρι. Κ?τι θα μας καλο?σε στην απ?ραντη αν?σταση— μπορε? του ?αρος η χ?ρη. Μα η καρδι? μας ?γρια τυφλ?θηκε π?ρασε στα φαιν?μενα του ?δη. Σιγ? και π?γος αδι?κοπα σκεπ?ζει στους αχυρ?νιους καιρο?ς το Ν?πιο Πνε?μα. Την ?ρα που ονειρε?ονται οι βυσσινι?ς και λ?μπουν αμυδρ? μεσ’ στο απλ?τατο σκοτ?δι τ?ποτα δε στοχ?ζονται οι βαβυλ?νιοι στα εργαστ?ρια με τ’ αυτ?ματα χρωματιστ? φ?τα. Π?ς να χαρο?με πια την π?μπτη μ?ρα του προβ?του; Φουρκ?σαμε τ’ αστ?ρια. Γ?ναμε σιγ?-σιγ? δ?θεν υπ?ροχοι με μαδημ?νες χ?μαιρες στα χ?ρια. Μας ν?μεται σκληρ? η επιστ?μη. From: Redwriter [Ερυθρογρ?φος], in Collected Works, Vol. II, Athens: Ikaros. 1993, 472. [Athens: Apopeira, 1990, 18.] DIALECTΙCS OF SPRING Christ the straight angle; Christ the Pythagorean theorem. Christ the infinitesimal calculus from above the blessed Sets Christ. Christ the tessellation of massive particles Christ the zero mass. Hence we spray numbers and fields of lust. We are carabineers of diseased logic plus something— Observed means observer and Hekate The darkness luminous and light in the dark Astarte banqueters demons from today. ΔΙΑΛΕΚΤΙΚΗ ΤΟΥ ΕΑΡΟΣ Χριστ?ς η ορθ? γων?α· Χριστ?ς το πυθαγ?ριο θε?ρημα. Χριστ?ς ο απειροστικ?ς λογισμ?ς ?νωθεν ?λβια Χριστ?ς τα Σ?νολα. Χριστ?ς η ψηφιδογραφ?α στα μαζικ? σωμ?τια Χριστ?ς η μ?ζα μηδ?ν. ?ρα ψεκ?ζουμε αριθμο?ς και πεδ?α λαγνε?ας. Ε?μαστε τυφεκιοφ?ροι νοσο?σης λογικ?ς και κ?τι-- παρατηρο?μενο σημα?νει παρατηρητ?ς και Εκ?τη σκ?τος το π?μφωτο και φως εν τ? σκοτ?? η Αστ?ρτη συνδαιτημ?νες δα?μονες απ’ ?ρτι. From: Redwriter [Ερυθρογρ?φος], Athens: Apopeira, 1990, 18. [1993, II, 472]. CREDO (as we are used to say in Latin) Α I believe in one Poet expelled from heavens / fugitive from the god and vagabond, Empedocles / and here on earth / exile on the earth etc. of Baudelaire /. Β I believe in one Computer inside thunder and through matter. Γ Suffering undefiled / substantive / the Poet uplifts himself slow-burning suicidal implying lengthy sleeps. Δ Cutting down on prospective mistakes. Ε Of all things visible and invisible officiating the onion-peelings. Ζ The Poet has nothing / see the departed /. Η I believe in one Poet that says: madness I enjoy; he ridicules existence; let me light up from my mana. Θ Syntax he doesn’t care about when musicality commands him. Along with still more licenses, and Ts are played according to the concept sound anywhere. E.g. the winters here, he winters there; it will not come—I will not rest, etc. etc. Ι The Poet exercises thought until it’s stripped down. Κ And if he’s Greek he must always study the fineness of Attica, in light, mountains, fields and sea. For this fineness teaches language. Λ And if he’s deeply destined the Poet expresses the unexplainable of the explained; he happens to be a rightful heir to the scientist and his predecessor. Μ On the froth he does not last; the Poet blusters at the bottom of the pot. Ν Flamebred and never redeemed. Ξ The Poet must sometimes say: what a consumption of presence — be a bit lonely for a change! Ο The Poet is twilit. Π He is susceptible of deaths and resurrections. Ρ He looks from the corner of his eye and exists in a glance. Σ He follows behind the mother. Τ Eveningless when it comes to age. Υ I believe in one Poet who says: let the purities coincide. Until the Corinth of the Universe or even further. Φ In a higher despair. Χ In a brighter quintessence. Ψ In one sensation that lifts off. Ω Forgiving everyone. CREDO (ως ε?θισται να λ?με λατινιστ?) Α Πιστε?ω εις ?να Ποιητ?ν εκτ?ς ουρανο? / φυγ?ς θε?θεν και αλ?της, Εμπεδοκλ?ς / και επ? της γης / εξ?ριστος π?νω στη γη κ.λπ. του Βωδελα?ρου /. Β Πιστε?ω εις ?να Υπολογιστ?ν εντ?ς κεραυνο? και δια της ?λης. Γ Υποφ?ροντας ?χραντα / ουσιαστικ?ν / ο Ποιητ?ς ανατε?νεται βραδυφλεγ?ς αυτ?χειρας εξυπακο?οντας πολ?ωρους ?πνους. Δ Τα υποψ?φια λ?θη λιγοστε?οντας. Ε Ορατ?ν τε π?ντων και αορ?των ιερουργ?ντας την αποκρομμ?ωση. Ζ Ο Ποιτ?ς ?χει τ?ποτα / βλ?πε τους αναχωρ?σαντες /. Η Πιστε?ω εις ?να Ποιητ?ν που λ?ει: η τρ?λα μ’ αρ?σει· γελοιοποιε? την ?παρξη· ας αν?ψω απ’ τη μ?να μου. Θ Συνταχτικ? δεν το γνοι?ζεται στην προσταγ? της μουσικ?τητας. Μαζ? και μ’ ?λλες ακ?μη λευτερι?ς, και τα νυ πα?ζονται κατ? την ?ννοια ?χος οπουδ?ποτε. Π.χ. τον χειμ?να εδ?, το χειμ?να εκε?· δε θ? ’ρθει – δεν θα καταλαγι?σουμε, κ. λπ. κ.λπ. Ι Ο Ποιητ?ς γυμν?ζει τη σκ?ψη σε απογ?μνωση. Κ Κι αν ε?ναι ?λληνας οφε?λει να σπουδ?ζει π?ντοτε της Αττικ?ς τη λεπτ?τητα, σε φως, βουν?, χωρ?φια και θ?λασσα. Διδ?σκει γλ?σσα η λεπτ?τητα το?τη. Λ Κι αν ε?ναι βαθι? πεπρωμ?νος ο Ποιητ?ς εκφρ?ζει το ανεξ?γητο του εξηγητο?· τυγχ?νει ν?μιμος δι?δοχος του επιστ?μονα και προκ?τοχ?ς του. Μ Στον αφρ? δεν ?χει δι?ρκεια· στο πατοκ?ζανο μα?νεται ο Ποιητ?ς. Ν Φλογοδ?αιτος και ποτ? ξελυτρωμ?νος. Ξ Ο Ποιητ?ς κ?ποτε πρ?πει να λ?ει: μεγ?λη καταν?λωση παρουσ?ας – γενε?τε και λ?γο μοναξι?ρηδες! Ο Ο Ποιητ?ς ε?ναι αμφ?φλοξ. Π Επιδ?χεται θαν?τους και αναστ?σεις. Ρ Ακροθωρ?ζει και υπ?ρχει σε ξαφνοκο?ταγμα. Σ Ε?ναι ουραγ?ς της μητ?ρας. Τ Αν?σπερος απ? ηλικ?α. Υ Πιστε?ω εις ?να Ποιητ?ν που λ?ει: να συμπ?σουν οι αγν?τητες. Μ?χρι την Κ?ρινθο του Σ?μπαντος ? μακρ?τερα. Φ Σε αν?τερη απελπισ?α. Χ Σε φαειν?τερη πεμπτουσ?α. Ψ Σε μια α?σθηση που πτηνο?ται. Ω Συγχωρ?ντας τους π?ντες. From Large Sized Logic [ Λογικ? μεγ?λου σχ?ματος ], Athens: Erato, 1989, n.p. [1993, II, 521-3]. (shrink)

It has been argued that Hume's denial of infinite divisibility entails the falsity of most of the familiar theorems of Euclidean geometry, including the Pythagorean theorem and the bisection theorem. I argue that Hume's thesis that there are indivisibles is not incompatible with the Pythagorean theorem and other central theorems of Euclidean geometry, but only with those theorems that deal with matters of minuteness. The key to understanding Hume's view of geometry is the distinction he (...) draws between a precise and an imprecise standard of equality in extension. Hume's project is different from the attempt made by Berkeley in some of his later writings to save Euclidean geometry. Unlike Berkeley, who interprets the theorems of Euclidean geometry as false albeit useful approximations of geometrical facts, Hume is able to save most of the central theorems as true. (shrink)

In this multidisciplinary book, mathematician Matthew He provides integrative perspectives of algebraic biology, cognitive informatics, and poetic expressions of the human mind. Using classical Pythagorean Theorem and contemporary Category Theory, the proposed matrix models of the human mind connect three domains of the physical space of objective matters, mental space of subjective meanings, and emotional space of bijective modes; draws the connections between neural sparks and idea points, between synapses and idea lines, and between action potentials and frequency (...) curves. (shrink)

This work analyzes two perspectives, Atomism and Infinite Divisibility, in the light of modern mathematical knowledge and recent developments in computer graphics. A developmental perspective is taken which relates ideas leading to atomism and infinite divisibility. A detailed analysis of and a new resolution for Zeno's paradoxes are presented. Aristotle's arguments are analyzed. The arguments of some other philosophers are also presented and discussed. All arguments purporting to prove one position over the other are shown to be faulty, mostly by (...) question begging. Included is a sketch of the consistency of infinite divisibility and a development of the atomic perspective modeled on computer graphics screen displays. The Pythagorean theorem is shown to depend upon the assumption of infinite divisibility. The work concludes that Atomism and infinite divisibility are independantly consistent, though mutually incompatible, not unlike the wave/particle distinction in physics. (shrink)

Mathematician. Philosopher. World traveler. Pythagoras was an intelligent and curious scholar and teacher. While he¿s best-known for the Pythagorean theorem, he shared ideas about numbers, animals, and many other areas of knowledge with his students. Since none of his writings were left behind, it¿s not always easy for historians to know what¿s true about Pythagoras and what may be legendary. What does seem apparent is that he was a vegetarian but not a trendy dresser. Some people saw him (...) as godlike. Others felt he made false claims about things. No matter what, Pythagoras¿s curiosity and willingness to grapple with complex issues have helped further the knowledge of mathematics and philosophy for thousands of years. (shrink)

This chapter contains sections titled: Euclid's Elements First Principles Aspects of Euclid's Plane Geometry Proportionality Greek Arithmetic and its History On the History of Greek Geometry Conclusion Bibliography.

The paper discusses Hilbert mathematics, a kind of Pythagorean mathematics, to which the physical world is a particular case. The parameter of the “distance between finiteness and infinity” is crucial. Any nonzero finite value of it features the particular case in the frameworks of Hilbert mathematics where the physical world appears “ex nihilo” by virtue of an only mathematical necessity or quantum information conservation physically. One does not need the mythical Big Bang which serves to concentrate all the violations (...) of energy conservation in a “safe”, maximally remote point in the alleged “beginning of the universe”. On the contrary, an omnipresent and omnitemporal medium obeying quantum information conservation rather than energy conservation permanently generates action and thus the physical world. The utilization of that creation “ex nihilo” is accessible to humankind, at least theoretically, as long as one observes the physical laws, which admit it in their new and wider generalization. One can oppose Hilbert mathematics to Gödel mathematics, which can be identified as all the standard mathematics until now featureable by the Gödel dichotomy of arithmetic to set theory: and then, “dialectic”, “intuitionistic”, and “Gödelian” mathematics within the former, according to a negative, positive, or zero value of the distance between finiteness and infinity. A mapping of Hilbert mathematics into pseudo-Riemannian space corresponds, therefore allowing for gravitation to be interpreted purely mathematically and ontologically in a Pythagorean sense. Information and quantum information can be involved in the foundations of mathematics and linked to the axiom of choice or alternatively, to the field of all rational numbers, from which the pair of both dual and anti-isometric Peano arithmetics featuring Hilbert arithmetic are immediately inferable. Noether’s theorems (1918) imply quantum information conservation as the maximally possible generalization of the pair of the conservation of a physical quantity and the corresponding Lie group of its conjugate. Hilbert mathematics can be interpreted from their viewpoint also after an algebraic generalization of them. Following the ideas of Noether’s theorem (1918), locality and nonlocality can be realized both physically and mathematically. The “light phase of the universe” can be linked to the gap of mathematics and physics in the Cartesian organization of cognition in Modernity and then opposed to its “dark phase”, in which physics and mathematics are merged. All physical quantities can be deduced from only mathematical premises by the mediation of the most fundamental physical constants such as the speed of light in a vacuum, the Planck and gravitational constants once they have been interpreted by the relation of locality and nonlocality. (shrink)

The previous Part I of the paper discusses the option of the Gödel incompleteness statement (1931: whether “Satz VI” or “Satz X”) to be an axiom due to the pair of the axiom of induction in arithmetic and the axiom of infinity in set theory after interpreting them as logical negations to each other. The present Part II considers the previous Gödel’s paper (1930) (and more precisely, the negation of “Satz VII”, or “the completeness theorem”) as a necessary condition (...) for granting the Gödel incompleteness statement to be a theorem just as the statement itself, to be an axiom. Then, the “completeness paper” can be interpreted as relevant to Hilbert mathematics, according to which mathematics and reality as well as arithmetic and set theory are rather entangled or complementary rather than mathematics to obey reality able only to create models of the latter. According to that, both papers (1930; 1931) can be seen as advocating Russell’s logicism or the intensional propositional logic versus both extensional arithmetic and set theory. Reconstructing history of philosophy, Aristotle’s logic and doctrine can be opposed to those of Plato or the pre-Socratic schools as establishing ontology or intensionality versus extensionality. Husserl’s phenomenology can be analogically realized including and particularly as philosophy of mathematics. One can identify propositional logic and set theory by virtue of Gödel’s completeness theorem (1930: “Satz VII”) and even both and arithmetic in the sense of the “compactness theorem” (1930: “Satz X”) therefore opposing the latter to the “incompleteness paper” (1931). An approach identifying homomorphically propositional logic and set theory as the same structure of Boolean algebra, and arithmetic as the “half” of it in a rigorous construction involving information and its unit of a bit. Propositional logic and set theory are correspondingly identified as the shared zero-order logic of the class of all first-order logics and the class at issue correspondingly. Then, quantum mechanics does not need any quantum logics, but only the relation of propositional logic, set theory, arithmetic, and information: rather a change of the attitude into more mathematical, philosophical, and speculative than physical, empirical and experimental. Hilbert’s epsilon calculus can be situated in the same framework of the relation of propositional logic and the class of all mathematical theories. The horizon of Part III investigating Hilbert mathematics (i.e. according to the Pythagorean viewpoint about the world as mathematical) versus Gödel mathematics (i.e. the usual understanding of mathematics as all mathematical models of the world external to it) is outlined. (shrink)

Logic in Action/Doron Avital Nothing is more difficult, and therefore more precious, than to be able to decide (Napoleon Bonaparte) Introduction -/- This book was born on the battlefield and in nights of secretive special operations all around the Middle East, as well as in the corridors and lecture halls of Western Academia best schools. As a young boy, I was always mesmerized by stories of great men and women of action at fateful cross-roads of decision-making. Then, like as today, (...) I felt the full weight of the moment of truth as it confronts the individual, the man or woman of action, with the imminent necessity to decide. Alongside, as I was climbing up the ladder of education, I found myself just as well captivated by the great intellectual stories of our times, in particular 20th century analytic philosophy and, at its core, the machinery of modern logic. How do then the cool contemplative modes of the philosophical and analytic outlooks, that are better fitting of the philosopher's armchair at the fireplace, hang together with the heat of the battlefield or the angst of the ticking clock of time left to the completion of a daring special operation? Growing up I was pondering exactly that. Not accidently, and in fact as consequence, I have chosen a course in life that challenged me to visit the poles of both theory and practice. I set myself on a mission to unravel the tension between the contemplative inquiry, supposedly theoretical, aiming at the logical structure of things, and the uncompromising test of life and action. The puzzle of the gap between theory and practice has become ever since the guiding question of my life. -/- Not long ago, I met two friends, Avner Shor, a veteran combat soldier of the special forces unit I eventually commanded, and Aviram Halevi, a fellow officer with whom I served. Why will we not put into writing the ideas that guided us during our years of military action, we wondered. At that time, I used to deliver the keynote lectures to the senior officers' course of Israel central intelligence organization, the Mossad. For this I already had to put some of these ideas in writing. Avner was quick to introduce me with a notable publisher and even helped me write the first chapter. With Aviram I sat to many meetings, where he would challenge me with key questions, and we followed them with memories of operations and battles, aiming to draw the required lessons. This served me well when I finally sat down to write the first draft of the book. The result was more extensive than me and my friends had expected. I had to add a particularly long chapter dealing with the story of modern philosophy and logic. This was required, I felt, as no decision-making process lives in an intellectual vacuum. Knowingly or subconsciously, the intellectual climate of the time guides us for better or worse with the decisions we finally take. The initial draft was naturally cumbersome and inaccessible. But here I had the privilege of meeting a particularly talented editor, Rami Rothholtz. I found in Rami both an editor and a critical reader. As an infantry combat soldier at youth and now a senior journalist and editor, Rami insisted on simplifying and refining my ideas. Had it not been for his work, this book would have been literally unreadable. *** I opened my book with a brief description of a single night of a special operation deep at the Lebanon Bekaa valley, far away from Israel northern border. The mission was to kidnap a senior terror leader known to be holding in captivity an Israeli pilot. This was also the last special operation I commanded in my military career. A single special operation night, but what had to precede it? For sure, many months of intense and skillful preparations. What is Planning? This is the question with which I open up chapter (2) of the book. I confront in this chapter two approaches to planning. The first is the one I saw as a mission to myself to cultivate in my many years of military service. The second is the one I saw as representing a deep-rooted fallacy that holds a firm grip on today's conventional wisdom about planning. To further examine the logic of the competing models, I was required in chapter (3) to analyze the concept of "The Standard". I argue that the false model of planning has its roots in an attitude of an uncompromising adherence to the power of literal obedience to standards and rules. The standard, or what is outlined by a rule as prescribing a future course of action, is seen as an answer to the fundamental tension between theory (planning) and practice (execution). I argue in contrast that the standard does not produce an answer but rather what I call: "A Well-Defined Question". In essence, it is a meticulously-well-planned invitation to exercise a fresh judgment. A judgement that has as an anchor the default answer that the standard produces but is open to extension as it confronts the new frame of reference of the test of reality. In chapter (4), I present a new tool for planning and execution: "The Polygon of Risks" or "The Polygon of Execution". The Polygon is a geometric illustration of the underlying tension that exists between all dimensions of the operational project. The construction and conceptualization of the trade-offs that exist between the various dimensions - segments of the polygon - is the first topic on the planner's agenda. I position the concepts of "Daring" and "Taking Proactive and Well-Informed Risks" as necessary logical constituents of the polygon and show how in contrast the false model of planning is responsible for an operational culture that is risk-averse. It encourages, in fact, the piling up of false securities at the segments' level that eventually lead to the collapse of the polygon. The necessary trade-offs relationships that exist between the segments of the polygon express the fundamental tension between partnership and competition: partnership in a joint goal and competition over resources, both material and abstract. When this tension is not managed well, that is, when the players' local optimizations overshadow the global optimization required for the success of the project, the execution polygon collapses. In this scenario, we may find the setting of the bar of "High Standards" in a players' spheres of responsibility to conceal a hidden agenda of personal insurances taken against a possible failure. Once the overall risk is not mitigated by way of collaborative joint executional dialogue, the process that leads to a collapse of the polygon is literally unstoppable. *** In chapter (5), I discuss "(Fateful) Decisions". I examine the way in which we should translate the polygon into sequential critical decisions on the timeline. The execution polygon is not a rigid scaffolding but a dynamic conceptual illustration of the -/- project. It will need now to be converted into successive decision junctions. We will find the content of a decision and its timing on the timeline to be inseparable. The right decision is the one that is taken at the right time. The moment of decision is in fact the last moment of deliberation, the first moment available for action, i.e. the first possible "Can" is also the latest possible "Must". Deciding too soon is analog to the "Jumping of the Gun". The decision maker is forcing a pre-conceived answer to a state of affairs not ripe for an answer, e.g. the target in the shooting range is not yet erected. Late to decide and the decision maker may envision what he or she should have done but reality is no longer available for them to harness it right, e.g. consider a tennis player's frustration when reviewing a too-late-to-react move in the video replay. I review in this chapter many fateful decisions, both historical, from Napoleon's Waterloo, to a Bridge Too Far of WWII, to the failed rescue attempt of American hostages at the time of the Iranian Revolution, as well as fateful decisions I had to take in the course of my military life, both in combat and in special operations, and show the sense in which they all answer to our analysis. The theme of chapter (6) is "The Moment of Truth". This is the moment when everything is on the line and from which there is no turning back. I discuss the formative repetitive structure of the schooling and simulation phase and contrast it with the "One Shot, One Opportunity" nature of the moment of truth. This is extremely helpful when we analyze the logical structure of failures as unwarranted repetitions of past lessons. The schooling curriculum indeed consists of important abstractions drawn from lessons of the past. However, it is when we project them in a fashion that is literal or mechanical that we cross path with the possibility of a failure. The paradox of education is that what we must repeat in school and simulation must be extended and not repeated in real life. What is required of us so we do not fall into the fallacy of repetition in face of the moment of truth is the core subject of the discussion of this chapter. In chapter (7), I took a lengthy philosophical detour surveying the intellectual drama of modern logic, and in general, the story of 20th century analytic philosophy. Readers who choose to bypass the somehow demanding narrative of this drama can do so and still keep the argumentative thread of the book. However, even by way of skipping the more technical parts, I would still urge the readers to delve into the discussion in this chapter as I believe this would grant them a deeper understanding of the overall conceptual framework. I explore here the work of two key figures: the logician Kurt Gödel and the philosopher Ludwig Wittgenstein, both were crowned in 2000 by TIME Magazine, as the leading Logician and Philosopher, respectively, of the 20th Century. I review the fascinating logical result of Gödel's Incompleteness Theorem and examine its relevance as setting a theoretical limit to the power of formalization. This amazing logical result - a result that shakes the logical foundations of 20th century thought in as much as the discovery of irrational numbers shook the foundations of the Pythagorean program of the ancient Greeks - bears significantly on the question of action and the blind trust we place in the power of formalization in the form of protocols and rules. It is here that Wittgenstein's philosophy enters in full force. We follow Wittgenstein as he dismantles our conception of rules as if there were "Railroads to Infinite", i.e. as prescribing for us in-advance their proper employments in all possible -/- future eventualities. Instead we propose that the dilemma of action demands of us an extension of rules rather than a mechanical or blind repetition of their literal imperative. In turn, this extension can serve as a new lesson added to the curriculum of school and simulation. The theme of chapter (8), the concluding chapter, is Strategy, Logic, and Ethics. The chapter comes to place the ethical dilemma in the conceptual map we presented in the book. I discuss here the tension between strategy as a process of setting goals and deriving backward as it were the steps that are necessary for achieving these goals, to the inescapable dictation of the ethical imperative. This unavoidable tension is particularly important against the backdrop of contemporary culture that is guided by a vision of personal success and achievement that may threaten our ability to do what is right. Most of all, I am troubled in this chapter, with the role logic may play, first as a liberating force enabling a change that is tuned to the ethical imperative, and second, as a force designed to justify on logical grounds, as it were, the existing power structure. It is in this junction that brave individuals will be called to challenge the governing logic of the zeitgeist - logic that operates now as an oppressive force that serves the vested interest of beneficials of the existing order. When success is favored over doing what is right, as we must admit is the prevailing cultural mood of our times, that the words of Emanuel Kant carry their full weight: "Do the Right thing and leave the Consequences to God". For this we need heroes and heroines that challenge the time by bearing the full weight of responsibility that comes with the doing of what is Right. *** This book was written more than anything from the perspective of the heroes and heroines of action. Sometimes they may be missing the exact words or the full insight into the solid logical complexity that stands between them and the doing of what is right. They feel what is right to do on the battlefield of life, whether in real battle, in political or economic life or whether in the life of creativity and the quest of the intellect. They do need, in analogy, proper logical and intellectual artillery. I hope they will find the logical cannons they require as they travel through the pages of this book. The rest is in their hands at the Moment of Truth. -/- Doron Avital, Tel Aviv 2011 -/- . (shrink)

Mathematics has had its share of historical shocks, beginning with the discovery by Hippasus the Pythagorean that the integers could not possibly be the elements of all things. Likewise with Kurt Gödel’s Incompleteness Theorems, which presented a serious (even fatal) obstacle to David Hilbert’s formalism, and Bertrand Russell’s own discovery of the paradox inherent in his intuitively simple set theory. More recently, Paul Benacerraf presented a problem for the foundations of arithmetic in “What Numbers Could Not Be” and “Mathematical (...) Truth.” Drawing out difficulties he found in mathematical realism, Benacerraf ended up proposing his own structuralist view of mathematics, according to which numbers could not be considered objects at all, but mere placeholders, wholly defined in terms of the structures within which they were found. While we do not intend to work through the intricacies of structuralism or other types of mathematical nonrealism, or even of any of the many other competing theories in contemporary philosophy of mathematics, our question concerns problems one finds in an earlier, and perhaps more naïve, view of mathematics which, though a realism of its own, begins with seemingly more problematic assumptions than these. Classically expressed, the question is: do numbers exist in the world, whether as substances or as attributes of things? These are among the things Benacerraf holds numbers could not be; still, we hope to dispel some of the confusion which might bring one prematurely to reject this account. Most precisely, we wish to address the problem whether there can be a science of arithmetic, as that term is understood by Aristotle and Thomas Aquinas. (shrink)

Following a long-standing philosophical tradition, impartiality is a distinctive and determining feature of moral judgments, especially in matters of distributive justice. This broad ethical tradition was revived in welfare economics by Vickrey, and above all, Harsanyi, under the form of the so-called Impartial Observer Theorem. The paper offers an analytical reconstruction of this argument and a step-wise philosophical critique of its premisses. It eventually provides a new formal version of the theorem based on subjective probability.

A search is under way for a theory that can accommodate our intuitions in population axiology. The object of this search has proved elusive. This is not surprising since, as we shall see, any welfarist axiology that satisfies three reasonable conditions implies at least one of three counter-intuitive conclusions. I shall start by pointing out the failures in three recent attempts to construct an acceptable population axiology. I shall then present an impossibility theorem and conclude with a short discussion (...) of how it might be extended to pluralist axiologies, that is, axiologies that take more values than welfare into account. (shrink)

This paper generalises the classical Condorcet jury theorem from majority voting over two options to plurality voting over multiple options. The paper further discusses the debate between epistemic and procedural democracy and situates its formal results in that debate. The paper finally compares a number of different social choice procedures for many-option choices in terms of their epistemic merits. An appendix explores the implications of some of the present mathematical results for the question of how probable majority cycles (as (...) in Condorcet's paradox) are in large electorates. (shrink)

Although the philosophical literature on the foundations of quantum field theory recognizes the importance of Haag’s theorem, it does not provide a clear discussion of the meaning of this theorem. The goal of this paper is to make up for this deficit. In particular, it aims to set out the implications of Haag’s theorem for scattering theory, the interaction picture, the use of non-Fock representations in describing interacting fields, and the choice among the plethora of the unitarily (...) inequivalent representations of the canonical commutation relations for free and interacting fields. (shrink)

Bell’s theorem has fascinated physicists and philosophers since his 1964 paper, which was written in response to the 1935 paper of Einstein, Podolsky, and Rosen. Bell’s theorem and its many extensions have led to the claim that quantum mechanics and by inference nature herself are nonlocal in the sense that a measurement on a system by an observer at one location has an immediate effect on a distant entangled system. Einstein was repulsed by such “spooky action at a (...) distance” and was led to question whether quantum mechanics could provide a complete description of physical reality. In this paper I argue that quantum mechanics does not require spooky action at a distance of any kind and yet it is entirely reasonable to question the assumption that quantum mechanics can provide a complete description of physical reality. The magic of entangled quantum states has little to do with entanglement and everything to do with superposition, a property of all quantum systems and a foundational tenet of quantum mechanics. (shrink)

We study the proof-theoretic strength and effective content of the infinite form of Ramsey's theorem for pairs. Let RT n k denote Ramsey's theorem for k-colorings of n-element sets, and let RT $^n_{ denote (∀ k)RT n k . Our main result on computability is: For any n ≥ 2 and any computable (recursive) k-coloring of the n-element sets of natural numbers, there is an infinite homogeneous set X with X'' ≤ T 0 (n) . Let IΣ n (...) and BΣ n denote the Σ n induction and bounding schemes, respectively. Adapting the case n = 2 of the above result (where X is low 2 ) to models of arithmetic enables us to show that RCA 0 + IΣ 2 + RT 2 2 is conservative over RCA 0 + IΣ 2 for Π 1 1 statements and that $RCA_0 + I\Sigma_3 + RT^2_{ , is Π 1 1 -conservative over RCA 0 + IΣ 3 . It follows that RCA 0 + RT 2 2 does not imply BΣ 3 . In contrast, J. Hirst showed that $RCA_0 + RT^2_{ does imply BΣ 3 , and we include a proof of a slightly strengthened version of this result. It follows that $RT^2_{ is strictly stronger than RT 2 2 over RCA 0. (shrink)

Kripke-completeness of every classical modal logic with Sahlqvist formulas is one of the basic general results on completeness of classical modal logics. This paper shows a Sahlqvist theorem for modal logic over the relevant logic Bin terms of Routley- Meyer semantics. It is shown that usual Sahlqvist theorem for classical modal logics can be obtained as a special case of our theorem.

Kripke-completeness of every classical modal logic with Sahlqvist formulas is one of the basic general results on completeness of classical modal logics. This paper shows a Sahlqvist theorem for modal logic over the relevant logic Bin terms of Routley-Meyer semantics. It is shown that usual Sahlqvist theorem for classical modal logics can be obtained as a special case of our theorem.

Here is an account of logical consequence inspired by Bolzano and Tarski. Logical validity is a property of arguments. An argument is a pair of a set of interpreted sentences (the premises) and an interpreted sentence (the conclusion). Whether an argument is logically valid depends only on its logical form. The logical form of an argument is fixed by the syntax of its constituent sentences, the meanings of their logical constituents and the syntactic differences between their non-logical constituents, treated as (...) variables. A constituent of a sentence is logical just if it is formal in meaning, in the sense roughly that its application is invariant under permutations of individuals.1 Thus ‘=’ is a logical constant because no permutation maps two individuals to one or one to two; ‘∈’ is not a logical constant because some permutations interchange the null set and its singleton. Truth functions, the usual quantifiers and bound variables also count as logical constants. An argument is logically valid if and only if the conclusion is true under every assignment of semantic values to variables (including all non-logical expressions) under which all its premises are true. A sentence is logically true if and only if the argument with no premises of which it is the conclusion is logically valid, that is, if and only if the sentence is true under every assignment of semantic values to variables. An interpretation assigns values to all variables. (shrink)

Attempts to articulate the real meaning or ultimate significance of a famous theorem comprise a major vein of philosophical writing about mathematics. The subfield of mathematical logic has supplie...

Quantified hybrid logic is quantified modal logic extended with apparatus for naming states and asserting that a formula is true at a named state. While interpolation and Beth's definability theorem fail in a number of well-known quantified modal logics , their counterparts in quantified hybrid logic have these properties. These are special cases of the main result of the paper: the quantified hybrid logic of any class of frames definable in the bounded fragment of first-order logic has the interpolation (...) property, irrespective of whether varying, constant, expanding, or contracting domains are assumed. (shrink)