The notion of linguistic geometry is defined in this book. It is pertinent to keep in the record that linguistic geometry differs from classical geometry. Many basic or fundamental concepts and notions of classical geometry are not true or extendable in the case of linguistic geometry. Hence, for simple illustration, facts like two distinct points in classical geometry always define a line passing through them; this is generally not true in (...) class='Hi'>linguistic geometry. Suppose we have two linguistic points as tall and light we cannot connect them, or technically, there is no line between them. However, let's take, for instance, two linguistic points, tall and very short, associated with the linguistic variable height of a person. We have a directed line joining from the linguistic point very short to the linguistic point tall. In this case, it is important to note that the direction is essential when the linguistic variable is a person's height. The other way line, from tall to very short, has no meaning. So in linguistic geometry, in general, we may not have a linguistic line; granted, we have a line, but we may not have it in both directions; the line may be directed. The linguistic distance is very far. So, the linguistic line directed or otherwise exists if and only if they are comparable. Hence the very concept of extending the line infinitely does not exist. Likewise, we cannot say as in classical geometry; three noncollinear points determine the plane in linguistic geometry. Further, we do not have the notion of the linguistic area of well-defined figures like a triangle, quadrilateral or any polygon as in the case of classical geometry. The best part of linguistic geometry is that we can define the new notion of linguistic social information geometric networks analogous to social information networks. This will be a boon to non-mathematics researchers in socio-sciences in other fields where natural languages can replace mathematics. (shrink)

The Aristotelian square of oppositions is a well-known diagram in logic and linguistics. In recent years, several extensions of the square have been discovered. However, these extensions have failed to become as widely known as the square. In this paper we argue that there is indeed a fundamental difference between the square and its extensions, viz., a difference in informativity. To do this, we distinguish between concrete Aristotelian diagrams and, on a more abstract level, the Aristotelian geometry. We then (...) introduce two new logical geometries, and develop a formal, well-motivated account of their informativity. This enables us to show that the square is strictly more informative than many of the more complex diagrams. (shrink)

We discuss in this paper the question of the scope of the principle of tolerance about languages promoted in Carnap's The Logical Syntax of Language and the nature of the analogy between it and the rudimentary conventionalism purportedly exhibited in the work of Poincaré and Hilbert. We take it more or less for granted that Poincaré and Hilbert do argue for conventionalism. We begin by sketching Coffa's historical account, which suggests that tolerance be interpreted as a conventionalism that allows us (...) complete freedom to select whatever language we wish—an interpretation that generalizes the conventionalism promoted by Poincaré and Hilbert which allows us complete freedom to select whatever axiom system we wish for geometry. We argue that such an interpretation saddles Carnap with a theory of meaning that has unhappy consequences, a theory we believe he did not hold. We suggest that the principle of linguistic tolerance in fact has a more limited scope; but within that scope the analogy between tolerance and geometric conventionalism is quite tight. (shrink)

Discussions of surface pattern in Islamic art resonate within broader tensions about the role of figural representation in communicating meaning. The question of whether geometric pattern communicates—whether it functions as a language without a code—reflects broader tensions about the relationship between secular and spiritual communication. Poised between discussions of modernism and Islam, the attribution of linguistic capacity to geometry serves as a measure for the possibility of abstracting pure reason from the religious roots of representationalism. This paper explores (...) this question in four parts. First, it examines the articulation of geometric meaning in Islamic discourses. It then explores European discussions of geometry between language and ornament poised between a search for universal reason and universal spiritualism, both in modernist painting and in modern exhibitions. The final section the expressive capacity of artistic responses to historical Islamic isometric geometries in the adaptive reuse of historical Iranian arts as contemporary abstraction in the sculptures of Monir Shahroudy Farmanfarmaian and in the popular transformation prints of M. C. Escher. It argues that the frequent recognition of intrinsic meaning in geometry, expressed in both premodern Islamic and modern secular interpretations, undermines art historical expectations of theorization as a necessary intermediary for communication. Geometry thereby functions as a language without a code. (shrink)

The concept of geometry may evoke a world of pure platonic shapes, such as spheres and cubes, but a deeper understanding of visual experience demands insight into the perceptual organization of naturalistic form. Japanese gardens excel as designed environments where the complex fractal geometry of nature has been simplified to a structural core that retains the essential properties of the natural landscape, thereby presenting an ideal opportunity for investigating the geometry and perceptual significance of such naturalistic characteristics. (...) Here, fronto-parallel perspective, asymmetrical structuring of the ground plane, spatial arrangement of garden elements, tuning of textural qualities and choice of naturalistic form, are presented as a set of physical features that facilitate a systematic analysis of Japanese garden design per se, as well as the geometry of the particular naturalistic features that it aims to enhance. Comparison with Western landscape design before and after contact between Western and Eastern hemispheres illustrates the degree of naturalness achieved in the Japanese garden, and suggests how classical Western landscape design generally differs in this regard. It further reveals how modern Western gardens culminate in a different naturalistic geometry, thus also a distinctly different vision of the natural landscape, even if these designs were greatly influenced by the gardens of China and Japan. (shrink)

This book extends the concept of linguistic coordinate geometry using linguistic planes or semi-linguistic planes. In the case of coordinate planes, we are always guaranteed of the distance between any two points in that plane. However, in the case of linguistic and semi-linguistic planes, we can not always determine the linguistic distance between any two points. This is the first limitation of linguistic planes and semi-linguistic planes.

The paper discusses the similarity between geometry, arithmetic, and logic, specifically with respect to the question of whether applied theories of each may be revised. It argues that they can - even when the revised logic is a paraconsistent one, or the revised arithmetic is an inconsistent one. Indeed, in the case of logic, it argues that logic is not only revisable, but, during its history, it has been revised. The paper also discusses Quine's well known argument against the (...) possibility of logical deviancy. (shrink)

The paper discusses the similarity between geometry, arithmetic, and logic, specifically with respect to the question of whether applied theories of each may be revised. It argues that they can - even when the revised logic is a paraconsistent one, or the revised arithmetic is an inconsistent one. Indeed, in the case of logic, it argues that logic is not only revisable, but, during its history, it has been revised. The paper also discusses Quine's well known argument against the (...) possibility of “logical deviancy”. (shrink)

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.

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

In lieu of an abstract, here is a brief excerpt of the content:Philosophy, Psychiatry, & Psychology 9.2 (2002) 127-129 [Access article in PDF] Linguistic Markers of Recovery:Underpinnings of First Person Pronoun Usage and Semantic Positions of Patients Patrick Suppes Keywords: association, freedom, habits, psychotherapy, roles, semantics. USING LINGUISTIC EVIDENCE to evaluate recovering psychotherapy patients is an attractive and useful idea. I agree with much of Dr. van Staden's proposals for doing so. The purpose of this commentary is to (...) give some comments and criticisms on points of detail. Roles, Not Semantics A fair amount of Dr. van Staden's article is concerned with the semantics of relations. The focus on relations is a good thing. Restricting discourse about patients to properties is not sufficient, for everyone regards how a person is relating to others, and even to other things, as an important feature of normal behavior and mental health. We do not want just to report, "John was upset," but consider it definitely more informative to say, "John was upset by his father's criticisms." The English language and all other natural languages are rich in verbs for expressing relations with emotional affect between two or more persons, or a person and an event or thing. Staden's article focuses on the interesting and important case of first person pronouns. To go from saying, "She made me do it" to "I did it" is, for reasons that need not be spelled out, a sign of progress. A way to describe this change is that the speaker has indicated that in the implicit context assumed he or she has shown signs of moving from the role of being a patient to that of being an actor, from being a recipient of actions to being a performer of acts.van Staden concentrates on the semantics of relations to bring out what I call the change of roles. I would certainly agree that such change of roles is important to the semantics of affective or action verbs and verb phrases. We have no disagreement at all about this. Yet I am critical of his wanting to range much wider in his analysis of semantics. The logical or formal treatment of relations is a large topic, full of many distinctions and subtleties that seem to me not too relevant to his main theme. In a very general way he wants to say that what is usually thought of as [End Page 127] the first position in a relation, what he calls the alpha position, is occupied by the owner of the relation, and the second position, in a binary relation—the main case discussed—is occupied by the accidental to that particular relation. This second position is called the omega position.This runs contrary to much of the mathematical literature on relations, which occupy an important place in many parts of mathematics such as geometry. So when we say that one triangle is larger than another, it seems positively weird to say the second triangle is somehow accidental to the relation. But this is true and confusing even for the familiar and classical kinds of actions often cited by van Staden. In analyzing the sentence "Brutus killed Caesar intentionally" it sounds really strange to say that Caesar occupied the accidental position in the expressed relation of killing.Moreover, in the conceptual and philosophical discussion of relations, it is common practice to stress that even if the natural grammar of a sentence has a subject-predicate structure, as in "Elizabeth is the mother of Charles," no special logical force is given to what van Staden calls the first or alpha position in most mathematical relations, such as "The number 2 is less than the number 3," expressed in ordinary mathematical notation as "2 < 3."On the other hand, I stress that it is natural to distinguish in a relational sentence expressing an action between the role of actor and the role of patient, no matter which is the grammatical subject of the sentence, as in the following pair "John hurt Mary" and "Mary was hurt by John." The roles of actor... (shrink)

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

Hans Reichenbach's 1928 thesis of the relativity of geometry has been misunderstood as the statement that the geometrical structure of space can be described in different languages. In this interpretation the thesis becomes an instance of trivial semantical conventionalism, as Grünbaum calls it. To understand Reichenbach correctly, we have to interpret it in the light of the linguistic turn, the transition from thought oriented philosophy to language oriented philosophy, which mainly took place in the first decades of our (...) century. Reichenbach — as Poincaré before him — is undermining the prejudice of thought oriented philosophy, that two propositions have different factual content, if they are associated with different ideas in our mind. Thus Reichenbach prepared the change to language oriented philosophy, which he also accepted later. (shrink)

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)

In this small book logician and mathematician Jens Erik Fenstad addresses some of the most important foundational questions of linguistics: What should a theory of meaning look like and how might we provide the missing link between meaning theory and our knowledge of how the brain works? The author’s answer is twofold. On the one hand, he suggests that logical semantics in the Montague tradition and other broadly conceived symbolic approaches do not suffice. On the other hand, he does not (...) argue that the logical approach should be discarded; instead, he opts for a methodological pluralism in which symbolic approaches to meaning are combined with geo- metric ones such as Conceptual Spaces [9] and discusses ways in which these geometric accounts could be hooked up with connectionist frameworks and dynamic systems approaches in neurophysiology. (shrink)

Muhyiddîn el-Kâfiyecî (öl. 879/1474), on beşinci yüzyıl Saruhanoğulları, Osmanlı ve Memlüklü alimlerindendir. Yüzden çok eser yazmıştır. Makale Kâfiyeci’nin hayatı ve eserlerini araştırır. Yazdığı eserler, onun, Arap dili, Arap grameri, belagat, tarih metodolojisi, hadis ve usulü, tefsir ve usulü, fıkıh ve usulü, kelâm, tasavvuf, dil felsefesi, semantik, metafizik meseleler, geometri, optik ve astronomi gibi konularda uzmanlaştığını göstermektedir. Kâfiyeci en önemli eserlerini Arap dili ve mantık sahalarında yazmıştır. Eserleri içerisinde yaptığı linguistik çözümlemeler, onun yetkin bir dil alimi olduğunu göstermektedir. Kâfiyeci eserlerini yazarken (...) linguistik, mantıki, ilmi ve felsefi bir yöntem kullanmıştır. Araştırmada, Kâfiyeci’nin, hayatı boyunca en az yüz on iki eser yazdığı tespit edilmiştir. Bu eserlerden, doksan altısının adı belirlenmiştir. Kâfiyeci’nin eserleri konularına göre şöylece tasnif edilebilir: Arap Dili ve Dil Felsefesi: Şerḥu’1-i‘râb ‘an Ḳavâ‘idi’l-i‘râb, el-Unmûzec fî baḥsi’l-isti‘âre, Şerḥu’l-isti‘âre, Risâle fî Kâne Zeydun Ḳâimen, Risâle fî Zeydun Ḳâimun, Risâle fi’l-mebâḥisi’l-mute‘alliḳa bi Emmâ, Nuzhetu’l-mu‘rib fi’l-Meşrıḳ ve’l-Maġrib, Nuzhetu’l-aṣhâb fî mes’eleti’l-kuhl, Remzu’l-esrâr, en-Niseb li ehli’l-edeb, el-İşrâḳ fî meratibi’ṭ-ṭıbâk, Ḥâşiye ‘ale’l-Muṭavvel, Risâle fî “men” eş-şartiyye. Mantık: et-Terġîb fî keşfi rumûzi’t-Tehẕîb. Tarih Metodolojisi: el-Muḫtaṣaru’l-mufîd fî ‘ilmi’t-târîḫ, en-Naṣru’l-ḳâhir ve’l-fetḥu’ẓ-ẓâhir. Akâid ve Kelâm: Envâru’ṣ-Ṣa ‘âde, Şerḥu’l-‘Aḳâidi’l-‘Aḍuḍiyye, Hâşiye ‘alâ Şerḥi’l-‘Aḳâidi’l-‘Aḍuḍiyye, Şerḥu’l-Mevâḳıf, Telḫîsu Şerḥi’l-Mevâḳıf, el-Hidâye li beyâni’l-ḫalḳ ve’t-tekvîn, Risâle fî beyâni’l-mu‘cizât, Nehru’l-hayât fî ma‘rifeti’ṣ-ṣıfât, ‘İlmu’l-kumûn ve’l-burûz, Risâle fî Kavlihi Teâlâ “Lâ ye’menu mekr Allâhi ille’l-kavmu’l-hâsirûn” [A‘râf, 7/99]. Kur’ân, Kırâat ve Tefsir Bilimleri: Şerḥu’l-İsti‘âẕe ve’l-Besmele, Ḥâşiye ‘ale’l-Keşşâf, Şerḥ fî muḥâkemât beyne’l-mutekellimîn ‘ale’l-Keşşâf, Ḥâşiye ‘alâ Tefsîri’l-Beyḍâvî, eẓ-Ẓafer ve’l-ḫalâṣ, et-Teysîr fî ḳavâ‘idi ‘ilmi’t-tefsîr, Risâle fî ‘ilmi’t-tefsîr ve vucûhi’l-ḳırâat, Keşfu’n-niḳâb li’l-aṣḥâb ve’l-aḥıbbe fî ‘icâzi’l-Ḳur’ân, et-Temhîd fî şerḥi’t-taḥmîd, Ḳarâru’l-vecd fi şerḥi’l-ḥamd, el-İlmâ‘ bi ifâdeti Lev li’l-imtinâ‘ fî ḳavlihi Te‘âlâ “Lev kâne fîhimâ âlihetun illallâh le fesedetâ” [Enbiyâ, 21/22], Cevâb ‘an suâl fî ḳavlihi Te‘âlâ “Ve’n-Necmi iẕâ hevâ” [Necm, 62/1], Nuzhetu’l-iḫvân fî tefsi¬ri ḳavlihi Te‘âlâ “Yâ Lûṭu İnnâ Rusülü Rabbike” [Hûd, 11/81], Neylu’l-merâm fi tefsîri ḳavlihi Te‘âlâ “Ve Mâ Rabbuke bi Ẓallâm” [Fussilet, 41/46], el-Bişâra fî ḳavlihi Te‘âlâ “Fe’tû bi sûratin min mislih” [Bakara, 2/23], Nuzhetu’l-aṣḥâb, el-Ġurratu’l-vâḍıḥa fî tefsîri’l-Fâtiḥa, Zeḫîratu’l-ḳaṣr fî tefsiri Sûrati’l-‘Aṣr. Hadis Bilimleri: Muḫtaṣar fî ‘ulûmi’l-ḥadîs, Menbe‘u’d-durer fî ‘ilmi’l-eser, Ḫulâṣatu’l-‘aḳvâl fî ḥadîsi “İnneme’l-a‘mâlu bi’n-niyyât”. Fıkıh, Usûl ve Fetva: Risale fî mesʹeleti’l-istisnâʹ, Ḥâşiye ‘ala şerḥi’l-Hidâye, Vecîzu’n-niẓâm fî ıẓhâri mevâridi’l-aḥkâm, el-Feraḥ ve’s-surûr fî beyâni’l-meẕâhîbi’l-erba‘ati fi’l-‘uṣûr, Neşâtu’ṣ-ṣudûr fî şerḥi Kitâbi’l-Feraḥ ve’s-surûr, et-Taḥrîr lima zukira fi’d-Dureri’l-manẓûm, Muhâveratu’l-fıḳhiyye, Risâle fi’stiḳbâli’l-Ḳıble, Telḫîṣu’l-Câmi‘i’l-kebîr ve’l-Mecma‘, Kitâbu’t-Teẕkira/Risâle fi’l-vaḳf, Ravḍatu’ẓ-ẓahira en-nâfi‘a fî’d-Dünya ve’l-Âḫira, Kitâbu’l-muḫtasar fî ‘ilmi’l-istibdâl, er-Remz fî ‘ilmi'l-istibdâl, el-Kâfî fî beyâni’ṣ-ṣaffi’ṭ-ṭavîli’l-mustaḳîm, Risâle fî ṭabaḳâti’l-butûn li beyâni’l-vaḳf ‘alâ evlâdi’l-evlâd, el-İḥkâm fî ma‘rifeti’l-Eymân ve’l-aḥkâm, Ferâid baḥri’l-fevâiḍ. Siyasetname, Adap ve Ahlak: Niẓâmu ḳalâidi’l-‘ıḳyân, Seyfu’l-kuḍât ‘ale’l-buġât, Seyfu’l-mulûk ve’l-ḥukkâm el-murşid lehum ilâ sebîli’1-Haḳḳ ve’1-aḥkâm, Seyfu’l-ḥaḳḳ ve’n-nuṣra ‘alâ riḳâbı ehli’l-baġy ve’l-fitne, ed-Durratu’l-‘âliyetu’l-ġâliye. Metafizik, Tasavvuf ve Felsefe: el-Kâfî eş-şâfî, Kitâbu’r-ravḥ fî ‘ilmi’r-rûḥ, Kitabu’r-raḥme fî beyânı aḥvâli ‘alemi’l-berzâḫ, en-Nuzhe fî ravḍati’r-rûḥ ve’n-nefs, Neşâtu’l-bişâra bi’l-işâra fî kıbleti’l-ervâh, Ḳıbletu’l-ervâḥ fi’t-taṣavvuf, Nuzhetu’l-ervâḥ ve ġıbṭatu’l-eşbâḥ fi’t-taṣavvuf, Menâzilu’l-ervâḥ fi’t-taṣavvuf, ‘Ikdu’l-ferâid min baḥri’l-fevâiḍ fî şerḥi Esmâillâhi’l-husnâ, el-Unsu’1-enîs fî ma‘rifeti şeʹni’n-nefsi’n-nefîs. Siyer, İrşat ve Fazâil: Zeynu’l-feraḥ bi mîlâdi’n-Nebî (s.a.v.), Muḫtasar fî ‘ilmi’l-irşâd, Risâle fi’l-maḥabbe, Ḳalâidu’l-‘ıḳyân fî baḥri faḍâili Raceb ve Şa‘bân: Fen Bilimleri: Ḥallu’l-işkâl fî mebâhisi’l-eşkâl fi’l-hendese, Şerḥu’l-caġmînî fi’l-heyʹe, er-Remz fî ‘ilmi’l-uṣturlâb. Diğer Eserleri: Remzu’l-ḫıṭâb bi raşḥi’l-‘ıyâb, Benâtu’l-efkâr fî şeʹni’l-‘i‘tibâr, Keşfu’l-vucûhi’l-ḥasene’l-mestûra bi’n-niḳâb li faḍlillâhi’l-Cevâdi’l-Vehhâb, el-Keşf ‘an şerafi Ehli’l-Beyti’l-kirâm li efḍali’r-rusuli ve’l-enâm, ‘Iḳdu’l-ferâid min taḥrîri’l-fevâid, Risâle mustaḳılle bi ‘ilmi’n-nefs ve vucûhi’l-ḳırâa, Cevâbu’l-enẓâr, Ḥusnu’l-ḫıtâm li’l-merâmi min haẕa’l-kelâm, Ḫıtâmu’l-misk. (shrink)

In this article, I revisit Landau and Jackendoff's () paper, “What and where in spatial language and spatial cognition,” proposing a friendly amendment and reformulation. The original paper emphasized the distinct geometries that are engaged when objects are represented as members of object kinds, versus when they are represented as figure and ground in spatial expressions. We provided empirical and theoretical arguments for the link between these distinct representations in spatial language and their accompanying nonlinguistic neural representations, emphasizing the “what” (...) and “where” systems of the visual system. In the present paper, I propose a second division of labor between two classes of spatial prepositions in English that appear to be quite distinct. One class includes prepositions such as in and on, whose core meanings engage force-dynamic, functional relationships between objects, with geometry only a marginal player. The second class includes prepositions such as above/below and right/left, whose core meanings engage geometry, with force-dynamic relationships a passing or irrelevant variable. The insight that objects’ force-dynamic relationships matter to spatial terms’ uses is not new; but thinking of these terms as a distinct set within spatial language has theoretical and empirical consequences that are new. I propose three such consequences, rooted in the fact that geometric knowledge is highly constrained and early-emerging in life, while force-dynamic knowledge of objects and their interactions is relatively unconstrained and needs to be learned piecemeal over a lengthy timeline. First, the two classes will engage different learning problems, with different developmental trajectories for both first and second language learners; second, the classes will naturally lead to different degrees of cross-linguistic variation; and third, they may be rooted in different neural representations. (shrink)

Origin of Geometry and Origin of the Work of Art: Reduction, Truth, and History in Husserl and Heidegger Husserl’s The Origin of Geometry and Heidegger’s The Origin of the Work of Art were surprisingly developed in parallel and deal with the same subject, the question of the origin: of a science, in one case, and of the artwork, in the other. From their comparison, the different conception that Husserl and Heidegger have of origin and truth clearly emerge. It (...) concerns what Heidegger identified as the real point of his divergence from Husserl, when they tried to write the entry “Phenomenology” for the Encyclopaedia Britannica, namely the method of reduction. For Husserl, the reduction opens up a material content, an essence, which ensures the persistence of truth in historical transmission, beyond linguistic variability, as happens in geometry; for Heidegger, reduction brings to light the ontological and, we might say, grammatical condition of truth, that is, its being an event and giving rise to an original syntax, to a world. This is the function of the work of art, whose essence consists in its mere being-made, in its being, that is, a work: the actualization of a possibility. (shrink)

What aspects of spatial relations influence speakers’ choice of locative? This article presents a study of static spatial descriptions from 24 languages. The study reveals two kinds of spatial terms evident cross‐linguistically: specific spatial terms and general spatial terms (GSTs). Whereas specific spatial terms—including English prepositions—occur in a limited range of situations, with concomitant specificity in their meaning, GSTs occur in all spatial descriptions (in languages that employ them). Because of the extreme differences in range of application, the two are (...) considered separately. A multidimensional scaling analysis is used with specific spatial terms to extract statistically valid similarities across the languages sampled. For GSTs, which have not been previously analyzed in the literature, a semantic analysis is proposed and experimentally validated. The results suggest the importance of geometry, function, and qualitative physics to the meanings of both kinds of spatial terms, although the details differ. (shrink)

Cross-linguistic strategies for mapping lexical and spatial relations from body partonym systems to external object meronymies (as in English ‘table leg’, ‘mountain face’) have attracted substantial research and debate over the past three decades. Due to the systematic mappings, lexical productivity and geometric complexities of body-based meronymies found in many Mesoamerican languages, the region has become focal for these discussions, prominently including contrastive accounts of the phenomenon in Zapotec and Tzeltal, leading researchers to question whether such systems should be (...) explained as global metaphorical mappings from bodily source to target holonym or as vector mappings of shape and axis generated “algorithmically”. I propose a synthesis of these accounts in this paper by drawing on the species-specific cognitive affordances of human upright posture grounded in the reorganization of the anatomical planes, with a special emphasis on antisymmetrical relations that emerge between arm-leg and face-groin antinomies cross-culturally. Whereas Levinson argues that the internal geometry of objects “stripped of their bodily associations” (1994: 821) is sufficient to account for Tzeltal meronymy, making metaphorical explanations entirely unnecessary, I propose a more powerful, elegant explanation of Tzeltal meronymic mapping that affirms both the geometric-analytic and the global-metaphorical nature of Tzeltal meaning construal. I do this by demonstrating that the “algorithm” in question arises from the phenomenology of movement and correlative body memories—an experiential ground which generates a culturally selected pair of inverse contrastive paradigm sets with marked and unmarked membership emerging antithetically relative to the transverse anatomical plane. These relations are then selected diagrammatically for the classification of object orientations according to systematic geometric iconicities. Results not only serve to clarify the case in question but also point to the relatively untapped potential that upright posture holds for theorizing the emergence of human cognition, highlighting in the process the nature, origins and theoretical validity of markedness and double scope conceptual integration. (shrink)

This is one of those cases to which Dr. 8 oodhouse's remark applies with all its force, that a method which leads to true results must have its logic — H.S Smith (" On Some of the Methods at Present in Use in Pure Geometry," p. 6) A goodly amount of modern metaphysics has concerned itself, in one form or another, with the question: what attitude should we take in regard to a language whose semantic underpinnings seem less than (...) certain? Since much of standard English is surely of this type, many philosophers have felt driven to extreme and unpalatable resolutions — wholesale rejection of attractive logical rules, radical "antirealism", "minimalism" about truth, and so forth, Scientists, however, have long dealt with analogous foundational insecurities and, under the duress of practicality, have framed a set of shifting attitudes towards language and reasoning that seem far more commonsensical than the extreme positions to which pure philosophy has often been led. Following this lead, this essay will recommend a more nuanced approach to linguistic insecurity that is neither naively optimistic nor mired in the gloomy sloughs of antirealism. (shrink)

Derrida's first book-length work, The Problem of Genesis in Husserl's Philosophy , was originally written as a dissertation for his diplôme d'etudes superieures in 1953 and 1954. Surveying Husserl's major works on phenomenology, Derrida reveals what he sees as an internal tension in Husserl's central notion of genesis, and gives us our first glimpse into the concerns and frustrations that would later lead Derrida to abandon phenomenology and develop his now famous method of deconstruction. For Derrida, the problem of genesis (...) in Husserl's philosophy is that both temporality and meaning must be generated by prior acts of the transcendental subject, but transcendental subjectivity must itself be constituted by an act of genesis. Hence, the notion of genesis in the phenomenological sense underlies both temporality and atemporality, history and philosophy, resulting in a tension that Derrida sees as ultimately unresolvable yet central to the practice of phenomenology. Ten years later, Derrida moved away from phenomenology entirely, arguing in his introduction to Husserl's posthumously published Origin of Geometry and his own Speech and Phenomena that the phenomenological project has neither resolved this tension nor expressly worked with it. The Problem of Genesis complements these other works, showing the development of Derrida's approach to phenomenology as well as documenting the state of phenomenological thought in France during a particularly fertile period, when Levinas, Sartre, Merleau-Ponty, Ricoeur, and Tran-Duc-Thao, as well as Derrida, were all working through it. But the book is most important in allowing us to follow Derrida's own development as a philosopher by tracing the roots of his later work in deconstruction to these early critical reflections on Husserl's phenomenology. "A dissertation is not merely a prerequisite for an academic job. It may set the stage for a scholar's life project. So, the doctoral dissertations of Max Weber and Jacques Derrida, never before available in English, may be of more than passing interest. In June, the University of Chicago Press will publish Mr. Derrida's dissertation, The Problem of Genesis in Husserl's Philosophy , which the French philosopher wrote in 1953-54 as a doctoral student, and which did not appear in French until 1990. From the start, Mr Derrida displayed his inventive linguistic style and flouting of convention."--Danny Postel, Chronicle of Higher Education. (shrink)

A complete recursive description of noetherian linear _KL_-algebras is given. _L_-algebras form a quantum structure that occurs in algebraic logic, combinatorial group theory, measure theory, geometry, and in connection with solutions to the Yang-Baxter equation. It is proved that the self-similar closure of a noetherian linear _KL_-algebra is determined by its partially ordered set of primes, and that its elements admit a unique factorization by a decreasing sequence of prime elements.

The chapters in this timely volume aim to answer the growing interest in Arthur Schopenhauer’s logic, mathematics, and philosophy of language by comprehensively exploring his work on mathematical evidence, logic diagrams, and problems of semantics. Thus, this work addresses the lack of research on these subjects in the context of Schopenhauer’s oeuvre by exposing their links to modern research areas, such as the “proof without words” movement, analytic philosophy and diagrammatic reasoning, demonstrating its continued relevance to current discourse on logic. (...) -/- Beginning with Schopenhauer’s philosophy of language, the chapters examine the individual aspects of his semantics, semiotics, translation theory, language criticism, and communication theory. Additionally, Schopenhauer’s anticipation of modern contextualism is analyzed. The second section then addresses his logic, examining proof theory, metalogic, system of natural deduction, conversion theory, logical geometry, and the history of logic. Special focus is given to the role of the Euler diagrams used frequently in his lectures and their significance to broader context of his logic. In the final section, chapters discuss Schopenhauer’s philosophy of mathematics while synthesizing all topics from the previous sections, emphasizing the relationship between intuition and concept. -/- Aimed at a variety of academics, including researchers of Schopenhauer, philosophers, historians, logicians, mathematicians, and linguists, this title serves as a unique and vital resource for those interested in expanding their knowledge of Schopenhauer’s work as it relates to modern mathematical and logical study. (shrink)

The book reconstructs the history of Western ethics. The approach chosen focuses the endless dialectic of moral codes, or different kinds of ethos, moral doctrines that are preached in order to bring about a reform of existing ethos, and ethical theories that have taken shape in the context of controversies about the ethos and moral doctrines as means of justifying or reforming moral doctrines. Such dialectic is what is meant here by the phrase ‘moral traditions’, taken as a name for (...) threads of moral discourse, made in turn of other interwoven threads, including transmission of shared codes, appeals to reform of prevailing custom, rational argument about the justification of some precept on the basis of some shared general teaching or principle, and rational argument about the ultimate basis for principles and justification of authoritative teaching. That is, the approach adopted to the reading of ethical texts depends on a firm belief in the fact that philosophers hardly created any ethical doctrine out of nothing. The main point this book tries to highlight is how different philosophical theories emerged and followed each other as a result of attempts at accounting for what was going on in the world of moral traditions. Changes were propelled by controversies between different schools, and highly abstract arguments were the unintended effects of moves made by controversialists forced to transform (and occasionally to turn upside down) their own doctrines in order to face the challenge posed by other arguments. This is the reason why the book examines not only texts that already enjoy pride of place in the history of philosophy (Aristotle, Kant, Hegel), but also other texts usually treated in the histories of religions (the Bible, the Talmud, the Quran), and others considered to be much less philosophical (Plutarchus, Pufendorf). -/- 1. Plato and a response to ethical scepticism. Two different traditions of morality in VI-V century Greece are described. The birth of philosophical questioning of traditional morality and temporal and spatial variation of custom is described within the context of the v century crisis, the demise of traditional aristocratic and tyrannical rule and the birth of democracy. Two conflicting answers to the challenge are reconstructed, namely conventionalist or immoralist theories formulated by the Sophists and the eudemonist and intellectualist Socratic theory. Plato’s own reformulation of Socrates answer to the Sophists is reconstructed. His psychological views, his classification of the four cardinal virtues and his political theory are described as parts of a unitary system, culminating in an extremely realist moral ontology identifying the idea of the good with the essence of the (moral and extra-moral) world itself. -/- 2. Aristotle and the invention of practical philosophy. Aristotle’s invention of practical philosophy as a field separated from first philosophy is shown to be an implication of his break with Plato moral ultra-realism. Aristotle’s agenda in his moral works is arguably dependent on a polemical intention, namely dismantling Socratic intellectualism. The semi-inductive or virtuously circular method of practical philosophy is illustrated, starting with the received opinions of the better and wiser individuals and trying dialectically to sift what is left of mistake and inconsistence in such opinions, finally trying to correct mistakes and make the overall practical science more consistent. The chapter illustrates then the relationship of individual ethics, or ‘monastics’, with the art and the science of the pater familias, or ‘economics’, and the science of the ruler and citizen, or politics. The nature of virtues, or better, excellences of character, is discussed, highlighting the basic role of hexis, or ‘disposition’. Prudence, or better practical wisdom, is the focus of the chapter. Its relationship with bouleusis or deliberation is examined, and its autonomous status vis-à-vis theoretical knowledge is stressed. -/- 3. Diogenes and philosophy as a way of life. The chapter provides an overview of Hellenistic ethics, which almost amounts to Hellenistic schools of philosophy, in so far as ‘philosophy’ became in these centuries primarily the name for a way of life. The typical character of the Cynical movement is highlighted, that of a school of life, not a school aimed at providing any kind of intellectual training was to be provided. -/- 4. Epicurus and ethics as self-care. The peculiar character of the Epicurean school is described, a combination of a science of well-being aiming, more than at pleasure as in the popular view, at reduction of useless suffering, of unnecessary needs, and at a balanced selection of pleasures of the best and most durable kind. -/- 5. Epictetus and ethics as therapy of the passions. The various phases of Stoicism are described, and the shifting place given to ethics in the Stoic system of idea, culminating in the paradoxical view of ethics, its impossibility in principle notwithstanding, as the only truly significant and necessary part of philosophy. Cicero is treated, showing how his own synthesis of various Hellenistic trends is as a truly philosophical enterprise, deserving serious consideration after one or two centuries when he was confined to the role of literate. Epictetus is chosen as the best example of what the Stoic tradition could yield, an art of living based on sophisticated introspection, in turn aimed at making a kind of cognitive therapy possible, dismantling obnoxious passions at their root by systematically correcting false representations. -/- 6. Philo and the reconciliation of Torah and Platonism. A reconstruction of basic ideas from a few books of the Hebrew Bible is provided, starting with the Prophetic tradition and the focus on God’s mercy as the source of motivation and standard for human behaviour. Then a comparative analysis is undertaken of a parallel tradition, namely the three codifications of the Torā (Law or, better, Instruction), highlighting how a core of moral ideas may be recognized as a basis and preamble of codification of civil law, cultural practice, and regulation of ritual purity. The importance of Leviticus is stressed as the turning point when emphasis mercy, typical of the Prophetic tradition, is combined with the legal tradition, yielding the change in sensibility of the Second Temple time. Philo of Alexandria is described as one of the three leading figures – together with Rabbi Hillel and Rabbi Yeshua – at the apex the emergence of the mentioned new sensibility, gradually including mercy as an essential part of justice and establishing the starting point of both the Rabbinic and the Christian tradition. This consists precisely in the precept of one’s neighbour’s love or of the golden rule, two eventually identical precepts whose meaning is arguably more sober and sensible than the long-lasting Christian tradition deriving from John and Augustine has made us believe and no novelty vis-à-vis so-called Ancient-Testament teachings. -/- 7. Augustine and Christianity as Neo-Platonism. The first section examines the moral doctrines of so-called 'ethical Treaties' from the Talmud, a group of treaties, among which the best known is Pirqé Avot, that were left out of the six "orders" of the canon as they did not fit in any of the six groups of issues ritual or legal on which the division was based. According to Maimonides, their peculiar theme is provided by the Deòt, 'opinions', i.e., mental dispositions, that is, the translation of the Greek term hexeis and Arab akhlak (in turn providing in this language the name for ethics as such). The three topics I reconstruct are: i) the notion of Torah: The Torah is understood as the world order itself, or as the ‘Wisdom’ that existed even before creation and was ‘the tool by which the world was built’; however, the Torah is an earthly and human entity, as it was "received" by humans, and from that very moment belongs to them; ii) the relationship between love of God and love of neighbour; the treaties require us to study and practice the Torah ‘for its own sake’, that is, require us to act out of love, not out of fear or hope of reward; iii) the idea of sanctification of daily life: having disappeared with the destruction of the Temple the possibility of any conflict between liturgical service and everyday life, the latter is assumed to be in itself divine service: to give food to the poor has the same value as sacrifices in the Temple, and as an implication, the insistence become recurrent on the goodness of created things in themselves along with a polemic against ascetic currents. The conclusions drawn are: i) the moral teachings of the Talmud and those of Yeshua are, rather than similar, virtually identical; one may safely say that the precept of love and the golden rule are central ones for all Talmudic rabbis, that mercy plays an indispensable role alongside with justice, and the latter is not a different thing from one’s neighbour’s love; ii) a peculiarity of Talmud rabbis facing Yeshua is the idea of study as worship, and knowledge as a source of justice; but this is an idea of Judaism after the Temple's destruction that cannot be attributed to the Pharisees of Yeshua’s time; iii) the relation of study and practice in the Talmud parallels that between faith and deeds in Paul's epistles, that is, respectively faith or learning are a necessary and sufficient condition to be recognized as righteous , but deeds are the inevitable effect of either faith or learning. The sayings ascribed to Yeshua are examined first, yielding the conclusion that a close equivalent may be found for every saying in Talmudic literature and yet the whole is ascribed to one rabbi, with rather consistent stress on God’s mercy and unconditional forgiving as the mark of true imitation of God. Thus, Yeshua’s teaching is pure Judaism. The third section describes briefly the galaxy of Gnostic currents and Manichaeism, trying to sketch the profile of moral teachings resulting from an encounter of Asian spiritual traditions, Hellenistic lore and sparks of teachings from apocalyptic Jewish currents. The last section summarizes the turbulent history of several encounters between Christian currents and Hellenistic philosophical schools. The first one was with late Cynicism. Recent, rather controversial, literature discovered the jargon and a few topics from the Stoic-Cynical popular philosophy in a few books from the New Testament itself. This, far from proving that Rabbi Yeshua had been influenced by cynic preachers, is a proof of the necessity felt by Christian preachers to translate the original ‘Christian’ teaching into a Greek lexicon deeply impregnated with cynic terminology. The second was with Platonism, yielding the mild and temperate moral teaching of Clemens of Alexandria, teaching the sanctity of nature and the human body, the joy of moderate fruition of ‘natural’ kinds of pleasures, and the beauty of the marital life – in short, the opposite of the standard picture of Medieval Christianity. Ambrose of Milan brings about a different kind of synthesis, namely with Middle Platonism, where Stoic themes prevail. The most shocking case is Augustine, where an early Manichean education is overcome in a former phase by a synthesis of Plotinian Neo-Platonism and Christian preaching, yielding a sustained polemic with the Manicheans and rather optimistic views on life and Creation, the body and sexuality, and Hebrew-Judaic tradition not far from Clemens of Alexandria. In a later phase, occasioned by controversy on the opposite front, with such Christian currents as the Pelagians and Donatists, Augustine comes back to heavily anti-Judaic and world-denying ascetic attitudes where is earlier Manichean upbringing seems to emerge again. The tragedy of medieval Christianity will be the later Augustine’s overwhelming influence. A final section is dedicated to the monastic tradition where a curious mixture of world-denying asceticism with an astonishingly penetrating ‘science’ of introspection emerges. -/- 8. Al Farabi and the reconciliation of Islam and Platonism. The Qurān and the qadit, that is, collections of sayings ascribed to the prophet Muhammad contain a wealth of precepts and catalogues of virtues mirroring moral contents from the Jewish and the Christian traditions, among them the basic notion of imitation of the Deity, where mercy is assumed to be its basic trait. An important tradition within Islam, namely Sufism, stressed to the utmost degree the role of mercy, turning Islam into a mystic doctrine centred on retreat from the world, abandonment to God’s will, and a peaceful and fraternal attitude to our fellow-beings. A secular literary tradition originating in the Sassanid Empire, the literature of advice to the Prince had for a time a widespread influence in the newly constituted Arabic-Islamic commonwealth. In a later phase, the legacy of Hellenic philosophy made its way into the intellectual elite of the Islamic world. The first important legacy was Platonic, and the Republic became the main text for Islamic Platonism. Al-Farabi wrote an enjoyable remake of the Platonic Republic, arguing that the citizen’s virtues were the basis on which the political building needs to rest. The secular lawgiver is enlightened by the light of reason in his everyday practice of governance, but room is made for the Prophet as the voice of revealed truth that confirms and completes the rational truth that philosophy may afford. In a second phase also Aristotelian and Stoic influence were assimilated. Miskawayh’s treatise was the masterwork at the time Arabic philosophy reached its Zenith. It is a treatment of the soul’s diseases and their remedies, combining the Aristotelian doctrine of the golden mean with the Stoic doctrine of the passions and elements of Galenic medicine. Towards the eleventh-twelve centuries a war raged among theologians and philosophers, finally won by the former with disappearance of philosophy as such. The newly established mainstream, yet, was no kind of intolerant fanaticism. It drew from the work of mystic theologian Al-Ghazālī, the best heir of Sufism, teaching a tolerant and peaceful attitude to our fellow-beings and a passive attitude to destiny as an expression of the Divine will. -/- 9. Moshe ben Maimon and the reconciliation of Torah and Aristotelian ethics. The encounter between the Talmudic tradition and Hellenic philosophy had taken place a first time in Alexandria at the time of Philo but the two traditions had parted way again. In fact, the kind of Platonic Judaism founded by Philo survived only within Christianity, in the fourth Gospel and then in writings by Clemens of Alexandria. The Talmudic literature had absorbed just less striking traits from the Hellenic Philosophy, namely an idea of ethics as care of the self and a role for the education of character as propaedeutic to theoretical knowledge. In the Arabic-Islamic world, a second round started when Jewish authors writing in Arabic undertook the task to prove the full compatibility of the tradition deriving from Torah and Platonic philosophy. The culmination of this attempt is provided by Moshe ben Maimon who tried to use Aristotelian ethics as a language into which the teaching from the Pirké Avot could be translated. -/- 10. Thomas Aquinas and the reconciliation of Christian morality and Aristotelian ethics. In the first section the fresh start is described of philosophical ethics in Latin Europe at the turn of the Millennium. In a first phase, Anselm and Peter Abelard built on a Platonic and Augustinian legacy. In this phase. remarkable innovations are introduced, including Abelard’s claim of the obliging character of erring consciousness. In a second phase, the rediscovery of Nicomachean Ethics thanks to Latin translation of Arabic versions gives birth to a new wave of ethical studies, recovering the very idea of Aristotelian practical philosophy, with the potential implication of full legitimization of natural morality, i.e. ethics without Revelation. Aquinas’s ethics is a theological ethics out of which it would be vain to try to extract a self-contained philosophical ethics. His treatment of topics in philosophical ethics, yet, does not boil down to repetition of Aristotelian arguments but is rather a creative reshaping of such arguments. For example, he introduces also in the practical philosophy a first principle parallel to the principle of non-contradiction; and he also carries out a synthesis of Aristotelianism, Stoicism, and neo-Platonism. Even though it is essentially moral theology, Aquinas’s doctrine - unlike Augustine – grants full citizenship to "natural" morality, firstly by rejecting the claim that the corruption of human nature due to the original sin is so radical as to leave "pure nature" incapable of moral goodness. The doctrine is presented in a more sophisticated formulation in a few of the Quaestiones, such as De Malo and De Veritate, in the Summa contra Gentes and in the commentaries to Aristotle than in the famous Summa Theologica, but the latter work includes the only or the largest exposition of some decisive part of the theory. Thus, the Summa Theologica should be read for what it is more than criticized for not being what it was not meant to be. It was not meant to be the brilliant synthesis of all that Reason he had been able to produce with what Revelation had added about which the Neo-Thomists used to dream, but rather a manual for the training of preachers and confessors, where theoretical claims are not too demanding and a few serious tensions are left. Besides a jump between the Prima Secundae and the Secunda Secundae, being the former an essay in virtue theory and the latter a handbook for confessors, the most serious tension is perhaps the one between the ethics of right reason presented in most of Ia-IIae and the eudemonistic ethics developed in quaestiones 1-5 of the same part; the alternative ethical theory which also may be found in Augustine, the Stoic view of a cosmic reason eventually coincident with the moral law, was believed by Anselm (followed by John Duns Scotus and William of Ockham) to be incompatible with eudemonism. It is questionable whether Thomas could reduce the tension proving that it is just an apparent tension, in so far as the right reason and bliss derived from knowledge of God tend to coincide, but this is just a conjecture. Thomas’s ethics is a virtue ethic, not a law-based one, and moral judgment focuses on the virtues, particularly charity, not on the commandments and even less on absolute prohibitions; Thomas, however, would not have considered a drastic alternative between law and the virtues such as the one which has been advanced in late twentieth-century philosophy to be justified. Nonetheless, when it discusses ‘special’ virtues, it ceases to be an ethics of virtue and becomes a disappointing and often contradictory discussion of legal and illegal acts. Such a discussion takes most of the time ‘reasonable’ middle positions on controversial issues but not the alternative approach that Aristotelianism would have made possible; even when some occasional Aristotelian claim shows up, such as money’s barrenness as a reason against usury, this seems to be made by an author who apparently ignores the Aristotelian Thomas of the Prima Secundae. It is an ethic of human autonomy which recognizes the binding character of the individual conscience and, potentially, even a duty to disobey unjust laws. It is true that what Thomas writes in his discussion of death penalty and persecution of heretics is simply disgusting, and yet we should blame Thomas the man, not Thomist ethical theory. Finally, Thomas’s ethics is not ethical ‘absolutism’, as both traditionalist Catholic theologians and secularist enemies of dogmatic morality tend to believe. It is instead an ethic of prudential judgment. Exceptions to this approach – or better results of logical fallacies – are such claims as the absolute character of negative precepts and of the existence of "intrinsically evil acts", claims that simply cease to make sense in the light of Thomas’s own distinction between human act and natural act, carrying consequences Thomas did not live long enough to draw or was not consistent enough to dare to draw. -/- 11. Francisco de Vitoria and casuistry. The first topic illustrated is the discussion about the notion of pura natura, namely the human condition after the original sin and before divine revelation. The core notion was already there in Aquinas but was later developed by Caietanus (Thomas de Vio) with a number of interesting implications: firstly a view of human nature as such much more positive than Augustine’s and his followers’ view, including both Calvinists and Jansenists; secondly, the implication that the philosopher’s morality, as opposed to the Christian’s morality, is a quite respectable kind of morality; thirdly, that theocracy and prosecution of the unfaithful lacked justification, with more far-fetching consequences than Aquinas himself had dared to draw. The second topic is casuistry, a new name for a comparatively older way of thinking, which was already there in late Stoicism and Cicero as well as in the Talmudic literature. This is an approach aimed at yielding probable enough solutions for doubtful cases even in absence of completely safe staring points. The genre developed from medieval reference books for confessors and became by the sixteenth-century a sophisticated tool-box for dealing with various fields of applied ethics. Francisco Vitoria, the main authority of Baroque Scholasticism, was a controversial figure, among the proponents of the new discipline of casuistry and a consistent proponent of more separation between the religious and the civil authority, stricter limits to the legitimacy of war, innate rights of non-Christian nations in the New World based on the notion of pura natura, providing a standard of ‘natural’ goodness, previous to revelation, on which the Indian nations were judged to live in a naturally good condition, provided with the institutions of family, justice, and religion had accordingly a right to full respect by Christian sovereigns. Bartolomé de Las Casas, arguing along similar lines, was the leader of a historical battle in defence of the rights of the Indios. -/- 12. Michel de Montaigne and the art of living. A short description of one of the Renaissance Phyrronism, one of the classical philosophies revived in the fifteenth and sixteenth century. Montaigne combines the sceptical epistemology with an understanding of ethics – indeed of philosophy as a whole – in terms of an art of living inspired by two basic ideas, sagesse, that is, practical wisdom as the only kind of rationality available after theology, science, and tradition have declared bankrupt, and an aesthetic ideal of self-transformation through the art of writing and introspection. -/- 13. Pierre Nicole and neo-Augustinianism. The chapter illustrates first the vicissitudes of Augustinianism newly born after the late medieval triumph of Thomist Aristotelianism in the alternative Protestant and Catholic versions processed by the Calvinists and the Jansenists. Then it illustrates the doctrines of Pierre Nicole, Blaise Pascal’s best disciple, on the impossibility of introspection, on the ubiquity of self-deception, on the incurably evil character of the passion of love, and on the two moralities, the one of the man of the world, the morality of honesty which is indispensable for granting peace and order but devoid of any true value for eternal salvation, since the same external behaviour may be prompted by opposite motivations, and the morality of charity, the only true one but also useless to society. The third topic is Pierre Malebranche’s view of self-deception as a ubiquitous phenomenon accounting for human action and responsibility and his reformulated theory of self-love, no less ubiquitous than for Nicole and yet not as incurably evil, given the distinction between morally indifferent and even necessary amour de soi and vicious amour proper, a distinction on which the whole of Rousseau’s moral and political theory rests. -/- 14. Samuel von Pufendorf and the new science of morality. The chapter is dedicated to the discovery of the idea of a ‘new moral science’. Hugo Grotius is discussed first highlighting the real character of his project, much closer to the Thomist idea of a natural law accessible to non-corrupted human reason when human beings are living in a state of pura natura. Then Hobbes’s combination of scepticism, voluntarism, and conventionalism is described and both the continuity with Grotius’s new science, in the search for non-revealed rational morality, and the break with him, in the adoption of voluntarism and refusal of an intellectualist view of natural law, are illustrated. Pufendorf’s work is illustrated as a synthesis of the two previous attempts and the – up to Schneewind underestimated – paradigmatic example of the new science of morality. -/- 15. Richard Cumberland and consequentialist voluntarism. The chapter gives an overview of eighteenth-century Anglican ethics, noticing how the Cambridge tradition gave special weight to natural theology as opposed to positive or revealed theology – and how two Cambridge fellows, John Gay and Thomas Brown, elaborated on Cumberland’s (and Malebranche’s, as well as Leibniz’s) strategy for finding a third way between intellectualist view and voluntarist view of the laws of nature. The result of their elaboration was a kind of a rational-choice account of the origins of natural laws, where a law-giver God chooses among a number possible sets of laws on a maximizing criterion, and God’s maximandum is happiness for his creatures. The chapter notices also how such a solution aimed at solving at once the problem of evil and that of the foundation of moral obligation by proving how God’s choice was justified as far as it was the one minimising the amount of suffering in the world. 16. Richard Price and intuitionism. The chapter describes first the doctrines of the Cambridge Platonists, an example of hyper-rationalist reaction to Calvinism. Secondly it describes the sophisticated and universally ignored – from Sidgwick to Anscombe –version of what was later labelled ‘ethical intuitionism’ – showing how it escapes familiar objections and misrepresentations of intuitionism – from Mill to Rawls – in grounding its argument on transcendental arguments while carefully avoiding recourse to introspection and psychological evidence, which has been taken as a too easy target by critics of intuitionism. Thirdly, the chapter discusses Whewell’s ‘philosophy of morality’, as opposed to ‘systematic morality’, not unlike Kant’s distinction between a pure and an empirical moral philosophy. Whewell worked out a systematization of traditional normative ethics as a first step before its rational justification; he believed that the point in the philosophy of morality is justifying a few rational truths about the structure of morality such as to rule hedonism, eudemonism, and consequentialism; yet a system of positive morality cannot be derived solely from such rational truths but requires consideration of the ongoing dialectics between idea and fact in historically given moralities. Whewell’s intuitionism turns out closer to Kantian ethics than commentators have made us believe until now, and quite different from what Sidgwick meant by intuitionism. -/- 17. Adam Smith and the morality of role-switch. The chapter describes first, Hutcheson’s attempt at basing the ‘new science’ of natural law on different bedrock than Pufendorf and the English Platonists, namely a moral sense, a faculty whose existence is assumed to be proved on an empirical basis. The second step is a reconstruction of Hume’s rejoinder in terms of a new science of man including morality on an ‘experimental’ basis, that is, a ‘Newtonian’ hypothetical-deductive approach, distinguished from Hutcheson’s allegedly uncritical descriptive approach to human nature. The third step is a reconstruction of Adam Smith theory of morality understood as emerging from a spontaneous interplay of exchanges of situations (the most basic meaning of the word ‘sympathy’ as construed by this author). -/- 18. Jeremy Bentham and utilitarianism. The bulk of the chapter is dedicated to the doctrines of Jeremy Bentham. The Enlightenment spirit that suggested the idea of a new morality, free from religion, is illustrated. The notion of utility is illustrated as well as the subsequent formulations of the principle of utility. The idea of felicific calculus is discussed, showing how its inner difficulties prompted several reformulations of the principle of utility in order to avoid undesirable implications of proposed formulations. The role of the thesis of spontaneous convergence between interest and duty is discussed, showing how it left numberless questions open, and the distinction between the virtues of prudence, justice and beneficence is described as a way out of the deadlocks of classical utilitarianism. After Bentham, Mill’s reformed utilitarianism is reconstructed, showing how it is a kind of mixed system – as closer to common sense as it gives up Bentham’s claims to consistency and simplicity – resulting as unintended consequences from the controversies in which Mill was too keen to engage. The hidden agenda of the controversy between Utilitarianism and Intuitionism, going beyond the image of the battle between Prejudice and Reason, is described, showing how both competing philosophical outlooks turn out to be more research programs than self-contained doctrinal bodies, and such programs appear to be implemented, and indeed radically transformed while in progress thanks to their enemies no less than to their supporters. -/- 19. Immanuel Kant, practical reason and judgment. The chapter argues that Kant took from Moses Mendelssohn the idea of a distinction between geometry of morals and a practical ethic. He was drastically misunderstood by his followers precisely on this point. He had learned from the sceptics and the Jansenists the lesson that men are prompted to act by deceptive ends, and he was aware that human actions are also empirical phenomena, where laws like the laws of Nature may be detected. His practical ethics made room for judgment as a holistic procedure for assessing the saliency of relevant moral qualities in one given situation; this procedure yields the same results as the geometry of morals does for abstract cases but does so immediately and without balancing conflicting duties with each other, since what makes for the salient quality of a situation is perceived from the very beginning. Kant's practical philosophy is richer than the received image, making room for an ‘empirical moral philosophy’ or moral anthropology including treatment of commerce, needs, value and price, happiness and well-being; the overall social theory and philosophy of history is less different from Adam Smith's than the received image makes believe; the paradox of happiness is central to Kant's philosophy; a distinction between happiness and well-being is clearly drawn; the distinction plays a basic role in establishing a link between the economic and the moral life. -/- 20. Georg Wilhelm Friedrich Hegel and the critique of abstract universalism. The chapter describes first the Romantic movement and the implications some of its concerns, rescuing passions, community, tradition, the individual as the bearer of a unique destiny, and the longing for organic unity between the individual, mankind and nature. Hegel’s contribution is discussed then, highlighting how, on the one hand, he shared a number of these concerns and on the other he had more rationalist leanings. The notion of morality is the pivotal point of the reconstruction, highlighting how Hegel construes this notion as a key to his own diagnosis of the malaise of modernity – the separation of individual and Gemeinschaft – and how his attack on Kant turns around this very idea. -/- 21. Friedrich Nietzsche against Christianity and the Enlightenment. The chapter illustrates first the idea of deconstruction of the back-world of values. Nietzsche claims to be legitimate heir of the French moralists, Leibniz, Kant and Hegel, in so far as he allegedly carries out to its deepest implications their discovery of what lies behind traditional naïve belief in the existence of an objective realm of values just waiting for description by the philosopher. The two exemplars form which genealogy draws inspiration are the classical philologist’s historical reconstructions of lost meanings and the chemist’s decomposition of elements. Then the genealogy of the notions of good and evil carried out in the first dissertation of Genealogy of morals is illustrated with its paradoxical conclusion that will to power is in fact the only ‘genuine’ kind of goodness. The third point illustrated is the dialectic of ascetism and self-realization with its ambiguous outcome. The suggested interpretation of such outcome is that Nietzsche’s normative ethics is a kind of virtue ethics taking an aesthetic ideal as a normative standard -/- 22. George Edward Moore and ideal utilitarianism. The chapter discusses the ideas of common sense and common-sense morality in Sidgwick. I argue that, far from aiming at overcoming common-sense morality, Sidgwick aimed purposely at grounding a consist code of morality by methods allegedly taken from the natural sciences to reach, also in ethics, the same kind of “mature” knowledge as in the natural sciences. His whole polemics with intuitionism was vitiated by the a priori assumption that the widespread ethos of the educated part of humankind, not the theories of the intuitionist philosophers, was what was worth considering as the expression of intuitionist ethics. In spite of a naïve positivist starting-point, Sidgwick was encouraged by his own approach in exploring the fruitfulness of coherentist methods for normative ethics. Thus, Sidgwick left an ambivalent legacy to twentieth-century ethics: the dogmatic idea of a “new” morality of a consequentialist kind, and the fruitful idea that we can argue rationally in normative ethics albeit without shared foundations. Then it reconstructs the background of ideas, concerns and intentions out of which Moore’s early essays, the preliminary version, and then the final version of Principia Ethica originated. I stress the role of religious concerns, as well as that of the Idealist legacy. I argue that PE is more a patchwork of rather diverging contributions than a unitary work, not to say the paradigm of a new school in Ethics. I add a comparison with Rashdall’s almost contemporary ethical work, suggesting that the latter defends the same general claims in a different way, one that manages to pave decisive objections in a more plausible way. I end by suggesting that the emergence of Analytic Ethics was a more ambiguous phenomenon than the received view would make us believe, and that the wheat (or some other gluten-free grain) of this tradition, that is, what logic can do for philosophy, should be separated from the chaff, that is, the confused and mutually incompatible legacies of Utilitarianism and Idealism. -/- 23. Edmund Husserl and the a-priori of action. The chapter illustrates first the idea of phenomenology and the Husserl’s project of a phenomenological ethic as illustrated in his 1908-1914 lectures. The key idea is dismissing psychology and trying to ground a new science of the a priori of action, within which a more restricted field of inquiry will be the science of right actions. Then the chapter illustrates the criticism of modern moral philosophy carried out in the 1920 lectures, where the main target is naturalism, understood in the Kantian meaning of primacy of common sense. The third point illustrate is Adolph Reinach’s theory of social acts as a key the grounding of norms, a view that basically sketches the very ideas ‘discovered’ later by Clarence I. Lewis, John Searle, Karl-Otto Apel and Jürgen Habermas. A final section is dedicated to Nicolai Hartman, who always refused to define himself a phenomenologist and yet developed a more articulated and detailed theory of ‘values’ – with surprising affinities with George E. Moore - than philosophers such as Max Scheler who claimed to Husserl’s legitimate heirs. -/- 24. Bertrand Russell and non-cognitivism. The chapter reconstructs first the discussion after Moore. The naturalistic-fallacy argument was widely accepted but twisted to prove instead that the intuitive character of the definition of ‘good’, its non-cognitive meaning, in a first phase identified with ‘emotive’ meaning. Alfred Julius Ayer is indicated as a typical proponent of such non-cognitivist meta-ethics. More detailed discussion is dedicated to Bertrand Russell’s ethics, a more nuanced and sophisticated doctrine, arguing that non-cognitivism does not condemn morality to arbitrariness and that the project of a rational normative ethics is still possible, heading finally to the justification of a kind of non-hedonist utilitarianism. Stevenson’s theory, another moderate version of emotivism is discussed at some length, showing how the author comes close to the discovery of the role of a pragmatic dimension of language as a basis for ethical argument. Last of all, the discussion from the Forties about Hume’s law is described, mentioning Karl Popper’s argument and Richard Hare’s early non-cognitivist but non-emotivist doctrine named prescriptivism. -/- 25. Elizabeth Anscombe and the revival of virtue ethics. The chapter discusses the three theses defended by Anscombe in 'Modern Moral Philosophy'. I argue that: a) her answer to the question "why should I be moral?" requires a solution of the problem of theodicy, and ignores any attempts to save the moral point of view without recourse to divine retribution; b) her notion of divine law is an odd one more neo-Augustinian than Biblical or Scholastic; c) her image of Kantian ethics and intuitionism is the impoverished image manufactured by consequentialist opponents for polemical purposes and that she seems strangely accept it; d) the difficulty of identifying the "relevant descriptions" of acts is not an argument in favour of an ethics of virtue and against consequentialism or Kantian ethics, and indeed the role of judgment in the latter is a response to the difficulties raised by the case of judgment concerning future action. A short look at further developments in the neo-naturalist current is given trough a reconstruction of Philippa Foot’s and Peter Geach’s critiques to the naturalist-fallacy argument and Alasdair MacIntyre’s grand reconstruction of the origins and allegedly unavoidable failure of the Enlightenment project of an autonomous ethic. -/- 26. Richard Hare and neo-Utilitarianism. The chapter addresses the issue of the complex process of self-transformation Utilitarianism underwent after Sidgwick’s and Moore’s fatal criticism and the unexpected Phoenix-like process of rebirth of a doctrine definitely refuted. A glimpse at this uproarious process is given through two examples. The first is Roy Harrod Wittgensteinian transformation of Utilitarianism in pure normative ethics depurated from hedonism as well as from whatsoever theory of the good. This results in preference utilitarianism combined with a ‘Kantian’ version of rule utilitarianism. The second is Richard Hare’s two-level preference utilitarianism where act utilitarianism plays the function of eventual rational justification of moral judgments and rule-utilitarianism that of action-guiding practical device. -/- 27. Hans-Georg Gadamer and rehabilitation of practical philosophy. The post-war rediscovery of ethics by many German thinkers and its culmination in the Sixties in the movement named ‘Rehabilitation of practical philosophy’ is described. Among the actors of such rehabilitation there were a few of Heidegger’s most brilliant disciples, and Hans-Georg Gadamer is chosen as a paradigmatic example. His reading of Aristotle’s lesson I reconstructed, starting with Heidegger’s lesson but then subtly subverting its outcome thanks to the recovery of the central role of the notion of phronesis. -/- 28. Karl-Otto Apel and the revival of Kantian ethics. Parallel to the neo-Aristotelian trend, there was in the Rehabilitation of practical philosophy a Kantian current. This started, instead than the rehabilitation of prudence, with the discovery of the pragmatic dimension of language carried out by Charles Peirce and the Oxford linguistic philosophy. On this basis, Karl-Otto Apel singled out as the decisive proponent of the linguistic and Kantian turn in German-speaking ethics, worked out the performative-contradiction argument while claiming that this was able to provide a new rational and universal basis for normative ethics. An examination of his argument is some detail is offered, followed by a more cursory reconstruction of Jürgen Habermas’s elaboration on Apel’s theory. -/- 29. John Rawls and public ethics as applied ethics. Rawls’s distinction between a “political” and a “metaphysical” approach to one central part of ethical theory, namely the theory of justice, is interpreted as a formulation of the same basic idea at the root of both the principles approach and neo-casuistry, both discussed in the following chapter, namely that it is possible to reach an agreement concerning positive moral judgments even though the discussion is still open – and in Rawls’ view never will be close – on the basic criteria for judgment. -/- 30. Beauchamp and Childress and bioethics as applied ethics. The chapter presents the revolution of applied ethics while stressing its methodological novelty, exemplified primarily by Beauchamp and Childress principles approach and then by Jonsen and Toulmin’s new casuistry. -/- . (shrink)

This dissertation concerns the nature of spacetime. It is divided into two parts. The first part, which comprises chapters 1, 2, and 3, addresses ontological questions: does spacetime exist? And if so, are there any other spatiotemporal things? In chapter 1 I argue that spacetime does exist, and in chapter 2 I respond to modal arguments against this view. In chapter 3 I examine and defend supersubstantivalism—the claim that all concrete physical objects (tables, chairs, electrons and quarks) are regions of (...) spacetime. Four-dimensional spacetime, we are often told, ‘unifies’ space and time; if we believe in spacetime, then we do not believe that space and time are separately existing things. But that does not mean that there is no distinction between space and time: we still distinguish between the spatial aspects and the temporal aspects of spacetime. The second part of this dissertation, comprising chapter 4, looks at this distinction. How is it made? In virtue of what are the temporal aspects of spacetime temporal, rather than spatial? The standard view is that the temporal aspects of spacetime are temporal because they play a distinctive role in the geometry of spacetime. I argue that this view is false, and that the temporal aspects are temporal because they play a distinctive role in the geometry of spacetime and in the laws of nature. (shrink)

An algebraization of the notion of topology has been proposed more than 70 years ago in a classical paper by McKinsey and Tarski, leading to an area of research still active today, with connections to algebra, geometry, logic and many applications, in particular, to modal logics. In McKinsey and Tarski’s setting the model theoretical notion of homomorphism does not correspond to the notion of continuity. We notice that the two notions correspond if instead we consider a preorder relation \( (...) \sqsubseteq \) defined by \(a \sqsubseteq b\) if _a_ is contained in the topological closure of _b_, for _a_, _b_ subsets of some topological space. A _specialization poset_ is a partially ordered set endowed with a further coarser preorder relation \( \sqsubseteq \). We show that every specialization poset can be embedded in the specialization poset naturally associated to some topological space, where the order relation corresponds to set-theoretical inclusion. Specialization semilattices are defined in an analogous way and the corresponding embedding theorem is proved. Specialization semilattices have the amalgamation property. Some basic topological facts and notions are recovered in this apparently very weak setting. The interest of these structures arises from the fact that they also occur in many rather disparate contexts, even far removed from topology. (shrink)

Inspired by Bermúdez’s notion of proto-logic, I would like to fathom what the true proto-logic could be like. But this will be approached only in a negative way of figuring out what it could not be. I shall argue that it could not be purely deductive by exploiting the recent researches in logic of maps. This will allow us to reorient the search for proto-logic, starting with animal abduction. I will also suggest that proto-logic won’t get off the ground without (...) proto-geometry. These negative results will shed some lights on some further conceptual and historical issues around the language of thought hypothesis to arrive at the true proto-logic. (shrink)

In this paper, we focus on the development of geometric cognition. We argue that to understand how geometric cognition has been constituted, one must appreciate not only individual cognitive factors, such as phylogenetically ancient and ontogenetically early core cognitive systems, but also the social history of the spread and use of cognitive artifacts. In particular, we show that the development of Greek mathematics, enshrined in Euclid’s Elements, was driven by the use of two tightly intertwined cognitive artifacts: the use of (...) lettered diagrams; and the creation of linguistic formulae. Together, these artifacts formed the professional language of geometry. In this respect, the case of Greek geometry clearly shows that explanations of geometric reasoning have to go beyond the confines of methodological individualism to account for how the distributed practice of artifact use has stabilized over time. This practice, as we suggest, has also contributed heavily to the understanding of what mathematical proof is; classically, it has been assumed that proofs are not merely deductively correct but also remain invariant over various individuals sharing the same cognitive practice. Cognitive artifacts in Greek geometry constrained the repertoire of admissible inferential operations, which made these proofs inter-subjectively testable and compelling. By focusing on the cognitive operations on artifacts, we also stress that mental mechanisms that contribute to these operations are still poorly understood, in contrast to those mechanisms which drive symbolic logical inference. (shrink)

Strict finitism is a minority view in the philosophy of mathematics. In this paper, we develop a strict finite axiomatic system for geometric constructions in which only constructions that are executable by simple tools in a small number of steps are permitted. We aim to demonstrate that as far as the applications of synthetic geometry to real-world constructions are concerned, there are viable strict finite alternatives to classical geometry where by one can prove analogs to fundamental results in (...) classical geometry. We consider this as one of many early steps investigating the extent to which strict finite foundations can be developed for the application of mathematics to the real-world. (shrink)

In this paper we first recall geometrical models of neurogeometical in Lie groups and we show that geometrical properties of horizontal cortical connectivity can be considered as a neural correlate of a geometry of the visual plane. Then we introduce a new model of non isotropic cortical connectivity modeled on statistics of images. In this way we are able to justify oblique phenomena comparable with experimental findings.