A standard interpretation of Plato’s metaphysics holds that sensible particulars are images of Forms. Such particulars are fairly independent, like Aristotelian substances. I argue that this is incorrect: Platonic particulars are not Form images but aggregates of Form images, which are property-instances. Timaeus 49e-50a focuses on “this-suches” and even goes so far as to claim that they compose other things. I argue that Form images are this-suches, which are tropes. I also examine the geometrical account, showing that the geometrical constituents (...) of the elements are also Form images. Thus everything in the sensible world is composed of tropes. (shrink)

Imagine an equilateral triangle “pointing upward”—its horizontal base under its apex angle. A semiotic triangle has the following three “vertexes”: (apex) an expression, (lower-left) one of the expression’s conceptual meanings or senses, and (lower-right) the referent or denotation determined by the sense [1, pp. 88ff]. One example: the eight-letter string ‘coleslaw’ (apex), the concept “coleslaw” (lower-left), and the salad coleslaw (lower-right) [1, p. 84f]. Using Church’s terminology [2, pp. 6, 41]—modifying Frege’s—the word ‘coleslaw’ expresses the concept “coleslaw”, the word ‘coleslaw’ (...) denotes or names the salad coleslaw, and the concept “coleslaw” determines the salad coleslaw—recalling Frege’s principle that sense determines denotation. Church [2, p. 6] wrote: -/- We shall say that a name denotes or names its denotation and expresses its sense. […] Of the sense we say that it determines its denotation, or is a concept of the denotation. -/- Aristotle seems cognizant of distinctions going beyond those in semiotic triangles. The expression Aristotle’s semiotic pyramids seem warranted by Aristotle’s Categories, 1a1: -/- When [two] things have a name (onoma) in common and the concept (logos) of being (ousia) which corresponds to the name in each case is different, they are called same-named (homonuma). Thus, for example, both a man and a picture [of an animal] are called animals. These have only a name in common. In each case the name’s concept of being [an animal] is different; for if one says what being an animal is for each of them, one will give two distinct concepts. -/- Semiotic triangles and pyramids in Aristotle’s logic are compared to those in Church’s [2]. [1] JOHN CORCORAN, Sentence, proposition, judgment, statement, and fact, Many Sides of Logic, College Publications, 2009. [2] ALONZO CHURCH, Introduction to Mathematical Logic, Princeton, 1956. -/- The semiotic pyramid in Categories, 1a1 has a square base under the vertex ‘animal’. On the corners of the square are: the concept “animal”, the concept “animal picture”, the animals, and the animal pictures. The animals are homonymous with the animal pictures. People find Aristotle’s example far-fetched or inept even if the experience of pointing to a picture while saying “That is Tarski” is familiar. Imagine looking at a painting while thinking “That is an animal”. Without putting too fine a point on this, notice that in Aristotle’s sense it is individual things that are homonymous, not words. It would be natural to say also in his sense that two things are homonyms if one is homonymous with the other. In contrast, we use the words homonym and homonymous to relate words that are spelled the same and pronounced the same but have different meaning. Consider the noun ‘center’ and the verb ‘center’. Consider the noun ‘smell’ and the verb ‘smell’. The spelling of two homonyms is an ambiguity, or an ambiguous spelling. We need appropriate adjectives to distinguish the Categorical senses of ‘homonym’ and ‘homonymous’ from the current English sense just mentioned. I propose ‘ontological’ for the sense relating things and ‘linguistic’ for that relating words. Given that all words are things but not all things are words, we ask are words that are linguistically homonymous also ontologically homonymous? END OF POST ABSTRACT. (shrink)

In the heart of southeast Texas, an industrial powerhouse often referred to as the 'Golden Triangle', the oil refineries and petrochemical plants stand as stalwart testaments to the region's economic evolution. Interestingly, before the discovery of oil at Spindletop, the lumber and cattle industries powered this region's economy. A profound shift occurred when the Lucas Gusher, a fountain of oil spurting thousands of feet into the air, struck the lands of Spindletop Hill on January 10, 1901. This remarkable discovery of (...) the Spindletop oilfield on a slat dome formation south of Beaumont, Texas, marked the birth of the modern petroleum industry and fundamentally transformed the region's geopolitical, economic, and cultural landscapes. People from all over the country relocated to Gladys City and Beaumont in search of work and new opportunities, creating a boomtown as the town's population swelled from 10,000 to 50,000. The newfound urbanization brought by the oil industry transformed the town and the surrounding areas, creating a ripple effect across the region that would be felt for generations to come. The surge in oil-driven economic activity transmuted the region into a vibrant nexus of commerce, with the oil industry emerging as a critical architect of growth and advancement. This newfound affluence incited by the petroleum sector ushered in an era of economic diversification, laying the groundwork for industrialization’s further evolution. The Spindletop Oil Fields not only underscored the United States’ ascent as the foremost petroleum-producing nation but also signaled the nation’s initiation into the Petroleum Age. It was Spindletop that solidified its position as America’s first significant oil field, becoming an enduring symbol of its industrial might. By 1985, over 153,000,000 barrels of oil had been produced and extracted from the Spindletop oil fields, underscoring the magnitude of this extraordinary resource. (shrink)

In this paper I contend that the 'superfluity' of triangles is only apparent; all those specified are indeed required for the smallest sub-units, so long as the symmetry of the final body to be constructed is taken into account at earlier stages.

In this paper titled 'The Intersect Point Theorem,' I had performed many mathematical operations on a figure formed by three non-collinear points called a triangle. In this paper Using a concept, when two lines intersect at a common point on one of the segments of the triangle, then their cause is defined. I had tried to keep my work in the ordinary language Of Geometry. All these principles keep me researching various geometrical concepts throughout the year.

Our conscious experiences are qualitative and unitary. The qualitative universals given in particular experiences, i.e. qualia, combine into the seamless unity of our conscious experience. The problematics of quality and cohesion are not unique to consciousness studies. In mathematics, the study of qualities (e.g., shape) resulting from quantitative variations in cohesive spaces led to the axiomatization of cohesion and quality. Using the mathematical definition of quality, herein we model qualia space as a categorical product of qualities. Thus modeled qualia space (...) is a codomain space wherein composite qualities (e.g. shape AND color) of conscious experiences can be valued. As part of characterizing the qualia space, we provide a detailed exemplification of the mathematics of quality and cohesion in terms of the categories of idempotents and reflexive graphs. More specifically, with qualities as commutative triangles formed of cohesion-preserving functors, first we calculate the product of commutative triangles. Next, we explicitly show that the category of idempotents is a quality type. Lastly, as part of showing that the category of reflexive graphs is cohesive, we characterize the adjointness between functors relating cohesive graphs to discrete sets. In conclusion, our category theoretic construction of qualia space is a formalization of the binding of qualitative features (colors and shapes) into the cohesive objects (colored-shapes) of conscious experiences. Compared to the feature-vector accounts of conscious experiences, our product-of-qualities account of consciousness is a substantial theoretical advance. (shrink)

The “language-communication-society” triangle defies traditional scientific approaches. Rather, it is a phenomenon that calls for an integration of complex, transdisciplinary perspectives, if we are to make any progress in understanding how it works. The highly diverse agents in play are not merely cognitive and/or cultural, but also emotional and behavioural in their specificity. Indeed, the effort may require building a theoretical and methodological body of knowledge that can effectively convey the characteristic properties of phenomena in human terms. New complexity approaches (...) allow us to rethink our limited and mechanistic images of human societies and create more appropriate emo-cognitive dynamic and holistic models. We have to enter into dialogue with the complexity views coming out of other more ‘material’ sciences, but we also need to take steps in the linguistic and psycho-sociological fields towards creating perspectives and concepts better fitted to human characteristics. Our understanding of complexity is different – but not opposed – to the one that is more commonly found in texts written by people working in physics or computer science, for example. The goal of this book is to extend the knowledge of these other more ‘human’ or socially oriented perspectives on complexity, taking account of the language and communication singularities of human agents in society. (shrink)

When you hallucinate an object, you are not in the normal sort of concurrent causal sensory interaction with that object. It's standardly further inferred that the hallucinated object does not actually exist. But the lack of normal concurrent causal sensory interaction does not imply that there does not exist an object that is hallucinated. It might be a past‐perceived object. In this paper, I argue that this claim holds for at least some interesting cases of hallucination. Hallucinations generated by misleading (...) cues (e.g. ‘seeing’ Kanizsa triangles), hallucinations of Charles Bonnet Syndrome patients, and dreams are experiences of past‐perceived objects. (shrink)

In the Fifth Meditation, Descartes makes a remarkable claim about the ontological status of geometrical figures. He asserts that an object such as a triangle has a 'true and immutable nature' that does not depend on the mind, yet has being even if there are no triangles existing in the world. This statement has led many commentators to assume that Descartes is a Platonist regarding essences and in the philosophy of mathematics. One problem with this seemingly natural reading is (...) that it contradicts the conceptualist account of universals that one finds in the Principles of Philosophy and elsewhere. In this paper, I offer a novel interpretation of the notion of a true and immutable nature which reconciles the Fifth Meditation with the conceptualism of Descartes' other work. Specifically, I argue that Descartes takes natures to be innate ideas considered in terms of their so-called 'objective being'. (shrink)

Kant argued that Euclidean geometry is synthesized on the basis of an a priori intuition of space. This proposal inspired much behavioral research probing whether spatial navigation in humans and animals conforms to the predictions of Euclidean geometry. However, Euclidean geometry also includes concepts that transcend the perceptible, such as objects that are infinitely small or infinitely large, or statements of necessity and impossibility. We tested the hypothesis that certain aspects of nonperceptible Euclidian geometry map onto intuitions of space that (...) are present in all humans, even in the absence of formal mathematical education. Our tests probed intuitions of points, lines, and surfaces in participants from an indigene group in the Amazon, the Mundurucu, as well as adults and age-matched children controls from the United States and France and younger US children without education in geometry. The responses of Mundurucu adults and children converged with that of mathematically educated adults and children and revealed an intuitive understanding of essential properties of Euclidean geometry. For instance, on a surface described to them as perfectly planar, the Mundurucu's estimations of the internal angles of triangles added up to ∼180 degrees, and when asked explicitly, they stated that there exists one single parallel line to any given line through a given point. These intuitions were also partially in place in the group of younger US participants. We conclude that, during childhood, humans develop geometrical intuitions that spontaneously accord with the principles of Euclidean geometry, even in the absence of training in mathematics. (shrink)

Mathematical realism asserts that mathematical objects exist in the abstract world, and that a mathematical sentence is true or false, depending on whether the abstract world is as the mathematical sentence says it is. I raise two objections against mathematical realism. First, the abstract world is queer in that it allows for contradictory states of affairs. Second, mathematical realism does not have a theoretical resource to explain why a sentence about a tricle is true or false. A tricle is an (...) object that changes its shape from a triangle to a circle, and then back to a triangle with every second. (shrink)

Equality and identity. Bulletin of Symbolic Logic. 19 (2013) 255-6. (Coauthor: Anthony Ramnauth) Also see https://www.academia.edu/s/a6bf02aaab This article uses ‘equals’ [‘is equal to’] and ‘is’ [‘is identical to’, ‘is one and the same as’] as they are used in ordinary exact English. In a logically perfect language the oxymoron ‘the numbers 3 and 2+1 are the same number’ could not be said. Likewise, ‘the number 3 and the number 2+1 are one number’ is just as bad from a logical point (...) of view. In normal English these two sentences are idiomatically taken to express the true proposition that ‘the number 3 is the number 2+1’. Another idiomatic convention that interferes with clarity about equality and identity occurs in discussion of numbers: it is usual to write ‘3 equals 2+1’ when “3 is 2+1” is meant. When ‘3 equals 2+1’ is written there is a suggestion that 3 is not exactly the same number as 2+1 but that they merely have the same value. This becomes clear when we say that two of the sides of a triangle are equal if the two angles they subtend are equal or have the same measure. -/- Acknowledgements: Robert Barnes, Mark Brown, Jack Foran, Ivor Grattan-Guinness, Forest Hansen, David Hitchcock, Spaulding Hoffman, Calvin Jongsma, Justin Legault, Joaquin Miller, Tania Miller, and Wyman Park. -/- ► JOHN CORCORAN AND ANTHONY RAMNAUTH, Equality and identity. Philosophy, University at Buffalo, Buffalo, NY 14260-4150, USA E-mail: [email protected] The two halves of one line are equal but not identical [one and the same]. Otherwise the line would have only one half! Every line equals infinitely many other lines, but no line is [identical to] any other line—taking ‘identical’ strictly here and below. Knowing that two lines equaling a third are equal is useful; the condition “two lines equaling a third” often holds. In fact any two sides of an equilateral triangle is equal to the remaining side! But could knowing that two lines being [identical to] a third are identical be useful? The antecedent condition “two things identical to a third” never holds, nor does the consequent condition “two things being identical”. If two things were identical to a third, they would be the third and thus not be two things but only one. The plural predicate ‘are equal’ as in ‘All diameters of a given circle are equal’ is useful and natural. ‘Are identical’ as in ‘All centers of a given circle are identical’ is awkward or worse; it suggests that a circle has multiple centers. Substituting equals for equals [replacing one of two equals by the other] makes sense. Substituting identicals for identicals is empty—a thing is identical only to itself; substituting one thing for itself leaves that thing alone, does nothing. There are as many types of equality as magnitudes: angles, lines, planes, solids, times, etc. Each admits unit magnitudes. And each such equality analyzes as identity of magnitude: two lines are equal [in length] if the one’s length is identical to the other’s. Tarski [1] hardly mentioned equality-identity distinctions (pp. 54-63). His discussion begins: -/- Among the logical concepts […], the concept of IDENTITY or EQUALITY […] has the greatest importance. -/- Not until page 62 is there an equality-identity distinction. His only “notion of equality”, if such it is, is geometrical congruence—having the same size and shape—an equivalence relation not admitting any unit. Does anyone but Tarski ever say ‘this triangle is equal to that’ to mean that the first is congruent to that? What would motivate him to say such a thing? This lecture treats the history and philosophy of equality-identity distinctions. [1] ALFRED TARSKI, Introduction to Logic, Dover, New York, 1995. [This is expanded from the printed abstract.] . (shrink)

The word ‘equality’ often requires disambiguation, which is provided by context or by an explicit modifier. For each sort of magnitude, there is at least one sense of ‘equals’ with its correlated senses of ‘is greater than’ and ‘is less than’. Given any two magnitudes of the same sort—two line segments, two plane figures, two solids, two time intervals, two temperature intervals, two amounts of money in a single currency, and the like—the one equals the other or the one is (...) greater than the other or the one is greater than the other [sc. in appropriate correlated senses of ‘equals’, ‘is greater than’ and ‘is less than’]. In case there are two or more appropriate senses of ‘equals’, the one intended is often indicated by an adverb. For example, one plane figure may be said to be equal in area to another and, in certain cases, one plane figure may be said to be equal in length to another. Each sense of ‘equality’ is tied to a specific domain and is therefore non-logical. Notice that in every cases ‘equality’ is definable in terms of ‘is greater than’ and also in terms of ‘is less than’ both of which are routinely considered domain specific, non-logical. The word ‘identity’ in the logical sense does not require disambiguation. Moreover, it is not correlated ‘is greater than’ and ‘is less than’. If it is not the case that a certain designated triangle is [sc. is identical to] an otherwise designated triangle, it is not necessary for the one to be greater than or less than the other. Moreover, if two magnitudes are equal then a unit of measure can be chosen and, no matter what unit is chosen, each magnitude is the same multiple of the unit that the other is. But identity does not require units. In this regard, congruence is like identity and unlike equality. In arithmetic, the logical concept of identity is coextensive with the arithmetic concept of equality. The logical concept of identity admits of an analytically adequate definition in terms of logical concepts: given any number x and any number y, x is y iff x has every property that y has. The arithmetical concept of equality admits of an analytically adequate definition in terms of arithmetical concepts: given any number x and any number y, x equals y iff x is neither less than nor greater than y. As Aristotle told us and as Frege retold us, just because one relation is coextensive with another is no reason to conclude that they are one. (shrink)

The original interest of this article lies in existential import. It provides a broader view on the problem by reference to modern, symbolic logic (ch.1). Gradually, however, our interest will change into the amalgamated expressions often used in logic; that is, why are such expressions as “x is a round triangle” applied in logic? We critically discuss this question from an extensional viewpoint, namely model theoretic semantics (ch.2). We also touch on Church’s λ-calculus in the appendix (app.2).

This largely expository lecture deals with aspects of traditional solid geometry suitable for applications in logic courses. Polygons are plane or two-dimensional; the simplest are triangles. Polyhedra [or polyhedrons] are solid or three-dimensional; the simplest are tetrahedra [or triangular pyramids, made of four triangles]. -/- A regular polygon has equal sides and equal angles. A polyhedron having congruent faces and congruent [polyhedral] angles is not called regular, as some might expect; rather they are said to be subregular—a word (...) coined for this lecture. To repeat, a subregular polyhedron has congruent faces and congruent [polyhedral] angles. A subregular polyhedron whose faces are all regular polygons is regular—using standard terminology. -/- Geometers before Euclid showed that there are “essentially” only five regular polyhedra: every regular polyhedron is a tetrahedron (4 faces), a hexahedron or cube (6 faces), an octahedron (8 faces), a dodecahedron (12 faces), or an icosahedron (20 faces). -/- The first question is whether there are subregular polyhedra that are not regular. For example, are there tetrahedra having congruent angles and congruent triangular faces but whose faces are not equilateral triangles? -/- Another question is the classification of subregular polyhedra if they exist. For example, considering the fact that the regular tetrahedra all have equilateral triangles as faces, we ask which triangles other than equilaterals are faces of subregular tetrahedra. Similarly, considering the fact that the regular hexahedra all have squares as faces, we ask which quadrangles other than squares are faces of subregular hexahedra. -/- After introductory remarks that include historical and philosophical points, we concentrate on tetrahedra. A triangle that is congruent to each of the four faces of a tetrahedron is called a generator of the tetrahedron. The main result proved is that every acute triangle is a generator of a subregular tetrahedron. The proof includes an algorithm –implementable with scissors and paper –that constructs from any given acute triangle a subregular tetrahedron whose faces are congruent to the given triangle. -/- Algorithm: Given any acute triangle. Construct a similar triangle whose sides are double the sides of the given triangle. Draw the three lines connecting the three midpoints of the sides (making four triangles congruent to the given triangle—a central triangle surrounded by three peripheral triangles). Make three “hinges” along the lines connecting the midpoints. “Fold” the peripheral triangles together (into a tetrahedron). [LIGHTLY EDITED VERSION OF PRINTED ABSTRACT] Acknowledgements: William Lawvere, Colin McLarty, Irvin Miller, Frango Nabrasa, Lawrence Spector, Roberto Torretti, and Richard Vesley. -/- . (shrink)

Berkeley in his Introduction to the Principles of Human knowledge uses geometrical examples to illustrate a way of generating “universal ideas,” which allegedly account for the existence of general terms. In doing proofs we might, for example, selectively attend to the triangular shape of a diagram. Presumably what we prove using just that property applies to all triangles.I contend, rather, that given Berkeley’s view of extension, no Euclidean triangles exist to attend to. Rather proof, as Berkeley would normally (...) assume, requires idealizing diagrams; treating them as if they obeyed Euclidean constraints. This convention solves the problem of representative generalization. View HTML Send article to KindleTo send this article to your Kindle, first ensure [email protected] is added to your Approved Personal Document E-mail List under your Personal Document Settings on the Manage Your Content and Devices page of your Amazon account. Then enter the ‘name’ part of your Kindle email address below. Find out more about sending to your Kindle. Find out more about sending to your Kindle. Note you can select to send to either the @free.kindle.com or @kindle.com variations. ‘@free.kindle.com’ emails are free but can only be sent to your device when it is connected to wi-fi. ‘@kindle.com’ emails can be delivered even when you are not connected to wi-fi, but note that service fees apply. Find out more about the Kindle Personal Document Service.Berkeley and Proof in GeometryVolume 51, Issue 3RICHARD J. BROOK DOI: https://doi.org/10.1017/S0012217312000686Your Kindle email address Please provide your Kindle [email protected]@kindle.com Available formats PDF Please select a format to send. By using this service, you agree that you will only keep articles for personal use, and will not openly distribute them via Dropbox, Google Drive or other file sharing services. Please confirm that you accept the terms of use. Cancel Send ×Send article to Dropbox To send this article to your Dropbox account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your account. Find out more about sending content to Dropbox. Berkeley and Proof in GeometryVolume 51, Issue 3RICHARD J. BROOK DOI: https://doi.org/10.1017/S0012217312000686Available formats PDF Please select a format to send. By using this service, you agree that you will only keep articles for personal use, and will not openly distribute them via Dropbox, Google Drive or other file sharing services. Please confirm that you accept the terms of use. Cancel Send ×Send article to Google Drive To send this article to your Google Drive account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your account. Find out more about sending content to Google Drive. Berkeley and Proof in GeometryVolume 51, Issue 3RICHARD J. BROOK DOI: https://doi.org/10.1017/S0012217312000686Available formats PDF Please select a format to send. By using this service, you agree that you will only keep articles for personal use, and will not openly distribute them via Dropbox, Google Drive or other file sharing services. Please confirm that you accept the terms of use. Cancel Send ×Export citation Request permission. (shrink)

David Chalmers’ two-dimensionalism is an ambitious philosophical program that aims to “ground” or “construct” Fregean meanings and restore “the golden triangle” of apriority, necessity, and meaning that Kripke seemingly broke. This paper aims to examine critically what Chalmers’ theory can in reality achieve. It is argued that the theory faces severe challenges. There are some gaps in the overall arguments, and the reasoning is in some places somewhat circular. Chalmers’ theory is effectively founded on certain strong philosophical assumptions. It is (...) concluded that it is unclear whether the theory can deliver all it promises. (shrink)

Tripura Upanishad is a minor Shakta or Tantra Upanishad explaining the structure of and meditation on Sri Chakra or Sri Yantra—a diagrammatic representation of the universe through nine interlocking triangles coming out of a central point. To date, there are two English translations of this Upanishad. The first and the earliest, by A G Krishna Warrier done in 1967, is a verse translation and because of the obvious constraints of such translation, fails to explain the intricacies and implied meanings (...) of this cryptic Upanishad. The second translation done many years later, is by Douglas Renfrew Brooks, who has also beautifully translated the commentary on this Upanishad by Bhaskararaya in the book 'The Secret of the Three Cities' published by the University of Chicago Press in 1990. While maintaining a high level of accuracy and bringing out the intricate nuances of this Upanishad, the emphasis of Brooks' translation seems to be more on the geometrical aspects of the Sri Chakra. The present translation emphasises the practice aspect of this Upanishad as derived from Bhaskararaya's commentary. (shrink)

In this article I critically re-examine Julia Kristeva's view that becoming a speaking subject requires psychical matricide: violent separation from the maternal body. I propose an alternative, non-matricidal conception of subjectivity, in part by drawing out anti-matricidal strands in Kristeva's own thought, including her view that early mother–child relations are triangular. Whereas she understands this triangle in terms of a first imaginary father, I re-interpret this triangle using Donald Winnicott's idea of potential space and Jessica Benjamin's idea of an intersubjective (...) space of thirdness. I argue that this space provides a maternal third term: a relation of connection and difference between two, a relation that inherits the affective, mobile, generative qualities of the maternal body as the infant (according to Kristeva) imagines it. This connecting space allows both mothers and children to emerge as subjects in their own right. I then suggest that potential-maternal space expands into language, so that language intrinsically allows the possibility of a speaking position of connection with the mother. Entrance into language need not entail separation or matricide: the problem is not language as such but the particular way that speech and logos have been defined historically. (shrink)

GILBERT JEAN FACCARELLO (Paris, 1950) is professor of economics at Université Panthéon-Assas, Paris, and a member of the Triangle research centre (École Normale Supérieure de Lyon and CNRS). He is presently chair of the ESHET Council (European Society for the History of Economic Thought). He completed his doctoral research in economics at Université de Paris X Nanterre. He has previously taught at the Université de Paris-Dauphine, Université du Maine and École Normale Supérieure de Fontenay/Saint-Cloud (now École Normale Supérieure de Lyon). (...) He is a co-founder of The European Journal of the History of Economic Thought, which he co-edited for 20 years with J. L. Cardoso, Heinz D. Kurz, and A. Murphy. With Alain Béraud, he edited the Nouvelle histoire de la pensée économique (La Découverte, 3 volumes, 1992-2000) and, together with Heinz D. Kurz, he is presently editing a Handbook of the History of Economic Analysis (3 volumes, forthcoming with Edward Elgar). -/- EJPE interviewed Gilbert Faccarello about his research career in the history of economic thought, where he has focused especially on old and new classical and Marxian political economy, and French political economy during the 18th and 19th centuries. G. Faccarello discusses his interest not only in the logical structure and context of the economic ideas of past thinkers but also the links between economic thought, philosophy, and religion. (shrink)

The concept of universal in Aristotle’s philosophy has several aspects. 1) Universal and plurality Aristotle posits universal (καθόλου) versus particular (καθ᾿ ἕκαστον) each covering a range of elements: some elements are universal while others are particulars. Aristotle defines universal as ‘that which by nature is predicated (κατηγορεῖσθαι) of many subjects’ and particular as ‘that which is not’ so. (OI ., I, 7, 17a38-b1) The plurality of possible subjects of universal is what Aristotle insists on. The inclusion of the notion of (...) ‘plurality’ in the definition of universal might make us expect to have ‘singularity’ in the definition of particular. So, when Aristotle says that universal is that which is naturally predicable of many subjects, we expect him to define the particular as ‘that which is predicable of one subject only.’ Nonetheless, Aristotle does not and indeed cannot define it this way. We cannot find a text in which he defines particular as such simply because particulars are not predicable of any subject unless we regard their predication of themselves a predication, which Aristotle does not, at least in a genuine sense. Thus, he defines a particular only negatively. Hence, the capability of predication of a plurality is indeed the capability of predication itself because the particular cannot be predicated of anything. A particular is that which cannot be predicated of anything (or: of anything else, if saying of a thing of itself is to be considered as predication). It is ‘numerically one’ and what of which the universal is predicated (Met., B, 999b34-1000a1). Those that cannot be predicated of anything, or particulars, are of two kinds: primary substances and individual accidents. Besides Categories we can hardly find a text where Aristotle discusses individual accidents maybe because they are of much less importance for him compared with substances. However, substances are what he mentions repeatedly so that we can confine particulars to substances. In fact, it is substance that Aristotle considers so repeatedly as what cannot be predicated. As the main particulars and individuals, substances are posited as the main things that are not universal. The closeness of the two concepts of ‘substance’ and ‘unpredicability’ is to the extent that he ignores individual accidents and makes these concepts equal and as the opposite of universal: ‘what is not predicated of a subject is said a substance (οὐσία λέγετα τὸ μὴ καθ᾿ ὑποκειμένου) but what is always said of some subject is called a universal.’ (Met., Z, 1038b15-16. cf. Met., B, 1003a7-9) Therefore, since particulars cannot be predicated of any subject and, thus, every predication is necessarily a predication of a plurality of subjects, the inclusion of ‘plurality’ in the definition of universal is either i) in the sense of ‘representation’ or it must be regarded as ii) an unnecessary addition mentioned just for clarification or iii) only for avoiding cases where something is predicated of itself. By the first we mean this: though a substance is not said of anything, it represents some one thing and there is some one thing that is that substance. A universal, on the other hand, can represent a plurality of things. There is, however, a third possibility that has no essential difference with the sense of representation. Aristotle might have ‘arbitrary predication’ in mind when he suses ‘plurality’ in the definition of universal: while a particular can be the arbitrary predicate of just one thing, a universal can be predicated, both really and arbitrarily, of many things. Whatever Aristotle’s intention was, what is important for our investigation is this: Aristotle uses the notion of plurality in the definition of universal in spite of the fact that it is not necessary. What this implies is that this notion is so important for Aristotle that albeit every predictability is a predictability of a possible plurality, he adds the notion of plurality. What makes this notion important, I believe, is that he has something like a class in mind when he defines a universal because the notion of plurality is indistinguishable from a class. 2) Universal and whole; particular and part In Aristotle, the concepts of universal and whole are so close: ‘That which is true of a whole class and is said to hold good as a whole (which implies that it is a kind of whole) is true of a whole in the sense that it contains many things by being predicated of each, and that each and all of them, e.g. man, horse, god, are one because all are living things (τὸ μὲν γὰρ καθόλου καὶ τὸ ὃλως λεγόμενον ὡς πολλὰ περιέχον τῷ κατηγορεῖσθαι καθ’ ἑκάστου καὶ ἓν ἃπαντα εἶναι ὡς ἑκαστον, ...). Phil Corkum points to Aristotle’s distinction between quantitative and integral wholes in Met., 5, 26, 1023b26-33 where a quantitative whole is called homoiomerous, as the sum of animal while an integral whole, e.g. a house, is called heteromerous. He links the notion of homoiomerous to the notion of indivisibility of individuals (in 1b6-9 and 3b10-18) and indivisibility of universals. He believes that in PrA., I, 4, 25b32-26a2 it is the transitivity of mereological containment that is discussed. 3) Universal is common between instances Universal is common (κοινόν) between all the plurality of subjects it can be predicated of because what belongs to more than one thing must be common between them. (Met., Z, 1038b10-12) In the Same way, what is common cannot be a particular and, thus, a substance. (Met., Z, 1040b23-24) An individual or substance is a ‘this’ and a ‘this’ cannot be what common indicates simply because it is here in ‘this’ and can be nowhere else while common must be common between several things. In fact, what can be indicated by a common is indeed a ‘such.’ (Met., B, 1003a7-9) 4) Universal is the same in its instances We have a universal i) in all of its instances and ii) in the same way (ταὐτὸ ἐπὶ πὰντων). (Met, Γ, 1005a9-10) While the first point is evident (otherwise how could it be their universal?), the second point might seem not only ambiguous but the cause of many problems. For Aristotle, therefore, a universal must be ‘one’ in number and not many. This numerically one universal is the very universal for all its instances. It is the same universal that is predicated on each of its instances. This sameness is not, however, a mere homonymous sameness or the sameness of a homonymous word. All these three points, namely oneness, sameness and rejection of mere homonymous sameness are asserted in Aristotle’s own words τι ἕν καὶ τὸ αὐτὸ ἐπὶ πλειόνων μη ὁμώνυμον. (PsA., A, 11, 77a8-9) This non-homonymous unity is asserted also in PsA., A, 24, 85b15-16. 5) Universal as predicate Contrary to substance that cannot be a predicate, universal is what cannot be prevented from being in the place of predicate. Therefore, Aristotle distinguishes universal from subject because while the latter must necessarily be capable of being a predicate, though it might take the position of subject as well, the latter does not necessarily have such a capability especially when it is a this because it cannot be a predicate in such a case: ‘For the subject and universal differ in being or not being a ‘this’; like man and body and soul are the subject of accidents while the accident is something like musical or white.’ (Met., Θ, 1049a27-30) This indicates that a universal is basically different from subject and although it can be posited in the place of subject, it is the position of predicate that is its position as a universal. In Metaphysics, Z, 13, Aristotle asserts that ‘no universal can be substance.’ The same is asserted in Met., I, 1053b16-17 cf. 1060b21. As James H. Lesher points out, Aristotle’s position is that ‘nothing predicated universally is a substance.’ 6) Universal in substance In Met., 10388b8-9 Aristotle says that no universal is a substance. While a universal cannot be a substance in the way essence (to ti en einai) is, it is, Aristotle asserts, ‘in’ it (ἐν τούτῳ δὲ ἐνυπάρχειν) (Met., Ζ, 1038b16-18). Aristotle’s examples are: animal in man and horse. A universal is ‘in’ the thing it is its universal. But in what sense a universal can be in a thing? It cannot be in it as ‘in a subject’, which is denied in Categories for secondary substances. If we check the senses of being in, we can find some other senses of ‘being in’ some of which are compatible with this sense of ‘being in.’ Michael J. Loux believes that contrary to his earlier works like Peri Ideaon and Organon in which the immanence of universals signals a reproduction of the platonic two worlds picture, in his later works like Physics and Metaphysics, when Aristotle tells us that universals are in particulars he means that they are ‘components of or ingredients in sensible particulars.’ In Metaphysics (Δ, 1014b3-9) Aristotle compares elements with universals and call them ‘the most universal things because elements are present either in all or in many things. 7) Logos is of universal Not only universals have logos (Met., Z, 1038b18-19) but ‘Every logos and every science is of universals and not of particulars.’ (Met., K, 1059b25-26) The reason is that they are the same: the logos of the unjiversal ‘circle’ is nothing but ‘being circle’ and the logos of the universal ‘soul’ is ‘being soul.’ (Met., Z, 1035b33-1036a2) The same is said about definition. (Met, Z, 1036a27-29) 8) Universal: in the soul Contrasting individuals, universals are in the soul (So., 3, 5, 417b22-23; cf. PsA., B, 19, 100a6-7). Aristotle also says that the form, i.e. the essence, of the artwork is in the soul. (met., Z, 1032a32-b2) Moreover, as Michael J. Loux points out, ‘the Peri Ideon tells us that we need universals to serve as the objects of noetic acts.’ 9) Universal: not beside individual In spite of the fact that demonstration creates the opinion that demonstrating is based on the existence of universals as existing among the existing things, they do not have existence besides individuals (PsA., A, 24, 85a31-35; Met, Λ, 1071a19-23). Aristotle believes that universals of a P-series (B203, 71) are not παρὰ τὰ εἴδη: 999a6ff. Also check De Anima, II, 3, 414b20-25 10) Primary or commensurate universal Aristotle distinguishes ‘πρῶτον καθόλού’ (PsA., B, 17, 99a33-35) literally meaning ‘primary universal’ but mostly, and truly, translated as commensurate universal. While a universal merely ‘μὴ ἀντιςτρέφει,’ a primary universal ‘ᾧ ἓκαςτον μὲν μὴ ἀντιςτρέφει’ (PsA., B, 17, 99a33-35). Aristotle mentions three conditions for a commensurate universal. It is an attribute that i) belongs to every instance of its subject (without exception!!!), and this belonging is ii) essentially and iii) qua that subject itself (ᾗ αύτό). (PsA., A, 4, 73b26-28) However, he insists that the second and the third conditions are indeed the same. (PsA., A, 4, 73b28-30) The first condition he paraphrases as ‘to belong to any random instance of that subject’ and the second and the third as ‘when the subject is the first thing to which it can be shown to belong.’ (PsA., A, 4, 73b32-74a3) Aristotle’s example is ‘the equality of its angles to two right angles’. This attribute is not a commensurate universal of figure due to the first condition: it cannot be demonstrated of any figure. Nonetheless, though it can be demonstrated of every isosceles because every isosceles has angles equal to two right angles, it is not a commensurate universal of isosceles due to the other conditions: it is not predicated of isosceles qua isosceles but qua triangle. These conditions seem to be like regulators: they organize everything to match to the right group. The method of finding commensurate universal is like test and error method based on elimination: you must eliminate each of the higher or lower universals and check if the attribute remains or not. The commensurate universal is that which remains in between eliminated universals. (cf. PsA., A, 5, 74a37-b4) Brad Inwoodpresents an understanding of commensurate universal that is different from what I have understood and, thus, must be checked: ‘These universals are propositions in which, for example, ‘all A are B’ is true and which are still true universal statements if converted: ‘all B are A’ is also true.’ -/- . (shrink)

This article explores aspects of Rosenzweig’s Star of Redemption from the perspective of systems theory. Mosès, Pollock, and others have noted the systematic character of the Star. While “systematic” does not mean “systems theoretic,” the philosophical theology of the Star encompasses ideas that are salient in systems theory. The Magen David star to which the title refers, and which deeply structures Rosenzweig’s thought, fits the classic definition of “system” – a set of elements (God, World, Human) and relations between the (...) elements (Creation, Revelation, Redemption). The Yes and No of the elements and their reversals illustrate the bridging of element and relation with the third category of “attribute,” a notion also central to the definition of “system.” In the diachronics of “the All,” the relations actualize what is only potential in the elements in their primordial state and thus remedy the incompleteness of these elements, fusing them into an integrated whole. Incompleteness is a major theme of systems theory, which also explicitly examines the relations between wholes and parts and offers a formal framework for expressing such fusions. In this article, the systems character of Parts I & II of the Star is explored through extensive use of diagrams; a systems exploration of Part III is left for future work. Remarkably, given its highly architectonic character, diagrams are absent in Rosenzweig’s book, except for the triangle of elements, the triangle of relations, and the hexadic star, which are presented on the opening page of each part of the book. While structures can be explicated entirely in words, diagrams are a visual medium of communication that supplements words and supports a nonverbal understanding that structures both thought and experience. (shrink)

"Jim would still be alive if he hadn't jumped" means that Jim's death was a consequence of his jumping. "x wouldn't be a triangle if it didn't have three sides" means that x's having a three sides is a consequence its being a triangle. Lewis takes the first sentence to mean that Jim is still alive in some alternative universe where he didn't jump, and he takes the second to mean that x is a non-triangle in every alternative universe where (...) it doesn't have three sides. Why did Lewis have such obviously wrong views? Because, like so many of his contemporaries, he failed to grasp the truth that it is the purpose of the present paper to demonstrate, to wit: No coherent doctrine assumes that statements about possible worlds are anything other than statements about the dependence-relations governing our world. The negation of this proposition has a number of obviously false consequences, for example: all true propositions are necessarily true (there is no modal difference between "2+2=4" and "Socrates was bald"); all modal terms (e.g. "possible," "necessary") are infinitely ambiguous; there is no difference between laws of nature (e.g. "metal expands when heated") and accidental generalizations (e.g. "all of the coins in my pocket are quarters"); and there is no difference between the belief that 1+1=2 and the belief that arithmetic is incomplete. Given that possible worlds are identical with mathematical models, it follows that the concept of model-theoretic entailment is useless in the way of understanding how inferences are drawn or how they should be drawn. Given that the concept of formal-entailment is equally useless in these respects, it follows that philosophers and mathematicians have simply failed to shed any light on the nature of the consequence-relation. Q's being either a formal or a model-theoretic consequence of P is parasitic on its bearing some third, still unidentified relation to P; and until this relation has been identified, the discipline of philosophical logic has yet to begin. (shrink)

This paper investigates the identity and function of τὰ μεταξύ in Aristotle and the Early Academy by focussing primarily on Aristotle’s criticisms of Xenocrates of Chalcedon, the third scholarch of Plato’s Academy and Aristotle’s direct competitor. It argues that a number of passages in Aristotle’s Metaphysics (at Β 2, Μ 1-2, and Κ 12) are chiefly directed at Xenocrates as a proponent of theories of mathematical intermediates, despite the fact that Aristotle does not mention Xenocrates there. Aristotle complains that the (...) advocates for mathematical intermediates produce theories that are ontologically and epistemologically inefficient (related to, but not confined by, the “Uniqueness Problem”); that their so-called “intermediates” feature properties opposed to those of the forms on which those intermediates are thought to depend; and that what is μεταξύ must be between objects of a different genus. In all three cases, Aristotle’s criticisms are shown to be reactions to the metaphysics and cosmology of Xenocrates, especially his doctrine of the intermediary demonic isosceles triangle souls. Xenocrates emerges as a prime candidate for Aristotle’s critique of Platonist τὰ μεταξύ theories – equal to, and possibly even exceeding, Plato as target. (shrink)

Hannah Arendt argues that a revolution must not only tear down, but build up a new government. That new government needs authority and it gets its authority from its founding moment, when peers come together in mutual promise, agreeing to treat each other as equals and obeying laws which they legislate for themselves. The paper then looks at the recent attempts of the U.S. government and its allies to bring democracy to Iraq. The paper argues that given the dynamics necessary (...) at the founding moment, U.S. heavy-handedness in setting up Iraq's new government was counterproductive. Also, while the U.S., through subcontracting with Research Triangle Institute, hoped to give Iraq a pyramid-shaped local governance structure, its diversions from Arendt's model make its success unlikely. (shrink)

Why should anyone be bound by cognitive norms, such as the norms of reason or mathematics? To become a mathematician is to learn to obey the norms of the mathematical community. A self becomes intentional by binding itself to communal norms. Only then can it have the freedom to think or make assertions about the community’s objects -- triangles or imaginary numbers, for example. Norms do not bind selves from the outside: being bound by norms is what constitutes a (...) self. (shrink)

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 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)

Phrase structure trees have a hierarchical structure. In many subjects, most notably in {\bf taxonomy} such tree structures have been studied using ultrametrics. Here syntactical hierarchical phrase trees are subject to a similar analysis, which is much simpler as the branching structure is more readily discernible and switched. The occurrence of hierarchical structure elsewhere in linguistics is mentioned. The phrase tree can be represented by a matrix and the elements of the matrix can be represented by triangles. The height (...) at which branching occurs is not prescribed in previous syntactic models, but it is by using the ultrametric matrix. In other words the ultrametric approach gives a complete description of phrase trees, unlike previous approaches. The ambiguity of which branching height to choose, is resolved by postulating that branching occurs at the lowest height available. An ultrametric produces a measure of the complexity of sentences: presumably the complexity of sentences increases as a language is acquired so that this can be tested. All ultrametric triangles are equilateral or isosceles. Here it is shown that \={X} structure implies that there are no equilateral triangles. Restricting attention to simple syntax a minimum ultrametric distance between lexical categories is calculated. A matrix constructed from this ultrametric distance is shown to be different than the matrix obtained from features. It is shown that the definition of {\sc c-command} can be replaced by an equivalent ultrametric definition. The new definition invokes a minimum distance between nodes and this is more aesthetically satisfying than previous varieties of definitions. From the new definition of {\sc c-command} follows a new definition of of the central notion in syntax namely {\sc government}. (shrink)

In the Timaeus, Plato’s Timaeus offers an account of the sensible world in terms of “images” of forms. Often, images are taken to be particulars: either objects or particular property instances (tropes). Contrary to this trend, I argue that images are general characteristics which are immanent in the receptacle, or bundles of such characteristics. Thus, the entire sensible world can be analysed in terms of immanent general characteristics, the receptacle, and forms. Hence, for Timaeus, fundamentally, there are no sensible particulars. (...) I end by arguing that my interpretation is compatible with Timaeus’ construction of sensible entities from triangles. (shrink)

A. Accident 1. We call an accident (συμβεβηκὸς) that which attaches to something and can be truly asserted, but neither of necessity nor usually.’ (Met. , Δ, 1025a14-16) 2. Whenever an accident attaches to a subject, it attaches to it not because it is that subject (μὴ διότι τοδὶ ἧν). (Met., Δ, 1025a21-24) 3. ‘There is no definite cause for an accident, but a chance cause, i.e. an indefinite one.’ (Met., Δ, 1025a24-25) 4. ‘The accident has happened or exists, -not (...) in virtue of itself, however, but of something else. (Met., Δ, 1025a28-29) 5. ‘What attaches to each thing in virtue of itself but is not in its substance is an accident.’ (Met., Δ, 1025a30-32) 6. ‘An accident is something which, though it is … neither a definition nor a property nor a genus, yet belongs to the thing.’ (To., I, 5, 102b4-) 7. An accident is ‘something which may either belong or not belong to any one and the self-same thing, as (e.g.) being seated may belong or not belong to some self-same thing.’ (To., I, 5, 102b4-) 8. ‘There is nothing to prevent an accident from becoming a temporary or relative property … [and] there is nothing to prevent an accident from becoming both a relative and a temporary property; but a property absolutely it will never be.’ Aristotle’s example is ‘being seated.’ That it is a temporary property is evident. It is also a relative property to those who are not seating when he is the only person sitting. Therefore, it is a temporary and also a relative property. 9. ‘In the case of accidents and in no other it is possible for something to be true in a certain respect and not universally’ and ‘there is nothing to prevent an attribute belonging in part’ so that it is open to dispute accidents e.g. being white or just about a man because he might be white or just in part only. Thus, ‘conversion is not a necessary process in the case of accidents.’ (To., B, 1, 109a9-) B. Essential versus accidental Aristotle distinguishes between two senses of being (τὸ ὄν): accidental sense (κατὰ συμβεβηκός) and essential sense (καθ᾿ αὑτό). The accidental sense happens when something is said to be another thing while there is in fact a third thing which is both of them. Thus, being the first thing is not by itself the second thing. a) Aristotle speaks of three ways in which one thing is said in an accidental sense to be another. (Met., Δ, 1017a8-22) i) When both the subject and the predicate are accidents of a third thing and it is in fact the third thing which ‘is’ by its own. Thus, when it is said e.g. that ‘The just is musical’ it is said accidentally because it is in fact a third thing, Socrates for example, which is just and musical. (Met., Δ, 1017a13-19) ii) When that to which the attribute belongs is. The example of this must be ‘the man is musical’ because it is in fact the man which is. iii) When the subject which has as an attribute that of which it is itself predicated, itself is. The example of this must be Aristotle’s third example: ‘The musical is man.’ b) Aristotle rejects the essential sense to categories: ‘Those things are said in their own right to be that are indicated by the figures of predication; for the senses of ‘being’ are just as many as these figures.’ (Met., Δ, 1017a22-24) Aristotle does not give us any example of the essential sense. Daniel W. Graham suggests that while essential predication is of the logical form of ‘S is P,’ the logical form of the accidental predication is ‘S has P.’ Although he concedes that the is/has contrast is not Aristotle’s, for which he thinks Aristotle prefers ‘said of’/’in’ terminology, he points to the discussion of ‘have’ in Cat. 15 as an evidence of its potential value for an analysis of predication. Kirwan analyses the distinction of essential and accidental predicate as such: while essential predicates ‘are identical with the subjects of which they are predicated; other predications are true in virtue of the fact that two distinct items, e.g. a substance and a quality, ‘coincide.’ Herman Weidemann believes ‘what divides a predicative statement which predicates essentially from a predicative statement which predicates accidentally is the fact that the former does- whereas the latter does not- answer the question what its subject cannot fail to be without ceasing to exist, as the somewhat its subject goes on existing, no matter what happens to be true of it during its existence, or, what kind of things its subject must be in order to be identifiable as one and the same object as long as it exists.’ C. Essential 1. Either the formula or the name of the subject of an essential attribute is involved in it. For example, animal is involved in female but not in white. (Met., Z, 1030b23-26) 2. Essential attributes, combined together, form a unity. Thus, e.g. animal and biped form a unity because they are both essential of man. (OI., II, 11, 21a14-16) 3. Aristotle distinguishes between two kinds of essential attributes (PsA., A, 4, 73a34-b3; 22, 84a12-17): a) Those attributes that belong to their subject as elements in its essential nature as e.g. line belongs to triangle or point to line. Here the very being of triangle is composed of line or that of line is composed of points: line is contained in the formula of triangle and point in that of line. b) Those attributes whose subjects are contained in their own defining formula as e.g. straight is an essential attribute of line because line is part of the defining formula of straight or odd belongs to number because number is part of the defining formula of odd. Every attribute that is related in neither of these two ways to its subject, it is not an essential but an accidental attribute. (PsA., A, 4, 73b3-5) 4. That attribute which is not predicated of a subject other than itself is an essential attribute. A substance cannot be predicated of anything but itself and is, thus, essential. (PsA., A, 4, 73b5-10) 5. A thing consequently co-stated with anything is essential. For example, if a beast dies when its throat is being cut, its death is also essentially connected with the cutting because the cutting was the cause of death, not death a coincidence of cutting. (PsA., A, 4, 73b10-15) 6. Essential attributes in both of this sense, either in the sense that their subjects are contained in them or in the sense that they are contained in their subjects, are necessary as well as consequently co-stated with their subjects. The reason is that it is impossible for them to inhere in their subjects either simply or in the qualified sense (that one or other of a pair of opposites must inhere in the subject as e.g. one of straight or curve must be necessarily predicable of a line or one of odd or even of number). (PsA., A, 4, 73b16-21; 6, 74b5-12; B, 96b1-5) 7. Only those attributes that are within a genus are essential and possessed by their respective subjects as such and, thus, are necessary. Thereupon, both the conclusion and the premises of demonstrations which produce scientific knowledge are essential. (PsA., A, 6, 75a28-31) Therefore, ‘the extreme and the middle terms must be drawn from the same genus; otherwise, as predicated, they will not be essential and will thus be accidents.’ (PsA., A, 6, 75b10-12) Also, the theorems of a science can be demonstrated by means of another science only when they are related as subordinate to superior. (PsA., A, 6, 75b13-) 8. Our knowledge of the connexion of an attribute with a subject is essential only when ‘we know that connexion through the middle term in virtue of which it inheres (καθ᾿ὃ ὑπἀρχει), and as an inference from basic premises essential and ‘appropriate’ to the subject.’ The essential belonging of the middle to the minor has a necessary connexion with its belonging to the same genus of the major and minor: ‘If that middle term also belongs essentially to the minor, the middle must belong to the same genus as the major and minor terms (ἀνάγκη τὸ μέσον ἐν τῇ αὐτῇ συγγενείᾳ εἶναι). (PsA., A, 9, 76a4-9) 9. In an essential predication, the predicate is signified of a subject identical with itself or with a species of itself. Since only predicates which signify substance signify that the subject is identical with the predicate or with a species of the predicate, predicates signifying substance are essentially predicated and any other predicate is accidental or coincidental. Thus, while animal is essentially predicated of man because man is identical with a species of animal, white is accidentally predicated of man because man is neither identical with white nor a species of it. (PsA., A, 22, 83a24-32) Weidemann discusses the confusion between essential predications and statements of identity. He warns that the expression ‘(to be) just what a man is’ (analogous to the Greek phrase ‘(εἶναι) ὃπερ ἀνθρώπον) ought not to be confused with the expression ‘just what it is for a man to be,’ which renders the phrase τὸ ὃπερ ἀνθρώπῳ εἶναι’ (1007a22, 23, 27-28); a confusion Kirwan discusses. 10. ‘Demonstration proves the inherence of essential attributes in things.’ (PsA., A, 22, 84a11-12) 11. ‘Nothing which does not happen to belong to the genus is essentially the genus; e.g. a white man is not essentially a color.’ Thus, while justice falls within the genus, a just man does not. (To., Γ, 1, ^116a24-) 12. There is no syllogism of essence (ti esti). Nevertheless, we come to know ti esti through a demonstration. (93b15-20; cf. 93b25-28) D. Accidental 1. An accidental attribute cannot be predicated of a subject universally. Thus, we cannot say ‘Every man is musical.’ The reason is that ‘universal attributes belong to things in virtue of their nature.’ Therefore, accidental predicates are predicated only of the individuals. (Met., Δ, 1017b33-1018a2) 2. An accidental is a mere name. (Met., E, 1026b13-) 3. An accidental is obviously akin to non-being. (Met., E, 1026b21) 4. ‘That which is neither always nor for the most part we call accidental.’ (Met., E, 1026b27-33) 5. The matter (ὕλη) is the cause of the accidental because it is matter which is capable of being otherwise than as it for the most part is. (Met., E, 1027a13-16) In fact, there is cause or principle of the incidental of the same kind as there are of the essential because if there were, everything would be of necessity. (Met., K, 1065a6-8) The causes of accidental are unordered and indefinite. (Met., K, 1065a25-26) 6. Since all science is either of that which is always or of that which is for the most part, there is no science of the accidental. (Met., E, 1027a19-24; K, 1064b30-1065a6) Therefore, since an accident may also not inhere and, thus, it is impossible to prove its inherence as a necessary conclusion, there is no demonstrative knowledge of accidents. (PsA., A, 6, 75a18-23) 7. The accidental is not necessary but indeterminate. (Met., K, 1065a24-26) 8. The co-positing of a subject and a predicate which are accidental either to the same subject or to one another, whiteness and being musical for instance, does not form a unity. (OI, II, 11, 21a7-14) 9. Negative predicates, i.e. predicates including a ‘not,’ are accidental predicates. Thus, if good be an essential predicate of a subject, ‘not bad’ cannot be essential but is accidental. (OI., II, 14, 23b15-20) 10. Every attribute which does not belong to its subject in either of two senses of essential attribute (cf. PsA., A, 4, 73a34-b3), namely, belonging to their subjects as an element of its formula or their subjects belonging to them as an element of them, is an accidental attribute. (PsA., A, 4, 73b3-5) 11. Those attributes that are predicated of a subject other than themselves are accidental attributes. Thus, everything except substance is an accidental in this sense. (PsA., A, 4, 73b5-10) 12. A thing not consequentially connected with anything is accidental. For example, in ‘while he was walking it lightened,’ the lightening was not due to his walking but was a coincidence. (PsA., A, 4, 73b10-15) 13. Those attributes that are not within a genus are accidental attributes. (PsA., A, 6, 75a28-31) Therefore, the extreme and middle terms that are not drawn from the same genus are accidents. (PsA., A, 6, 75b10-12) Moreover, to apply an attribute to subjects in different genera afford knowledge of the attribute, only as inhering accidentally, not as belonging to its subject as such.’ (PsA., A, 9, 75b38-76a3) . (shrink)

John Keane’s book is an important intervention in the debate on the persistent proliferation of violence and its role in political life, especially in democracies.

The fine-structure constant, which determines the strength of the electromagnetic interaction, is briefly reviewed beginning with its introduction by Arnold Sommerfeld and also includes the interest of Wolfgang Pauli, Paul Dirac, Richard Feynman and others. Sommerfeld was very much a Pythagorean and sometimes compared to Johannes Kepler. The archetypal Pythagorean triangle has long been known as a hiding place for the golden ratio. More recently, the quartic polynomial has also been found as a hiding place for the golden ratio. The (...) Kepler triangle, with its golden ratio proportions, is also a Pythagorean triangle. Combining classical harmonic proportions derived from Kepler’s triangle with quartic equations determine an approximate value for the fine-structure constant that is the same as that found in our previous work with the golden ratio geometry of the hydrogen atom. These results make further progress toward an understanding of the golden ratio as the basis for the fine-structure constant. (shrink)

Some people believe that critical thinking is not a modern science, but its roots are old and deeply rooted in the history of philosophy. Its roots date back to Aristotle, the inventor of logic and who was called the first teacher by virtue of this invention. Aristotle was impressed by the language of mathematics and wanted to invent a language to logic similar to the language of Mathematics. What encouraged Aristotle to do so is that Math language is quite different (...) from daily life language. For example , when you say one in Mathematics language, what is this one you mean or intend? That is what is one? Is it a specific man or an animal or something abstract or any creature of the creatures of the natural world? The one we are talking about is neither this nor that and you can use it to talk about one human being or one animal or one creature. Therefore, the concept of the one is something abstract and it is so far from the creatures of the exterior world. Geometry has the same case. Triangle, quarter and rectangle as such are mere geometrical shapes and are far away from real life language. But when we use them in our daily life language we give them specific life meanings. -/- As I said, Aristotle was impressed by the language of Mathematics and wanted to build a similar language for logic .The rationale behind this is to get away from the language of ambiguity caused by the use of daily life language .One word or utterance in a language has many usages in one language or when translated into other languages , and therefore it causes many problems in usage instead of simplifying it and this is what Aristotle tried to avoid. Aristotle chose letters instead of words, the letter is part of the word, and the letter is meaningless unless it is a part of the word. The meaning of a letter depends on its occurrence in the word. If you compose a word from a group of letters and use it , it becomes ambiguous because it becomes part of the used language. -/- However, Aristotle did not combine letters in a word but he used them as abstract letters and gave them meanings in the language of logic where there is no divergence about its use. Aristotle used them in syllogism and they mean the limits of reason . Until recently critical thinking was part of formal logic and learning it means that you have logical background or at least you are familiar with the rules and fundamentals of logic. But critical thinking theorists soon took the initiative to separate it from logic because the latter started to suffer from its rules which are difficult to understand and which require some awareness and thoughts on the part of the scholar to understand. Hence , critical thinking is not supposed to possess a background of logic and it adopts a specific questioning approach where it suggests a number of questions the practitioner of critical thinking has to successively answer since he has to make sure that the opinion he reaches does not rely on any subjective considerations. This is how the skills that are acquired on a continuum of related critical questions are established ,this is not to mention that the concept of critical thinking itself is usually defined in this context as the awareness of such kinds of questions and the ability to arouse and answer them at appropriate times. My paper explains, How to be able to implement these skills in high education and what are the benefits of these skills? The paper includes; Critical Thinking and Logic Socrates `s Approach The impact of critical thinking on teaching methods Questioning How questions are organized in the light of critical thinking? Using Critical Thinking in Problem-Solving Use of Non-Oral language in Critical Thinking. (shrink)

Thought is the primary realm in which truth and falsity may occur and speech the secondary realm of this occurrence while the realm of external being has no truth and falsity in itself. The first and last points are directly asserted by Aristotle in one text: ‘Falsity and truth are not in things-it is not as if the good were true, and the bad were in itself false- but in thought.’ (Met., E, 1027b25-27; cf. Met., K, 1065a22-23) The second point (...) is also somehow implied: ‘As there are in the mind thoughts which do not involve truth and falsity, and those which must be either true or false, so it is in speech.’ (OI, I, 1, 16a9-11) 1) Truth and falsity belong only to co-positings and positing aways and not to incomposites Truth and falsity do not belong to the realm of external being but only to the realms of thought and language. However, they do not belong to all the elements in these realms. Neither do they belong to incomposite elements in thought (‘With regard to simple things and essences, falsity and truth do not exist even in thought.’ (Met., E, 1027b26-28; So., Γ, 6, 430a26-b4)), nor to such elements in language. Consequently, all incomposites are incapable of being true or false. (cf. Met., Θ, 1051b17-22 and b26-27) Where we have no co-positing or positing away, of the kind made by copula or verb, we cannot have truth and falsity though we can have significance: ‘“Man” and “white”, as isolated terms, are not yet either true or false. In proof of this, consider the word “good-stag.” It has significance, but there is no truth or falsity about it, unless “is” or “is not” is added, either in present or in some other tense.’ (OI., I, 1, 16a14-18) Therefore, any other kind of co-positing cannot be the subject of truth and falsity. One good example of co-positings not made by copula or verbs are simple negations: ‘He that uses the expression ‘not-man,’ if nothing more be added, is not nearer but farther from making a true or a false statement than he who uses the expression “man.”’ (OI., I, 10, 20a34-36) It is indeed a co-positing or a positing away that is the subject of truth and falsity: ‘For it is co-positing (σύνθεσιν) and positing away (διαίρεσιν) that truth and falsity is about.’ (OI., I, 1, 16a12-13) Co-positing is so vital for truth-falsity that ‘of the things that are never said based on co-positing, nothing is true or false.’ (Cat., 10, 13b9-11) This co-positing can be both in thought (‘where the alternative of true or false applies, there we always find a sort of combining of objects of thought in a quasi-unity … For falsehood always involves a combining’ (So., Γ, 6, 430a26-b2) and ‘truth or false is a co-positing of concepts of thought’ (So., Γ, 6, 430a11-12) and in language because they are produced in thought: ‘There is not only the true or false assertion that Cleon is white but also the true or false assertion that he was or will be white. In each and every case that which unifies is thought.’ (So., Γ, 6, 430b4-6) 2) Correspondence to external world Although truth and falsity belong to thought and language and not to external world, their basis is in the external world. It is, in fact, the correspondence of thought and language to external world that makes what is true, true and what is false, false: ‘To say of what is that it is, or of what is not that it is not, is false, while to say of what is that it is, and of what is not that is not, is true.’ (Met., Γ, 1011b26-27; cf. Met., Θ, 1051b2-5) When the two realms are correspondent, we can transfer from one to the other: ‘If it is true to say that a thing is white, it must necessarily be white; if the reverse proposition be true, it will of necessity not be white. Again, if it is white, the proposition stating that it is white was true; if it is not white, the proposition to the opposite effect was true. And if it is not white, the man who states that it is making a false statement, it follows that it is not white.’ (OI., I, 9, 18a39-b3) This possible mutual transformation does not, however, imply that each realm is the basis of the other. In fact, it is only the external world that is the basis of truth and falsity in thought or language and not vice versa: ‘it is not because we think that you are white, that you are white, but because you are white we who say this have the truth.’ (Met., Θ, 1051b6-9; Cat., 5, 4b8-10) Aristotle even calls the external world the ‘cause’ of truth in the realm of language while the converse causation is not approved: ‘Whereas the true statement is in no way the cause of the actual thing’s existence, the actual thing does seem in some way the cause of the statement’s being true: it is because the actual thing exists or does not that the statement is called true or false.’ (Cat., 12, 14b18-22) This one-sided relation Aristotle calls ‘reciprocation as to implication of existence’: ‘For there being a man reciprocates as to implication of existence with the true statement about it: if there is a man, the statement whereby we say that there is a man is true, and reciprocally- since if the statement whereby we say that there is a man is true, there is a man. And whereas the true statement is in no way the cause of the actual thing’s existence, the actual thing does seem in some way the cause of the statement’s being true: it is because the actual thing exists or does not that the statement is called true or false,’ (Cat., 12, 14b14-22) Aristotle attaches falsity to not-being: ‘We cal things false in this way, then, either because they themselves do not exist, or because the appearance which results from them is that of something that does not exist.’ (Met., Δ, 1024b24-26) Or: ‘A false formula is the formula of non-existent objects, in so far as it is false.’ (Met., Δ, 1024b26-27) This correspondence to ‘non-being’ is, however, more a miscorrespondence: ‘Hence every formula is false when applied to something other than that of which it is true, e.g. the formula of a circle is false when applied to a triangle.’ (Met., Δ, 1024b27-28) . (shrink)

Although symmetry has been discussed in terms of a major law of perceptual organization since the early conceptual efforts of the Gestalt school (Wertheimer, Metzger, Koffka and others), the first quantitative measurements testing for effects of symmetry on processes of Gestalt formation have seen the day only recently. In this study, a psychophysical rating study and a “foreground”-“background” choice response time experiment were run with human observers to test for effects of bilateral symmetry on the perceived strength of figure-ground in (...) triangular Kanizsa configurations. Displays with and without bilateral symmetry, identical physically-specified-to-total contour ratio, and constant local contrast intensity within and across conditions, but variable local contrast polarity and variable orientation in the plane, were presented in a random order to human observers. Configurations with bilateral symmetry produced significantly stronger figure-ground percepts reflected by greater subjective magnitudes and consistently higher percentages of “foreground” judgments accompanied by significantly shorter response times. These effects of symmetry depend neither on the orientation of the axis of symmetry, nor on the contrast polarity of the physical inducers. It is concluded that bilateral symmetry, irrespective of orientation, significantly contributes to the, largely sign-invariant, visual mechanisms of figure-ground segregation that determine the salience of figure-ground in perceptually ambiguous configurations. (shrink)

Aristotle’s points about taking ‘more or less’ (μᾶλλον καὶ ἧττον) are as following: 1. Substances do not admit of the more and the less. This is supposed to mean that a man is not more a man either than itself or than another man. This is not, however, the case between different substances because one substance can be more a substance than another. (Cat., 5, 3, 3b33-4a2) In Metaphysics, Aristotle tells us that substance in the sense of form does not (...) admit of the more and the less ‘but if any substance does, it is only the substance which involves matter.’ (Met., H, 1044a9-11) 2. Quantities, including numbers, do not admit of a more and a less. (Cat., 6, 6a19-25; Met., H, 1044a9-10) 3. Some qualities, but not all of them, admit the more and the less. For example, one thing can be called paler or more beautiful than another. This is not, however, true about the shape: neither things that admit nor those that do not admit the definition of a shape, e.g. triangle or circle, is called more a triangle or a circle. (Cat., 11a5-) Moreover, a thing can admit of more or less of a quantity than itself in a different time. (Cat., 5, 4a3-9) 4. Some, but not all, relatives admit of a more and a less. For example, while a thing is called more or less similar or more or less unequal to another thing, what is double is not spoken more double or less double. (Cat., 7, 6b20-27) 5. Doing and being affected admit of a more and a less. It is possible, for example, to heat more or to be heated more or less. (Cat., 9, 11b1-7) 6. Aristotle speaks of the possibility of being more false or less false. (Met., Γ) . (shrink)

Our object of study in this dissertation is the paralogism due to the concomitant ( which we’ll call abreviated “PDC”). This paralogism is analysed by Aristotle in the Sophistical Refutations as a false argument. Our study about this paralogismo is divided into four chapters. Trough the first chapter we pretend to answer an important question about the PDC’s appearance. Although this paralogism is a false argument, it possess certain similarity with arguments recognized as good by the one refuted. This similarity (...) would be the reason why debaters are refuted by the PDC. Identifying the nature of this similarity is relevant to comprehend how this paralogism’s appearance can be deceitful. The content of the second chapter can be better undestood as a deepening of our interest for the subjective or psychological dimension present in the studies about the paralogisms in the Sophistical Refutations. This deepening results in the description of a mental phenomenon which we call “parallel computation”. The parallel computation consists in a rational effort (although it is a misguided one) by the one who is refuted when he tries to make sense of the logical and semantical traits of an argument. This interpretation is motivated by a set of false presumptions held by the refuted one. In the third chapter we present a case study, turning our attention to an instance of the PDC, the triangle paralogism. By studying the relevant passages for the interpretation of this paralogism we put forward what we understand to be exegetical constraints. Afterward, making use of these constraints as evaluative parameters, we analyse some reconstitution proposals of the triangle paralogism. In the final part of our dissertation we analyse the content of the chapter 24 of SE. In this chapter Aristotle presents us with a inferential principle which isn’t satisfied by any of the PDC’s instance. The interpretation of this principle is a cause of huge controversy in the secondary literature. In order to make a contribution to the specialized debate of this point we test two interpretative hyphotesis of this principle’s content, the referential and the intensional interpretation. (shrink)

Contrary to Bell’s theorem it is demonstrated that with the use of classical probability theory the quantum correlation can be approximated. Hence, one may not conclude from experiment that all local hidden variable theories are ruled out by a violation of inequality result.

Ordinary perceiving relies heavily on our sensing the spatial properties of objects, e.g., their shapes, sizes, and locations. Such spatial perception is central in everyday life. We safely cross a street by seeing and hearing the locations of oncoming vehicles. And we often identify objects by seeing and feeling their distinctive shapes. -/- To understand how we perceive spatial properties, we must explain the nature of the mental states figuring in spatial perception. The experience one has when seeing a cube, (...) e.g., differs from the experiences one has when seeing other shapes, e.g., spheres and pyramids. We must explain how such experiences differ to fully understand how we perceive differences in the spatial properties of objects. This presents a challenge often overlooked in philosophy and cognitive science. Whereas we can differentiate physical objects by their spatial properties, we cannot differentiate the experiences involved in perception in respect of their own spatial properties. Experiences are mental states, not physical objects, so they do not themselves have spatial properties; a visual experience of a 50 ft. tall cube, e.g., isn’t itself 50 ft. tall or cubical. So we must differentiate our perceptual experiences of those objects some other way, in terms of their own properties. -/- I argue the experiences figuring in spatial perception have mental properties distinct from, but analogous to, the spatial properties we perceive. The experience one has when seeing a square, e.g., has a property that resembles and differs from other such mental properties in ways parallel to the ways physical squares resemble and differ from other shapes. Just as squares are more similar to rectangles than triangles, the mental property of an experience of a square is more similar to that of an experience of a rectangle than that of an experience of a triangle. -/- I show how this theory helps solve several problems in philosophy and cognitive science; explaining change blindness, accounting for our ability to perceive combinations of distinct properties, e.g., color and shape, and determining whether the properties of experiences pertaining to the same spatial properties in different sensory modalities are themselves the same. (shrink)

A natural extension of Heron's 2000 year old formula for the area of a triangle to the volume of a tetrahedron is presented. This gives the fourth power of the volume as a polynomial in six simple rational functions of the areas of its four faces and three medial parallelograms, which will be referred to herein as "interior faces." Geometrically, these rational functions are the areas of the triangles into which the exterior faces are divided by the points at (...) which the tetrahedron's in-sphere touches those faces. Part I presents an overview of these results and some necessary but little-known background in areal geometry. Part II derives the promised extension, and ends with a conjecture as to how the formula extends to n-dimensional simplices for all n>3. Part III explains how, for n=3, the zeros of the polynomial constitute a five-dimensional semi-algebraic variety consisting almost entirely of collinear tetrahedra with vertices separated by infinite distances, but with generically well-defined distance ratios; it further proves that these unconventional Euclidean configurations can be identified with a quotient of the Klein quadric by an action of a group of reflections isomorphic to ℤ42, wherein four-point configurations in the affine plane constitute a distinguished three-dimensional subvariety. Part IV consists of five appendices which show, among other things, that the algebraic structure of the zeros in the affine plane naturally defines the associated four-element, rank 3 chirotope, aka affine oriented matroid. (shrink)

ABSTRACT of The Flying Termite by L.L. Katona -/- In this book I would like to show the term “intelligence“ has a universal, non-anthropomorphic meaning. We can perceive intelligence in dogs, dolphins or gorillas without understanding of it, but intelligence can be also seen in many other things from insects and the Solar System to elementary particles or the rules of a triangle. But that doesn’t mean intelligence comes from Intelligent Design, yet alone a Designer, they seems to be the (...) “state of things” in the Universe. One of the most known law of the triangle is “the measures of the interior angles of a triangle in Euclidean space always add up to 180 degrees.” The question is this law exists without any existing triangle, or not? If there is an encrypted program in DNA, what determines which living being will be a chameleon or a cat, where this program lies? Whether in the DNA itself? But DNA also programmed. Where is the program of a computer? You may say in the hard drive or in the head of the programmer. But the hard drive or the head of the programmer is just the vehicle of the program; when you dissect a computer or a programmer’s head you will not find the program, just the diodes or cells connected by pack of orders. And you won’t see the orders in them. In my book I want to show the works of this elusive intelligence with some interesting contradictions between causality, determinism, teleology, and dependent formation through biology, mathematics and logic. If I need to summarize my idea I can use the following parable. A priest and an atheist are playing with a beach ball in a pool, in a two-dimensioned plane. They are rod-people with a circle. What they can’t perceive is this plane is a part of a three-dimensioned world, where the wind is blowing, so they can’t understand why the ball disappearing continuously, and why it comes back at another point randomly. Both of them have a hint, the ball is a sphere, and the space is three-dimensioned, but instead of trying to work in team to understand this curious phenomena of the Beach Ball, they’re using the old method: insulting each other. (shrink)

Sudan is the largest state in Africa geographically and has a population of about 41 million people, of which 70% are Sunni Muslims by religion. (CIA, 2006). The non-Muslim population resides mostly in Southern Sudan and is either Christian or of indigenous religions. Darfur consists of three provinces in the western part of Sudan: North ("Shamal") Darfur, West ("Gharb") Darfur, and South ("Janub") Darfur. The greater Khartoum province, an area nicknamed the "Arab Triangle," has become a developed and industrialized part (...) of the country. This financial development has been driven by Sudan's oil trade. However, oil has brought war, death, and hunger for a vast majority of the Sudanese living in the squalor of Africa. This paper outlines the historical background of Sudan, the history of the war in Sudan, the Darfur conflict, and the causes and effects of the Darfur crisis. It also examines the interests and interventions aimed at de-escalating the Darfur crisis. The conclusion reached is that solutions to the decade-old conflict in Sudan must adapt and change as necessary if they are to remain relevant and effective. -/- . (shrink)

History witnesses alternative approaches to “the proposition”. The proposition has been referred to as the object of belief, disbelief, and doubt: generally as the object of propositional attitudes, that which can be said to be believed, disbelieved, understood, etc. It has also been taken to be the object of grasping, judging, assuming, affirming, denying, and inquiring: generally as the object of propositional actions, that which can be said to be grasped, judged true or false, assumed for reasoning purposes, etc. The (...) proposition has also been taken to be the subject of truth and falsity: generally as the subject of propositional properties, that which can be said to be true, false, tautological, informative, inconsistent,etc. It has also been taken as the subject and object of logical relations, e.g. that which can be said to imply, be implied, contradict, be contradicted, etc. Prima facie, such properties and relations are non-mental and objective. It has also been taken to be the resultants or products of propositional operations, usually mental or linguistic; e.g. judging, affirming, and denying have been held to produce propositions called judgments, affirmations, and negations, respectively. Propositions have also been taken to be certain declarative sentences. Finally,propositions have been taken to be meanings of certain declarative sentences. This essay is an informal, selective, and incomplete survey of alternative approaches to “the proposition” with special attention to the views of the late American philosopher Peter Hare (1935–2008)and of those who influenced him. DOI: 10.5007/1808-1711.2011v15n1p51. (shrink)

The Five Greatest Kinds discussed in Plato‘s Sophist are taken to be just one instance of a fivefold structure found in various related texts. Contemporary linguistic theories are a source for ideas about its functioning.

Dialectica: Mathematica and Physica, Truth and Justice, Trick and Life. Mathematica as the Constructive Metaphysica and Ontology. Mathematica as the constructive existential method. Сonsciousness and Mathematica: Dialectica of "eidos" and "logos". Mathematica is the Total Dialectica. The basic maternal Structure - "La Structure mère". Mathematica and Physica: loss of existential certainty. Is effectiveness of Mathematica "unreasonable"? The ontological structure of space. Axiomatization of the ontological basis of knowledge: one axiom, one principle and one mathematical object. The main ideas and concepts (...) of the ontological construction/ "Point with a vector germ" and "heavenly triangle". "Ordo geometricus" and "Ordo onto-topological". Architecture of the onto-topological basis of knowledge: general framework structure, carcass and foundation. The absolute space and the absolute field. The absolute (natural) system of coordinates of Universum. Eidos of "idea of ideas", the symbol and the "formula of Justice". (shrink)

A condensed summary of the adventures of ideas (1990-2020). Methodology of evolutionary-phenomenological constitution of Consciousness. Vector (BeVector) of Consciousness. Consciousness is a qualitative vector quantity. Vector of Consciousness as a synthesizing category, eidos-prototecton, intentional meta-observer. The development of the ideas of Pierre Teilhard de Chardin, Brentano, Husserl, Bergson, Florensky, Losev, Mamardashvili, Nalimov. Dialectic of Eidos and Logos. "Curve line" of the Consciousness Vector from space and time. The lower and upper sides of the "abyss of being". The existential tension of (...) being. Five reference “points” (existential-extremum) - world events in the evolution of Consciousness from “homo habilis” to “homo sapiens sapiens”. “Prometheus Effect". Inversion and reversion of Сonsciousness. "Open" and "closed" Сonsciousness. Consciousness and Self-Consciousness. Protogeometer: fall in the future, in the "EgoLand". The problem of justification/substantiation of mathematics (knowledge) is an ontological problem. The crisis of ontology and ontological limits of cognition. Ontological structure of space. The project of constructive dialectic ontology. The methodology of dialectical-ontological construction (modeling): conceptual-figure synthesis, total unification of matter, coincidence of ontological opposites, primordial Meta-Axioma and Superprinciple, vector (bivector) of the absolute state (states) of matter (absolute forms of existence). The ontological "celestial triangle" (Plato). Ontological invariants of the Universum and their ontological paths. Ontological and gnoseological dimensions of space. Triune (ontological) space of nine gnoseological dimensions: absolute rest (linear state, "Continuum")+ absolute motion (absolute vortex, "Discretuum") + absolute becoming (absolute wave, "Dis-Continuum"). Primordial generating structure: ontological framework, ontological carcass and ontological foundation. Ontological (structural, cosmic) memory - the semantic core of the conceptual construction of the Universum being as an eternal process of generation of new meanings and structures. Consciousness is an absolute (unconditional) attractor of meanings, a univalent phenomenon of ontological memory, which manifests itself at a certain level of the Universum being. Ontological space-MatterMemory-Ontological time (S-MM-T). Ontological Rhythm and Cycle. The nature and structure of ontological time. Natural (absolute) determinism. (shrink)

Emotions often misfire. We sometimes fear innocuous things, such as spiders or mice, and we do so even if we firmly believe that they are innocuous. This is true of all of us, and not only of phobics, who can be considered to suffer from extreme manifestations of a common tendency. We also feel too little or even sometimes no fear at all with respect to very fearsome things, and we do so even if we believe that they are fearsome. (...) Indeed, instead of shunning fearsome things, we might be attracted to them. Emotions that seem more thought-involving, such as shame, guilt or jealousy, can also misfire. You can be ashamed of your big ears even though we can agree that there is nothing shameful in having big ears, and even though you judge that having big ears does not warrant shame. And of course, it is also possible to experience too little or even no shame at all with respect to something that is really shameful. Many of these cases involve a conflict between one’s emotion and one’s evaluative judgement. Emotions that are thus conflicting with judgement can be called ‘recalcitrant emotions’. The question I am interested in is whether or not recalcitrant emotions amount to emotional illusions, that is, whether or not these cases are sufficiently similar to perceptual illusions to justify the claim that they fall under the same general heading. The answer to this depends on what emotions are. For instance, the view that emotions are evaluative judgments makes it difficult to make room for the claim that emotional errors are perceptual illusions. Fearing an innocuous spider would simply amount to making the error of judging that the spider is fearsome while it is in fact innocuous. This might involve an illusion of some sort, but it certainly does not amount to anything like a perceptual illusion. In this chapter, I argue that recalcitrant emotions are a kind of perceptual illusion.. (shrink)