AS SOME PHILOSOPHERS KNOW, the paradox about inquiry at 80d-e of Plato’s Meno is more than a tedious sophism. Plato is one such philosopher. The puzzle is an obstacle to his project of discovering definitions, and is introduced as such. And it is met with an elaborate response: the theory of recollection, explicitly presented as an answer to the obstacle. But then what of the famous conversation in which Socrates coaxes a geometrical theorem from a slave boy Is (...) the theory not designed to explain the boy’s ability to respond to the coaxing? It is, certainly, but that is not its only purpose. The structure of the passage is this: the theory is there to disarm the paradox, and the conversation is there to support the theory. To see this structure is to understand a notorious and otherwise troubling fact, that Plato is so very quick to take the slave’s behavior—which he might have tried to explain in some other way-to be clear evidence for recollection. The reason why he so takes it is that the paradox has led him to think that only if recollection occurs is fruitful inquiry possible—and he is very anxious indeed to be assured that it is possible. He would not have been so enticed by explanations of the boy’s behavior which did not also seem to him to dispose of the puzzle. Here now is Plato’s setting of the paradox. (shrink)

It is a well-known fact that although the poset of open sets of a topological space is a Heyting algebra, its Heyting implication is not necessarily stable under the inverse image of continuous functions and hence is not a geometric concept. This leaves us wondering if there is any stable family of implications that can be safely called geometric. In this paper, we will first recall the abstract notion of implication as a binary modality introduced in Akbar Tabatabai (Implication via (...) spacetime. In: Mathematics, logic, and their philosophies: essays in honour of Mohammad Ardeshir, pp 161–216, 2021). Then, we will use a weaker version of categorical fibrations to define the geometricity of a category of pairs of spaces and implications over a given category of spaces. We will identify the greatest geometric category over the subcategories of open-irreducible (closed-irreducible) maps as a generalization of the usual injective open (closed) maps. Using this identification, we will then characterize all geometric categories over a given category $${\mathcal {S}}$$ S, provided that $${\mathcal {S}}$$ S has some basic closure properties. Specially, we will show that there is no non-trivial geometric category over the full category of spaces. Finally, as the implications we identified are also interesting in their own right, we will spend some time to investigate their algebraic properties. We will first use a Yoneda-type argument to provide a representation theorem, making the implications a part of an adjunction-style pair. Then, we will use this result to provide a Kripke-style representation for any arbitrary implication. (shrink)

BackgroundPrenatal and early postnatal choline supplementation reduces cognitive and behavioral deficits in animal models of Fetal Alcohol Spectrum Disorder. In a previously published 9-month clinical trial of choline supplementation in children with FASD, we reported that postnatal choline was associated with improved performance on a hippocampal-dependent recognition memory task. The current paper describes the neurophysiological correlates of that memory performance for trial completers.MethodsChildren with FASD who were enrolled in a clinical trial of choline supplementation were followed for 9 months. Delayed (...) recall on a 9-step elicited imitation task served as the behavioral measure of recognition memory. Neurophysiological correlates of memory were assessed via event-related potentials.ResultsDelayed recall on EI was correlated with two ERP components commonly associated with recognition memory in young children: middle latency negative component and positive slow wave. No significant ERP differences were observed between the choline and placebo groups at the conclusion of the trial.ConclusionAlthough the small sample size limits the ability to draw clear conclusions about the treatment effect of choline on ERP, the results suggest a relationship between memory performance and underlying neurophysiological status in FASD. This trial was registered.1. (shrink)

The discussion of mathematical knowledge and its relation to the construction of an appropriate diagram in Aristotle's Metaphysics Θ 9. 1051 a21—33 is an important, if compressed, account of Aristotle's most mature thoughts on mathematical knowledge. The discussion of what sort of previous knowledge one must have for understanding a theorem recalls the discussion at An. Post. A 1. 71 a 17–21, where the epistemological point is similar and the examples the same. The first example, that the interior angles (...) of a triangle equal two right angles, appears no less than thirty times in the corpus . The example of the angle inscribed in a semicircle being a right angle also occurs at An. Post. B 11. 94a 27–34, but in a very different context from its two companions. Illustrations of both theorems provided clear stock examples for Aristotle. (shrink)

The energy-momentum tensor for a particular matter component summarises its local energy-momentum distribution in terms of densities and current densities. We re-investigate under what conditions these local distributions can be integrated to meaningful global quantities. This leads us directly to a classic theorem by Max von Laue concerning integrals of components of the energy-momentum tensor, whose statement and proof we recall. In the first half of this paper we do this within the realm of Special Relativity and in the (...) traditional mathematical language using components with respect to affine charts, thereby focusing on the intended physical content and interpretation. In the second half we show how to do all this in a proper differential-geometric fashion and on arbitrary space-time manifolds, this time focusing on the group-theoretic and geometric hypotheses underlying these results. Based on this we give a proper geometric statement and proof of Laue's theorem, which is shown to generalise from Minkowski space to space-times with significantly less symmetries. This result, which seems to be new, not only generalises but also clarifies the geometric content and hypotheses of Laue's theorem. A series of three appendices lists our conventions and notation and summarises some of the conceptual and mathematical background needed in the main text. (shrink)

As with the dialogue, so with the slave-boy episode within it, two questions are handled, one of them substantive, the other a question of method. The substantive question is how to double the square of a side of 2 units; the procedural question is how, if at all, can an answer be found by one who does not know it. It develops that the answer must be sought exclusively among opinions which the boy already holds, by means of questioning. What (...) I want to argue is that the boy's lesson is not an unqualified success any more than is Meno's. I shall maintain that, exactly parallel to the main discussion, Plato means to cast suspicion on the substantive answer which Socrates and the boy arrive at, and for an exactly parallel reason: the inquiry has been conducted improperly. On the other hand the lesson is far from an unqualified failure, since despite its lack of substantive result, it contains an important message about method. Just as it becomes clear in the discussion with Meno that any disciplined inquiry into virtue must refrain from asking what virtue is like until it has settled the more fundamental question of what virtue itself is, so in the slave-boy passage it is learned that we must refrain from asking what the solution to our problem is like until we have settled on what the solution itself is. (shrink)

Plato invokes the Theory of Recollection to explain both ordinary and philosophical learning. In a new reading of Meno's Paradox and the Slave‐Boy Interrogation, I explain why these two levels are linked in a single theory of learning. Since, for Plato, philosophical inquiry starts in ordinary discourse, the possibility of success in inquiry is tied to the character of the ordinary comprehension we bring to it. Through the claim that all learning is recollection, Plato traces the knowledge achievable through inquiry (...) back to our pretheoretical comprehension, showing not just that knowledge is in us, but that it is inchoate in the grasp of a property—akin to a concept—that enables us to speak and think about it ordinarily. Plato acknowledges in the Meno that a second step of argument, and a second application of Recollection, is needed to explain how knowledge comes to be inchoate in our ordinary grasp of a property. Though this second argument is provided most fully in the Phaedo, the evidence of the Meno is sufficient to outline Recollection as a two‐stage theory of learning, beginning in ordinary speech and thought and extending, through philosophical reflection, to knowledge. (shrink)

This paper argues that a well known passage from Plato’s Meno exemplifies how to employ scaffolded learning in the philosophy classroom. It explores scaffolded learning by fully defining it, explaining it, and gesturing at some ways in which scaffolding has been implemented. We then offer our own model of scaffolded learning in terms of four phases and eight stages, and explicate our model using a well known example from Plato’s Meno as an exemplar. We believe that any practical concerns one (...) might have against the employment of scaffolded learning in the philosophy classroom ought not serve as an impediment to adopting our model. (shrink)

In lieu of an abstract, here is a brief excerpt of the content:99 HUME AND DERRIDA ON LANGUAGE AND MEANING "...Language itself is menaced in its very life, helpless, adrift in the threat of limitlessness, brought back to its own finitude at the very moment when its limits seem to disappear, when it ceases to be self-assured, contained, and guaranteed by the infinite signified which seemed to exceed it." Is this true? What does it mean? Derrida is making a contrast (...) between two views of language. On the view that Derrida believes correct, the linguistic sign, whether spoken or written, acquires its meaning, its significance, from conventions. On this view, meaning is a matter of nomos or institution rather than physis or nature. Since meaning is a matter of convention rather than nature, the sign on this view is arbitrary. And in this respect, there is no distinction between the linguistic or phonic sign and the written or graphic sign. The other view of language denies that all significance is a matter of nomos. There are, rather, at least some signs the signification of which is natural. This view of language is one that Derrida, quite correctly, locates in Plato. It is there already, perhaps especially, in the Meno. Socrates distinguishes knowledge from true opinions :...true opinions, as long as they remain, are a fine thing and all they do is good, but they are not willing to remain long, and they escape from a man's mind, so that they are not worth much until one ties them down by (giving) an account of the reason why. And that, Meno my friend, is recollection, as we previously agreed. After they are tied down, in the first place they become 100 knowledge, and then they remainin place. That is why knowledgeis prized higher than correct opinionin being tied down. Knowledge is thus certain, or incorrigible. There is no knowledge that lacks a guarantee of its truth. This knowledge is obtained through a form of knowing that Socrates argues is akin to "recollection." The details of what recollection is need not detain us; what is important is what recollection is a knowledge of. There are in fact two cases with which the Meno deals. The first is the case of geometrical 4 propositions. In the course of Socrates' interrogation, Meno' s slave boy discovers that a certain geometrical proposition is true. Recollection is, thus, in the first place a discovery of facts that makes propositions true. That a proposition is true is, in general, not a matter of convention: truth depends upon the facts that the proposition is about, and whether those facts do or do not obtain will in general not be dependent upon any social institutions or conventions. On the other hand, it is compatible with this to hold that what a proposition means is a matter of convention. That 'Hume ist tot' means in German that Hume is dead is a matter of convention, just as it is a matter of convention that 'Hume is not dead' means in English that Hume is not dead; but it is not a matter of convention, but of non-linguistic fact, that the former is true and the latter false. Importantly, Socrates first obtains from Meno an affirmative answer to the question "Does he speak Greek?" before he begins his examination of the slave boy. The boy can thus be taken to understand the conventions that determine the meaning of the proposition the truth of 101 which he discovers. Grasping meaning may in this way be a matter of grasping conventions; but to grasp the truth is not a matter of convention. However, this is not Plato's view, as is made clear by the second example of the sort of thing of which recollection is supposed to yield knowledge. What Socrates seeks is a definition of 'virtue;' he even gives Meno a brief course on how to give good definitions. And this search is, for Socrates, analogous to the search for the truth of the slave boy's geometrical proposition. As the slave boy searches for the truth with respect to a geometrical proposition, so Socrates searches for the truth with respect to the definition of 'virtue.' In each case the... (shrink)

This dissertation concerns the central role of aporia in philosophical thought and Platonic philosophy. In contrast with the standard sense of aporia as a perplexity that clears away an interlocutor's ignorance and pretension, I argue that aporia is a necessary step in the movement from ignorance to knowledge. Aporia thus involves a kind of understanding that in principle leads one out of perplexity to knowledge. This conception of aporia also reveals, I argue a connection between Platonic metaphysical doctrines, such as (...) the Forms, and Socratic knowledge of ignorance, which is central to the practice of Socratic dialectic. ;I show how this conception of aporia plays out in a specific Platonic dialogue, the Meno, and provides a basis for understanding the dialogue as a whole. Specifically, I argue that the Meno consists of three attempts on Socrates' part to bring his interlocutor, Meno, into aporia. These attempts correspond to the main divisions of the dialogue---the opening section that involves ordinary, Socratic elenchus; the myth of recollection and the slave-boy argument; and the third section containing the hypothetical method. Following Brague , I argue that the dialogue deviates from the model of rational discourse, Socratic dialectic. This deviation is the consequence of Socrates' efforts to respond constructively to the state of Meno's soul. Meno is unable to participate in dialectic proper due to his geometric, or dianoetic, cast of mind, as well as the eros of his soul. By and large these character traits render Meno incapable of understanding the spirit of Socratic dialectic. ;The central importance of aporia to philosophy, which is worked out in the Meno, illuminates a number of features about philosophical thought. In particular, I argue that there is an essential connection between the soul's cognitive and moral powers, i.e., between learning and virtue. Because learning depends upon the soul's desire to learn, philosophical understanding is both promoted and limited by the perspective of the soul that attempts to learn. As we seen in the slaveboy argument, through aporia the perspective of the soul can broaden by becoming more knowledgeable. While this success provides rational confidence in the possibility of philosophical understanding, it falls short of a privileged standpoint from which comprehensive and certain knowledge could be attained. (shrink)

The compulsion of proofs is an ancient idea, which plays an important role in Plato’s dialogues. The reader perhaps recalls Socrates’ question to the slave boy in the Meno: “If the side of a square A is 2 feet, and the corresponding area is 4, how long is the side of a square whose area is double, i.e. 8?”. The slave answers: “Obviously, Socrates, it will be twice the length” (cf. Me 82-85). A straightforward analogy: if the area is double, (...) the side is double. Nevertheless, the answer is wrong. Socrates wants to lead the slave to the right conclusion. The boy should reach the truth through steps that are all “his own”, performed with full conviction. To this aim, Socrates addresses a series of short pressing questions to the slave boy. Simple questions provoke equally simple replies, though the boy is sometimes puzzled and surprised by the answers he feels compelled to give. In the first part of the exchange the boy is gradually forced to admit that a square B with double side (i.e. side of length 4) has an area which is not double, but four times as big, i. e. 16. In the second part the boy hazards the guess that the square with double area has a side of length 3, since 3 is between 2 and 4. But Socrates easily drives him to acknowledging that if the side is 3, the area of the resulting square C is 9, not 8. The boy can only exclaim: “By Zeus, Socrates, I do not know!”. In the third part, at Socrates’ prompting, the diagonal of the original square A is drawn within the second fourfold square B. Responding to Socrates’ questions, the boy is led to conclude that the square D whose side is the diagonal of A is precisely the required square: its area is 8. (shrink)

In this paper we extend the NeutroAlgebra & AntiAlgebra to the geometric spaces, by founding the NeutroGeometry & AntiGeometry. While the Non-Euclidean Geometries resulted from the total negation of one specific axiom (Euclid’s Fifth Postulate), the AntiGeometry results from the total negation of any axiom or even of more axioms from any geometric axiomatic system (Euclid’s, Hilbert’s, etc.) and from any type of geometry such as (Euclidean, Projective, Finite, Affine, Differential, Algebraic, Complex, Discrete, Computational, Molecular, Convex, etc.) Geometry, and the (...) NeutroGeometry results from the partial negation of one or more axioms [and no total negation of no axiom] from any geometric axiomatic system and from any type of geometry. Generally, instead of a classical geometric Axiom, one may take any classical geometric Theorem from any axiomatic system and from any type of geometry, and transform it by NeutroSophication or AntiSophication into a NeutroTheorem or AntiTheorem respectively in order to construct a NeutroGeometry or AntiGeometry. Therefore, the NeutroGeometry and AntiGeometry are respectively alternatives and generalizations of the Non-Euclidean Geometries. In the second part, we recall the evolution from Paradoxism to Neutrosophy, then to NeutroAlgebra & AntiAlgebra, afterwards to NeutroGeometry & AntiGeometry, and in general to NeutroStructure & AntiStructure that naturally arise in any field of knowledge. At the end, we present applications of many NeutroStructures in our real world. (shrink)

In lieu of an abstract, here is a brief excerpt of the content:Journal of the History of Philosophy 39.1 (2001) 137-139 [Access article in PDF] Gareth B. Matthews. Socratic Perplexity and the Nature of Philosophy. New York: Oxford University Press, 1999. Pp. 137. Cloth, $29.95 Thomas C. Brickhouse and Nicholas D. Smith. The Philosophy of Socrates. Boulder, CO: Westview Press, 2000. Pp. x + 290. Paper $22.00. Matthews' little book tracks the course of Socrates' perplexity, which, Matthews contends, starts out (...) as an obstacle to successful inquiry, but becomes, eventually, itself the target of inquiry. Matthews regards as quite genuine the perplexity that Socrates professes in certain of the early dialogues—Laches, Charmides, Euthyphro—largely, however, because Matthews himself finds Socrates' questions so stubbornly persistent. Yet, couldn't Socrates, at least with respect to some of his questions, experience no state of confusion, no perplexity, but, nevertheless, profess perplexity? Perhaps Socrates' confessions of perplexity signal only his awareness that he lacks knowledge of virtue; perhaps they are but rhetorical devices for encouraging Socrates' interlocutors to admit to their own ignorance.As the book continues, Matthews, following a line fairly standard by now, thinks that, in the "transitional" Meno, Socrates discovers two ways out of the perplexity: recollection, and the method of hypothesis; and that, in the "middle" dialogues—the Republic and Phaedo—the Theory of Forms comes to his rescue. In the Parmenides and, perhaps later, Matthews maintains, the Theory of Forms becomes itself the object of inquiry, as Socrates recognizes the perplexities with which it is fraught. Matthews mentions, but takes no stand on, the controversies surrounding the late dialogues, moving on at last to Aristotle and the different kind of perplexity one encounters there.For the briefest moment it looks as if Matthews might resist the standard line, when, in discussing the Meno, he notices not only that the geometrical problem of the Meno's slave-boy passage causes no perplexity comparable to that caused by Socrates' philosophical problems, but that Socrates is fully in charge in the geometrical demonstration, stinging another without simultaneously stinging himself. Yet, not only does Matthews fail to pursue his insight that moral or philosophical questions are, for Socrates, not like mathematical ones; on the contrary, he suggests that, in the Phaedo, philosophical problems are viewed by Socrates as mathematical ones.In The Philosophy of Socrates, Brickhouse and Smith undertake the ambitious task of expounding and defending a coherent and cohesive Socratic philosophy, one that encompasses Socrates' method, his wisdom and ignorance, his understanding of the nature of knowledge, his moral philosophy, psychology, politics, and religion. For a book meant to be a popular introduction, it is remarkably comprehensive—even [End Page 137] exhaustive—and serves to acquaint the reader with the current scholarly controversies surrounding Socratic philosophy as it is debated in the analytic tradition associated most frequently with the late and great Gregory Vlastos.In many respects, The Philosophy of Socrates is but an updated version of Brickhouse and Smith's earlier book, Plato's Socrates (Oxford University Press, 1994)—though without the rich and copious notes. The earlier book is the better one. The new book, although it purports to discuss the philosophy of the historical Socrates, is, from Chapter 2 on, a book about the Socrates of Plato's "early" dialogues. Yet, it is precisely this Socrates who is largely absent from its pages. The Socrates of the early dialogues is a character: both in the sense that he is a protagonist in the dramas that are Plato's dialogues, and in the sense that he is sarcastic, funny, makes bad puns, strikes at his interlocutors where they are most vulnerable, teases, baits, and annoys. But, most importantly, this Socrates is no solitary thinker. He talks to people; and what he says and how he says it is influenced both by his interlocutor and by the context of their conversation. Brickhouse and Smith, however, take such factors into account but rarely—and then only to save their own take on a particular Socratic doctrine.The two interpretive principles that Brickhouse and Smith invoke most often... (shrink)

In this paper we extend Neutro-Algebra and Anti-Algebra to geometric spaces, founding Neutro/Geometry and AntiGeometry. While Non-Euclidean Geometries resulted from the total negation of a specific axiom (Euclid's Fifth Postulate), AntiGeometry results from the total negation of any axiom or even more axioms of any geometric axiomatic system (Euclidean, Hilbert, etc. ) and of any type of geometry such as Geometry (Euclidean, Projective, Finite, Differential, Algebraic, Complex, Discrete, Computational, Molecular, Convex, etc.), and Neutro-Geometry results from the partial negation of one (...) or more axioms [and without total negation of any axiom] of any geometric axiomatic system and of any type of geometry. Generally, instead of a classical geometric Axiom, one can take any classical geometric Theorem of any axiomatic system and of any type of geometry, and transform it by Neutrosophication or Antisofication into a Neutro-Theorem or Anti-Theorem respectively to construct a Neutro-Geometry or Anti-Geometry. Therefore, Neutro-Geometry and Anti-Geometry are respectively alternatives and generalizations of Non-Euclidean Geometries. In the second part, the evolution from Paradoxism to Neutrosophy, then to NeutroAlgebra and Anti-Algebra, then to Neutro-Geometry and Anti-Geometry, and in general to Neutro-Structure and Anti-Structure that arise naturally in any field of knowledge is recalled. At the end, applications of many Neutro-Structures in our real world are presented. (shrink)

I state and prove, in the context of a space having only the metrical structure imposed by the geometrized version of Newtonian gravitational theory, a theorem analagous to that of Weyl's in a Lorentzian space. The theorem, loosely speaking, says that a projective structure and a suitably defined compatible conformal structure on such a space jointly suffice for fixing the metrical structure of a Newtonian spacetime model up to constant factors. It allows one to give a natural, physically (...) compelling interpretation of the spatiotemporal geometry of a geometrized Newtonian gravity spacetime manifold, in close analogy with the way Weyl's Theorem allows one to do in general relativity. (shrink)

The CPT theorem of quantum field theory states that any relativistic (Lorentz-invariant) quantum field theory must also be invariant under CPT, the composition of charge conjugation, parity reversal and time reversal. This paper sketches a puzzle that seems to arise when one puts the existence of this sort of theorem alongside a standard way of thinking about symmetries, according to which spacetime symmetries (at any rate) are associated with features of the spacetime structure. The puzzle is, roughly, that (...) the existence of a CPT theorem seems to show that it is not possible for a well-formulated theory that does not make use of a preferred frame or foliation to make use of a temporal orientation. Since a manifold with only a Lorentzian metric can be temporally orientable—capable of admitting a temporal orientation—this seems to be an odd sort of necessary connection between distinct existences. The paper then suggests a solution to the puzzle: it is suggested that the CPT theorem arises because temporal orientation is unlike other pieces of spacetime structure, in that one cannot represent it by a tensor field. To avoid irrelevant technical details, the discussion is carried out in the setting of classical field theory, using a little-known classical analog of the CPT theorem. (shrink)

We establish a criterion for deciding whether a class of structures is the class of models of a geometric theory inside Grothendieck toposes; then we specialize this result to obtain a characterization of the infinitary first-order theories which are geometric in terms of their models in Grothendieck toposes, solving a problem posed by Ieke Moerdijk in 1989.

This paper has two specific goals. The first is to demonstrate that the theorem in MetaphysicsΘ 9, 1051a24-27 is not equiva-lent to Euclid’s Proposition 32 of book I (which contradicts some Aristotelian commentators, such as W. D. Ross, J. L. Heiberg, and T. L. Heith). Agreeing with Henry Mendell’s analysis, I ar-gue that the two theorems are not equivalent, but I offer different reasons for such divergence: I propose a pedagogical-philosoph-ical reason for the Aristotelian theorem being shorter than (...) the Euclidean one (and the previous Aristotelian versions). Aristotle wants to emphasize the deductive procedure as a satisfactory method to discover scientific knowledge. The second objective, opposing some consensus about geometrical deductions/theo-rems in Aristotle, is to briefly propose that the theorem, exactly as we found it in Metaphysics and without any emendation to the text (therefore opposing Henry Mendell’s suggested amend-ments), allows the ancient philosopher to demonstrate that universal mathematical knowledge is in potence in geometrical figures. This tentatively proves that Aristotle emphasizes that geometrical deduction is sufficient to actualize mathematical knowledge. (shrink)

Sen's classic social choice result supposedly demonstrates a conflict between Pareto and even minimal forms of liberalism. By providing the first direct mathematical proof of this seminal result, we underscore a significantly different interpretation: rather than conflicts among rights, Sen's result occurs because the liberalism assumption negates the assumption that voters have transitive preferences. This explanation enriches interpretations of Sen's conclusion by including radically new kinds of societal conflicts, it suggests ways to sidestep these difficulties, and it explains earlier approaches (...) to avoid the difficulties. (shrink)

SummaryIn 1693, Gottfried Wilhelm Leibniz published in the Acta Eruditorum a geometrical proof of the fundamental theorem of the calculus. It is shown that this proof closely resembles Isaac Barrow's proof in Proposition 11, Lecture 10, of his Lectiones Geometricae, published in 1670. This comparison provides evidence that Leibniz gained substantial help from Barrow's book in formulating and presenting his geometrical formulation of this theorem. The analysis herein also supports the work of J. M. Child, who (...) in 1920 studied the early manuscripts of Leibniz and concluded that he had frequently copied his diagrams from Barrow's book, but without acknowledgement. It is also shown that the diagram of Leibniz associated with his 1693 proof has often been reproduced with errors that make some aspects of his text difficult to comprehend. (shrink)

Geometric theories are presented as contraction- and cut-free systems of sequent calculi with mathematical rules following a prescribed rule-scheme that extends the scheme given in Negri and von Plato. Examples include cut-free calculi for Robinson arithmetic and real closed fields. As an immediate consequence of cut elimination, it is shown that if a geometric implication is classically derivable from a geometric theory then it is intuitionistically derivable.

We study the idea of implantation of Piron's and Bell's geometrical lemmas for proving some results concerning measures on finite as well as infinite-dimensional Hilbert spaces, including also measures with infinite values. In addition, we present parabola based proofs of weak Piron's geometrical and Bell's lemmas. These approaches will not used directly Gleason's theorem, which is a highly non-trivial result.

Large portions of mathematics such as algebra and geometry can be formalized using first-order axiomatizations. In many cases it is even possible to use a very well-behaved class of first-order axioms, namely, what are called coherent or geometric implications. Such class of axioms can be translated to inference rules that can be added to a sequent calculus while preserving its structural properties. In this work, this fundamental result is extended to their infinitary generalizations as extensions of sequent calculi for both (...) classical and intuitionistic infinitary logic. As an application, a simple proof of the infinitary Barr’s theorem without the axioms of choice is shown. (shrink)

The aim of this paper is to provide a logic-based conceptual analysis of the twin paradox (TwP) theorem within a first-order logic framework. A geometrical characterization of TwP and its variants is given. It is shown that TwP is not logically equivalent to the assumption of the slowing down of moving clocks, and the lack of TwP is not logically equivalent to the Newtonian assumption of absolute time. The logical connection between TwP and a symmetry axiom of special (...) relativity is also studied. (shrink)

This paper corrects the common misconception that Meno's slave (in Plato's dialogue of that name) is a boy. The first part of the paper shows how long-standing and widespread that misconception is. The description of Meno's slave as a "slave-boy" goes back at least to Benjamin Jowett, and the phrase is still commonly seen today in books and journal articles in philosophy and classics generally, even in presses and journals with the highest reputation. The paper then shows that the Greek (...) term pais, often translated as "boy", is when addressed to slaves used to indicate their condition, not their age. When the text of the Meno is examined carefully, it is clear that there is no evidence that Meno's slave is a boy. In fact, it is clear that the expression "boy" is used in relation to his condition, not in relation to his age. It thus demeans us to refer to Meno's slave as a "slave-boy" or just "boy", since it either displays our ignorance about the use of the term pais or, worse, makes us complicit in using a term of condescension. The paper concludes by suggesting that the proposed correction is philosophically significant, since it opens an investigation into Plato's depiction of slaves that is otherwise blocked by supposing the slave to be a boy. (shrink)

Historically, Ehrenfest’s theorem is the first one which shows that classical physics can emerge from quantum physics as a kind of approximation. We recall the theorem in its original form, and we highlight its generalizations to the relativistic Dirac particle and to a particle with spin and izospin. We argue that apparent classicality of the macroscopic world can probably be explained within the framework of standard quantum mechanics.

We give a self-contained geometric proof of the completeness theorem for the infinite-valued sentential calculus of Łukasiewicz.

Consider the following well-known result from the theory of normed linear spaces ([2], p. 80, 4(b)): (g) the unit ball of the (continuous) dual of a normed linear space over the reals has an extreme point. The standard proof of (~) uses the axiom of choice (AG); thus the implication AC~(w) can be proved in set theory. In this paper we show that this implication can be reversed, so that (*) is actually eq7I2valent to the axiom of choice. From this (...) we derive various corollaries, for example: the conjunction of the Boolean prime ideal theorem and the Krein-Milman theorem implies the axiom of choice, and the Krein-Milman theorem is not derivable from the Boolean prime ideal theorem. (shrink)

The theorem that the arithmetic mean is greater than or equal to the geometric mean is investigated for cardinal and ordinal numbers. It is shown that whereas the theorem of the means can be proved for n pairwise comparable cardinal numbers without the axiom of choice, the inequality a2 + b2 ≥ 2ab is equivalent to the axiom of choice. For ordinal numbers, the inequality α2 + β2 ≥ 2αβ is established and the conditions for equality are derived; (...) stronger inequalities are obtained for finite and infinite sequences of ordinals under suitable monotonicity hypotheses. MSC: 03E10, 04A10, 03E25, 04A25. (shrink)

We take a fresh look at voting theory, in particular at the notion of manipulation, by employing the geometry of the Saari triangle. This yields a geometric proof of the Gibbard/Satterthwaite theorem, and new insight into what it means to manipulate the vote. Next, we propose two possible strengthenings of the notion of manipulability (or weakenings of the notion of non-manipulability), and analyze how these affect the impossibility proof for non-manipulable voting rules.

The connections between mathematical logic and combinatorics have a rich history. This paper focuses on one aspect of this relationship: understanding the strength, measured using the tools of computability theory and reverse mathematics, of various partition theorems. To set the stage, recall two of the most fundamental combinatorial principles, König's Lemma and Ramsey's Theorem. We denote the set of natural numbers by ω and the set of finite sequences of natural numbers by ω<ω. We also identify each n ∈ (...) ω with its set of predecessors, so n = {0, 1, 2, …, n − 1}. (shrink)

Mathematical theorems are cultural artifacts and may be interpreted much as works of art, literature, and tool-and-craft are interpreted. The Fundamental Theorem of the Calculus, the Central Limit Theorem of Statistics, and the Statistical Continuum Limit of field theories, all show how the world may be put together through the arithmetic addition of suitably prescribed parts (velocities, variances, and renormalizations and scaled blocks, respectively). In the limit — of smoothness, statistical independence, and large N — higher-order parts, such (...) as accelerations, are, for the most, part irrelevant, affirming that, in the end, most of the world's particulars may be averaged over (a very un-Scriptural point of view). (We work out all of this in technical detail, including a nice geometric picture of stochastic integration, and a method of calculating the variance of the sum of dependent random variables using renormalization group ideas.) These fundamental theorems affirm a culture that is additive, ahistorical, Cartesian, and continuist, sharing in what might be called a species of modern culture. We understand mathematical results as useful because, like many other such artifacts, they have been adapted to fit the world, and the world has been adapted to fit their capacities. Such cultural interpretation is in effect motivation for the mathematics, and might well be offered to students as a way of helping them understand what is going on at the blackboard. Philosophy of mathematics might want to pay more attention to the history and detailed technical features of sophisticated mathematics, as a balance to the usual concerns that arise in formalist or even Platonist positions. (shrink)

Using Butz and Moerdijk's topological groupoid representation of a topos with enough points, a ‘syntax-semantics’ duality for geometric theories is constructed. The emphasis is on a logical presentation, starting with a description of the semantic topological groupoid of models and isomorphisms of a theory. It is then shown how to extract a theory from equivariant sheaves on a topological groupoid in such a way that the result is a contravariant adjunction between theories and groupoids, the restriction of which is a (...) duality between theories with enough models and semantic groupoids. Technically a variant of the syntax-semantics duality constructed in 1 for first-order logic, the construction here works for arbitrary geometric theories and uses a slice construction on the side of groupoids—reflecting the use of ‘indexed’ models in the representation theorem—which in several respects simplifies the construction and the characterization of semantic groupoids. (shrink)

The formulas for the Lie covariant differentiation of spinors are deduced from an algebraic viewpoint. The Lie covariant derivative of the spinor connection is calculated, and is given a geometric meaning. A theorem about the Lie covariant derivative of an operator in spin space that was stated in Part I of this work is discussed.

The class of geometric surgical theories is examined. The main theorem is that every stable theory that is interpretable in a geometric surgical theory is superstable of finite U-rank.

We consider the set of all matrices of the form pij = tr[W (Ei ⊗ Fj)] where Ei, Fj are projections on a Hilbert space H, and W is some state on H ⊗ H. We derive the basic properties of this set, compare it with the classical range of probability, and note how its properties may be related to a geometric measures of entanglement.

We study the theory of lovely pairs of geometric structures, in particular o-minimal structures. We use the pairs to isolate a class of geometric structures called weakly locally modular which generalizes the class of linear structures in the settings of SU-rank one theories and o-minimal theories. For o-minimal theories, we use the Peterzil–Starchenko trichotomy theorem to characterize for a sufficiently general point, the local geometry around it in terms of the thorn U-rank of its type inside a lovely pair.

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

The Bell inequalities of the metric form are introduced. The quantum-mechanical correlations of the particles with s=1/2 and photons are described using the relative measure of probability on the concave surfaces. The relation of the proposed scheme with the Bayes theorem about conditional information entropy and J. von Neumann's postulates is discussed.

In this paper we present a general theory of normal forms, based on a categorial result for the free monoid construction. We shall use the theory mainly for proposictional modal logic, although it seems to have a wider range of applications. We shall formally represent normal forms as combinatorial objects, basically labelled trees and forests. This geometric conceptualization is implicit in and our approach will extend it to other cases and make it more direct: operations of a purely geometric and (...) combinatorial nature will be introduced in order to give a mathematical description of the basic logical/algebraic constructions .We begin by recalling the above-mentioned categorial construction: we need a careful inspection of it because in the various examples considered later we plan to deduce from it in a uniform way the normal forms and the description of finitely generated free algebras. This method always works whenever we can describe the category of algebras corresponding to the logic under consideration as a T-objects category. When this simple description seems not to be available, still the general theory might be of some interest, because a description of the category of algebras as a T-objects category plus equation is possible .The central part of the paper is more advanced and specific: we show how the general approach presented here can provide some insights even in the basic case of the modal system K. Section 4 contains a contribution to the theory of normal forms, namely the description of the uniform substitution. This result will enable us to give a duality theorem for the category of finitely generated free modal algebras and in Section 5 to provide a characterization of the collections of normal forms which happen to be normal forms for a logic, thus giving a description of the lattice of modal logics.Section 6 deals with some applications: we shall show how to use normal forms in order to prove for the modal system K the definability of higher-order propositional quantifiers and of the tense operator F .As to the prerequisites, the paper is almost self-contained. The reader is only assumed to have familiarity with standard techniques in algebraic logic ). Knowledge of the basic facts about adjoint functors is required too, see e.g. McLane or the appendix. (shrink)

This paper seeks to demonstrate that propositions 23–27 of the Euclidian Optics originated in the context of geometrical astronomy. These entries, which deal with the geometry of spheres and rays, present material that overlaps considerably with propositions 1–3 of Aristarchus of Samos’ On the Sizes and Distances of the Sun and the Moon. While all these theorems deal with material that could conceivably be native to celestial illumination, the proofs do not work for binocular vision. It therefore seems probable (...) that the proofs were borrowed from Aristarchus or, more likely, some common astronomical predecessor. As an extension of this observation, this paper argues that the Optics displays a far less unified purpose than has typically been assumed. Instead, it reflects a conglomerate of concerns, goals and dependencies, insofar as it presents multiple proofs that emerge from a set of geometrical concerns not directly related to vision as a physical phenomenon. These entries complicate the idea that the Optics is about explaining light, sight or false appearances in any straightforward sense. Instead, the text is oriented toward a somewhat more diffuse goal: simply to articulate the geometry of vision and rays. Optics as a discipline should should therefore be understood not as a purely mathematical explanation of vision, but as a target-field of explanation constituted by features and boundaries derived from extra-visual geometric concerns. (shrink)

ArgumentThe aim of this paper is to employ the newly contextualized historiographical category of “premodern algebra” in order to revisit the arguably most controversial topic of the last decades in the field of Greek mathematics, namely the debate on “geometrical algebra.” Within this framework, we shift focus from the discrepancy among the views expressed in the debate to some of the historiographical assumptions and methodological approaches that the opposing sides shared. Moreover, by using a series of propositions related toElem.II.5 (...) as a case study, we discuss Euclid's geometrical proofs, the so-called “semi-algebraic” alternative demonstrations attributed to Heron of Alexandria, as well as the solutions given by Diophantus, al-Sulamī, and al-Khwārizmī to the corresponding numerical problem. This comparative analysis offers a new reading of Heron's practice, highlights the significance of contextualizing “premodern algebra,” and indicates that the origins of algebraic reasoning should be sought in the problem-solving practice, rather than in the theorem-proving tradition. (shrink)