In this paper, we would show how the logical object “square of opposition”, viewed as semiotic object, has been historically transformed since its appearance in Aristotle’s texts until the works of Vasiliev. These transformations were accompanied each time with a new understanding and interpretation of Aristotle’s original text and, in the last case, with a transformation of its geometric configuration. The initial textual codification of the theory of opposition in Aristotle’s works is transformed into a diagrammatic one, based on a (...) new “reading” of the Aristotelian text by the medieval scholars that altered the semantics of the O form. Further, based on the medieval “Neo-Aristotelian” reading, the logicians of the nineteenth century suggest new diagrammatic representations, based on new interpretations of quantification of judgements within the algebraic and the functional logical traditions. In all these interpretations, the original square configuration remains invariant. However, Nikolai A. Vasiliev marks a turning point in history. He explicitly attacks the established logical tradition and suggests a new alternation of semantics of the O form, based on Aristotelian concepts that were neglected in the Aristotelian tradition of logic, notably the concept of indefinite judgement. This leads to a configurational transformation of the “square” of opposition into a “triangle”, where the points standing for the O and I forms are contracted into one point, the M form that now stands for particular judgement with altered semantics. The new transformation goes beyond the Aristotelian logic paradigm to a new “Non-Aristotelian” logic, i.e. to paraconsistent logic, although the argumentation used in support of it is phrased in Aristotelian style and the context of discovery is foundational. It establishes a bifurcation point in the development of logic. No unique logic is recognized, but different logics concerning different domains. One branch of logic remains to be the “Neo-Aristotelian” one, while the new logic is “Non-Aristotelian”. (shrink)

This is the first interdisciplinary exploration of the philosophical foundations of the Web, a new area of inquiry that has important implications across a range of domains. - Contains twelve essays that bridge the fields of philosophy, cognitive science, and phenomenology. - Tackles questions such as the impact of Google on intelligence and epistemology, the philosophical status of digital objects, ethics on the Web, semantic and ontological changes caused by the Web, and the potential of the Web to serve as (...) a genuine cognitive extension. - Brings together insightful new scholarship from well-known analytic and continental philosophers, such as Andy Clark and Bernard Stiegler, as well as rising scholars in “digital native” philosophy and engineering. - Includes an interview with Tim Berners-Lee, the inventor of the Web. (shrink)

The Web may critically transform the way we understand the activity of proving. The Web as a collaborative medium allows the active participation of people with different backgrounds, interests, viewpoints, and styles. Mathematical formal proofs are inadequate for capturing Web-based proofs. This article claims that Web provings can be studied as a particular type of Goguen's proof-events. Web-based proof-events have a social component, communication medium, prover-interpreter interaction, interpretation process, understanding and validation, historical component, and styles. To demonstrate its claim, the (...) article discusses the Kumo and Polymath projects, both of which employ Web-based communication as part of proving. Web proving is a novel type of proving activity that may have a serious impact on the change in mathematical practices, despite the fact that it is not currently a universally acceptable methodology. (shrink)

The style of arithmetic in the treatises the Neo-Pythagorean authors is strikingly different from that of the "Elements". Namely, it is characterised by the absence of proof in the Euclidean sense and a specific genetic approach to the construction of arithmetic that we are going to describe in our paper. Lack of mathematical sophistication has led certain historians to consider this type of mathematics as a feature of decadence of mathematics in this period [Tannery 1887; Heath 1921]. The alleged absence (...) of originality in these works has also given grounds to believe that “the arithmetic presented in these works derives substantially from an ancient, primitive stage of Pythagorean arithmetic” and to use them as “an index of the character of arithmetic science in the 5th century” [Knorr 1975]. In this paper, we take Nicomachus’ Introduction to Arithmetic for point of departure, because it is the richest and most well organised treatise representing this tradition. However, we also take into account the works of other Neo-Pythagorean authors. We are going to show that the Neo-Pythagorean arithmetic might have been developed in a natural, self-contained manner as a simple theory of counting over a domain of concrete initial objects, designated by fixed signs. This approach can be comfortably realised without appealing to assumptions of axiomatic character, but relying upon some ‘genetic’ constructions intented to be carried out by means of the designated entities. (shrink)

The paper examines Andrei A. Markov’s critical attitude towards L.E.J. Brouwer’s intuitionism, as is expressed in his endnotes to the Russian translation of Heyting’s Intuitionism, published in Moscow in 1965. It is argued that Markov’s algorithmic approach was shaped under the impact of the mathematical style and values prevailing in the Petersburg mathematical school, which is characterized by the proclaimed primacy of applications and the search for rigor and effective solutions.

The concept of proof can be studied from many different perspectives. Many types of proofs have been developed throughout history such as apodictic, dialectical, formal, constructive and non-constructive proofs, proofs by visualisation, assumption-based proofs, computer-generated proofs, etc. In this paper, we develop Goguen’s general concept of proof-events and the methodology of algebraic semiotics, in order to define the concept of mathematical style, which characterizes the proofs produced by different cultures, schools or scholars. In our view, style can be defined as (...) a semiotic meta-code that depends on the underlying mode of signification (semiosis), the selected code and the underlying semiotic space and determines the individual mode of integration (selection, combination, blending) into a narrative structure (proof). Finally, we examine certain historical types of styles of mathematical proofs, to elucidate our viewpoint. (shrink)

The lack of specific arithmetical axioms in Book VII has puzzled historians of mathematics. It is hardly possible in our view to ascribe to the Greeks a conscious undertaking to axiomatize arithmetic. The view that associates the beginnings of the axiomatization of arithmetic with the works of Grassman [1861], Dedekind [1888] and Peano [1889] seems to be more plausible. In this connection a number of interesting historical problems have been raised, for instance, why arithmetic was axiomatized so late. This question (...) was first posed and quite conclusively answered by Yanovskaja [1956]. Her major and quite conclusive argument was that “algorithms in arithmetic have absolute character, while in geometry we have to do with reducibility algorithms”. In this paper we are going to draw attention on certain peculiarities of the construction of the arithmetical Books of Euclid’s "Elements" and show that Euclidean arithmetic is constructed by effective procedures. In spite of the use of inference by reductio ad absurdum and methods equivalent to mathematical induction, Euclidean arithmetic retains its finitary character. It is not full arithmetic, but a finitary fragment of classical arithmetic, and thereby there was not any internal reason for its axiomatization. (shrink)

In Plato’s Parmenides 132a-133b, the widely known Third Man Paradox is stated, which has special interest for the history of logical reasoning. It is important for philosophers because it is often thought to be a devastating argument to Plato’s theory of Forms. Some philosophers have even viewed Aristotle’s theory of predication and the categories as inspired by reflection on it [Owen 1966]. For the historians of logic it is attractive, because of the phenomenon of self-reference that involves. Bocheński denies any (...) possibility of correct logical reasoning before Aristotle. In particular, he flatly declares of Plato that “correct logic we find none in his work” [1951, 15]. In this line, many papers have been written that call attention to the violation of a metalogical principle – the type rules – because of the Third Man Paradox. The problem of interpretation of the paradox raised many discussions, since the 50’s, and the literature devoted to this topic is today enormous. Nevertheless, many points in Plato’s reasoning remain obscure. Hitherto, it is commonly believed that the paradox was simply stated by Plato in his "Parmenides", but not actually solved. Even more, it is believed that it is hardly possible for Plato to have suggested a solution, because he confused the categories of substance and attribute. B. Russell has also argued that Plato violates in his arguments the restrictions imposed on language by the theory of logical types. This view is encouraged by the linguistic difficulties, which Plato has faced in his attempt to formulate an ontology of abstract entities, i.e. that in Greek language abstract and concrete terms are formally indistinguishable [Kneales 1984]. In this paper, we analyse the logical structure of the argument in an attempt to give a systematic consistent reconstruction of the text appealing to methods and concepts of modern logic and semantics. Already in 1954 G. Vlastos had noted in his seminal paper “The ‘Third Man’ Argument in the Parmenides” that “if any progress … is to be made at this juncture it must come from some advance in understanding the logical structure of the Argument”. Accordingly, our analysis is primarily focused on the line of logical reasoning, rather than the metaphysical underpinnings and philosophical implications of the paradox for Plato’s theory of Forms. We divide the Platonic text into three parts, presenting apparent thematic coherence: a) Formulation of the Third Man Paradox (132a-132b). The central issue of this part is a step-like generation of Forms that can continue ad infinitum. b) Discussion of the paradoxical situation (132b-c), by passing to the use of terminology echoing Eleatic philosophy. The concept of “thought” and the underlying semantics of Eleatic origin are central in this passage. c) Solution of the paradox (132d-133a). In our view, this part contains not only a resemblance regress, as most interpretators con-sider, but also the solution of the paradox by the introduction of a sound definition of the concept of “similarity”. We should note that the name ‘Third Man Paradox’ never occurs in Plato, who, strictly speaking, formulates a ‘Third Large Paradox’. The name ‘Third Man Paradox’ appears in the works of Aristotle and the commentators of the Peripatetic tradition. Later commentators have identified Aristotle’s Third Man with Plato’s Third Large Argument. In our paper, we also examine other testimonies about the Third Man Paradox found in the works of commentators in order to illuminate the logical structure of the argument. Although the vocabulary used in these versions is different, they do not affect its logical structure, but rather reveal different understandings by the ancient authors. These testimonies are divided into two main categories: those met in Neo-Platonic authors, notably in Proclus’ "Commentary on Plato’s Parmenides", and the versions of the Third Man Paradox found in texts of the Peripatetic tradition (Eudemus, Aristotle, Alexander). From our discussion, it becomes clear that the approaches to the paradox by the various scholars of antiquity are different, depending also on their participation in the one or the other campus of philosophical thought. We show that the Peripatetic authors are aware of the source of the paradox. However, the first scholar of antiquity who explicitly ascribes a solution to Plato seems to be Proclus. (shrink)

ExtractDuring the second half of the twentieth century an attention of historians of mathematics shifted to mathematics of the Late Antiquity and its subsequent development by mathematicians of the Arabic world. Many critical editions of works of mathematicians of the Hellenistic era have made their appearance, giving rise to a new, more detailed historical picture. Among these are the critical editions of the works of Diophantus, Apollonius, Archimedes, Pappus, Diocles, and others.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.The Readings of Apollonius' On the Cutting off of a RatioVolume 22, Issue 1Ioannis M. Vandoulakis DOI: https://doi.org/10.1017/S0957423911000130Your 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. The Readings of Apollonius' On the Cutting off of a RatioVolume 22, Issue 1Ioannis M. Vandoulakis DOI: https://doi.org/10.1017/S0957423911000130Available 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. The Readings of Apollonius' On the Cutting off of a RatioVolume 22, Issue 1Ioannis M. Vandoulakis DOI: https://doi.org/10.1017/S0957423911000130Available 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)