By reconsidering all the available sources (from Simon du Val’s inquisitorial summons of November 1276 to the Continuatio brabantina, from the Fiore to the X canto of Dante’s Paradiso, from William of Tocco to Peckham’s letters) the article calls into question the thesis – still widely shared – according to which Siger of Brabant died in Italy, and more precisely in Orvieto, at the Papal Court, before November 1284. Above all, from a doctrinal point of view, it shows how (...) it is completely implausible that Peckham could refer to Siger of Brabant when, in his letter to the University of Oxford of November 10, 1284, he mentions two Seculars as the main defenders or ‘inventors’ of the thesis of the uniqueness of the substantial form in the human compound. The end of Siger in Italy could thus be, for the most part, a ‘black legend’, like that of the poisoning of Thomas Aquinas on behalf of Charles of Anjou. The article closes by showing some possible implications, even of a methodological nature, of this different approach. (shrink)

Metaphors allow us to come to terms with abstract and complex information, by comparing it to something which is structured, familiar and concrete. Although modern science is “iconoclastic”, as Gaston Bachelard phrases it, scientists are at the same time prolific producers of metaphoric images themselves. Synthetic biology is an outstanding example of a technoscientific discourse replete with metaphors, including textual metaphors such as the “Morse code” of life, the “barcode” of life and the “book” of life. This paper focuses (...) on a different type of metaphor, however, namely on the archetypal metaphor of the mandala as a symbol of restored unity and wholeness. Notably, mandala images emerge in textual materials related to one of the new “frontiers” of contemporary technoscience, namely the building of a synthetic cell: a laboratory artefact that functions like a cell and is even able to replicate itself. The mandala symbol suggests that, after living systems have been successfully reduced to the elementary building blocks and barcodes of life, the time has now come to put these fragments together again. We can only claim to understand life, synthetic cell experts argue, if we are able to technically reproduce a fully functioning cell. This holistic turn towards the cell as a meaningful whole also requires convergence at the “subject pole”: the building of a synthetic cell as a practice of the self, representing a turn towards integration, of multiple perspectives and various forms of expertise. (shrink)

The paper presents a preliminary estimation of the extent of dissemination of optical texts, ideas, and issues among the masters connected with the Prague faculty of arts in the late 14th and early 15th century. Investigation of this topic, so far rather neglected, is based chiefly on manuscript research. The paper brings evidence that perspectiva was taught in Prague at least since the 1370s. It suggests that investigation of Prague quodlibetal disputations (ca. 1390s – 1410s) and consideration of perspectivist authorities (...) employed in these texts can also shed some light on the issue. However, the paper is mainly devoted to the question on the mechanism of vision disputed by Bohemian arts master John of Borotín (Iohannes de Borotin, 1378 – after 1458) in the quodlibet organised by John Hus in 1411. It is suggested that the single extant exemplar of the question (preserved in the codex Prague, National Library, X.H.18) probably is Borotín’s autograph. The structure and sources of the question are analysed, with the conclusion being that Borotín proposes a compromise between intromission and extramission influenced by John Peckham’s Perspectiva communis. A critical edition of Borotín’s question is appended to the paper. / / / Please note this article is published with Brepols Publishers as a Gold Open Access article under a Creative Commons CC 4.0: BY-NC license. The article is also freely available on the website of Brepols Publishers under this same license. (shrink)

This paper investigates how later medieval intellectuals dealt with perspectiva – the medieval discipline of optics, which had seen considerable popularity in Latin Europe since the 13th century and was epitomized in several “books of knowledge” of differing scopes, levels of difficulty and intended audience. This paper is focused narrowly on one of these intellectuals – Reimbotus de Castro (fl. 1350s–1380s), who was not only personal physician to the Roman Emperor Charles IV but was also a diligent copyist and abbreviator (...) of many quadrivial and medical texts, and the owner of several codices now kept in the Bibliotheca Palatina. One of these codices, Pal. lat. 1380, includes two optical treatises copied by Reimbot himself. A closer reading of these texts, hitherto unnoticed by historians of medieval science, provides fresh insight into the reception of optical knowledge in the intellectual milieu of Paris in the 1360s, when Reimbot resided here. The first text is Reimbot’s reportatio of lectures on the famous optical textbook Perspectiva communis by John Peckham; the second is Reimbot’s redaction of the unknown optical compendium Perspectiva cum sit una. Finally, this paper addresses the issue of why optics was interesting at all for people at the intersection of the scholarly and courtly communities like Reimbot (and late medieval scholars generally). It is suggested that medieval optics, being useful for astronomical observations, could be considered a sort of auxiliary discipline for astrology and astrological medicine. (shrink)

The incompleteness of set theory ZF C leads one to look for natural nonconservative extensions of ZF C in which one can prove statements independent of ZF C which appear to be “true”. One approach has been to add large cardinal axioms.Or, one can investigate second-order expansions like Kelley-Morse class theory, KM or Tarski-Grothendieck set theory T G or It is a nonconservative extension of ZF C and is obtained from other axiomatic set theories by the inclusion of Tarski’s (...) axiom which implies the existence of inaccessible cardinals. See also related set theory with a filter quantifier ZF (aa). In this paper we look at a set theory NC# ∞# , based on bivalent gyper infinitary logic with restricted Modus Ponens Rule In this paper we deal with set theory NC# ∞# based on bivalent gyper infinitary logic with Restricted Modus Ponens Rule. Nonconservative extensions of the canonical internal set theories IST and HST are proposed. (shrink)

According to Cantor (Mathematische Annalen 21:545–586, 1883 ; Cantor’s letter to Dedekind, 1899 ) a set is any multitude which can be thought of as one (“jedes Viele, welches sich als Eines denken läßt”) without contradiction—a consistent multitude. Other multitudes are inconsistent or paradoxical. Set theoretical paradoxes have common root—lack of understanding why some multitudes are not sets. Why some multitudes of objects of thought cannot themselves be objects of thought? Moreover, it is a logical truth that such multitudes do (...) exist. However we do not understand this logical truth so well as we understand, for example, the logical truth $${\forall x \, x = x}$$ . In this paper we formulate a logical truth which we call the productivity principle. Rusell (Proc Lond Math Soc 4(2):29–53, 1906 ) was the first one to formulate this principle, but in a restricted form and with a different purpose. The principle explicates a logical mechanism that lies behind paradoxical multitudes, and is understandable as well as any simple logical truth. However, it does not explain the concept of set. It only sets logical bounds of the concept within the framework of the classical two valued $${\in}$$ -language. The principle behaves as a logical regulator of any theory we formulate to explain and describe sets. It provides tools to identify paradoxical classes inside the theory. We show how the known paradoxical classes follow from the productivity principle and how the principle gives us a uniform way to generate new paradoxical classes. In the case of ZFC set theory the productivity principle shows that the limitation of size principles are of a restrictive nature and that they do not explain which classes are sets. The productivity principle, as a logical regulator, can have a definite heuristic role in the development of a consistent set theory. We sketch such a theory—the cumulative cardinal theory of sets. The theory is based on the idea of cardinality of collecting objects into sets. Its development is guided by means of the productivity principle in such a way that its consistency seems plausible. Moreover, the theory inherits good properties from cardinal conception and from cumulative conception of sets. Because of the cardinality principle it can easily justify the replacement axiom, and because of the cumulative property it can easily justify the power set axiom and the union axiom. It would be possible to prove that the cumulative cardinal theory of sets is equivalent to the Morse–Kelley set theory. In this way we provide a natural and plausibly consistent axiomatization for the Morse–Kelley set theory. (shrink)

The incompleteness of set theory ZFC leads one to look for natural extensions of ZFC in which one can prove statements independent of ZFC which appear to be "true". One approach has been to add large cardinal axioms. Or, one can investigate second-order expansions like Kelley-Morse class theory, KM or Tarski- Grothendieck set theory TG [1]-[3] It is a non-conservative extension of ZFC and is obtaineed from other axiomatic set theories by the inclusion of Tarski's axiom which implies the (...) existence of inaccessible cardinals [1].Non-conservative extension of ZFC based on an generalized quantifiers considered in [4]. In this paper we look at a set theory NC_{∞^{#}}^{#},based on bivalent gyper infinitary logic with restricted Modus Ponens Rule [5]-[8]. In this paper we deal with set theory NC_{∞^{#}}^{#} based on gyper infinitary logic with Restricted Modus Ponens Rule. Set theory NC_{∞^{}}^{} contains Aczel's anti-foundation axiom [9]. We present a new approach to the invariant subspace problem for Hilbert spaces. Our main result will be that: if T is a bounded linear operator on an infinite-dimensional complex separable Hilbert space H,it follow that T has a non-trivial closed invariant subspace. Non-conservative extension based on set theory NC_{∞}^{#} of the model theoretical nonstandard analysis [10]-[12] also is considered. (shrink)

Crime and Culpability, by Larry Alexander, Kimberly Kessler Ferzan (with Stephen Morse) is a visionary work of moral and legal philosophy. Nonetheless, it is fundamentally morally misguided. In seeking to free criminal law from what the authors take to be the distorting influence of outcome luck, they arrive at a position that is overly exculpatory. It fails to hold actors liable for the harms they cause when they have taken less care they should. -/- I argue, first, that the (...) authors’ attempt to strip criminal law of outcome luck is incoherent in its own terms, and that any attempt to follow through on their program would require a much more severe loss of accountability than the authors acknowledge. I then argue that outcome luck is pervasive in our moral lives, and that, given the choice of (a) not being able to hold agents accountable for much, or (b) holding them accountable for the results of their choices, including outcomes that result in part from moral luck, the right choice is (b). I wrap up with a concluding observation about the implausible moral psychology that underlies the authors’ view of culpable action. (shrink)