We prove that Standardization fails in every nontrivial universe definable in the nonstandard set theory BST, and that a natural characterization of the standard universe is both consistent with and independent of BST. As a consequence we obtain a formulation of nonstandard class theory in the ∈-language.
We develop an axiomatic set theory — the Theory of Hyperfinite Sets THS— which is based on the idea of the existence of proper subclasses of large finite sets. We demonstrate how theorems of classical continuous mathematics can be transfered to THS, prove consistency of THS, and present some applications.
We present an axiomatic framework for nonstandard analysis-the Nonstandard Class Theory (NCT) which extends von Neumann-Gödel-Bernays Set Theory (NBG) by adding a unary predicate symbol St to the language of NBG (St(X) means that the class X is standard) and axioms-related to it- analogs of Nelson's idealization, standardization and transfer principles. Those principles are formulated as axioms, rather than axiom schemes, so that NCT is finitely axiomatizable. NCT can be considered as a theory of definable classes of Bounded Set Theory (...) by V. Kanovei and M. Reeken. In many aspects NCT resembles the Alternative Set Theory by P. Vopenka. For example there exist semisets (proper subclasses of sets) in NCT and it can be proved that a set has a standard finite cardinality iff it does not contain any proper subsemiset. Semisets can be considered as external classes in NCT. Thus the saturation principle can be formalized in NCT. (shrink)
It would be no exaggeration to say that our art is presently undergoing a revolutionary stage in its development. The revolutionary renewal that has seized all spheres of life is being ever more forcefully represented in artistic culture as well. That culture is beginning to free itself from bureaucratic regimentation, incompetent pressure from above, inflated authorities, lifeless cliches, and opportunism of the moment. Art is actively seeking its place in perestroika, and is in turn in many respects defining the moral (...) climate of perestroika. Recent works such as Fire [Pozhar] by V. Rasputin, the Sad Detective Story [Pechal'nyi detektiv] by V. Astaf'ev, The Execution Block [Plakha] by Ch. Aitmatov, the paintings of G. Korzhev and V. Sidorov, the sculpture of V. Klykov, etc., are actively working for renewal. (shrink)
The year is 1901. Two minor celebrities from opposite corners of the globe share an evening meal in Chicago. Both are politically left-leaning, both are evolutionists of a sort, both are concerned with the plight of the poor in the face of the escalation of the Industrial Revolution. The Russian man has been giving a series of lectures to the people of Chicago; he is staying at the American woman's settlement house-Hull House. They are Jane Addams, Chicago's activist social worker (...) and Petr Kropotkin, Russian nobleman by birth, anarchist in politics, and naturalist by inclination. Each awaits publication of their first full-length book concerning politics and moral development: Democracy and Social Ethics (1902) on .. (shrink)
In this article the most important text of twentieth-century Russian intellectual history, Landmarks (Vekhi) (1909) comes under reexamination. Looking at the rivalry of the volume''s two organizers, Mikhail Gershenzon and Petr Struve, Professor Brian Horowitz explains why Landmarks succeeded in offering such a biting critique of radical ideology, while lacking its own internal intellectual unity.
[Sensation, Causality, and Attention: Roger Bacon and Peter Olivi] This paper investigates what conditions are to be met for sensory perception to occur. It introduces two diff erent theories of perception that were held by two medieval Franciscan thinkers — namely, Roger Bacon (1214/1220–1292) and Peter Olivi (ca. 1248–1298). Bacon analyses especially the causal relation between the object and the sensory organ in his doctrine of the multiplication of species. In his view, a necessary condition of perception is the reception (...) of the species in a fully disposed sensory organ. On the contrary, Olivi stresses the active role of the sensory power. A necessary condition of sensation is the aspectus — i.e. the focus of our power’s attention on the object. Furthermore, the paper investigates whether and how each of the two thinkers can deal with the arguments proposed by his opponent — namely whether Bacon’s theory is able to explain attention and what the causal role of the object in Olivi’s theory is. (shrink)
[What is the Human Being? Peter Auriol and the Role of Cognitive Psychology in the Medieval Definition of the Human Being: ] This paper explores how medieval philosophers used cognitive psychology in defining what the human being is, paying special attention to the Franciscan thinker Peter Auriol (c. 1280 – 1322). First, I examine the motivations of Auriol’s claim that the property of being alive is bound to the property of being cognitive (i. e. being capable of cognition). Then, the (...) foundations of medieval faculty psychology and Auriol’s conception of cognition are introduced. I also argue that the emphasis which Auriol puts on the activity of soul’s faculties leads him to the conclusion (unusual in his days) that the distinction among these faculties is established from the first person perspective. Finally, Auriol’s cognitive definition of the human being is introduced – human beings are human beings precisely because their cognitive experience differs from the way the cognition works in animals on the one hand and in God on the other hand. Whereas animals have only sensory soul’s faculties, humans have the intellect in addition and, therefore, they are capable of universal cognition. Moreover, since humans have not only intellect but also the inner senses (particularly, the phantasy), the universals appear to the human intellect only as blended with the individual that the universal was abstracted from, and the human intellect is not capable of paying attention to no more than just one object at the same moment. Since God and angels have only intellect, these distinctively human features are absent from their cognitive experience. (shrink)
Centre and Periphery in the Historiography of Philosophy: Peter Olivi and Medieval Psychology The paper inquiries into the (historiographical) question what does it mean to be a “marginal thinker” in the context of the medieval philosophy. The question is investigated on the example of Franciscan philosopher and theologian Peter Olivi (1248/49–1298) and his philosophical psychology. First, a preliminary option is introduced: for a thinker, being “marginal” depends on his relation to who is considered to be canonical. Since the most famous (...) thinker of the Middle Ages is Thomas Aquinas (at least according to the traditional canon of medieval philosophy), Olivi’s positions in psychology are compared with these of Aquinas. It is revealed that Olivi’s psychology is very different from the Aquinas’ one. (E.g. Olivi stresses the activity of perception, proprioceptual nature of the sense of touch, and direct access of the intellect to its own acts.) Moreover, Olivi is very critical towards the Aristotelian philosophy as is done by some thinkers of his time. Nevertheless, it does not follow that Olivi is a marginal thinker only because of his dissimilarity from the more Aristotelian-minded ones. It is argued that “centre” and “periphery” in the history of medieval philosophy depends not on the canon (which is rather a historians’ construct and instrument), but rather on the tradition. Hence, although Olivi can be considered as a marginal thinker if we take into the account the Aristotelian tradition of medieval philosophy, he is definitely a central and important thinker, if considered as a member of the Augustinian tradition. (shrink)
I present and argue for twotheses: the first concerns the degree to whichChaadaev''s thought represents a breakthrough inthe development of Russian social philosophyand the second concerns the Hegelian characterof this thinking. I also show that Chaadaev''stheory retained an open character closely tiedto the crisis character of the social realityof his time and that it depended for itsjustification on the further course of thehistorical process, which is impossible topredict. All this leads to an interpretation ofChaadaev''s view according to which the standardopposition (...) of Chaadaev''s two best-known texts,The Philosophical Letters, with theirpredominantly pessimistic picture of Russia,and the Apology of a Madman, whichrefutes this evaluation, is rejected. (shrink)
Nová kniha Petra Koláře, Pravda a fakt (Filosofia, Praha, 2002) je věnována tématu, kterým se Kolář částečně zabýval již ve své předchozí knize: teoriím pravdivosti a zejména teorii korespondenční. Diskuse o tom, jak explikovat pojem pravdy či pravdivosti se analytickou filosofií táhnou od jejích počátků, a rozdmychány byly zejména výsledky Tarského matematických analýz tohoto pojmu1. Kolář v první části knihy probírá a srovnává hlavní kategorie těch teorií, které jsou výsledky těchto diskusí (některé z nich samozřejmě tak či onak existovaly dávno (...) před Tarskim): dělí je na korespondenční, koherenční, pragmatistické, minimalistické, redundanční a sémantické (tou poslední je de facto původní Tarského varianta). Způsobu, jak Kolář tyto teorie vykládá, lze, myslím, vytýkat skutečně pramálo: jeho výklad je systematický, jasný a názorný. Také shrnutí na koncích jednotlivých kapitol jsou velice užitečná a dokumentují autorovu pedagogickou pečlivost. Přesto bych si k této části dovolil mít dvě drobné a jednu zásadnější připomínku. Zaprvé se mi nezdá příliš adekvátní, jak Kolář používá termíny deflacionismus a minimalismus: za deflacionistickou prohlašuje každou teorii, která rezignuje na explikaci pojmu pravdy, takže mu do této kolonky spadne jak horwichovský minimalismus, tak davidsonovský názor, že pojem pravdy nelze explikovat ne proto, že by byl triviální, ale naopak proto, že je příliš fundamentální (tady se dere na jazyk termín maximalismus). To se ale těžko slučuje s Davidsonovým výslovným odmítáním deflacionismu2. Zdá se mi, že termín deflacionismus je spíše víceméně synonymní s termínem minimalismus a ty oba se mi zdají do velké míry krýt i s redundanční teorií. Druhou drobnou výhradou je, že mi připadá škoda, že Kolář v rámci pojednávání pragmatistických teoriích zcela pominul současný pragmatismus. Škoda to je pro to, že pragmatismus v současné době zažívá velký revival a i různé pragmatistické teorie pravdivosti se proto dostávají na pořad dne (včetně například kontextu matematiky). Za.... (shrink)