The sum of all objects of a science, the objects’ features and their mutual relations compose the reality described by that sense. The reality described by mathematics consists of objects such as sets, functions, algebraic structures, etc. Generally speaking, the use of terms reality and existence, in relation to describing various objects’ characteristics, usually implies an employment of physical and perceptible attributes. This is not the case in mathematics. Its reality and the existence of its objects, leaving aside its application, (...) are completely virtual and yet clearly organized. This organization can be recognized in the creation of axioms or in the arrival at new theorems and definitions. It results either from a formalization of an intuitive idea or from a combinatorics that has not been guided by intuition. In all four possible cases—therefore, also in the two in which there is no intuitive “lead”, we can plausibly talk about a discovery of mathematical facts and thus support the Platonist view. (shrink)

This discussion note points to some verbal imprecisions in the formulation of the Enhanced Indispensability Argument. The examination of the plausibility of alternative interpretations reveals that the argument’s minor premise should be understood as a particular, not a universal, statement. Interpretations of the major premise and the conclusion oscillate between de re and de dicto readings. The attempt to find an appropriate interpretation for the EIA leads to undesirable results. If assumed to be valid and sound, the argument warrants the (...) rationality of the belief in an unusual variant of Platonism. On the other hand, if taken as it stands, the argument is either invalid or is unsound or does not support the mathematical Platonism. Thus, the EIA in its present form cannot serve as a useful device for the Platonist. (shrink)

In this paper, we study the genesis and evolution of geometric ideas and techniques in investigations of movable singularities of algebraic ordinary differential equations. This leads us to the work of Mihailo Petrović on algebraic differential equations and in particular the geometric ideas expressed in his polygon method from the final years of the nineteenth century, which have been left completely unnoticed by the experts. This concept, also developed independently and in a somewhat different direction by Henry Fine, generalizes the (...) famous Newton–Puiseux polygonal method and applies to algebraic ODEs rather than algebraic equations. Although remarkable, the Petrović legacy has been practically neglected in the modern literature, although the situation is less severe in the case of results of Fine. Therefore, we study the development of the ideas of Petrović and Fine and their places in contemporary mathematics. (shrink)

In 2020, Daniele Molinini published a paper outlining two types of mathematical objectivity. One could say that with this paper Molinini not only separated two mathematical concepts in terms of terminology and content, but also contrasted two mathematical-philosophical contexts, the traditional-idealistic and the modern-practical. Since the first context was the theoretical basis for a large number of analyses that we find in the framework of the philosophy of mathematics, the space was now offered to re-examine such analyses in the second (...) context. More specifically, with respect to the second context, we will analyse the strength of one of the few explicit Platonic arguments that seeks to justify the ontological status of mathematical objects and the central claim of mathematical Platonism about the existence of mathematical objects. (shrink)

Platonists in mathematics endeavour to prove the truthfulness of the proposal about the existence of mathematical objects. However, there have not been many explicit proofs of this proposal. One of the explicit ones is doubtlessly Baker’s Enhanced Indispensability Argument, formulated as a sort of modal syllogism. We aim at showing that the purpose of its creation – the defence of Platonist viewpoint – was not accomplished. Namely, the second premise of the Argument was imprecisely formulated, which gave space for various (...) interpretations of the EIA. Moreover, it is not easy to perceive which of the more precise formulations of the above-mentioned premise would be acceptable. For all these reasons, it is disputable whether the EIA can be used to defend Platonist outlook. At the beginning of this century, Baker has shown that the so-called Quine-Putnam Indispensability Argument can not provide “full” platonism - a guarantee of the existence of all mathematical objects. It turns out, however, that the EIA has a similar disadvantage. (shrink)

Alan Baker argues that mathematical objects play an indispensable explanatory role in science. There are several examples cited in the literature as solid candidates for such a role. We discuss two such examples and show that they are very different in their strength and (im)perfection, although both are recognized by the scientific community as examples of the best scientific explanations of particular phenomena. More specifically, it will be shown that the explanation of the cicada case has serious shortcomings compared with (...) the explanation of the case of Königsberg’s bridges. We will argue that the latter is a perfectly reliable scientific explanation that employs mathematical reasoning whereas the former is not. (shrink)

Spinoza’s rejection of Aristotelian final causation seems to create a difficulty for his account of individuation. If causation is indeed blind, how do finite modes come to assume complex, differentiated forms? And why do we find in nature a great regularity of such forms? Several recent commentators have proposed that Spinoza maintains something of the Aristotelian conception of causation where the formal essences of individuals guide the process of individuation toward certain desirable outcomes. But this sort of approach introduces other (...) difficulties that threaten to undermine Spinoza’s naturalistic framework and his ontology of immanence. This paper outlines a mechanistic and probabilistic account of individuation whereby modes are individuated by entering into relations that increase their mutual power of enduring. Together with conatus as the principle of individuation, this mechanistic account suffices to explain the individuation of finite bodies without introducing additional kinds of causation into Spinoza’s philosophy. (shrink)

It is shown that, in some non-QCD theories, there are effects shared by QCD: (i) in SU(2) Yang–Mills theory containing a nonlinear spinor field, there is a mass gap; (ii) in SU(3) Proca–Higgs theory, there are flux tube solutions with a longitudinal electric field required for producing a force binding quarks; (iii) in non-Abelian Proca–Higgs theories, there exist flux tube solutions with a momentum directed along the tube axis and particlelike solutions with a nonvanishing total angular momentum created by crossed (...) color electric and magnetic fields; in QCD, such configurations may contribute to the proton spin. We discuss the conjecture that such non-QCD theories might be a consequence of approximate solution of an infinite set of Dyson–Schwinger equations describing the procedure of nonperturbative quantization. The phenomenon of dimensional transmutation for nonperturbative quantization and the analogy between nonperturbative quantization and turbulence modeling are also discussed. (shrink)

In this paper we prove the following theorem: Any one-dimensional definably connected group G over an o-minimal structure is, as an abstract group, isomorphic to either pPp∞δ or δ.

Physical theories are complex and necessary tools for gaining new knowledge about their areas of application. A distinction is made between abstract and practical theories. The last are constantly being improved in the cognitive activity of professional physicists and studied by future physicists. A variant of the philosophy of physics based on a modified structural-nominative reconstruction of practical theories is proposed. Readers should decide whether this option is useful for their understanding of the philosophy of physics, as well as other (...) philosophies of particular sciences. (shrink)

In this paper, I argue (1) that the contents of some delusions are believed with sufficient confidence; (2) that a delusional subject could have a conscious belief in the content of his delusion (p), and concurrently judge a contradictory content (not-p) – his delusion could be transparent, and (3) that the existence of even one such case reveals a problem with pretty much all existing accounts of lying, since it suggests that one can lie by asserting what one consciously and (...) confidently believes is true, and (4) sincerity, since it suggests that asserting a proposition you believe is true is neither sufficient nor necessary for sincerity. If I am right about (1) and (2), then (3) and (4) follow easily. Therefore, the paper is mainly devoted to an analysis of transparent delusion and defending (1) and (2). (shrink)

The short exposition of the triplet model of concepts and some definitions connected with it are given. In this model any concept may be depicted as having three characteristics: a base, a representing part and the linkage between them. The paper introduces the fuzzification of concepts in terms of the triplet model.

In 1995 N. C. A. da Costa and F. Doria proposed the modaltype elegant axiomatization of Jaśkowski’s discussive logic D2. Yet his ownproblem which was formulated in 1975 in a following way: Is it possible toformulate natural and simple axiomatization for D2, employing classical disjunction and conjunction along with discussive implication and conjunctionas the only primitive connectives? — still seems left open. The matter of factis there are some axiomatizations of D2 proposed, e.g., by T. Furmanowski, J. Kotas and N. (...) C. A. da Costa , G. Achtelik, L. Dubikajtus,E. Dudek and J. Konior , satisfying da Costa’s conditions, but they arerather looking very complicated and unnatural. An attempt is made to solveda Costa’s problem. The new axiomatization of D2 is proposed essentiallybased on da Costa’s-Doria axiomatization from 1995. (shrink)

The scheme for classifications of concepts is introduced. It has founded on the triplet model of concepts. In this model a concept is depicted by means of three kinds of knowledge: a concept base, a concept representing part and the linkage between them. The idea of triplet classifications of concepts is connected with a usage of various specifications of these knowledge kinds as classification criteria.

ABSTRACT Two models of the Aristotelian syllogistic in arithmetic of natural numbers are built as realizations of an old Leibniz idea. In the interpretation, called Scholastic, terms are replaced by integers greater than 1, and s.Ap is translated as “s is a divisor of p”, sIp as “g.c.d. > 1”. In the interpretation, called Leibnizian, terms are replaced by proper divisors of a special “Universe number” u < 1, and sAp is translated as “s is divisible by p”, sIp as (...) ‘l.c.m. < u”. Both interpretations are proved to be adequate to the Aristotelian syllogistic. They are extended to syllogistic including term negation and term conjunction as well. (shrink)

We consider games over a finite alphabet with Gurevich-Harrington's winning conditions and restraints as in Yakhnis-Yakhnis . The game tree, the Gurevich-Harrington's kernels of the winning condition and the restraints are defined by finite automata. We give an effective criterion to determine the winning player and an effective presentation of a class of finite automata defined winning strategies.Our approach yields an alternative solution to the games considered by Büchi and Landweber . The BL algorithm is an important tool for solving (...) effectively presented games. The differences between the BL and our algorithms are as follows. Our algorithm is uniformly applicable to winning sets that correspond to boolean combinations of BL winning conditions, while the BL algorithm requires a transformation of such boolean combinations into a single BL condition. Our algorithm directly applies to a larger class of game trees for a given game alphabet than does the BL one. We describe Büchi-Landweber's class of winning conditions in a more transparent way by using finite automata and Gurevich-Harrington's form of winning conditions. Our algorithms that determine the winner of the game and find a winning strategy, are based on a translation of the events in the game in terms of the transition table of the automaton representing the game. This automaton differs from the ones that represent BL winning conditions. Both algorithms need further investigation.The algorithm determining the winner in the game, can be used as a verifier of concurrent programs. This is because programs can be viewed as strategies in Gurevich-Harrington's games and program specifications could be regarded as winning conditions for such games. The algorithm which finds a winning strategy could be used as a basis for automated concurrent program design.The present paper is a sequel to Annals of Pure and Applied Logic 48 277–297. (shrink)

The triplet model treats a concept as complex structure that expresses three kinds of information. The first is about entities subsumed under a concept,their properties and relations. The second is about means and ways of representing the first information in intelligent systems. The third is about linkage between the first and second ones and methods of its constructing. The application of triplet models to generalization and development of concept models in philosophy, logic, cognitive psychology, cognitive science, linguistics, artificial intelligence has (...) been demonstrated. The componential structure of the triplet model and extended concept classification have been given. Theoretical considerations are combined with historical case-studies of concept history. For scientists and students with an interest in problems of knowledge structure and representation. Abstract and Extended Summary in English. (shrink)

The paper has three objectives: to expound a set-theoretical triplet model of concepts; to introduce some triplet relations (symbolic, logical, and mathematical formalization; equivalence, intersection, disjointness) between object concepts, and to instantiate them by relations between certain physical object concepts.

It is proposed to analyze fuzzy concepts and relations between them in the frame of triplet concept modeling. Fuzzy concepts are introduced by means of the so-called fuzzification of dichotomous concepts. The cognitive and psychological aspects of concept possession are separated and studied.

It is proposed to analyze the unity of scientific knowledge in terms of its structure-nominative reconstruction. -/- .

There are no universally adopted answers to the natural questions about scientific concepts: What are they? What is their structure? What are their functions? How many kinds of them are there? Do they change? Ironically, most if not all scientific monographs or articles mention concepts, but the scientific studies of scientific concepts are rare in occurrence. It is well known that the necessary stage of any scientific study is constructing the model of objects in question. Many years logical modeling was (...) dominant in the concept studies. Last decades, concepts came to be regarded as the subject of mathematical modeling. However, different authors take different features of concepts as independent variables of their models. Our objective is to characterize informally the spectra of relevant variables for the modeling of scientific concepts. (shrink)

The changes of concepts are described in the frame of concept triplet model. -/- .

. We demonstrate that a comprehensive nonstandard set theory can be developed in the standard $\displaystyle{\in}$ -language. As an illustration, a nonstandard ${\sf Law of Large Numbers}$ is obtained.

Již letmý pohled na filosofii 20. století, která se snaží promýšlet podstatu lidského jednání, budí dojem, že patří k její metodě vrátit se k antickým teoriím a představit je jako historické východisko, jehož nové uchopení povede k lepšímu porozumění aktuálního problému. Tyto návraty, jak je lze sledovat u Heideggera, Gadamera, Ricoeura nebo Arendtové, směřují – nakolik se jedná o teorii jednání – především k Aristotelovi a jeho základnímu rozlišení mezi poiésis a praxis. Cílem následující stati je představit základ teorie jednání, (...) jež byla dílem prvních stoiků ve 3. stol. př. Kr. V naznačeném kontextu si však taková interpretace může činit větší ambice, než jak by se mohlo na první pohled zdát. Zůstává-li totiž Aristotelés pro novodobou filosofii jednání myslitelem, k němuž je třeba se vrátit a s nímž je třeba se vyrovnat, pak je dobré si uvědomit, že právě ve stoicismu lze vidět první a bezprostřední kritickou reakci na Aristotelovu filosofii. Navíc byl stoicismus naukou, v níž se jednání stalo gravitačním bodem všech ostatních úvah a která proto musela svoje vyrovnání s Aristotelem soustředit do pojmů souvisejících úzce právě s jednáním. Lze ovšem snadno vytušit, jaký byl hlavní důvod, proč zůstal stoicismus stranou pozornosti současných filosofů . Přesný vztah stoické nauky k aristotelismu má totiž stále tendenci se skrývat. Navzdory malé četnosti explicitních konfrontací s Aristotelovými argumenty je aristotelismus ve stoicismu přítomen všude – je v něm přítomen natolik, že snaze odlišit oba systémy občas hrozí sklouznutí do povrchních kategorizací, z nichž lze podezírat doxografickou tradici hellénistické a římské doby, která se snaží o uchopení téhož rozdílu. Přesto stoicismus není peripatetická filosofie a lze oprávněně předpokládat, že nahlédnutí odlišnosti mezi obojím vrhne nové světlo na moderní přístup k Aristotelovi. (shrink)

PurposeThe purpose of this paper is to present implications of the seller's ability to bid in the four classical auction forms, with independent private values: English, Dutch, first‐ and second‐price auctions.Design/methodology/approachUnder each auction form, the identity of the winning bidder and the expected winning bid are compared between the case when the seller may bid and when he cannot, using equilibrium bidder strategies. The seller's incentive to bid is evaluated.FindingsThe strategies and the welfare results differ with auction type and underlying (...) information assumptions – bidders are either aware or unaware of the seller's ability to bid. In the Dutch and the first‐price auctions, seller‐bidding does not affect any classical results. In the English and the second‐price auctions, it leads to no lower expected prices than without it and higher prices with positive probability. In the English and the second‐price auctions, the seller bids above his reservation value and may unintentionally win the auction. These auctions result in inefficiency with positive probability.Practical implicationsThe English and the second‐price auctions are the most common real‐world auctions. In these auctions, the seller's ability to bid – secretly or publicly – redistributes welfare among participants and introduces a possibility of inefficiency. Making this ability publicly known does not solve the latter problem. Auctioneers must prevent the seller from bidding, or must select a different auction form when seller‐bidding is anticipated.Originality/valueThe paper clarifies to regulators, auction designers, bidders, and other readers which auction forms are susceptible to subversion by seller‐bidding and what the potential damages are. (shrink)

A hyperfinitely long coin flipping game between the Gambler and the Casino, associated with a given set A, is considered. It turns out that the Gambler has a winning strategy if and only if A has Loeb measure 0. The Casino has a winning strategy if and only if A contains an internal subset of positive Loeb measure.

Можно ли сказать, что в центре христианской культуры лежит искушение - как о том свидетельствуют в евангельских повествованиях искушения Христа Дьяволом? Так, кажется, думает Достоевский, который - в особенности в «Братьях Карамазовых» - стремится показать, что именно невинные становятся жертвами дьявольских козней. Такова для него и ситуация русского народа, который он идеализирует, противопоставляя его интеллигенции. В реальности, как показывает недавняя история, эта претензия на «невинность» русского народа была лишь фантазмом Достоевского. La tentation serait-elle au centre de la culture chrétienne comme (...) le donneraient à penser les tentations du Christ par le Démon que racontent les évangiles ? C’est ce que semble croire Dostoïevski qui, en particulier dans Les frères Karamazov, s’efforce de montrer que ce sont justement des innocents qui sont soumis à ces manoeuvres démoniaques. Telle serait, d’ailleurs, d’après lui, la situation du peuple russe, qu’il idéalise en l’opposant à l’intelligentsia. En réalité - l’histoire récente l’a montré - cette prétendue « innocence » du peuple russe n’était qu’un fantasme de Dostoïevski. Is temptation at the center of Christian culture, as suggested by the temptations Christ had been exposed to by the Devil, according to the Gospels? This is what Dostoyevsky seems to believe, most notably in The Brothers Karamazov, in which he aims at showing that only innocent people are subject to these demonic maneuvers. This would also be, according to him, the situation of the Russian people, which he idealizes, contrasting it to the intelligentsia. In fact, recent history has shown that this alleged « innocence » of the Russian people was a fantasy of Dostoyevsky. (shrink)