Boris Čulina - Academia.edu (original) (raw)

Papers by Boris Čulina

Science & Philosophy, 2024

Contemporary semantic description of logic is based on the ontology of all possible interpretatio... more Contemporary semantic description of logic is based on the ontology of all possible interpretations, an insufficiently clear metaphysical concept. In this article, logic is described as the internal organization of language. Logical concepts-logical constants, logical truths and logical consequence-are defined using the internal syntactic and semantic structure of language. For a first-order language, it has been shown that its logical constants are connectives and a certain type of quantifiers for which the universal and existential quantifiers form a functionally complete set of quantifiers. Neither equality nor cardinal quantifiers belong to the logical constants of a first-order language.

arXiv (Cornell University), May 28, 2021

According to Georg Cantor [Can83, Can99] a set is any multitude which can be thought of as one ("... more According to Georg Cantor [Can83, Can99] 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 ∀x x = x. In this paper we formulate a logical truth which we call the productivity principle. Bertrand Rusell [Rus06] 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 ∈-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 ZF C 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

arXiv (Cornell University), May 28, 2021

On the basis of elementary thinking about language functioning, a solution of truth paradoxes is ... more On the basis of elementary thinking about language functioning, a solution of truth paradoxes is given and a corresponding semantics of a truth predicate is founded. It is shown that it is precisely the two-valued description of the maximal intrinsic fixed point of the strong Kleene threevalued semantics.

arXiv (Cornell University), May 28, 2021

In this article I develop an elementary system of axioms for Euclidean geometry. On one hand, the... more In this article I develop an elementary system of axioms for Euclidean geometry. On one hand, the system is based on the symmetry principles which express our a priori ignorant approach to space: all places are the same to us (the homogeneity of space), all directions are the same to us (the isotropy of space) and all units of length we use to create geometric figures are the same to us (the scale invariance of space). On the other hand, through the process of algebraic simplification, this system of axioms directly provides the Weyl's system of axioms for Euclidean geometry. The system of axioms, together with its a priori interpretation, offers new views to philosophy and pedagogy of mathematics: (i) it supports the thesis that Euclidean geometry is a priori, (ii) it supports the thesis that in modern mathematics the Weyl's system of axioms is dominant to the Euclid's system because it reflects the a priori underlying symmetries, (iii) it gives a new and promising approach to learn geometry which, through the Weyl's system of axioms, leads from the essential geometric symmetry principles of the mathematical nature directly to modern mathematics.

Узданица

While there are satisfactory answers to the question “How should we teach children mathematics?”,... more While there are satisfactory answers to the question “How should we teach children mathematics?”, there are no satisfactory answers to the question “What mathematics should we teach children?”. This paper provides an answer to the last question for preschool children (early childhood), although the answer is also applicable to older children. This answer, together with an appropriate methodology on how to teach mathematics, gives a clear concep- tion of the place of mathematics in the children’s world and our role in helping children develop their mathematical abilities. Briefly, children’s mathematics consists of the world of children’s internal activities that they eventually purposefully organize in order to understand and control the outside world and organize their overall activities in it. We need to support a child in math- ematical activities that she does spontaneously and in which she shows interest, and we need to teach her mathematics that she is interested in developing...

The thesis deals with the concept of truth and the paradoxes of truth. Philosophical theories usu... more The thesis deals with the concept of truth and the paradoxes of truth. Philosophical theories usually consider the concept of truth from a wider perspective. They are concerned with questions such as - Is there any connection between the truth and the world? And, if there is - What is the nature of the connection? Contrary to these theories, this analysis is of a logical nature. It deals with the internal semantic structure of language, the mutual semantic connection of sentences, above all the connection of sentences that speak about the truth of other sentences and sentences whose truth they speak about. Truth paradoxes show that there is a problem in our basic understanding of the language meaning and they are a test for any proposed solution. It is important to make a distinction between the normative and analytical aspect of the solution. The former tries to ensure that paradoxes will not emerge. The latter tries to explain paradoxes. Of course, the practical aspect of the solution is also important. It tries to ensure a good framework for logical foundations of knowledge, for related problems in Artificial Intelligence and for the analysis of the natural language. Tarski’s analysis emphasized the T-scheme as the basic intuitive principle for the concept of truth, but it also showed its inconsistency with the classical logic. Tarski’s solution is to preserve the classical logic and to restrict the scheme: we can talk about the truth of sentences of a language only inside another essentially richer metalanguage. This solution is in harmony with the idea of reflexivity of thinking and it has become very fertile for mathematics and science in general. But it has normative nature | truth paradoxes are avoided in a way that in such frame we cannot even express paradoxical sentences. It is also too restrictive because, for the same reason we cannot express a situation in which there is a circular reference of some sentences to other sentences, no matter how common and harmless such a situation may be. Kripke showed that there is no natural restriction to the T-scheme and we have to accept it. But then we must also accept the riskiness of sentences | the possibility that under some circumstances a sentence does not have the classical truth value but it is undetermined. This leads to languages with three-valued semantics. Kripke did not give any definite model, but he gave a theoretical frame for investigations of various models | each fixed point in each three-valued semantics can be a model for the concept of truth. The solutions also have normative nature | we can express the paradoxical sentences, but we escape a contradiction by declaring them undetermined. Such a solution could become an analytical solution only if we provide the analysis that would show in a substantial way that it is the solution that models the concept of truth. Kripke took some steps in the direction of finding an analytical solution. He preferred the strong Kleene three-valued semantics for which he wrote it was "appropriate" but did not explain why it was appropriate. One reason for such a choice is probably that Kripke finds paradoxical sentences meaningful. This eliminates the weak Kleene three valued semantics which corresponds to the idea that paradoxical sentences are meaningless, and thus indeterminate. Another reason could be that the strong Kleene three valued semantics has the so-called investigative interpretation. According to this interpretation, this semantics corresponds to the classical determination of truth, whereby all sentences that do not have an already determined value are temporarily considered indeterminate. When we determine the truth value of these sentences, then we can also determine the truth value of the sentences that are composed of them. Kripke supplemented this investigative interpretation with an intuition about learning the concept of truth. That intuition deals with how we can teach someone who is a competent user of an initial language (without the predicate of truth T) to use sentences that contain the predicate T. That person knows which sentences of the initial language are true and which are not. We give her the rule to assign the T attribute to the former and deny that attribute to the latter. In that way, some new sentences that contain the predicate of truth, and which were indeterminate until then, become determinate. So the person gets a new set of true and false sentences with which he continues the procedure. This intuition leads directly to the smallest fixed point of strong Kleene semantics as an analytically acceptable model for the logical notion of truth. However, since this process is usually saturated only on some transfinite ordinal, this intuition, by climbing on ordinals, increasingly becomes a metaphor. This thesis is an attempt to give an analytical solution to truth paradoxes. It gives an analysis of why and how some sentences lack the classical truth value. The…

Cornell University - arXiv, Mar 29, 2021

The concept of inertial frame of reference in classical physics and special theory of relativity ... more The concept of inertial frame of reference in classical physics and special theory of relativity is analysed. It has been shown that this fundamental concept of physics is not clear enough. A definition of inertial frame of reference is proposed which expresses its key inherent property. The definition is operational and powerful. Many other properties of inertial frames follow from the definition, or it makes them plausible. In particular, the definition shows why physical laws obey space and time symmetries and the principle of relativity, it resolves the problem of clock synchronization and the role of light in it, as well as the problem of the geometry of inertial frames.

Uzdanica, 2022

While there are satisfactory answers to the question "How should we teach children mathematics?",... more While there are satisfactory answers to the question "How should we teach children mathematics?", there are no satisfactory answers to the question "What mathematics should we teach children?". This paper provides an answer to the last question for preschool children (early childhood), although the answer is also applicable to older children. This answer, together with an appropriate methodology on how to teach mathematics, gives a clear conception of the place of mathematics in the children's world and our role in helping children develop their mathematical abilities. Briefly, children's mathematics consists of the world of children's internal activities that they eventually purposefully organize in order to understand and control the outside world and organize their overall activities in it. We need to support a child in mathematical activities that she does spontaneously and in which she shows interest, and we need to teach her mathematics that she is interested in developing through these activities. In doing so, we must be fully aware that the child's mathematics is part of the child's world of internal activities and is not outside of it. We help the child develop mathematical abilities by developing them in the context of her world and not outside of it. From the point of view of this conception, the standards established today are limiting and too focused on numbers and geometric figures: these topics are too prominent and elaborated, and other mathematical contents are subordinated to them. Adhering to the standards, we drastically limit the mathematics of the child's world, hamper the correct mathematical development of a child, and we can turn her away from mathematics.

Science & Philosophy, 2022

Zbornik radova, 9. međunarodna konferencija "Dani kriznog upravljanja", Veleučilište Velika Gorica, 2016

Sažetak. U radu je izložena matematička povijest modernog računala o kojoj se izvan matematičke s... more Sažetak. U radu je izložena matematička povijest modernog računala o kojoj se izvan matematičke struke malo zna, a koja je vrlo važna za razumijevanje matematičkih i logičkih ideja koje su u osnovi računarstva i nastanka modernog računala (dijagonalna metoda, formalni sustavi, kodiranje, poluizračunljivost i izračunljivost, ekvivalentne formulacije izračunljivosti, univerzalni stroj s uskladištenim programom, neizračunljivost), kao i za razumijevanje važne uloge matematike u računarstvu. Ključne riječi: dijagonalna metoda; Hilbertov program; formalni sustavi; Gödelovi teoremi, Turingovi strojevi; univerzalni Turingov stroj; problem dijagonalnog zaustavljanja; Church-Turingova teza; Von Neumannov draft Danas, kad računalska tehnologija napreduje brzinom od koje zastaje dah, i kad se divimo uistinu izvanrednim dostignućima inženjera, lako se mogu previdjeti logičari čije su ideje učinile sve to mogućim. Martin Davis [1] I. Uvod Općepoznato je da je moderno računalo, koncepcijski gledano, digitalno univerzalno računalo s uskladištenim programom. Njegov logički opis dao je 1945. godine John von Neumann u radu First Draft of a Report on the EDVAC [2]. Kao što i sam naslov rada kaže, u pitanju je bila prva skica za izvještaj o projektu EDVAC, projektu izgradnje novog računala pri Moore School of Electrical Engineering u SAD. Taj draft je brzo " procurio" izvan projekta i po njemu su napravljena prva moderna računala, počevši s britanskim računalom Manchester Baby iz 1948. godine. Draft nikad nije prerastao u konačni izvještaj. Između ostalog, u draftu nisu citirani izvori niti navedeni doprinosi članova grupe. Tako je taj ključni događaj u povijesti računarstva ostao sporan. Najviše je prijepora bilo oko revolucionarnog koncepta uskladištenog programa. Koliko su tom konceptu pridonijeli glavni inženjeri na projektu John Presper Eckert i John Mauchly, a koliko matematičar John von Neumann koji je kao konzultant uključen u projekt? Najmjerodavniji za odgovor trebao bi biti sam voditelj projekta Herman Goldstine koji je napisao sljedeće: " Prije njegova [von Neumannova] dolaska Moore School grupa bila je koncentrirana primarno na tehnološke probleme koji su bili vrlo veliki. Nakon njegova dolaska on je preuzeo vodstvo u logičkim problemima. ... Svakako, i prije von Neumanna ljudi su znali da se

arXiv: Logic, May 28, 2021

Tesis (Lima)

Se analiza el papel esencial del lenguaje en la cognición racional. El enfoque es funcional: solo... more Se analiza el papel esencial del lenguaje en la cognición racional. El enfoque es funcional: solo se consideran los resultados de la conexión entre lenguaje, realidad y pensamiento. El lenguaje científico se analiza como una extensión y mejora del lenguaje cotidiano. El análisis ofrece una visión uniforme del lenguaje y la cognición racional. Las consecuencias para la naturaleza de la ontología, la verdad, la lógica, el pensamiento, las teorías científicas y las matemáticas son derivadas

Sažetak. U našem sustavu poučavanja matematike dominira analitički pristup, dok je zanemaren kval... more Sažetak. U našem sustavu poučavanja matematike dominira analitički pristup, dok je zanemaren kvalitativni i numerički pristup. To ima negativne posljedice za ispravan matematički razvoj učenika. Ograničava mu se pristup rješavanju problema i opseg problema koje može riješiti. Gledano konceptualno, učenik ne razvija kvalitativan način razmišljanja i ne usvaja bitne ideje numeričke matematike. Kvalitativan i numerički pristup podrazumijevaju odgovarajući software, što dodatno potencira nužnost uvođenja adekvatnog matematičkog softwarea u nastavu matematike. Na primjeru rješavanja jednadžbi pokazana je prednost izbalansiranog kvalitativnog, analitičkog i numeričkog pristupa.

Croatian Journal of Philosophy 23 (67): 1-31. 2023., 2020

This article informally presents a solution to the paradoxes of truth and shows how the solution ... more This article informally presents a solution to the paradoxes of truth and shows how the solution solves classical paradoxes (such as the original Liar) as well as the paradoxes that were invented as counterarguments for various proposed solutions ("the revenge of the Liar"). This solution complements the classical procedure of determining the truth values of sentences by its own failure and, when the procedure fails, through an appropriate semantic shift allows us to express the failure in a classical two-valued language. Formally speaking, the solution is a language with one meaning of symbols and two valuations of the truth values of sentences. The primary valuation is a classical valuation that is partial in the presence of the truth predicate. It enables us to determine the classical truth value of a sentence or leads to the failure of that determination. The language with the primary valuation is precisely the largest intrinsic fi xed point of the strong Kleene three-valued semantics (LIFPSK3). The semantic shift that allows us to express the failure of the primary valuation is precisely the classical closure of LIFPSK3: it extends LIFPSK3 to a classical language in parts where LIFPSK3 is undetermined. Thus, this article provides an argumentation, which has not been present in contemporary debates so far, for the choice of LIF-PSK3 and its classical closure as the right model for the truth predicate. In the end, an erroneous critique of Kripke-Feferman axiomatic theory of truth, which is present in contemporary literature, is pointed out.

This article analyses the essential role of language in rational cognition. The approach is funct... more This article analyses the essential role of language in rational cognition. The approach is functional I only look at the e ects of the connection between language, reality and thinking. I begin by analysing rational cognition in everyday situations. Then I show that the whole scienti c language is an extension and improvement of everyday language. The result is a uniform view of language and rational cognition which solves many epistemological and ontological problems. I use some of them the nature of ontology, truth, logic, thinking, scienti c theories and mathematics, to demonstrate that the view of language and rational cognition developed in this article is fruitful and e ective. keywords: rational cognition, language, ontology, truth, logic, thinking, scienti c theory, mathematics

Science & Philosophy, 2024

arXiv (Cornell University), May 28, 2021

Узданица

Cornell University - arXiv, Mar 29, 2021

Uzdanica, 2022

Science & Philosophy, 2022

Zbornik radova, 9. međunarodna konferencija "Dani kriznog upravljanja", Veleučilište Velika Gorica, 2016

arXiv: Logic, May 28, 2021

Tesis (Lima)

Croatian Journal of Philosophy 23 (67): 1-31. 2023., 2020

In this handbook, I put into practice my philosophical views on children's mathematics. The handb... more In this handbook, I put into practice my philosophical views on children's mathematics. The handbook contains brief instructions and examples of mathematical activities. In the INSTRUCTIONS section, instructions are given on how, and in part why that way, to help preschool children in their mathematical development. In the ACTIVITIES section, there are examples of activities through which the child develops her mathematical abilities.

In this article, modern standards of early years mathematics education are criticized and a propo... more In this article, modern standards of early years mathematics education are criticized and a proposal for change is presented. Today's early years mathematics education standards rest on a view of mathematics that became obsolete almost two hundred years ago, while the spirit of children's mathematics is precisely the spirit of modern mathematics. The proposal for change is not a return to the "new mathematics" movement, but something different.