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O pojęciu prawdy w matematyce

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Murawski R.
Zagadnienia Naukoznawstwa
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2011
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vol. 47
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issue 4(190)
471-481
EN
The paper is devoted to the concept of truth in mathematics. The starting point is Tarski's definition of truth. The philosophical background of this definition is discussed, its meaning for the language of mathematics and for philosophy, its relation to various definitions of truth. The relation provability vs. truth is also considered. With use of some results of mathematical logic, it is shown that the conditions from Tarski's definition are too weak to assure the uniqueness of interpretation of truth predicate. It is also shown that semantic notions such as satisfaction and truth are not finitistic and require the concept of infinity.
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On Development of the Awareness of the Difference between Truth and Provability in Mathematics

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Murawski R.
Filozofia Nauki
|
2001
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vol. 9
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issue 1
7-28
PL
The author shows in his article how the awareness of the difference between truth and provability in mathematics has developed. He points out the role played in this process by Gödel's results concerning incompleteness of formalised theories and also indicates the attempts at overcoming these limitations by giving up the finitistic condition and by allowing infinitary methods in the notion of mathematical proof. The philosophical assumptions that one accepts are important for the problem under discussion. For strict formalists and intuitionists the problem of distinguishing between truth and proof does not exist at all. For them a mathematical statement is true if it is provable, where proofs are considered to be our own constructions - syntactic or mental. The situation is entirely different for the proponents of platonism (realism) in the philosophy of mathematics. It can be said that it is just the platonist approach to mathematics that made it possible for Gödel to both pose the problem and to understand and show the difference between provability and truth.
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Some Words on a Proof in Mathematics

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Murawski R.
Filozofia Nauki
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2013
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vol. 21
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issue 1
7-22
PL
The aim of the paper is to study the role and features of proofs in mathematics. Formal and informal proofs are distinguished. It is stressed that the main roles played by proofs in mathematical research are verification and explanation. The problem of the methods acceptable in informal proofs, in particular of the usage of computers, is considered with regard to the proof of the Four-Color Theorem. The features of in-formal and formal proofs are compared and contrasted. It is stressed that the concept of an informal proof is not precisely defined, it is simply practised and any attempts to define it fail. It is - so to speak - a practical notion, psychological, sociological and cultural in character. The second one is precisely defined in terms of logical con-cepts. Hence it is a logical concept which is rather theoretical than practical in char-acter. The first one is - in part at least - semantical in nature, the second is entirely syntactical. A proof-theoretical thesis, similar to the Turing-Church Thesis in the re-cursion theory, is formulated. It says that both concepts of a proof in mathematics are equivalent. Arguments for and against it are formulated.
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Proof in Mathematics: Today and Tomorrow

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Murawski R.
Filozofia Nauki
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2013
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vol. 21
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issue 2
201-203
PL
The paper is a review of the book by Krzysztof Wójtowicz, O pojęciu dowodu w matematyce [On the Concept of Proof in Mathematics]. It presents the main theses of the book and evaluates them.
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Philosophy of Logic and Mathematics in the Cracow Circle

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Murawski R.
Filozofia Nauki
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2014
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vol. 22
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issue 2
21-35
PL
The paper is devoted to the presentation and analysis of the philosophical views concerning logic and mathematics put forward by the leading members of the Cracow Circle, i.e., of Jan Salamucha, Jan Franciszek Drewnowski, and Józef (Innocenty) Maria Bocheński. It also discusses their views on the problem of applicability of logical tools in metaphysical and theological research.
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Between Theology and Mathematics. Nicholas of Cusa’s Philosophy of Mathematics

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Murawski R.
Studies in Logic, Grammar and Rhetoric
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2016
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vol. 44
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issue 1
97-110
EN
The paper is devoted to the philosophical and theological as well as mathematical ideas of Nicholas of Cusa (1401–1464). He was a mathematician, but first of all a theologian. Connections between theology and philosophy on the one side and mathematics on the other were, for him, bilateral. In this paper we shall concentrate only on one side and try to show how some theological ideas were used by him to answer fundamental questions in the philosophy of mathematics.
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Nieskończoność w matematyce. Zmagania z potrzebnym, acz kłopotliwym pojęciem

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Murawski R.
Zagadnienia Filozoficzne w Nauce
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2014
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issue 55
5-42
PL
Infinity has appeared in mathematics since the very beginning. Moreover the mathematical concept of infinity was and is connected with philosophical and theological concepts. The aim of the paper is to show how mathematicians struggled with this concept and how they tried to bring it under control.
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Wiara a wychowanie

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Murawski R.
Colloquia Theologica Ottoniana
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2014
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issue 1
59-74
PL
Autor poszukuje odpowiedzi na kluczowe pytanie: czym jest wychowanie w wierze, zamiast dociekać, jak wychować dzieci do wiary. Zadanie to wydaje się szczególnie trudne ze względu na brak profesjonalnych badań w tej dziedzinie, zarówno w literaturze krajowej, jak i zagranicznej. Brak fachowej literatury jest motywacją do prowadzenia badań na ten temat. Inną trudnością jest to, że wiara jest złożonym zjawiskiem. Z jednej strony jest to dar od Boga, z drugiej zaś – to akt wolnej woli człowieka. Mimo że wiara jest łaską, pod uwagę brane są wszystkie czynniki ludzkie, albowiem wiara jest zawsze „wcielona” wszystkim ludziom. Ponadto wiara może być analizowana w różnych wymiarach. Wszystko to sprawia, że proces wychowania w wierze jest niezwykle skomplikowany. W swojej pracy autor skupia się na wybranych aspektach problemu. W pierwszej części odnosi się do terminologicznych niejasności związanych z omawianym tematem. Następnie przechodzi do katechezy w służbie wiary. Trzecia część jest refleksją na temat relacji między wiarą a wychowaniem.
EN
The author is looking for an answer to a pivotal question: what is upbringing in faith and not how to bring children up in faith. This task seems to be especially difficult due to lack of professional research in the field, both in Polish and foreign literature. On the other hand this fact obliges researchers to get down to this topic. Another diffi culty is that faith is complex phenomenon. On the one hand it is God’s gift and on the other a totally free human act. Although it is a grace the act of faith manifests itself taking into account all human factors, for such is the law of the incarnation and faith is always „incarnated” into all human. Moreover faith may by analysed in various dimensions. All this makes the process of upbringing in faith extremely complex. In his paper the author focuses on chosen apects of the problem. The first part refers to terminological ambiguities related to the analysed topic. Afterwards the author turns to a catechesis in service of faith. The third part is refl\exion on a relations between faith and upbringing.
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O czym rozprawiają matematycy, c zyli o statusie bytowym przedmiotów matematyki

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Murawski R.
Przestrzenie Teorii
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2004
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issue 3-4
253-260
EN
In this article the author considers various conceptions concerning ontological status of the objects of mathematics. The relation is show between these problems and a dispute over universals, and then Platonism, conceptualism and nominalism as the fundamental standpoints on the question o f mathematical objects are presented. Examples are given of concrete (specific) theories w ith in these standpoints and their implications (also epistemological ones) are shown.
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Karolińskie "Ordo de catechizandis rudibus"

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Murawski R.
Vox Patrum
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2012
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vol. 57
419-426
EN
The document presented in the article dates from the time of Charlemagne and was part of his reform program of religious and intellectual life. Its origins must be sought in Alcuins’ letter of 796, addressed to Charlemagne (a copy of this letter was also sent to Arno, the Archbishop of Salzburg), which – after a failed mission to the Saxons – offered rulers a new form of evangelising the Avars. Recalling the disappointment of attempts to evangelise in Saxony, he warns against making too rapid and violent actions. They have to be taught first, and then baptised. At the end of the letter, referring to St. Augustine’s De catechizandis rudibus, he states what teachings in the truths of the faith they must receive before baptism. This letter sent to both the king and the bishop Arno gave the unknown author (editor) a frame­work and plan for the drafting of Ordo de catechizandis rudibus. The document appeared in the surroundings of the bishop Arno about the year 800. The word Ordo used in the title indicates that it is a kind of guide with guidelines for teaching adults before baptism. Although the title of the document was taken from the work of Saint Augustine of the same title, it differs substantially from its original. The Carolingian Ordo can be divided into four parts or sections. The first part (1-13) is a comprehensive introduction showing the need to instruct catechu­mens and the overall characteristics of the process of Christian initiation. The second part (14-44) includes the initiatory catechesis and provides topics covered in teaching converts. The content consists of biblical texts and the writings of various Fathers of the Church and practical instructions derived from the work of St. Augustine. The third part (45-74) presents the liturgical rites of baptism. In the fourth part (75-94), the document explains the baptismal rite itself and the other celebrated sacraments of Christian initiation – the Eucharist and Confirmation. The Ordo text ends with a call to neophytes to faithfully fulfil in practice what has been communicated to them.
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Pojęcie „katechetyki” w refleksji współczesnych autorów polskich

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Murawski R.
Studia Pastoralne
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2005
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vol. 1
68-81
DE
Im Ausgangspunkt der Analyse steht die Feststellung, dass die neue Situation in der Katechese, die 1990 mit der Riickkehr des Religionsunterrichts (der Katechese) von den Pfarrgemeinden in die Schulen entstand, eine präzisere Bestimmung des Begriffs der Schulkatechese erfordert. Das ist aber kaum möglich ohne ein Nachdenken iiber den Begriff der Katechese selbst. Der Verfasser dieses Beitrags hat die Versuche der Begriffsbestimmung der Katechetik in den polnischen Publikationen, insbesondere in den Handbiichern der Katechetik, die nach Vaticanum II erschienen sind, aufmerksam verfolgt. Er bedauert, dass diesem Thema sehr wenig Aufmerksamkeit geschenkt wird. Im analysierten Zeitraum haben nur zwei Autoren in Polen Versuche unternommen, ihre eigenen Konzepte der Katechetik zu entwickeln: Franciszek Blachnicki und Mieczysław Majewski. Der erste von ihnen suchte die Ergebnisse der kerygmatischen Erneuerung in der Katechese anzuwenden; der zweite tat ähnliches auf dem Boden der anthropologischen Orientierung. Gegenwärtig strömt in die polnische Katechetik eine neue, hauptsächlich aus Deutschland stammende Tendenz, die Probleme der heutigen Katechese nicht im Rahmen der als ein Teil der Pastoraltheologie betrachteten Katechetik, sondern der Religionspadagogik zu erwägen. Der vorliegende Artikel ist ein Versuch, einen Anstoß zur Diskussion iiber den aktuellen Stand der polnischen Katechetik und über die Perspektiven ihrer Wei-terentwicklung zu geben.
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Wokół pojęcia katechezy według Jana Pawła II

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Murawski R.
Studia Pastoralne
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2007
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vol. 3
167-177
IT
L'articolo sopra presentato propone una riflessione dedicata al tema delia catechesi, tirato fuori dall'abbondante insegnamento teologico di Giovanni Paolo II. Dell'argomento preso in esame viene trattato in modo da poter dare una risposta a ąuesta domanda: qual è la concezione delia catechesi nell'insegnamento lasciatoci dal papa che durante il suo pontificato non soltanto insegnava autorevolmente su di essa, ma la praticava in maniera affascinante ed esemplare. Giovanni Paolo II non si stancava mai di radicare nei cuori dei credenti la convinzione che la catechesi è un atto ecclesiastico. Perciò, tutti coloro che si dedicano ad essa agiscono nel nome delia Chiesa ed esercitano il suo mandato. E' il motivo, per cui devono acąuisire un'esatta comprensione delia catechesi, senza la ąuale rischiano di non afferrarne tutto il significato e la portata (cfr. Catechesi tradendae, nr 18). L'esattezza di tale comprensione consiste nella necessità di intendere la catechesi, come la intende la Chiesa che manda i catechisti ad evengelizzare il mondo. Il contenuto presentato nell'articolo ha il desiderio di aiutare i cristiani di oggi nella riscoperta di ąuesto segreto sottile e cosi importante. Lo facciamo basando sul contenuto delle catechesi dedicate al tema da noi trattato, svolte da Giovanni Paolo II durante alcune udienze del mercoledì. Prendiamo anche in considerazione appositi documenti dati alla luce durante il suo potificato.
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The Conception of Logic in the Cracow Circle: Salamucha, Drewnowski, Bocheński

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Murawski R.
Studia Philosophiae Christianae
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2021
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vol. 57
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issue 1
109-125
EN
The aim of this paper is to present and analyse the views on logic of the members of the so-called Cracow Circle, namely the Dominican Father Józef (Innocenty) M. Bocheński, Rev. Jan Salamucha, and Jan Franciszek Drewnowski. They tried to apply the methods of modern formal/mathematical logic to philosophical and theological problems. In particular, they attempted to modernise contemporary Thomism (the trend which was then prevailing) by employing logical tools. The influence of Jan Łukasiewicz, the co-founder of the Warsaw School of Logic will be also discussed.
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Badania logiczne prowadzone w Uniwersytecie Poznańskim w latach 1945 - 1955

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Murawski R., Pogonowski J.
Investigationes Linguisticae
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2006
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vol. 14
68-77
EN
This paper is a synopsis of a more comprehensive work presented at the conference 50 years of Studia Logica in 2003. The full English version of the text will be published elsewhere. The text below is devoted to the presentation of logical investigations carried out at the University of Poznań in the first decade after its reopening in 1945, i.e. at the time when and at the place where Studia Logica came into existence. 
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