Drozdek, Adam. 1998. "Infinity and Bolzano’s Eschatology." Axiomathes no. 9:275-286.
"Bernard Bolzano is known today for his work in logic and mathematics.
Paradoxes of the infinite is the only book from a vast body of his writings that was translated into many languages. The rest of his books
and articles have only relatively recently being republished in a collected edition of his works. In his time, however, Bolzano was known mainly for his
theological works which were widely read and which stirred controversies because of the many unorthodox views they contained. One problem was that Bolzano was
a Roman Catholic priest and many of his views flew in the face of orthodox Christianity in general and Catholicism in particular. One of the most interesting
works in this respect is his Athanasia, or the grounds for immortality of the soul , which had two editions during Bolzano's life (in 1827 and 1838).
The goal of Athanasia is to prove, with the power of reason and "truths that observation of nature offers," that the soul is immortal (A
17)." (p. 275)
George, Rolf, and Rusnock, Paul. 2006. "Bolzano’s Political Philosophy." In The Austrian Contribution to Analytic
Philosophy , edited by Textor, Mark, 264-292. New York: Routledge.
"Bolzano’s major work on political philosophy is a book called On the Best State , which was written around 1830.(2) As its
title suggests, the work is concerned not with suggestions for reforms of existing institutions, but rather the elaboration of an ideal, namely, an
organisation of civil society that maximises the well being of its members. The approach of this work has earned Bolzano the reputation of a utopian in
political philosophy, someone unconcerned with the practical, the here and now, devoted instead to the passive contemplation of the unattainable. This
impression might easily be confirmed by the knowledge that he never sought to publish his book, and towards the end of his life actually resisted the attempts
of others to publish it for him.
But this view of Bolzano the political philosopher is completely mistaken.
Indeed, it would be difficult to find an example of a philosopher who had a greater impact on the political culture of his country. An
examination of the exhortations (Erbauungsreden ) Bolzano read weekly to the university students and educated public of Prague shows him to have been
one of the most prominent advocates for reform of the time.
It is worth noting, too, that after a little over 150 years, a great many ideas very like Bolzano’s have been implemented somewhere or other
– for better and for worse." (pp. 265-266)
(2) Von dem besten Staate , ed. by A. Kowalewski, Prague 1932. New edition by Jaromir Louzil in the Bernard Bolzano
Gesamtausgabe (hereafter BBGA) Series IIA, Vol. 14.
Livingston, Paisley. 2014. "Bolzano on Beauty." British Journal of Aesthetics no. 54:269-284.
Abstract: "This paper sets forth Bolzano’s little-known 1843 account of beauty. Bolzano accepted the thesis that beauty is what rewards
contemplation with pleasure. The originality of his proposal lies in his claim that the source of this pleasure is a special kind of cognitive process, namely,
the formation of an adequate concept of the object’s attributes through the successful exercise of the observer’s proficiency at obscure and confused
To appreciate this proposal we must understand how Bolzano explicated a number of concepts (especially clarity, confusion, and intuition) in
his Wissenschaftslehre . I argue that Bolzano was ahead of his time and anticipated some of the results of recent empirical psychological research on
the relations between beauty, affect, and processing fluency. Bolzano’s remarks on ugliness and on relations between pure and mixed beauty are also of
contemporary interest. The upshot is that Bolzano’s account of beauty is neither as derivative nor ‘dark’ as some of his commentators have claimed."
———. 2015. "An Introduction to Bolzano's Essay on Beauty." Estetika: The Central European Journal of Aesthetics no.
Abstract: "A neglected gem in the history of aesthetics, Bolzano’s essay on beauty is best understood when read alongside his other
writings and philosophical sources.(1)
This introduction is designed to contribute to such a reading. In Part I, I identify and discuss three salient ways in which Bolzano’s
account can be misunderstood. As a lack of familiarity with Bolzano’s background assumptions is one source of these misunderstandings, in Part II, I elucidate
some of his ideas about the psychological processes involved in the contemplation and enjoyment of beauty. In Part III, I situate Bolzano’s discussion of
beauty within the more general framework of his ideas about the nature of philosophy, the relation between philosophy and aesthetics, and the place of the
concept of beauty within the latter. Part IV is devoted to Bolzano’s discussion of some of his philosophical antecedents, including Kant. In Part V, I raise
some objections to Bolzano’s account and indicate how his advocates might respond to them."
(1) Bernard Bolzano, Abhandlungen zur Ästhetik: Erste Lieferung; Über den Begriff des Schönen. Eine Philosophische Abhandlung
(Prague: Borrosch et André, 1843). The first half of the essay (§§ 1–25) is translated as ‘On the Concept of the Beautiful: A Philosophical Essay ’ in
this issue of Estetika. In what follows I cite this Borrosch et André edition as B, giving paragraph and then page numbers, followed by those in the Estetika
translation. For background information about Bolzano’s life and works, see Edgar Morscher, ‘Bernard Bolzano’, in Stanford Encyclopedia of Philosophy
, ed. Edward N. Zalta, fall 2014 ed. (Stanford University, 1997–).
———. 2016. "Bolzano on Art." British Journal of Aesthetics no. 56:333-345.
Abstract: "In his little-known essay published posthumously in 1849, Über die Eintheilung der schönen Künste (On the
Classification of the Beautiful Arts ), Bernard Bolzano proposes an explication of the concept of beautiful art as well as a classification of these arts.
Bolzano’s divisions allowed him not only to provide a principled and comprehensive classification of actual, well-established arts, but also to anticipate
kinds of beautiful art that would not exist or be widely recognized until decades after his death, such as moving pictures, abstract paintings, and what he
called ‘the arts of pure thought’."
———. 2021. "Bernard Bolzano: (1781-1848)." In Aesthetics: The Key Thinkers , edited by Giovannelli, Alessandro, 117-128.
New York: Bloomsbury.
"If asked to say which philosophers did significant work in aesthetics in the nineteenth century, very few people-including experts in
the field-would be likely to mention Bernard Bolzano, who is best known for his findings in mathematics and logic. This is unfortunate, as Bolzano also made
important contributions to aesthetics. In an early sermon (Bolzano 1810), he eloquently defended the traditional thesis that in appreciating the beauty of
nature we worship God and His creation.
In the early 1840s, he planned to write a series of five interrelated studies on topics in aesthetics, yet a fatal pulmonary illness
prevented him from realizing this ambition (Winter 1949, 46-7). He did manage to complete the first part of his planned series, an essay on beauty (Bolzano
2015) that was published in his native Prague in 1843 as Über den Begriff des Schonen. Eine Philosophische Abhandlung (On the Concept of the
Beautiful. A Philosophical Essay ; see also Bolzano 1972 and 2017). Bolzano also finished an essay on the fine arts, which was published posthumously in
1849 as Über die Eintheilung der schonen Kunste. Eine ästhetische Abhandlung ( On the Division of the Beautiful Arts. An Aesthetic Essay ).
These two published essays are the primary focus of this entry, although some of Bolzano's other work will be mentioned where relevant." (p. 117)
Bolzano, Bernard. 1810. "Uber denn Sinn für die Naturschonheiten." (On the sense for natural beauty ) In
Erbauungsreden für Akademike (Exhortations for academics ), 262-285. Prague: Widtmann.
--- 1972. Untersuchungen zur Grundlegung der Asthetik (Investigations into the foundations of aesthetics ). Dietfried
Gerhardus (ed.). Frankfurt am Main: Athenäum.
--- 2015. "On the Concept of the Beautiful: A Philosophical Essay." Adam Bresnahan (trans.). Estetika: The Central European
]oumal of Aesthetics , 52: 229-266. A translation of the first half of the essay.
--- 2017. Ecrits ésthetiques. Carole Maigne, Nicolas Rialland, and Jan Sebestik (trans. and eds.). Paris: Vrin.
Winter, Eduard. 1949. Leben und geistige Entwicklung des Sozialethikers und Mathematikers Bernard Bolzano . Halle: Niemeyer.
Löffler, Winfried. 2013. "Bernard Bolzano's Analytic Philosophy of Religion." Philosophy no. 28:53-61.
"In this short paper, I want to give you an overview to the thought of the perhaps most original, yet still widely unknown Austrian
philosopher: Bernard Bolzano (1781-1848)." (p. 53)
"Was he also a pioneer in modern analytic philosophy of religion? There is surely no direct influence to current analytic philosophy of
religion, and Bolzano founded no “school” in that field, but we find in his works some really excellent pieces of analytic philosophy of religion, and all that
around 1810. I will just mention four of them: Firstly, Bolzano was a pioneer of the “logic of religion”, i.e. the clarification of the structure, meaning and
justification of religious belief-systems. 130 years later, Joseph Bochenski in The Logic of Religion of 1965 published a similar approach.
Secondly, Bolzano formulated a logically sophisticated proof for God’s existence. It follows the pattern of proof from the contingency of
things to God as their cause, but it is in detail highly original and has no parallel in the literature, as far as I can see. Thirdly, Bolzano brings a
probabilistic analysis of our reasoning about the credibility of historical testimony. This was important for Christian theology, since the Christian religion
is a revealed religion: It claims to go back to certain extraordinary historical events, and hence its rationality hangs on the historical credibility of the
testimonial accounts of these events. Testimony is a huge topic also in current epistemology, but Bolzano presents an early and sophisticated probabilistic
treatise on that – just a few years after Laplace. Fourthly, in some of his sermons (“exhortations”) which he had to give each Sunday and holiday for his
students, he presents detailed analyses of various speech-acts, e.g. of the exact difference between “lying” and “deceiving”. In these sermons, Bolzano
anticipates Austin’s and Searle’s work on speech-act theory by more than 100 years." (pp. 53-54)
"After explaining Bolzano’s overall project in his textbook in section 2, I will pick out three interesting points in sections 3 to 5:
His definition of religion, the question whether he was a total non-cognitivist in religion, and whether he argued for a sort of self-deception or
self-persuasion in religious matters." (p. 55)
Lukasiewicz, Dariusz. 2007. "Logical and Metaphysical Assumptions of Bernard Bolzano's Theodicy." Forum Philosophicum:
International Journal for Philosophy no. 12:33-56.
Abstract: "Bolzano's theodicy is a very good example of Platonism in the philosophy of religion. Above all, Bolzano believes that
there obtains an ideal realm of truths in themselves and mathematical objects, which are independent of God. Therefore, we are allowed to conclude that God is
only a contractor; true, more powerful than Plato's demiurge because He created substances (and matter) and sustains them in existence, but God must
follow a project which is independent of Him. Since the world is determined, by the program and God follows the program, then in fact the program is
a god, or better, there is no God (at least in the sense of the classical Christian tradition).
Bolzano's project is not related to God's essence, since it is external to God, and is not made by God. Thus, Bolzano's theodicy is also the
absolute opposite of the Cartesian theodicy. God in the Cartesian theodicy can change all rules, all scientific laws and, in consequence, He can create any
world He wants. Bolzano's God cannot change anything and cannot create a different world than the world determined by the project, a world different
than the one He has created. The responsibility of Bolzano's God for the evil in the world is limited by the project of the world."
McCormick, Peter. 1981. "Bolzano and the Dark Doctrines: An Essay on Aesthetics." In Structure and Gestalt: Philosophy and
Literature in Austria-Hungary and Her Successor States , edited by Smith, Barry. Amsterdam: John Benjamins.
§1. Kant and the subjectivisation of aesthetics; §2. Bolzano and the theory of the beautiful; §3. Appraising Bolzano's aesthetics.
"In this paper however I want to look sympathetically yet critically at the major feature of his most important reflections on
aesthetics, his theory of beauty.
In order to situate these reflections in the development of modern aesthetics and to make their expression more understandable, I propose in
the first section of the paper to review briefly several of the central difficulties in Kant's aesthetic. In §2 I will then make use of Bolzano's criticisms of
Kant as an initial means of presenting Bolzano's own philosophical concerns, before sketching further those themes which are peculiar to Bolzano's work alone.
Finally in §3 I will suggest critical questions about Bolzano's theory which call for further attention on the part of his interpreters and also on the part of
those interested in a certain strain of post-Kantian aesthetics.
Although my purposes here remain largely critical ones, I have tried to call attention to textual, historical and interpretative points in
some of the accompanying notes.
My larger purpose is to suggest in a general way only that Bolzano's aesthetics, specifically his investigations into the concept of the
beautiful, represent more than just an important and almost completely overlooked criticism of Kant's well-known views; these investigations provide the basic
link between Kantian aesthetics and the work of the later Brentano, Meinong, Husserl, and Ingarden. In short, Bolzano's aesthetics include the initial elements
of what might be called the realist tradition in modern aesthetics." (pp. 69-70)
Mezei, Balasz. 2009. "Two Models of Radical Revelation in Austrian Philosophy." European Journal for Philosophy of
Religion no. 1:99-120.
Abstract: "In this paper I highlight two opposing models of the notion of divine revelation: the propositional and the radical. The
propositional understanding of revelation was central to theology and philosophy until the 19th century. Since then, a number of other models of revelation
have emerged. I define as radical the understanding of revelation which emphasizes two features of revelation: 1) God’s existence is *per se* revelatory; 2)
God’s revelation is *per se* self-revelation. I propose too an assessment of the notion of propositional revelation as presented by Richard Swinburne. And I
offer detailed analyses of two representatives of the early understanding of divine revelation as self-revelation: the views of Bernard Bolzano and Anton
Günther. Bolzano, the renowned mathematician, was also a philosopher of religion; and Günther, one of the most ingenious writers in Austrian philosophy, was
not only a theologian but also a philosopher comparable to the important figures of 19th century German thought."
Reicher, Maria E. 2006. "Austrian Aesthetics." In The Austrian Contribution to Analytic Philosophy , edited by Textor,
Mark, 293-323. New York: Routledge.
"In this chapter, however, the focus is on a number of lesser known Austrian contributions to aesthetics. These contributions concern
the following, partly interrelated, central problems of philosophical aesthetics:
I. The problem of the definition of beauty (i.e. What is beauty? What does it mean to say of an object that it is beautiful?)
II. The problem of the ontological status of works of art (i.e. What kinds of objects are works of art?)
III. The problem of the objectivity of aesthetic values (i.e. Do we claim objective validity for aesthetic value judgements and, if so, is
this claim justified?)
This chapter will consider the answers of Bolzano, Meinong, Witasek and Ehrenfels to these questions." (p. 294)
"The task of the essay ‘On the concept of the beautiful’ is to find a definition for the concept of beauty. In the preface to this
treatise Bolzano states in a few sentences his views on what philosophical aesthetics should do and how it should be done:
That I have filled so many pages with the analysis of a single concept will necessitate an excuse in the eyes of some. I cannot state
anything but that I found this concept to be of particular importance and that analysis of concepts is a business that commonly demands somewhat lengthy
investigations, if it shouldn’t be just stated that one thinks of the concept as consisting of these parts, but rather shown to the reader in an at least
fairly convincing way, which entails that one has to demonstrate that the previously suggested, other explanations have been more or less erroneous.(3) (CB,
This passage shows clearly that Bolzano does not take philosophical aesthetics to be a discipline that demands less rigour than, say,
epistemology and ontology. The task of philosophical aesthetics is, according
to Bolzano, to clarify the basic concepts of aesthetic discourse by means of meticulous analysis. The two perhaps most prominent concepts of
aesthetics are the concept of beauty and the concept of art.
Consequently, Bolzano considers the clarification of these concepts to be the most important task of philosophical aesthetics." (p.
(3) All translations in this chapter are mine.
Bolzano, B.: ‘Über den Begriff des Schönen. Eine philosophische Abhandlung’, in D. Gerhardus (ed.) Bernard Bolzano. Untersuchungen zur
Grundlegung der Ästhetik , Frankfurt/Main: Athenäum, 1972, 1–118. (Originally published 1843.) (CB)
Rosenkoetter, Timothy. 2014. "Kant, Bolzano, and Moore on the Value of Good Willing." In New Anti-Kant , edited by
Lapointe, Sandra and Tolley, Clinton, 235-271. London: Palgrave Macmillan.
"Bolzano’s many achievements in mathematics, logic, metaphysics, and epistemology, as well as the monumental scope and ambition of his
Theory of Science and Theory of Magnitudes , make it easy to overlook his relatively brief treatments of morality. Yet Bolzano took moral
theory quite seriously. (1) This carries over to Bolzano’s engagement with Kant. Though only a small fraction of The New Anti-Kant (192–202) is
concerned with moral philosophy, Bolzano thinks that the correction he is able to provide there is every bit as important as his critical engagement with the
central topics of the first Critique . Thus, one purpose of this chapter is to place that brief comparison with Kant’s moral theory within the context
of Bolzano’s most thorough defence of utilitarianism, which is found in lectures that his students published as Treatise on the Science of Religion
.(2)" (p. 235)
(1) For historical background on how Bolzano endeavoured to guide his life as a philosopher and public intellectual by utilitarian
principles, see Rusnock’s and George’s introduction to Bolzano (2007).
(2) This present essay is based on four sources: The New Anti-Kant (“ NAK ”), Theory of Science , Science of
Religion (Bolzano 1994/1995, hereafter “ R ”), and “On the Right of the Clergy”. The Rusnock/George translation of relevant sections of the last two are
available in Bolzano (2007). I frequently depart from their translation of Science of Religion , using Bolzano (1994/1995) as my basis. See Künne
(1996, 325ff.) for a list of inaccuracies in the latter edition.
Bolzano, Bernard. (1994/1995). Lehrbuch der Religionswissenschaft , Erster Teil. In Bernard Bolzano-Gesamtausgabe , Series
1, Vol. 6/1 and 6/2. Jaromir Louzil (ed.). Stuttgart-Bad Canstatt: Frommann-Holzboog.
Bolzano, Bernard. (2007). Selected Writings on Ethics and Politics . Paul Rusnock and Rolf George (eds), Amsterdam: Rodopi.
Künne, Wolfgang. (1996). ‘Bolzanos Philosophie der Religion und der Moral’, Archiv für Geschichte der Philosophie , 78, 309-328.
Rusnock, Paul, and Šebestik, Jan. 2019. Bernard Bolzano: His Life and Work . New York: Oxford University Press.
Chapter 2. Ethics 83-104; 3. Political Philosophy 105-138; 4. Philosophy of Religion 139-166; 5. Catholicism and the Catholic Church 167-186;
10. Aesthetics, the Science of Beauty 544-594-
Šebestik, Jan. 1994. "Prague Mosaic. Encounters with Prague Philosophers." Axiomathes no. 2-3:205-223.
"In 1814, a young priest, professor of the "science of religion" at Charles University and author of two mathematical
pamphlets, presented another mathematical booklet asking for the admission to the Royal Society. His name was Bernard Bolzano (1781-1848) and I am happy that I
have the opportunity to speak about him in the city which bears his name. He was, as I think, the outstanding figure of Prague philosophy." (p. 209)
"Bolzano can be called the father of Austrian - or Central-European - philosophy. Almost all Austrian philosophers have acquired
something of his method, of his way of thinking and of his fundamental ideas. Austrian philosophy in the broad sense, or rather Central-European philosophy,
has always tried not to separate philosophy and science (a large number of Austrian philosophers are at the same time scientists or scientifically trained), to
identify scientific and philosophical method, to concentrate on logical analysis, to avoid the tcmptation of idcalism, to be attentive to language, and, last
but not least, to consider the public activity of a philosopher as a social rclbrmcr to bc an integral part of his task. In this respect, Masaryk and the
Vienna Circle arc the true heirs of Bolzano." (p. 212)
Sirůček, Pavel, and Jaroslav, Šetekm. 2019. "Bernard Bolzano." Acta Oeconomica Pragensia no. 27:95-104.
Abstract: "Bernard Bolzano was not merely a mathematician of the European format. He was also a critical social thinker whose work has
distinct economic dimensions. This is why he rightly appears in the overviews of the beginnings of Czech economic thought. He began writing his work “About the
Best State” 210 years ago, which made him the most famous Czech utopian. The half-forgotten Bolzano legacy, which is still alive and ahead of its time in many
respects, recalls this text."
Stern, J. P. 1989. "Language Consciousness and Nationalism in the Age of Bernard Bolzano." Journal of European Studies no.
"From the end of the eighteenth century an especially close, complex and ultimately disastrous interrelationship develops in Central
Europe between language consciousness and nationality. In this region, national and natural borders do not often coincide for very long; and in this age,
greater value is ascribed to political notions of "natural" or "organic" han to politics based on tradition and agreement bureaucratically
enforced. Here linguistic borders are endowed with the pathos of national borders, national consciousness is identified with language
consciousness." (p. 169)
"The intensity of the Romantic apotheosis of das Volk together with its new system of values is bound to escape Bolzano - this is the
pathos that will dominate the next hundred years of Czech and Central European history. This omission is all the more surprising since Bolzano (in contrast to
Frege,47 and to the young Wittgenstein) has a very clear understanding of the social function of language; but since he fails to provide a theoretical
foundation for them, his insights do not go beyond the practical concerns of daily life. He seeks to solve the problem of the two languages in social and
economic terms leading to moral and spiritual improvement. But the subsequent history of Bohemia as well of most other parts of Central Europe has taught us
that the appeal to a threatened language consciousness can be politically effective even when it entails social and economic deprivations; that nationalism is
stronger than Marxism. Languages - the two languages of Bohemia - are indeed what Wittgenstein calls them: "forms of life". The common humanity to
which Bolzano appealed lies behind this insight, buried beneath the antagonisms of two nations once united by their common roots in one soil." (p.
(47) See Dummett [Ursprunge der analytischen Philosophie , translated from the English by Joachim Schulte (Frankfurt/M, I988)],
Świętorzecka, Kordula. 2014. "An Argument for the Existence of God by Bolzano. A Formalization with a DFistinction between
Menge and Inbregriff ." Bulletin of the Section of Logic no. 43:155-172.
Abstract: "Bernard Bolzano is the author of a text considered by him as an argument for the existence of God, originally published in
the "Lehrbuch der Religionwissenschaft" (1834). His disquisition was formalized by H. Ganthaler and P. Simons in 1987 who based their approach on the
well-known set theoretical interpretation of Bolzano's idea of multiplicity. The authors formulated a theory expressed in set theoretical language extended by
some specific terms and constructed a proof as a formalization of the reasoning given by Bolzano. A problematic aspect of the proposal of Ganthaler and Simons
is that it passes over the distinction of two different types of multiplicities which enable to point some interesting formal and material details of Bolzano's
revised argumentation. In the frame of the presented proposal we invent two special meanings of Bolzano's notion of multiplicity: distributive and collective.
In connection with this distinction we reconstruct the argument analyzed by Ganthaler and Simons and discuss the formal and philosophical power of the obtained
formalization. The inspiration for our approach comes from the fact that Bolzano used in his text two terms Menge and Inbegriff in probably
different meanings. However the main aim of this proposal is to take a new look at Bolzano's argument, rather than to provide historical
H. Ganthaler, P. Simons, Bernard Bolzanos kosmo-logischer Gottesbeweis , Philosophia Naturalis 24 (1987), pp. 469-475.
Synytsia, Andrii. 2021. "Bernard Bolzano’s Philosophical Conception of Education from a Modern Standpoint." Journal of
Education Culture and Society no. 12:32-44.
Abstract: "Aim. The aim of the article is to analyse Bernard Bolzano's philosophico-educational approach, given the importance of his
pedagogical legacy for further progress in the held of education and training.
Methods. The research uses methods of historico-philosophical reconstruction (to determine the foundations and features of Bolzano's
pedagogico-philosophical work), interpretation (to characterise the still relevant ideas of the thinker in the held of education) and critical analysis (to
assess the significance of Bolzano's legacy from the standpoint of modern educational theory and to point out what remained out of the researchers'
Results. The study showed that Bolzano's pedagogical ideas were not only in demand in the context of European culture of the early XIX
century, but also proved relevant today, especially given the need for harmonious personal development, finding sources of intrinsic motivation in students and
improving their logical thinking skills, the importance of the spiritual qualities of the teacher and - in general - the assistance of the state in the
direction of the progress of education as the most effective means of qualitative change in society.
Conclusion. Bernardo Bolzano's philosophical conception of education was strongly influenced by religion, the ideas of the late Enlightenment
and the late Josefinist. This conception was an integral part of his socio-political theory. Bolzano attached great importance to the study of philosophy and
the acquisition of logical skills in education. He considered writing textbooks to be one of the cornerstones of the educational process. This made it possible
to intensify communication between teachers and students, to gradually involve everyone in education, and thus to promote social progress in general."
Tapp, Christian. 2019. "Bolzano’s Concept of Divine Infinity." In The Infinity of God: New Perspectives in Theology and
Philosophy , edited by Göcke, Benedikt Paul and Tapp, Christian, 150-166. Notre Dame: University of Notre Dame Press.
"Infinity is central to each of the three areas in which Bernard Bolzano (1781–1848) had expertise: mathematics, philosophy, and
theology. The Bohemian priest dealt with all the respective concepts in these disciplines: quantitative infinity in mathematics, qualitative infinity in
philosophy and theology, and some of their mutual interrelationships.
I focus on “infinite” used as a divine predicate. Since this use of “infinite” is related to a quantitative use in several ways, we will also
dwell on these." (p. 150)
"Divine infinity is an integral part of Bolzano’s concept of God. It is a test case for his broader claim that the concept of
quantitative infinity is more basic than all other concepts of infinity; that all meaningful discourse about infinity presupposes quantitative infinity. This
thesis from §11 of the Paradoxes of the Infinite must not be understood as reductivist, as though Bolzano would only accept quantitative infinity. He
claims only that in each case in which we are entitled to use qualitative concepts of infinity, which are vaguer and more opaque, we must also be able to find
aspects of the subject matter that exhibit the more precise quantitative infinity.
Ascribing to Bolzano a formal background conception of infinity as limitlessness or unrestrictedness, as suggested by his treatment of
omnipotence, helps us to interpret his doctrine of God in a coherent way.
However, what exactly it means to be unlimited, to suffer from no limitations in a certain respect, remains a topic for further philosophical
analysis. Does every conception of “limits” require a metric or at least a
total linear ordering? How can one align such a concept with the mathematical way of speaking, according to which there are unlimited finite
magnitudes and also limited infinite magnitudes? Finally, what “respects” are to be considered, in the case of divine infinity, as unrestricted in certain, if
not all, respects?" (p. 166)