428 • Philosophia Mathematica

Menzel, Christopher [1991] ‘The true modal ’, Journal of Philosophical Logic 20, 331–374. ———[1993]: ‘Singular and modal logic’, Philosophical Topics 21, 113–148. ———[2008]: ‘Actualism’, in Edward N. Zalta, ed., Stanford Encyclopedia of (Winter 2008 Edition). Stanford University. http://plato.stanford.edu/archives/win2008/entries/actualism, last accessed June 2015. Nelson, Michael [2009]: ‘The contingency of ’, in L.M. Jorgensen and S. Newlands, eds, and the Good: Themes from the Philosophy of Robert Adams, Chapter 3, pp. 95–155. Oxford University Press. Downloaded from https://academic.oup.com/philmat/article/23/3/428/1449457 by guest on 30 September 2021 Parsons, Charles [1983]: in Philosophy: Selected Essays. New York: Cornell University Press. Plantinga, Alvin [1979]: ‘Actualism and possible worlds’, in Michael Loux, ed., The Possible and the Actual, pp. 253–273. Ithaca: Cornell University Press. ———[1983]: ‘On ’, Philosophical Studies 44, 1–20. Prior, Arthur N. [1956]: ‘Modality and quantification in S5’, The Journal of Symbolic Logic 21, 60–62. ———[1957]: and Modality. Oxford: Clarendon Press. ———[1968]: Papers on Time and Tense. Oxford University Press. Quine, W.V. [1951]: ‘Two dogmas of empiricism’, Philosophical Review 60, 20–43. ———[1969]: Ontological Relativity and Other Essays. New York: Columbia University Press. ———[1986]: Philosophy of Logic. 2nd ed. Cambridge, Mass.: Harvard University Press. Shapiro, Stewart [1991]: Foundations without Foundationalism: A Case for Second-order Logic.Oxford University Press. Stalnaker, Robert [2012]: Mere Possibilities: Metaphysical Foundations of Modal Semantics. Princeton, N.J.: Princeton University Press. Turner, Jason [2005]: ‘Strong and weak possibility’, Philosophical Studies 125, 191–217. Williamson, Timothy [2010]: ‘Modal logic within counterfactual logic’, in Bob Hale and Aviv Hoffmann, eds, Modality: Metaphysics, Logic, and , pp. 81–96. Oxford University Press. ———[2012]: Modal Logic as Metaphysics. Oxford University Press. Zalta, Edward N. [1983]: Abstract Objects: An Introduction to Axiomatic Metaphysics. Dordrecht: D. Reidel. ——— [1993]: ‘Twenty-five basic theorems in situation and world theory’, Journal of Philosophical Logic 22, 385–428. doi:10.1093/philmat/nkv017 Advance Access publication June 26, 2015

Bernard Bolzano. Theory of Science. Volumes I–IV. Paul Rusnock and Rolf George, trans. Oxford: Oxford University Press, 2014. ISBN: 978-0-19-968438-0. Pp. 2044.†

Reviewed by Jan Sebestik∗

After 177 years, thanks to Paul Rusnock and Rolf George, we finally have the com- plete English translation of Bolzano’s monumental work Theory of Science, the great

†Volume I contains a general introduction and a translation of key terms; each volume has a special introduction, a bibliography, and an index of names and an index of subjects. All quotations in foreign languages are translated. ∗Institut d’Histoire et de Philosophie des Sciences de l’Université de Paris I Panthéon-Sorbonne and CNRS, UMR 8590, Paris, France. E-mail: sebestik@flu.cas.cz Critical Studies/Book Reviews • 429 unknown classical work in logic and epistemology. The translation is excellent, both in keeping the flavor of nineteenth-century books and in eminently readable. The fundamental innovations brought about by his work were not really understood by his contemporaries, which accounts for the initial neglect of his work until, about a hundred years after its publication, logicians of the twentieth century arrived at similar . Downloaded from https://academic.oup.com/philmat/article/23/3/428/1449457 by guest on 30 September 2021 1. BOLZANO’S LIFE AND WORK This author lived in a Europe shaken to its foundations by the French revolution and Napoleonic wars. At the end of the eighteenth century, his native country Bohemia was a multicultural society with Germans politically dominating the Czech majority. The third component, the mainly German-speaking Jewish community, had been settled in the Czech lands since the Middle Ages. Symbolically, his life went through revolutions, both political and philosophical. Bernard Bolzano (1781–1848) was born into a Catholic family; his father was Italian, his mother German. Bolzano knew some Czech, but his native language was German and all his publications were in that language. In the same year 1781 the Austrian Emperor Joseph II initiated a series of reforms which put an end to the forced labor of peasants, modernized justice and forbade torture, abolished inactive religious orders, introduced partial religious freedom (Protestants could reopen their churches) and tolerance, resulting in the modernisation of the society and awakening of the sci- ences. Also in 1781 was published Kant’s Critique of Pure Reason which revolutionized philosophy. During his studies at the Prague Charles-Ferdinand University, Bolzano became interested in mathematics and philosophy. His mathematical notebooks are an excep- tionally rich source for the history of mathematics: they show his extensive knowledge of the literature of the seventeenth and eighteenth centuries, contain entries about cur- rent mathematical problems, and also reflections about methodology and critique of some fundamental concepts. In 1800, he decided to study , but at the same time he prepared his first publication on the Euclidean postulate of parallels (published in 1804). Having finished his studies, he participated in the contest for the chair of the ‘science of [the Catholic] religion’, newly founded by the Emperor Franz II to fight atheism and the ideals of the French revolution, and also for the chair of mathemat- ics. He won both and the commission appointed him to the ‘science of religion’. As a university professor of this discipline, he had to be a Catholic . Bolzano did not perceive the larger political context and that this chair could provide a forum for spreading his own on the reform of society in the spirit of tolerance and equal- ity of all citizens. At the end of 1819, the between the of the Emperor and Bolzano’s own representation of an ideal society led to his dismissal from the University. At the same time, he was deprived of the right to publish even scientific articles and his name was not to be mentioned in the Austrian press. Nevertheless, the Prague Royal Society of Sciences did not suspend his membership, and whenever he was in Prague he attended its meetings, where in the 1840s he presented his theory and his theory of real numbers. With the help of his friends, he succeeded in his works in Germany, mainly in Bavaria, sometimes without mention of their author. When he died, he left important mathematical manuscripts, and his young friend Robert Zimmermann was supposed to publish them. Instead he simply transferred 430 • Philosophia Mathematica them to the Viennese Imperial Library so that, except for Bolzano’s last and best known mathematical work The Paradoxes of the Infinite (1851), his , his theory of real numbers and of (real) functions, which are parts of the great treatise of mathematics called The Theory of Magnitudes, were published only in the twentieth century. During his tenure as professor of religion he continued his work on mathematics. In 1810, he published a small book Contributions to a Better Founded Presentation of

Mathematics which contains his first sketch of logic and of the methodology of mathe- Downloaded from https://academic.oup.com/philmat/article/23/3/428/1449457 by guest on 30 September 2021 matics. In 1817, four years before Cauchy’s Cours d’analyse, there appeared one of his best works, Purely Analytical Proof of the Intermediate Theorem.Init,heelabo- rated a sequence of concepts and theorems necessary to prove it, those of the and of continuity of real functions and demonstrated the Bolzano-Cauchy criterion of convergence of sequences and the least-upper-bound theorem. All this can be found in Cauchy’s Cours, and this circumstance gave birth to a controversy whether Cauchy plagiarized Bolzano. In a way, his forced departure from the University was a blessing both for his sci- entific work and for his personal life. He had a chance to become the friend of the Hoffmanns, who lived on their estate in the countryside and who invited him to stay with them, first for summers, then between 1827 and 1842 for the whole time, interrupted only by several weeks’ sojourns in Prague. An extraordinary friendship developed between the priest Bolzano and Mrs Anna Hoffmann, his ‘benefactress and so to speak savior (Lebenserhalterin)’, who became his secretary, taking care of his correspondence with friends (Bolzano always was watched by the police). It was in the small village of Techobuzˇ that he wrote the Theory of Science and the unfinished and unpublished Theory of Magnitudes. But neither his logic nor his mathematics was understood during his lifetime. Once during his life, he met somebody who did his best to understand his logical theories and who partly understood them: Franz Exner, a professor of philosophy at the University in Prague. When Bolzano learned about his living in Prague, he sent him the manuscript of On the Mathematical Method,a summary of his logic extracted from the Introduction to the Theory of Magnitudes (it was probably the first time in the history of mathematics that a long chapter on logic became part of a mathematical treatise). A lively correspondence followed, and they became friends, but Bolzano did not succeed in convincing Exner of the soundness of his theory of propositions in themselves. After the death of the Emperor Franz in 1835, censorship decreased and Bolzano was able to publish scientific articles. In the meantime, his manuscripts circulated among friends and the general public, especially his lectures on religion and his social utopia On the Best State, published only in the twentieth century. These were the happiest years of his life. In 1842 Anna Hoffmann died and Bolzano returned to Prague, a living shadow. He nevertheless continued to attend the meetings of the Royal Society. He abandoned his work on the Theory of Magnitudes hoping that Zimmermann would finish it and began thinking about a shorter text that would contain a summary of his mathematical and physical theories crowned by the of the infinite. The result was the Paradoxes of the Infinite, not completely revised by its author, but at least quickly published after his death. Bolzano died on December 19, at the end of the revolutionary year 1848. Neither the Prague nor the Viennese took Bolzano’s mathemati- cal work seriously. In 1833 Bolzano learned of Cauchy’s arrival in Prague. They met Critical Studies/Book Reviews • 431 there several in 1833–1834 when Cauchy was for a time tutor of the grandson of Charles X, the exiled king of France. Nothing is known about their conversations. When Bolzano’s disciple Michael Joseph Fesl reported the negative reaction of Viennese mathematicians, Bolzano replied proudly, ‘Do not be disconcerted by the unfavorablejudgmentofthesetwoprofessors[...]Iknowthatthefirstrankingmath- ematicians judged differently and Legendre and Cauchy confirmed the correctness of my views in a reassuring way, having arrived at the same ideas without ever having Downloaded from https://academic.oup.com/philmat/article/23/3/428/1449457 by guest on 30 September 2021 known any of my writings’ (Letter to Fesl, 23 March 1834). At the beginning of the 1860s, Weierstrass spoke about Bolzano’s Purely Analytical Proof, and its method of proof became a hot in the Weierstrass school and in discussions with Kronecker. On the other hand, Cantor and Dedekind exchanged their views about Bolzano’s ideas of the infinite. Bolzano’s logic, expounded in detail in the Theory of Science, remained for a longer time unnoticed and misunderstood. Except for Hussserl, who read and annotated all four volumes, most of the readers did not go further than the first volume, while the most important innovations can be found only in the second volume.

2. THE THEORY OF SCIENCE Why does Bolzano treat logic in the framework of the theory of science? Why does sci- ence become the subject of another science at a certain moment of its development? From Descartes to Kant and German , philosophy was essentially a study of the human , of its capacities, its limitations, and its products. Among those products, one is directed especially towards , towards systematic exploration of among true propositions about world. How do we acquire them? How do we connect them in order to form a well-organized whole? The answer is in the exami- nation of the logical structure of theories, and this is why logic is at the heart of the Bolzanian venture. Against Kant, who considered logic as completed and finished, Bolzano showed its progress in modern times due to Bacon, Leibniz, and Condillac; he could also add his own theories. He renovated the Leibnizian style of thinking which so strongly contrasts with that of German idealism. Leibniz belongs to the analytic tra- dition, which was interrupted by Kant and his followers. Bolzano is at the origin of the analytic stream which has lasted without interruption up to our times. In the first half of the nineteenth century, he worked alone in an atmosphere hardly favorable to formal logic; his intentions and in his realizations were not understood by his contem- poraries. Even Husserl declared that he would not have understood Bolzano’s logic if he had not studied contemporary logical calculi. After a hundred years of slow growth if not stagnation, with Bolzano logic became a central philosophical discipline again, and the clarification of concepts and propositions one of the main tasks of philosophy. For the first time since Leibniz, philosophy was wedded to creative scientific work, above all in mathematics. Let us take a closer look at the four beautiful volumes of his Theory of Science.

3. THE STRUCTURE OF THE THEORY OF SCIENCE (§15) The introduction begins with the definitions of science and the theory of science and of their justification. The first is that of the sum of all human knowledge, but it 432 • Philosophia Mathematica is too large for the capacities of a single man. We are thus compelled to divide this domain into ‘several areas, and then to take the most noteworthy of their in each area and compile them in special books in the most comprehensible and convincing manner possible’ (§1). Bolzano defines science in the proper objective sense as ‘a col- lectionoftruthsofa certainkind[...],ifwhatisknownofitisimportantenoughto be set forth in a special book called treatise of that science’ (ibid.). Here an impor- tant division immediately appears between the objective truths (called also truths in Downloaded from https://academic.oup.com/philmat/article/23/3/428/1449457 by guest on 30 September 2021 themselves, in German Wahrheiten an sich) of a science and its known truths, which occupies Part I of the ‘Theory of fundamentals’. Thinking of the sciences, Bolzano has in mind their presentations in textbooks. This idea guides him through the whole book, so that by definition, theory of science ‘is the science which teaches us how to present other sciences (actually only their treatises). Thus, by theory of science I mean the collectionofallruleswemustfollow[...]whenwedividethetotaldomainoftruths into individual sciences, and present them in their respective treatises’ (ibid.). Husserl found that definition strange; but we have to remember Bolzano’s method of defining a discipline by its ultimate goal. The goal of logic (in the largest sense) is to give rules for the classification of sciences and the rules of writing scientific textbooks. Bolzano embraces the encyclopedic ideas of Comenius, Leibniz, and of the Grande Encyclopédie of d’Alembert and Diderot, imagining a series of scientific textbooks that would bring scientific progress to all students and to the public interested in science. The rest of the introduction is devoted to traditional matters such as the uses of logic, whether it is an art or a science, whether it is a purely formal science, and whether it depends on other sciences. The goal of logic ‘is to teach us rules by which our knowledge can be orga- nized into a scientific whole’ (§13). Logic must therefore teach us how to find truths and how to discover error, which implies ‘reference to the faculty of representation, to memory, to association of ideas, imagination’ (ibid.). But all this is contained in empir- ical psychology. Logic is therefore dependent on psychology. One should immediately add that these subjects are treated in the section ‘Theory of knowledge’ (epistemol- ogy), which follows after the formal logic whose laws and rules consist in objective truths. Formal logic (‘Theory of elements’) is thus independent of psychology. In the ‘Fundamentals’ (Book One, Volume I),1 we learn essentially Bolzano’s ontol- ogy of logic: what Bolzano means by truth in itself or objective truth (true in itself), that there is an actual infinity of them (the proof is not really convincing), and that we can know a (potential) infinity of them. ‘The theory of elements’, the most important part of the whole work, contains logic proper (Book Two, Volumes I and II), namely:

1. the properties of ideas in themselves, the distinction between intuitions and concepts, the concept of objectless (empty) ideas, and the extensional logic of ideas in themselves (logic of classes) and of their relations (Volume I): subordination, inclusion, intersection, equivalence, opposition, negation, etc., and other topics, e.g., the concepts of space and time (§79), the

1The work is divided into five books by the author but is physically divided into four volumes of the same size. Critical Studies/Book Reviews • 433

of the difference between equality and (§91), the concepts of collection, of multitude (Menge), of sum, unity, multiplicity and totality, the concept of magnitude both finite and infinite (§§82–87), and the first treatment of infinite collections (§§100–102); 2. the core of the theory of science, namely the complete system of extensional relations between propositions in themselves (Volume II, §§164–168) with

the same headings as for classes except for deducibility, which replaces Downloaded from https://academic.oup.com/philmat/article/23/3/428/1449457 by guest on 30 September 2021 inclusion, and logic of probability. Before, he explicitly defined the important concept of propositional form (Satzform, §147), which had been used by logicians since but which nobody except Leibniz tried to explain. Following the interpretation of certain linguistic forms, the concept of grounding (the ground-consequence relation holding only among true propositions; §221 treats the criteria of the relations of dependence between truths), further the theory of axiomatic systems under the heading ‘Of true propositions’, the theory of inferences (enlarged syllogistic and some Boolean forms), and a large Appendix about the ‘Previous treatments of the subject of this part’.

The organisation of knowledge presupposes that we are able to know and to dis- cover it; this is why the ‘Theory of elements’ leads to the ‘Theory of knowledge’ and to the ‘Art of discovery’ (Volume III). The ‘Theory of knowledge’ (Book Three) explains the notions of subjective idea and ‘how we arrive at our most general judgments of ’ (§303). Here Bolzano presents the scheme of the logical construction of the world, starting with tempo- ral relations (immediate recognition of an idea, simultaneity of ideas, temporal order of ideas, etc.), moving on to spatial relations, solidity, movement, shape and size, etc. Bolzano also analyses traditional notions such as knowledge and its limits, ignorance and error, and confidence, and improves their definitions. In Book Four, ‘Art of discovery’ or heuristic, various traditional rules are assembled. This is the least-known part of the Theory of Science. As in the whole work, asterisks indicate the most important sections. I recommend especially §§322–329 (concept of reflection; determination of the truth we actually seek; direct and indirect methods) and §332 (testing the truth of a given proposition). Book Five (Volume IV) is wholly dedicated to the art of writing scientific textbooks, i.e., to the theory of science proper. It is governed by the highest : ‘One must proceed here in such a way that, in addition to the originally specified end, namely of the Good, as many goods (i.e., ends indicated by the moral law) as possible may also be achieved’ (§395). Applied to the theory of science proper, this highest principle becomes: ‘In dividing the entirety of truth into individual sciences and in presenting these sciences in special treatises, everything must be done in the way required by the laws of morality, thus in such a way that the greatest possible sum of good (the great- est possible promotion of the general well-being is thereby produced’ (ibid.). This is simply the application of Bolzano’s highest moral law to the theory of science proper. In addition to instructions about the determination of the domain of the sci- ences, about the of a class of readers, about the kinds of propositions which should appear in a treatise, etc., other sections of Volume IV complete the exposition 434 • Philosophia Mathematica of logic from Volume II. Here Bolzano writes on basic propositions or (axioms, §483–490) and determinations. Especially important are the sections on proofs (§§512–537), on definitions (§§554–559), on the important concept of objec- tive connection of truths (§§576–578) which governs Bolzano’s conception of an axiomatic system, and on the order of presentation of the propositions in a treatise. Part VII (§§637–648 and further sections) contains Bolzano’s important theory of signs and of language. The last section of the whole work is dedicated to the devastating Downloaded from https://academic.oup.com/philmat/article/23/3/428/1449457 by guest on 30 September 2021 criticism of the Hegelian dialectical method. The whole book is written, according to its complete title, ‘with constant attention to earlier authors’, which makes it a unique, irreplaceable source for the and philosophy, especially with respect to Kant. Clever students will find in Bolzano enough objections to Kant to put their professors on the spot. What are Bolzano’s merits? It is true, he had no symbolic language and never con- sidered logic as a calculus. Nevertheless, he largely anticipated logical semantics. The most original part of his work is to be found in the first two volumes. He delimited the domain of formal logic by separating it from psychology and theory of knowledge (the latter belongs to the theory of science as one of its disciplines, but comes after for- mal logic). He defined logic not as a theory of thought, but as an objective theory of propositions in themselves. He explained fundamental concepts of propositional form and of validity. He created an original logical system that has no equivalent in the past. Moreover, he anticipated relevance logic and Gentzen’s method of natural deduction. To quote Jean Cavaillès, his theory of science is ‘an of his own, original in its and autonomous in its movement’. We have also to take into account the pedagogical aspect of his works. In logic as well as in mathematics, he takes the reader by the hand and guides one through the most obscure corners of thought. Bolzano clarified hundreds of concepts — not only the logical ones — often disguised by other authors under learned and unintelligible jargon.

4. HOW TO READ THE THEORY OF SCIENCE ‘Take it and read’, says the logician . Begin with the first sections, then go the §42 to read the delicious dialogue with a skeptic in order to appreciate the subtlety of Bolzano’s argumentation. His phrases are often long, reminiscent of Latin syntax, but his terms are defined with extreme precision and sentences almost always have a unique , as opposed to Kant’s or Hegel’s. Bolzano is the heir of Aris- totle, Leibniz, and of what is the best in scholasticism. He explains every concept, and rather twice than only once. Follow his method: confront the concept of proposition in itself with the objections of Exner, Quine, and Patocka.˘ Study his logical system with a pencil in your hand, learn by heart the most important passages, e.g., some parts of §147 about validity, §148 about analytic and synthetic propositions, §223 about purely logical deduction and ‘impure’ deduction. A digression in §483 offers an explanation of pantheism in relation to Spinoza and Hegel, an example of Bolzano’s treatment of religious matters. Take a glance at the diagrams showing the relation of ground and consequence among truths. It is preferable to read him in a small group of students, to read him slowly, to keep in mind and to meditate on his definitions, to assimilate the essential concepts, to verify his examples and to construct counter-examples, to test his arguments. Those who Critical Studies/Book Reviews • 435 succeed in mastering Bolzano’s method of thinking will have an exceptional analyti- cal tool at their disposal. I know no better introduction to philosophy than Bolzano’s Theory of Science.‘Itisthebestthatwehave’(PerMartin-Löf).

REFERENCES Bolzano, Bernard [2004]: On the Mathematical Method and Correspondence with Exner. Paul Rusnock Downloaded from https://academic.oup.com/philmat/article/23/3/428/1449457 by guest on 30 September 2021 and Rolf George, trans. Amsterdam and New York: Rodopi. ———[2011]: Premiers écrits — Philosophie, logique, mathématique. Paris: Vrin. Behboud, Ali [2000]: Bolzano’s Beiträge zur Mathematik und ihrer Philosophie. Bern: Bern Studies in the History and . Berg, Jan [2008]: Bolzano’s Logic. Stockholm: Almqvist & Viksell. Cavaillès, Jean [2008]: Sur la logique et la théorie de la science. Paris: Vrin. Husserl, Edmund [2001]: Logical Investigations I–II. J.N. Findley, trans. London: . Konzelmann Ziv, Anita [2010]: Kräfte, Wahrscheinlichkeit und ‘Zuversicht’. Sankt Augustin: Academia. Künne, Wolfgang [2008]: Versuche über Bolzano/Essays on Bolzano. Sankt Augustin: Academia. Künne, W., M. Siebel, anf M. Textor, eds [1997]: Bolzano and .Amsterdam: Rodopi. And Grazer philosophische Studien 53, 1–266 (whole volume). Morscher, Edgar [1973]: Das Logische An-Sich bei Bernard Bolzano. Salzburg and München: Verlag Anton Pustet. Roski, Stefan [2014]: Bolzano’s of Grounding and the Classical Model of Science. Zutphen: CPI Wöhrmann Printservice. Rumberg, A. [2013]: ‘Bolzano’s concept of grounding (Abfolge) against the background of normal proofs’, Review of Symbolic Logic 6, 424–459. Rusnock, Paul [2000]: Bolzano’s Philosophy and the Emergence of Modern Mathematics.Amsterdam and New York: Rodopi. Rusnock, Paul, and Rolf George [2004]: ‘Bolzano as logician’, in D.M. Gabbay and J. Woods, eds, Handbook of the History of Logic. Vol. 3, pp. 177–205. Amsterdam: Elsevier. Russ, Steve [2004]: The Mathematical Works of Bernard Bolzano. Oxford: Oxford University Press. Sebestik, Jan [1992]: Logique et mathématique chez Bernard Bolzano. Paris: Vrin. ———[2011]: ‘Bolzano’s logic’, The Stanford Encyclopedia of Philosophy. ———[2013]: Review of Sandra Lapointe, Bolzano’s Theoretical Philosophy: An Introduction, Journal for the History of Analytical Philosophy 2, No. 2, 1–8. Siebel, Mark [1996]: Der Begriff der Ableitbarkeit bei Bolzano. Sankt Augustin: Academia. Textor, Mark [1996]: Bolzanos Propositionalismus. Berlin and New York: Walter de Gruyter. doi:10.1093/philmat/nkv001 Advance Access publication February 19, 2015

Pavel Pudlák. Logical Foundations of Mathematics and Computational Complex- ity: A Gentle Introduction. Springer Monographs in Mathematics. Springer, 2013. ISBN: 978-3-319-00118-0 (hbk); 978-3-319-00119-7 (ebook). Pp. xiv + 695.

Reviewed by Alasdair Urquhart ∗

This monograph by the outstanding Czech logician Pavel Pudlák provides a broad but also deep survey of work in logic and computer science relevant to foundational issues,

†Department of Philosophy, University of Toronto, Toronto, Ontario M5R 2M8, Canada. E-mail: [email protected]