BOOK REVIEW

Javier Echeverria, Andoni Ibarra, and Thomas Mormann (eds.), The Space

of Mathematics. Philosophical, Epistemological, and Historical Explorations, Walter de Gruyter, Berlin/New York, 1992, xvi + 422 pp., DM 228.00 (cloth). This book is the main outcome in print of the symposium on Structures in Mathematical Theories held at the University of the Basque Country in Donostia/San Sebastian, Spain, in September 1990. It was a huge conference, attended by more than two hundred people, mainly mathematicians, philosophers, and historians. The thirty-five invited and seventy-five contributed papers covered a big number of topics related to philosophy and history of mathematics. (There is available also a volume of contributed papers: Structures in Mathematical Theories. Reports of the San Sebastian International Symposium, ed. A. Diez, J. Echeverria~ A. Ibarra~ Bilbao (Spain), University of the Basque Country Press, 1990.) The twenty-three papers were chosen by the editors so as to form a conference volume with a reasonable coherence. (The thematic spectrum of the conference itself was considerably broader.) The editors formed four groups of papers under the headings

1. 2. 3. 4.

Structural Dimensions, Dimensions of Applicabili~', Historical Dimensions, Global Dimensions of Knowledge: Information, Implementation, and Intertheoretic Relations.

Of course, this division is arbitrary to a certain degree. In their Introducto~ afterthoughts" the editors claim that there is some "New" philosophy of mathematics on the way, the pattern of which is still difficult to visualize (p. IX). An important trait of this gestalt is seen in the "concern with historical and sociological aspects of mathematical cognition" (p. XI). The traditional schools of Logicism, Formalism, and Intuitionism, all adhering to the foundational task, "have turned out to be dead ends" (p. XII), while the contributions to the present volume would belong to what, still in 1988, has been called the "maverick tradition" by Aspray and Kitcher (ibid.). Erkennmis 4$: 119-122, 1996. (~ 1996 Kluwer Academic Publishers. Printed in the Netherlands,

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Several contributions deal with what mathematics is about or traditionally speaking with the ontolog), of mathematics. One fashionable answer seems to be the seemingly anti-ontological one: there is no mathematical subject matter at all! Thus Saunders MacLane, in the opening contribution to the first part, argues for what he calls the protean character of mathematics, i.e. that "one and the same mathematical structure has many different empirical realizations" (p. 3). If one does not take the qualification "empirical" too seriously (as MacLane himself evidently does not), tiffs is a well-known trait of mathematics, impressingly documented by MacLane with a diversity of examples. Somehow hastily he concludes "that mathematics does not need a 'Foundation' ", because "mathematics is not about things but about form" (p. 9). Thus he at least keeps the idea that there is something, if not a thing, what mathematics is about! -- I only mention in passing F. William Lawvere's Categories of Space and oJ'OuanliO' with its Hegelian outlook. The third paper, by Andoni Ibarra and Thomas Mormann, stresses Structural Analogies Between Mathematical and Empirical Theories, as opposed to gakatos's methodological and Quine's functional ones. These structural analogies are said to include such moves as swings (Margenau's term) or "hermeneutic circles" between data and symbolic constructions (p. 38), as is illustrated for Group Theory {pp. 40f0. In the next paper, Veikko Rantala compares science and mathematics with respect to reduction and explanation. Also Ilkka Niiniluoto (fifth paper) refers to notions of the philosophy of (empirical) science when reflecting on the "quasi-empiricist" (kakatos) programme. Next is Herbert Breger's paper Tacit Knowledge in Mathematical Theo~?'. He differentiates between various types of "'tacit knowledge" (Polanyi) or "'know-how'" as opposed to "know-that" (H. and St. Dreyfus), including knowledge of relevance, pre-axiomatic knowledge, and knowledge regarding "naturality'2 Interesting examples are taken from the development of algebraic topology, Descartes's analytic geometry, Fermat's extreme-value problems, Felix Klein's grlangen programme, the theory of categories, and other fields. I think one could add the knowledge which one holds back while strictly operating within Hilbert's formalist programme. In the first part's last paper, lvor Grattan-Guinness takes Structure-similarity as a Cornerstone oj" the Philosophy of Mathematics. After a range of case-studies he addresses the question o f " h o w this philosophy of mathematics ... means" (p. 108), The second section is opened by Michael Resnik's paper on the Indispensability Argument of Quine and Putnam, as opposed to Hartry Field's Science Without Numbers. Resnik argues that "the approaches that appear to undercut the Quine~putnam argument ... in so far that each succeeds it still presupposes substantial commitments to mathematical objects and

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principles" (p. 116). Included in his discussion are views of how mathematics applies in empirical science by N. Cartwright and H. Kyburg. Especially devoted to mathematical physics is the next essay, Roberto Torretti's Mathematical Structures and Physical Necessi~. So is Erhard Scheibe's The Role of Mathematics in Physical Science, although in a more general way: he is mainly concerned with the "miracle" that Wigner has called "the unreasonable effectiveness of mathematics in the natural sciences" (p. 141 ). By a careful analysis- using, inter alia, set theoretic devices for the reconstruction of physical theories - he attempts to show "that we really have reasons to marvel at that effectiveness" (after blaming various attempts to explain this effectiveness as "not ... very successful") (p. 153t"). The conclusion he draws is: "The overladenness of physics with mathematics is strong evidence that physical theories are non-conservatively embedded in mathematics" (p. 155).- In a similar vein evidently, Heinz-Jfirgen Schmidt analyzes The Status of Set-theoretic Axioms in Empirical Theories. Like Scheibe he refers to set theoretic conceptions of physical theory (Ludwig, Suppes, Sneed, etc.). So do Newton C. A. da Costa and F. Antonio Doria in their contribution Suppes Predicates for Classical Physics. Their main focus, however, is the transfer of metamathematical undecidability and incompleteness results to set theoretic axiomatizations of classical physical theories. In the section's last contribution, Colin Howson reflects the role of Mathematics in Philosophy, more specifically: in (empiricist) epistemology. The third, historical, part starts with Joseph W. Dauben's Are there Revolutions in Mathematics?, a question denied by M. J. Crowe, but answered by Dauben with "an emphatic 'yes' " (p. 229). Indeed, Dauben gathers impressive examples of radical changes from, inter alia, the development of the calculus, Cauchy's rigorization of analysis, and A. Robinson's nonstandard analysis. But one may reasonably doubt that these examples fit the concept of scientific revolution as used in the philosophy and history of (empirical) science. Dauben admits that "revolutions in mathematics may not involve crisis" (p. 228). If this seems all right one still may wonder why some discoveries should "inevitably lead to revolutionary new theories" (ibid., my italics). - In a second historical paper Javier Echeverria questions the a priori character of Number Theoo'. In a third one Eberhard Knobloch discusses Historical Aspects of the Foundations of Error Theo~'. Hans Niels Jahnke gives A Structuralist View of Lagrange 's Algebraic Analysis and the German Combinatorial School, and Michael Otte's paper focuses on Constructivism and Objects of Mathematical Theoo'. The recent development of recursion theo O' is the topic chosen by Solomon Feferman. He makes "the case for the prima~ significance for practice of

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the various notions of relative (rather than absolute) computability, but not of most methods or results obtained thereto in recursion theory" (p. 315). The last historical contribution is Michael S. Mahoney's Computers' and

Mathematics: The Search for a Discipline ofiComputer Science. The shortest and yet least coherent part of the book is the fourth and last one, as is already indicated by its somewhat clumsy title. Jesus Mosterin discusses some mathematical aspects of InJormation. In his essay Structuralism and Scientific Discove~ Joseph D. Sneed applies the idea of "a procedure for discovering scientific theories" by "genetic modification" (referring to D. E. Goldberg and J. H. Holland)"to a 'population' ofstmcturalist representations of scientific theories" (p. 379). Last, but not least, C. Ulises Moulines sketches a Typology of Intertkeoretical Relations "based on the structuralist metatheory of empirical science" (p. 403). Although a "wealth ofintenheoretical relations" has been studied over the last decades, it "has not been investigated from a unifying, systematic point of view" (p. 404). This is done, in the present paper, by proposing the concept of intertheoretical link as "a fundamental unit of relationship" (ibid.). For a reviewer the incoherence of the whole book is overwhelming: no chance to evaluate the various contributions with respect to a common field of inquiry. However, the specific value of the collection may well lie just in its diversity, thereby demonstrating the richness of what can be labelled only metaphorically: The Space of Mathematics. Manuscript receivedOctober 12, 1995 Philosophisches Seminar Universit/itHamburg Von-Melle-Park6 20146 Hamburg Germany

WERNER DIEDERICH

BOOK REVIEW

Barry Smith, Austrian Philosophy. The Legacy of Franz Brentano, Open Court Publishing Company, Chicago and La Salle (Illinois), 1994, 381 pp., $36.95 (cloth). The theme around which the 10 chapters of this book revolve is the uncovering of a distinctive pattern exhibited in the writings of thinkers, either born in Austria or active at Universities of the Habsburg Empire between 1860 and 1914, whose interests ranged from psychology to linguistics, from aesthetics to value theory and economics. According to Barry Smith, the salient features of this pattern are to be traced back to the philosophy of Franz Brentano: and it is this specific sense in which the cross-section of "Austrian philosophy" displayed by Smith's impressive book can be seen as the legacy ofFranz Brentano. Indeed, it is Smith's conviction that, although much work has been done on certain eminent Austrians, such as Mach, Wittgenstein and some members of the Vienna Circle, "the central axis" of Austrian philosophy is constituted by the work of Brentano and his school (above all Meinong and Twardowski), and that this part is still poorly understood (p. 2). It was Franz Brentano who taught his pupils a new way of asking questions, thus creating a matrix which could be employed also by those who did not share his specific theses, nor his fervent commitment to Aristotelian doctrines. Brentano's Aristotelianism was decisively influenced by F. A. Trendelenburg, who, as Smith points out (p. 326), had also been a teacher of Dilthey, Kierkegaard, Feuerbach and Marx. It may come as a surprise to some readers, that the roots of Austrian philosophy go back to the work of a Rheinliinder who learned his Aristotle from Trendelenburg and taught at the University of Wfirzburg until 1873. It is arguable, however, that the new way of philosophizing initiated by Brentano could find a better terrain at Austrian Universities. which had remained relatively untouched by the philosophy of German Idealism, and where the tradition of scholastic philosophy fostered by the Catholic Church had solid roots. What bitter fruits such roots might bear was experienced by Bernard Bolzano in his own time and place (Prague) and then by Brentano himself. Erkenntnis 45: 123-127, 1996. (~) 1996 Kluwer Academic Publishers. Printed in the Netherlands.

124 Brentano's best-known pupils were Carl Stumpf, Edmund Husserl and Alexius Meinong, but only the latter is relevant to Austrian philosophy proper. However, it is not Meinong himself who occupies the center of the stage in Smith's book, but rather Meinong's co-workers (e.g. Christian von Ehrenfels, Stephan Witasek, Vittorio Benussi) on the one hand, and Anton Marry and Kasimir Twardowski on the other. Through the latter, the influence of Brentano's philosophy came to be felt in the Polish school of logic. Accordingly, Chapters 2 and 3 of Smith's book present the tenets of Brentano's doctrines of intentionality and his interpretation of Aristotle; this theme is further elaborated in Chapter 7 through a comparison between Kotarbinski's and Brentano's varieties ofreism. Chapters 4 and 5 deal, respectively, with Anton Marty's conception of truth and Witasek's contribution to the aesthetics of the Graz School; Chapters 6 and 7 are devoted to the discussion of Twardowski's book of 1894, Zur Lehre yore Inhalt und Gegenstand der l~btvvtellungen, and of Twardowski's influence on Polish logicians. Chapter 8 discusses Gestalt theories (Ehrenfels's conception of Gestalt qualities, Graz production theory and the Berlin School); Chapters 9 and 10 deal with Ehrenfels's conception of value and sketch the peculiarity of the subjectivist approach characteristic of Austrian economics. What are the tenets of Brentano's philosophy which can be seen to make it valuable even today? According to Smith, "the power of Brentano's philosophy consists precisely in having set forth a means by which psychology and ontology can develop in tandem with each other in fruitful ways" (p. 245); however, in order fully to appreciate the fruitfulness of this approach, a larger shift is needed, i.e. a shift from the early Brentano's immanentistic (or psychologistic) conception of judgement to an "objectivistic conception of propositions and states of affairs. This shift was effected on the one hand in the work of Frege. But it was effected independently by Brentano's disciples, involving in their case an hard-fought struggle for both ontological and philosophical clarification" (p. 185). An assumption which is taken fbr granted by Smith is that to do full justice to mental acts and their contents a bicategorial ontology is required (pp. 225---229), i.e. an ontology which contains states of affairs alongside objects. While Smith is somewhat suspicious of the profligate ontologies encouraged by a certain reading of Meinong's doctrines and has mixed feelings over Twardowski's positing of general objects (pp. 162-171), he is adamant that only an ontology which encompasses Sachverhalte can do full justice to the correspondence theory of truth (e.g.p. 201). Now, quite independently of slingshot arguments from G6del to Davidson and Barwise, many a reader would like to know (a) what would speak in favour of such an ontology (we all know plenty

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of reasons which speak against it), and, more to the point, (b) why this specific concern is supposed to be crucial to an understanding of Austrian philosophy. Except for Stumpf(pp. 86--88 and p. 186), none of the Austrian philosophers discussed by Smith gave much thought to this specific ontological enrichment. In fact, the concern with the notion of truth which is so prominent in the writings of Frege arises from his aim to provide a semantic account for the language of Grundgesetze, a concern which is nowhere to be found in Brentano or his pupils. Frege's ontology of saturated and unsaturated entities (individuals, events, truth-values, classes, numbers, on the one hand, functions of first and second level, on the other) is dictated by his new conception of semantics. Bolzano, who, before Frege, had been keenly interested in accounting for the way in which Vorstellungen an sich combine into Siitze an sich, viewed the latter as non-actual entities which are either true or false independently of acts of judgement in which they may be involved. They are not "made" true by anything, nor do they correspond (though some of their parts do) to anything: they are truths. The craving for an ontology of states of affairs is therefore a specific concern of Barry Smith, and not of Brentano and his pupils. This is why one feels that in Chapter 4 Anton Marry is not given his due, for he is insistently tapped for remarks on issues of truth and correspondence, which, although he did address them in his writings, were by no means at the center of his interests- let alone among his most valuable contributions to the philosophy of language. Many, following in the footsteps of his teacher, tackled the problem of impersonal judgements in accordance with Brentano's conception of the import of universal and particular judgements. Brentano's "existential" paraphrase of the contents of such judgements was distinctly adverse to an ontology of states of affairs; also in his later works, as Smith correctly points out (pp. 235-239), Brentano's ontology was never a dualist one. Now, whatever the value of this Brentano-style paraphrase may be for the study of acts and contents of presentations, it represents no advance from a logical point of view; and the same must be said of Bolzano's view that the logical form of all propositions can be paraphrased into an union of two Vorstellungen by means of the constant Vorstellung "to have" (Wissenschafislehre I, ~ 127, referred to by Smith on p. 190, footnote 81 ). This is perhaps one of the least interesting ideas of the whole Wissenschafislehre, as a moment's reflection on sentences with multiple generality shows. What Smith could possibly mean when he suggests (p. 190) that this part of Bolzano's doctrine is superior to Frege's is hard to make out. In Barry Smith's opinion the Aristotelian spirit which Brentano brought to philosophy is not only an essential feature of "Austrian philosophy"

126 (pp. 320-22), but goes hand in hand with the scientific spirit. In t:act, according to Smith, "[t]he relation of philosophy to matters of scientifically established fact is therefore not normally a subject lbr investigation in post-Kantian philosophy ( . . . ) there are exceptions: for example Bauch, Natorp and Cassirer" (p. 4). This is indeed an extraordinary statement to make, and it is belied, moreover, by Smith's own work: just to remain in the narrow field of psychology, it suffices to think of the Berlin school of Gestalt Psychology (discussed in Chapter 8, one of the best of this book), the Wfirzburg and Munich schools of Denkpsychologiepromoted by Oswald Kfilpe and Karl Biihler, the Fries X'cheSchule, revived by Leonard Nelson, Wundt's scientific and philosophical work - not to mention H. Lotze, C. Sigwart, T. Lipps, B. Erdmann. What Smith probably means is that the landscape of German philosophy is so rich and varied that the "scientific" or "positivistic" trend, though very substantial, never became mainstream philosophy. But then, neither Meinong nor Twardwoski ever became mainstream philosophy otherwise Barry Smith's excavations would not be so badly needed. Luckily, this judgement does not mar the substance of Smith's valuable book. It might, however, give the reader the impression that Austrian philosophers had little commerce with what was going on in the philosophical community across the border. Nothing could be further from the truth. Even a cursory reading of Twardowski's monograph of 1894, on which Smith lays much stress, shows how conversant the lbrmer was with the works of Ueberweg, Erdmann and Sigwart. Twardowski's work centers on an exquisitely Bolzanian question: Are all the parts of the content of a Vorstellung an sich also parts of the object which is represented by means of it, and if they are not Merkmale of the object in question, what other function do they fulfil? Very probably the immediate inspiration for asking such a curious question came to Twardowski both from his teacher Robert Zimmermann and from a series of eight articles Ober Anschauung und ihrepsychische Verarbeitung, published between 1885 and 1891 in the Vierteljahrsschr~ J~ir wissenschafiliche Philosophie, where the Austrian philosopher Benno Kerry discusses in detail not only Bolzano's, Cantor's, Kronecker's, etc. but also Frege's views on concept and object as put forward in his Grundlagen derArithmetik of 1884. Although in his monograph Twardowski refers to the article which Kerry specifically devoted to Frege, he apparently felt no need to read (or mention) Frege's book, nor Frege's article Uber Begriffund Gegenstand published in 1892 in the same journal as a reply to Kerry's criticisms. Predictably enough, Twardowski finds a source of inspiration in the logic books of Benno Erdmann and Christoph Sigwart: the former is credited with having anticipated the remarkable

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notion "unbestimmt bestimmter Gegenstand" (Twardowski 1894, p. 96 and p. 106) and the latter with having had a hunch that "wholeness" is an essential feature of presentations (ibidem, p. 92). (Incidentally, at p. 171 of Smith's book the reference to Twardowski's work should be to page 91, and not 88. At the page indicated by Smith there is a long quotation from Sigwart's Logik, which is indeed very similar in spirit and wording to the view which Twardowski will then put forward, and which he describes as a compromise between Erdmann's and Sigwart's.) Twardowski's lack of curiosity for Frege's work is easily understood: unlike Kerry, he at the time had little interest in the new work which was being done in logic and mathematics, and as regards the conceptual distinctions which he wanted to draw, he felt he could follow Bolzano's lead. With respect to Stephan Witasek one is naturally curious to know whether and how the notions of empathy and sympathy which he employed in his writings on music connect with those put forward by Theodor Lipps in his celebrated Jtsthetik and further elaborated in that little gem which is Wilhelm Worringer's book on Abstraktion und Einfuhlung of 1908. One feels that a better understanding could be gained by placing works such as Witasek's in the larger context of European culture to which they belonged and contributed. In sum, a lot of exciting work still remains to be done! It is one of the greatest assets of Barry Smith's book that it works as a powerful stimulus to know more and better. There can be no doubt that during the last 15 years Barry Smith has contributed like few others - notably Rudolf Haller, Kevin Mulligan and Peter Simons - with his articles, editions and translations to making the work of Austrian philosophers of the turn of the century available to a larger audience: the present book is an outstanding contribution to the subject. Manuscript receivedJanuary 17, 1996 Dipartimento di Filosofia Universit~tdi Bologna via Zamboni 38 140126 Bologna

EVA PICARDI