Wednesday, 13 February 2013

Review of Mancosu's 'The Adventure of Reason'

I've been asked to review Paolo Mancosu's book The Adventure of Reason -- Interplay between Philosophy of Mathematics and Mathematical Logic, 1900-1940 (OUP, 2010) for Mind. What follows is a draft of my review, so comments and suggestions are most welcome!

The first half of the 20th century witnessed the birth and flourishing of a radically novel subdiscipline: mathematical logic. While mathematics and logic had been on friendly terms at least since the 17th century (Mugnai 2011), and while there were 19th century precursors to the idea of applying mathematics to logic (e.g. Boole) and logic to mathematics (e.g. Frege), it is only in the first half of the 20th century that mathematical logic became a fully mature subdiscipline/research program.

The Adventure of Reason offers a compelling narrative of this exciting chapter of the history of logic and mathematics. Paolo Mancosu is one of the world’s leading experts on these developments, and while many others have made important contributions to the topic, it is fair to say that Mancosu has gone a step beyond with his painstaking work on sources other than the canonical, printed versions of articles and books. He has done extensive research at a number of archives both in Europe and North America, examining documents such as letters, minutes of meetings, informal reports, unpublished transcriptions of lectures – in short, unpublished material of various sorts. This approach has enabled him to produce new insights and at times to provide novel answers to interpretive open questions concerning some of the towering figures in this tradition (such as the controversy on whether Tarski did or did not accept domain variation in his definition of logical consequence – chapter 16).

The book essentially consists of re-prints of several of Mancosu’s articles (some of them co-authored pieces) on a wide range of topics and authors, but all within the framework of the emergence of mathematical logic in the 20th century and its tight connections with philosophical discussions on the nature of mathematics (as the subtitle of the book indicates). It is divided into five parts: Part I is composed of a long chapter surveying the developments from 1900 to 1935; Part II focuses on the foundations of mathematics program, in particular Russell, Hilbert, Bernays and Gödel; Part III is less ‘mainstream’ and is dedicated to the contacts between phenomenology and the philosophy of the exact sciences, pioneered by Husserl and pursued by less well-known figures such as Weyl, Becker and Mahnke; Part IV is rather short and concerns Tarski and Quine on nominalism; Part V focuses on Tarski and the Vienna Circle on truth and logical consequence, and includes a hitherto unpublished lecture by Tarski on completeness and categoricity (more on which below).

Among the many innovations introduced by mathematical logicians in the first half of the 20th century, the emergence of formal systems stands out: these are collections of axioms and rules of transformation, typically formulated in a specially designed language (a formal language), which would allow for the precise investigation of whole portions of mathematics by means of axiomatizations/formalizations. Starting with the system presented in Whitehead and Russell’s 1910 Principia Mathematica (which in turn was heavily inspired by Frege’s Begriffsschrift and by the notation introduced by Peano), there was a crescendo of enthusiasm and optimism concerning what could be accomplished with this novel tool, which then culminated in Hilbert’s program in the 1920s (the topic of Part II). The basic idea was that meta-mathematical questions such as the consistency of arithmetic would become mathematical questions once adequately formulated within a convenient formal system, and could thus be investigated and settled by purely mathematical means.

At first, it seemed that these powerful tools, formal systems, offered almost limitless possibilities for the investigation of such foundational issues. (Hilbert: “We must know. We will know.”) But it also became increasingly clear that the desired meta-properties of these systems, in particular in view of the goal of describing portions of mathematics completely (in different senses of ‘completely’ – see chapter 1 of the book and Awodey and Reck 2002), could not be taken for granted. A seemingly well-designed, plausible collection of axioms, say the axioms of Peano arithmetic, could still fail to deliver a complete description of the mathematical theory in question. So the properties of these very systems also had to be investigated in a systematic way – the meta-meta-level of investigation, so to speak. Now, once thoroughly formalized, formal systems become mathematical objects themselves, thus allowing for the application of the same general methodology. So one remarkable feature of mathematical logic in this period is that it set out to establish its own limits by means of the very methodology it had inaugurated.

As is well known, the most astonishing limitative results were proved by Gödel, who in the early 1930s showed (the First Incompleteness Theorem) that any consistent system containing arithmetic (in a suitable sense of ‘containing’) is inherently incomplete in that there is a sentence which can neither be proved nor refuted by the system, though true according to the standard model of arithmetic. The Second Incompleteness Theorem established that Hilbert’s dream of proving the consistency of arithmetic within arithmetic itself could not be realized. And while it became generally accepted that Gödel’s results represented a fatal blow to Hilbert’s program, this by no means meant the end of mathematical logic as a research program. In fact, it seemed to count as a victory rather than as a failure, as the methodology was thus shown to be able to investigate and delineate its own limits. The emergence of model-theory with Tarski’s work on truth and logical consequence (Part V) was also a response to Gödel’s results and inaugurated a new approach in mathematical logic, alongside the proof-theoretical approach originally envisaged by Hilbert and collaborators.

Ultimately (and as noted by M. Potter in his review of the book in Philosophia Mathematica, 2012), Mancosu does not (and does not intend to) offer a radically novel picture of the development of mathematical logic in the first half of the 20th century. The broad lines of his narrative are very much the ones largely agreed-upon in the literature. But starting from the general picture, he fills in the gaps, thus offering a very detailed account of these developments. Moreover, Mancosu is to be praised for bringing to the fore the contribution of less well-known figures such as Becker, Behmann and others, and for his keen eye for how these developments unfolded historically. His is a narrative focusing on actors and processes, not only on the results of their endeavors (theories, theorems etc.). (One of my favorite chapters in the volume is his piece on the immediate reception of Gödel’s incompleteness theorems (chapter 7), which describes how, after initial shock and some skepticism, consensus emerged concerning the correctness of the results and their wide-ranging implications.)

One may wonder what justifies the publication of a collection of previously published papers, which moreover have not been substantially altered for their inclusion in this volume. Now, the book does contain updated bibliographical lists of works on the topics that have appeared since the original publication of the articles. It also includes helpful summaries for each of its five parts, which thus make manifest that the different individual articles are all part of a larger, general project. At any rate, the fact that these articles are all conveniently assembled in just one volume represents a useful resource for all those interested in these topics.

It must be said, however, that the book is not exactly a page-turner, which may well be due to the fact that, in the end, it remains a collection of articles rather than a book conceived as such. In fact, it could (but need not be) viewed as a reference work, with its impressive collection of ‘facts and figures’ (accompanied by a rather detailed index of names, but alas no index of terms). It is for the most part also a very useful pedagogical resource, where some difficult concepts and results are explained in an accessible way (especially in the long survey in chapter 1).

To close this review, let me briefly discuss the two thus far unpublished chapters of the book, namely a transcription of Tarski’s lecture “On the Completeness and Categoricity of Deductive Systems”, and Mancosu’s own discussion of the lecture and its content (chapters 18 and 17, respectively, the last two chapters of the book). Completeness and categoricity are both crucial desiderata for formal systems, and are in a sense each other’s duals; while completeness ensures that all the relevant facts about a given portion of mathematics can be captured by a deductive system, categoricity ensures that only these relevant facts are captured by the system, i.e. that it is not also a description of structures falling outside the targeted mathematical domain/theory (‘alien intruders’, in Dedekind’s terms). When completeness fails, not enough fish are caught; when categoricity fails, too many fish are caught.

In his lecture, Tarski remarks that, in view of Gödel’s results, absolute completeness is a rare phenomenon; there are few deductive theories that can be complete (i.e. only those not sufficiently expressive so as not to allow for a formalization of arithmetic). He then introduces two weaker concepts of completeness, namely relative completeness and semantic completeness (the interested reader will have to consult the text itself for details, as limitations of space prevent me from offering further technical elaborations), and asks himself what methods could be used to show a given system to be relatively/semantically complete. Lest one should despair, Tarski announces the grand news: “We shall see, namely, that the concepts of relative and semantical completeness are closely related to the concept of categoricity (due to Veblen) and the investigation of the latter concept does not require in general any special and subtle methodological investigations.” (p. 490) Tarski then proves two theorems relating (relative and semantic) categoricity to (relative and semantic) completeness, and in view of the abundance of categorical systems, he concludes: “in opposition to absolute completeness, relative or semantical completeness occurs as a common phenomenon.” (p. 492) We thus have yet another display of Tarski’s uncanny ability to convey difficult technical concepts in an accessible way, while at the same time drawing sophisticated general conclusions from the results he discusses. A great way to end the book, and accordingly an appropriate way to end this review.


  1. Nice review, Catarina. Two small points:

    6th par: What Gödel proved is that if T is omega-consistent then there is a sentence G which is neither provable nor refutable. Simple consistency of T is sufficient to see that G is not provable; omega-consistency is required to see that G is not refutable. The strengthening to simple consistency is due to Rosser (using a different sentence).

    In the last par., probably 'interpretation' is more precise characterization than 'formalization'.

    1. Thanks, Aldo! You are of course right, and it never hurts to be as precise as one can on such matters. I'll take care of these points for the final version.

  2. I think it's been common practice in English-speaking philosophy to collect published articles by major philosophers in collections, see eg many of Quine's books (Ways of Paradox, From a Logical Point of View, etc). Oxford has been doing this extensively recently. So I don't think Paolo should be blamed for that.

    1. I didn't mean to suggest there is anything wrong with publishing collections of previously published articles, and certainly not in this case (as I said, it's extremely useful to have them all bundled together). The only point I was making is that after reading one chapter, one does not feel compelled to start on the next chapter right away.

  3. Göran Sundholm28 May 2013 19:52

    An interesting read ,but I have some cavils.
    (1) How was Principia "heavily influenced" by Frege? It is a formal language, but one that is patterned on Peano's Formulairo. Its axiomatization is different: we have types, and the "assertion-sign" is not built up from two parts. In Frege the truth-values play a totally different role from PM. Frege defines propositional truth via truth-values. A true Gedanke is a sense that determines Das Wahre. Whitehead and Russell go in the opposite direction and start the all-pervasive tradition where truth-values are defined from truth and falsity. So where is the "heavy influence"?
    (2) The works of Gödel and Tarski ±1930, and their codification by Paul Bernays, served to alter the logical paradigm. "Object languages" are no longer epistemological tools for foundational research. They are turned into objects of study. One cannot speak "object language"; one can only speak about it. You go too far, I think, at least as a description of how things are done in actual research. "Thoroughly formalized" formal systems become "mathematical objects themselves", it is true, which allows for the application of "the same general methodology". Here I cavil. The methodological study of these systems is not carried out by means of a FORMALIZED meta-theory: that would be hellish. The ONLY treatise known to me using a "thoroughly formalized" meta-theory is Heinrich Scholz's posthumous masterpiece Grundzüge der mathematischen Logik. Like Church's Introduction to Mathematical Logic it contains an engrossing philosophical introduction that still after fifty years offers the best discussion of second-order Platonism versus Constructivism in logic and the foundations of mathematics. The book is almost unreadable owing to its remorseless formalization. As the enduring, practice-based, methodological precept has it: study formalizations as mathematical objects, but do not formalize the study unless necessary.

  4. Göran Sundholm28 May 2013 19:54

    (3) Your peroration praises Tarski for his "uncanny ability to convey difficult technical concepts" and "drawing sophisticated general conclusions". I do not see that on display here. Trivially the theory, whether first- or second-order, of any relation structure is semantically ("negation"-) complete. There, with respect to complete theories, be they axiomatizable or not, we can accordingly speak truly of "an abundance". The harvest of categorical theories on the other hand, is by comparison singularly meagre, and hardly constitutes "an abundance". By their Hintikka (or Scott) sentences whichever terminology one prefers, finite structures may have categorical as well as axiomatizable first-order theories. Interesting first-order theories (with infinite models) are never categorical, owing the Löwenheim-Skolem theorems. They may be complete, as Tarski showed in a number of cases by means of successful quantifier-eliminations, for instance regarding the theory of real closed fields. In the matching "full" second-order cases we only have a formal language, but NO FORMAL SYSTEM, since categoricity together with Tarski's theorem on arithmetical truth rules that out. The "deviant" concepts Tarski discusses have, as far as I know, played little role in the development of logic, whereas categoricity and negation-completeness are crucial. So why the abundant praise for the Master?
    On the contrary, Paolo deserves a measure of praise for having sorted out what Tarski was doing here. Logical theory coalesced into something reasonably firm only after WWII. Tarski was one of the main architects of that development. No doubt previous musings were crucial in getting things right later. Two decades after Van der Waerden's Moderne Algebra, and with Bourbaki getting fully into his stride, mathematicians were ready to accept structuralism, and their readiness for satisfaction was rewarded model-theoretically. By comparison, the Tarski of the Polish period appears much more outré than we have good reason to think of him today.