Agazzi’s Impact on the Philosophy of Mathematics and Logic

The first stage of Agazzi’s professional work was almost entirely devoted to logic, philosophy of logic and philosophy of mathematics. Only four years elapsed between Agazzi’s graduation in philosophy (1957) and the publication of his first book, Introduzione ai problemi dell’assiomatica (Introduction to the problems of axiomatics) of 1961, and only four more years later he published La logica sim- bolica (Symbolic logic), another very fortunate book that had four successive edi­tions and was also translated into Spanish (Agazzi 1964).

Agazzi himself has explained in some autobiographic interviews (the most extended and accessible of which is Agazzi 2008¹) the reasons for this beginning of

¹See also Agazzi and Alai (2012c). ^[106]

his philosophical journey: mathematical logic, after receiving pioneering develop­ments by Peano^[107] and his school at the beginning of the twentieth century, had remained nearly ignored in Italy during almost 40 years, so that the only books avail­able in Italian until 1960 were the old handbook by Peano’s disciple Burali-Forti (2nd edition 1919) and the booklet Nove lezioni di logica simbolica (Nine lectures on sym­bolic logic) by Bochenski and Jozef (1938). Therefore, when Agazzi, still an under­graduate, began to enter (practically self-taught) the field of mathematical logic, he was obliged to study several of the most reputed books of this discipline, especially in German and English, coming into contact with different approaches and contents.

At the same time, Agazzi was inevitably influenced by the fact that, during the first half of that century, the developments of mathematical logic have been strictly interwo­ven with the research on the foundations of mathematics. As a consequence of this fact, logic and foundations of mathematics became the core of Agazzi’s intellectual interests for a few years.

The first part of this work is devoted to a historical-philosophical reconstruc­tion of the axiomatic method, considered as the most natural explicit realization of the idea of a “demonstrative knowledge” proposed since the time of Plato and Aristotle. This classical view was deeply modified towards the end of the 19 th century owing to the crisis of mathematical intuition that produced the so-called “foundational crisis” of mathematics, and led to the formal conception of this method with the related metatheoretical problems.

The second part of the book is devoted to Godel’s Incompleteness Theorems, with a careful progressive introduction of all the necessary notions and technical prerequisites, that culminates in a simplified but completely rigorous reproduction of the different theorems of the original Godelian paper of 1931.

An Appendix contains the Italian translation of the original German paper, that actually was the first translation appeared in any other language and had the privilege of adding a footnote communicated by Godel himself. The clarity of exposition and the pedagogic skill that allowed also readers with no technical background to understand the complex structure of this famous set of theorems was undoubtedly the main ground for the great success of this book.

The conclusion of this book specifically attracted the interest and appreciation by the philosophers: here Agazzi, though underscoring all the merits and advan­tages of formalization, argues against the purely formal conception of logic and mathematics by critically discussing several metalogical results, like those con­cerning syntactic and semantic completeness and incompleteness, undecidability and categoricity, and by considering their philosophical implications.

In particular, Agazzi analyzes Godel’s Incompleteness Theorems, that have always remained a constant point of reference for his research. This is confirmed by a number of papers (Agazzi 1966, 1978, 1994, 1997, 2007, 2010, 2011, 2012a, 2012b, 2014) and by the fact that in 2006 he was invited at various international symposia to celebrate the hundred years of Godel’s birth (for example at University of Giessen and the University of Urbino). The just mentioned titles constitute an evidence of the persistent interest in the philosophy of logic and in particular in Godel’s Incompleteness Theorems that characterizes Agazzi’s think­ing from his first book for more than half a century.

From a historical point of view, we can say that Agazzi was the philsopher who, with Ludovico Geymonat and Ettore Casari, did most to launch and stimulate the study of logic and philosophy of mathematics in Italy. A complete account of Agazzi’s research in these areas should offer details at least on four topics of his groundlaying work: (a) the relationships between form and matter^[108]; (b) philosoph­ical reflections on Henkin Theorem^[109]; (c) philosophical reflections on Lowenheim- Skolem Theorem^[110]; (d) the study of Godel’s Incompleteness Theorems and their implications. Such a complete account would show how the suggestions and cues coming from Agazzi’s work in the 60’s in the following years have been devel­oped, more ore less independently, by other Italian and foreign scholars. Unfortunately it is not possible for me to develop in this paper the whole project, and I will limit my analysis only to point (d).

Here I will just propose some reflections on Agazzi’s study of the consequences of Godel’s theorems. In a review of I Problemi dell’Assiomatica, Ludovico Geymonat^[111] emphasized the importance of this publication, which filled a gap in the Italian cul­ture. In fact, Agazzi’s publications devoted to the problems of the axiomatic, and in particular to Godel’s Incompleteness Theorems, enabled many Italian scholars not only to approach and understand these important results, but also to problematize them theoretically, developing for example the so called Godelian arguments, that is, arguments using Godel’s Incompleteness Theorems to show that minds cannot be explained in purely mechanist terms.

Turing himself understood such implications of these theorems^[112]; beside him, in the 1950s Nagel and Newman (1956, 1958),^[113] developed argumentations hinged upon the idea that Godel’s Incompleteness Theorems could provide a logical tool to refute the philosophical thesis of mechanism. Despite this tradition, Godelian anti-mechanists argument is associated with the name of the English philosopher J. R. Lucas. In fact, in 1961, he developed an argumentation aimed at demonstrat­ing, on the basis of Godel’s theorems, that it is not possible to represent human intelligence by a Turing machine.^[114] Agazzi developed his Godelian Argument^{[115] [116]} almost simultaneously with Nagel and Newman, and Lucas,¹¹ but independently from them. Obviously, just as any other scholar, at that time Agazzi couldn’t know Godel’s own ideas on this topic, which were published later on^[117]).

In 2006, during a conference in Cesena (Italy), Agazzi explained his thoughts on Godel’s Incompleteness Theorems, inviting the audience to continue to reflect on the great results in logic and mathematics, because they are a great gymnasium for our minds and theories, teaching us important lessons about our cognitive capacities and our representation of the world. In that occasion, Fano and I decided to take up this challenge and write a survey on the implications of Godel’s Incompleteness Theorems in the philosophy of mind. We published our papers in 2011 and 2013, the latter in Epistemologia, a journal founded by Agazzi in 1978. This little story captures an important side of Agazzi’s thought and intellectual life: his capacity to propose important problems and stimulate scholars to their solution.^[118]

Following this example, the tribute I wish to offer to Agazzi’s mastership will be an analysis of Godel’s thought on the consequences of his Incompleteness Theorems for the philosophy of mind. That is part of the research developed after Agazzi’s conference in Cesena and it is for me a way to honour Evandro Agazzi’s impulse to the researches in philosophy of logic.

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