Michel BITBOL (Paris)
Why is quantum mechanics a probabilistic theory?
The probabilistic status of quantum mechanics partly accounts for its bad reputation of being an "unintelligible" theory. In fact, this negative judgment might well derive from a representationalist prejudice about what is and what should be a physical theory. By contrast, if physical theories are seen as "bets on action" (Jean Cavaillès) rather than images of the world, then quantum mechanics no longer appears as a deviant case, but as an archetype. Its intelligibility can even surpass that of any previous theory, provided the reason of its irreducible probabilistic nature is clarified. An inquiry into the origins of quantum probabilities is then undertaken, by first using the insights of two founders of probability theory: Girolamo Fracastoro and Blaise Pascal. The structural connection between contextuality and probability, already suggested by these early authors, is then confirmed in microphysics. Finally, an epistemological justification of the Born-Luders rule is found to be close at hand. I conclude that the right phenomenological attitude, in order to understand quantum mechanics, is reflective rather than directly intentional.

Umberto BOTTAZZINI (Milan)
Between the real and the imaginary: from 'sophistic' roots to complex numbers
Once the formula for the solution of the cubic equation had been settled by Scipione dal Ferro, Tartaglia and Cardano, its application led to a kind of paradox. This was called the 'casus irriducibilis', arising when all the roots of the given equation are real. In this case the formula involved 'impossible roots', i.e. square roots of negative numbers. Tartaglia was unable to deal successfully with it as was Cardano, who tried to overcome the difficulty by a number of algebraic transformations. Contemporary to him, Bombelli succeeded by resorting to 'a new kind' of numbers, and established the rules for them. He also provided a geometrical explanation, but apparently in his days his work had not the influence it deserved. A deep understanding of the nature of the 'numbers' occurring in the 'casus irriducibilis', and the related proof of the fundamental theorem of algebra, had to wait until the early 19th century.

Argante CIOCCI
Pacioli and Dürer in Bologna: Ars secreta and Divina proportione
This survey aims to investigate the conceptual and historical links between Pacioli and Dürer's mathematical works through an analysis of the confluence of two cultural traditions which regarded both the idea of mathematical sciences and their applicative use in a very different way. One of those cultural traditions was based on the scholars' knowledge expressed in Latin and cultivated in the universities, the courts, and the humanistic circles of the Renaissance, the other reflected the practical culture of the so called "middle cultural class" placed between the scholars and the illiterates: the class of craftsmen, traders, architects, painters, abacus teachers, algebraists, engineers, cartographers, artillery experts, in short, the technicians who used the vulgar language and produced a great deal of practical mathematics. In this respect the Bolognese journey of Pacioli and Dürer is a perfect case study of the relations between learned mathematics and artists and technicians' practical mathematics.

Maria CONFORTI (Rome)
Visual and non-visual in Renaissance medicine
As Nancy Siraisi has shown, the construction of the 'objective' and visual language of autopsies and anatomical observation has been a long and devious process, accumulating for centuries. In the medical Renaissance interpreting the unseen and the invisible in bodies and diseases – reading signa - was a matter of logic and scientific knowledge, and training in these subjects, more than the result of observation or ‘experience’. The autopsy, on the contrary, despite referring to auctoritates, build on observed “facts” (a notion whose ambiguity is well-known). Anatomy and its language and practice, far from substituting more traditional notions of medicine, of its methods and aims, found a place among other parts of the medical science. The operations of dissecting dead corpses and constructing coniecturae about what was going on in living ones were two very different parts of medical practice. Anatomy was in itself no simple knowledge, as shown by recent contributions on its relationship to geometry and geography, on non-visual anatomies, and on the discussions on the anatomy of the female body and of pathological processes.

Jean DHOMBRES (Paris)

From the fundamental theorem of algebra to the Heisenberg relations:
the role of complex numbers in physics

The introduction of complex numbers in algebra during the XVI century in Italy for the solution of equations of degree two, three and four, is well documented as an abstract internal step in mathematics. It rose the famous question of the fundamental theorem of algebra, with its four different proofs provided by Gauss. The introduction of complex quantities in Analysis by Euler (1748) is also well known, and reckoned on pure mathematical inspiration. But few historians are aware of the emergence of complex numbers in physics a bit earlier, with the Dissertation on winds by d'Alembert (1746) and the famous problems de vibratione chordarum. It is this "physical" line of development for complex numbers that I would like to follow, as it shows an epistemological turn in the mathematical sciences. It led to the definition of what was called in French "le mathématico-physique", a rather biased expression for mathematical physics. To this line of development as much Fourier analysis as the Heisenberg relations in quantum mechanics may be assigned. What is more, and perhaps not well known, even Argand’s geometric representation (1806) can be viewed as a discovery of a physical nature (as explained by Maxwell speaking about quaternions).

Artur EKERT (Oxford & Singapore)
Why quantum theory needs complex numbers
Can we describe the world without complex numbers? I will argue that any good statistical framework theory, a meta-level description of the world, requires complex numbers. In particular, I will show that once we request continuity of admissible physical evolutions we will end up with quantum theory, and if this requirement is dropped we obtain classical probability theory. Thus quantum theory can hardly be any different from what it is, and Cardano's "useless" discoveries can hardly be avoided.

J. V. FIELD (London)
Algebra and the art of getting answers
The foundation text of algebra, written by al-Khwarizmi in the ninth century AD, was transmitted to the West under the Latin title Dixit algorismus... The work is short but impeccably rigorous, giving geometrical proofs and establishing that algebraic methods can also give rigorous proof. But what Latin authors took over was merely the division of equations (up to the second degree) into six kinds, and worked examples. These formed the beginnings of the tradition of abacus mathematics, written largely in the vernacular. However, almost all the problems considered are ones in which the answer can be checked. If all you want is an answer, checking removes the need for a rigorous method for obtaining the answer; in effect you have an experimental method in mathematics. Cardano's silent use of complex numbers is an example of this kind of method. (Bombelli's work, though admired by historians today seems to have been without influence in its own day.) This new kind of mathematics, in which traditional Euclidean style is combined with the fine-if-you-can-check-it method, is important because in the sixteenth and seventeenth century an increased use of mathematics is characteristic of significant changes in natural philosophy. It seems likely that the changed nature of mathematics itself, and the changed capabilities that went with it, played a part in this process.

Veronica GAVAGNA (Salerno)
The meaning of Ars magna in Cardano's mathematical work
Printed in 1545, the Ars magna sive de regulis algebraicis became famous for the solving formulas of third and fourth degree equations, but it contained other interesting issues, treating, for example, the "structure" of the equations and the relations between coefficients and roots. The Ars magna represented the concluding work of a path, developed in the environment of the abacus tradition, started in 1539 with the publication of Practica arithmetice and carried on successively by the unpublished Ars magna arithmeticae (printed only in Opera omnia, 1663). At the same time, the Ars magna was just a tessera - actually, the only one printed - of a much more ambitious work, the Opus perfectum - an encyclopedia in itself able to sum up all arithmetical knowledge. This twofold key let us to suggest a new interpretation of the role of Ars magna both in Cardano's scientific work and in the XVI century mathematics.

Martin KEMP (Oxford)
"Continuous Quantities": the primacy of geometry in Renaissance art from a Leonardesque perspective
The Renaissance theory of perspective was deeply associated with notions of mathematical proportion, and thus with aspects of musical harmonics expressible in numbers. Yet its greatest power was seen as lying in incommensurable relationships that were not reducible to numbers. Even Piero della Francesca, the painter-theorist who was most alert to arithmetic and algebra, recognised their limits in the arena of geometrical relationships. For Leonardo, the superiority of geometry became an issue of principle, driven not least by a powerful aesthetic sense of the primacy of "continuous quantities" over the "discontinuous quantities" of number in the beauty of nature. He enlisted time in this quest, redefining the moment of time in narrative painting in terms of what I will be calling a "synthetic instant". Although Leonardo expressed the primacy of geometry over numbers in his own particular way, his basic preference holds true on a broad basis across the fields of painting and architecture in the Renaissance.

George SPARLING (Pittsburgh)
A primordial approach to unifying quantum mechanics and relativity
Roger Penrose has pointed out that the number "i", or the square root of minus one, occurs naturally in physics in two places: in the spherical geometry of the null cone of space-time and in quantum mechanics. Using the concept of triality due to Elie Cartan and the related octonions of John Graves, we have put forward the beginnings of a theory, the primordial theory, whose goal is to explain this coincidence. Recent work has revealed that this problem is connected with the idea of non-commutative geometry and with the concept of Fermi-Bose transmutation. We will discuss our approach and explain how we are led to suspect that there are dual ways of describing our universe: two six-dimensional space-times, connected through seven dimensions in much the same way that the Schwarzschild space-time forms a four-dimensional bridge from its horizon to its conformal infinity.

John STILLWELL (San Francisco)
From perspective drawing to the eighth dimension
The discovery of the "costruzione legittima" for perspective drawing led to interest in a new kind of geometry - projective geometry - in which points and lines are the main ingredients. Even with this simple subject matter there are some surprises, where three points fall on the same line or three lines pass through the same point, seemingly for no good reason.
The big surprises, or "coincidences", of projective geometry are known as the Pappus theorem, Desargues theorem, and the little Desargues theorem. Even more surprising, these purely geometric theorems were found (by David Hilbert and Ruth Moufang) to control what kind of *algebra* is possible in two, four, and eight dimensions.

Lambert WILLIAMS (Harvard)
Chance, complex numbers, and the heuristic art of Girolamo Cardano
Sketching the relation between the cultural idiosyncrasies of the Renaissance, on the one hand, and the embodied idiosyncrasy named Girolamo Cardano, on the other, has been a prominent feature of recent literature both in history of medicine and of astrology. The relation is not so well drawn, however, when one turns to Cardano’s mathematical corpus. Both the Ars Magna and the Liber de Ludo Aleae have been taken at multiple times and places to provide an object lesson in what poor mental hygiene might look like. Critics have found the works at best prolix and disunified, and at worst simply wrong. Even where Cardano’s mathematical corpus has been more favourably treated, very often it is an appreciation that I will show to be heavily wrapped in historiographical presentism.
My own aim, drawing mostly on the Liber de Ludo Aleae and only to a lesser degree on the Ars Magna, is to sketch a context for his work that brackets such standard pejoratives and presentism. At multiple levels, and ranging over a surprisingly wide area of Cardano’s mathematical pursuits, I will insist on a crucial role for questions of participation and competition, be that hidden away at the gaming table or grandstanding publicly at the mathematical contest. Relatedly, I will highlight Cardano’s frequent construction of ‘artful’ heuristics rather than ‘rigorous’ demonstrations. Reading parts of his mathematical work more as vade mecum than as treatise, I argue, serves to restore some much-needed scholarly symmetry between Cardano’s work on the formal, the medical, and the astrological.

19-20 November
Bologna, Scuola di Studi Umanistici