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.
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.
Pacioli and Dürer in Bologna: Ars secreta and
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.
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.
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).
(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
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...
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.
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
represented the concluding work of a path, developed in the environment of
the abacus tradition, started in 1539 with the publication of
and carried on
successively by the unpublished Ars magna arithmeticae
(printed only in Opera omnia
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.
"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.
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.
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.
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.