1,721,192 research outputs found
Cuts, Qubits, and Information
No full textIn his search for the ‘essence’ of continuity, Richard Dedekind (1872) discovered the notion of cut. Epistemologically speaking, a cut produces a separation of a simply infinite system into two parts (Stücke) such that all the elements of one part are screened off all the elements of the other. The distinct continuity of a two-state quantum system is encapsulated in the notion of qubit, the basic ‘unit’ of quantum information. A qubit secures an infinite amount of information, which, however, appears to be only penetrable through ‘sections’ of classical bits. Whereas Dedekind’s cuts dwell on the discrete of number theory, the theory of nature is primarily concerned with continuous transformations. In contrast with Dedekind’s line of thought, could the notion of information be derived from a ‘principle’ of continuity
Ways of Abstraction. Artistic Vision and the "Ideality" of Mathematics
No full textWhen I run into the Needham’s question – “What was it that happened in Renaissance Europe when mathematics and science joined in a combination that was qualitatively new and destined to transform the world?” – I could not help thinking of Brunelleschi and Leonardo. The “exact fantasy” of the design for the cupola of Santa Maria del Fiore together with the grace that emanates from The Virgin and Child with Saint Anne immediately came to my mind. Is there an answer here? Cassirer’s philosophical reflection helped me focus on the issue of abstraction, whereas Hilbert’s vision of geometry helped me grasp the ideality of mathematics and art
Hilbert’s Axiomatics as “Symbolic Form”?
No full textAccording to Ernst Cassirer, symbolic forms “are not different modes in which an independent reality manifests itself to the human spirit but roads by which the spirit proceeds towards its objectivization, i.e., its self-revelation” (1923, p. 78). Can Hilbert’s axiomatics be viewed as a symbolic form? The aim of this introductory essay is to frame the question in a ‘map’ and to provide a few signs to orient it. Each of the essays collected in this volume may contribute a different set of instructions to reach an answer
L'inimitabile intelligenza del vuoto
No full textThe idea of a «thinking» machine, i.e., a machine able to stand comparison with a person in Turing's
«imitation game», emerges from a deeper investigation into strengths and weaknesses of a universal Turing machine.
If one judges intelligence from observable performances, it seems perfectly reasonable to expect a machine to
perform like a human, once endowed with a suitable program (which might involve random elements). However,
by taking «mechanical procedures» to their limits, incompleteness and undecidability focus on a crucial difficulty
involved in a search for a non-theorem through the class of theorems. Gödel maintained that our understanding
of abstract terms requires some effective non-mechanical procedures. Turing continued to struggle over possible ways
to «copy» human initiative into a program. Going back to Dedekind's reflection on mathematical thought, we
encounter an «effective definition of the essence of continuity» in his explanation of the irrational numbers as
«cuts». Dedekind observed that even if we were sure that space is discontinuous, there would be nothing to
hinder us from making it continuous by intellectually filling in its holes; this filling in would consist in the creation of
new individual points. Analogously, the creation of new irrational numbers will make the system of numbers
continuous. Such an intellectual act, however, requires the vision of the void, the vision of a non-observable. How
can a machine, which operates on symbols «immediately recognisable», see and fill in the void
Taking it on trust: Reason and observability in quantum computing
No full textIf quantum computing is located somewhere between physics and theoretical computing,
a basic question concerns which characteristic features are derived from the latter. From
a logical point of view, the concept of computation provides a definition of the natural
process of calculare. It rests on trust that a procedure of reason can be reproduced
mechanically. Turing argues for the adequacy of the concept by introducing a requirement
of “observability,” which is expressed through finiteness and locality conditions.
However, according to the uncertainty principle, no computational path can be observed.
How does quantum computing contend with Turing’s constraints? What observables are
relevant to the computation? This is an attempt to sharpen such questions
Seeking Quantum Waves
No full textQuantum theory has thrown new light on our understanding of physical properties. Notably the epistemological reflection on the nature of quantum waves has made clear to what extent our knowledge of the external world is dependent on the process of measurement. Gino Tarozzi has been deeply engaged in this reflection; not solely he has contributed to the philosophical debate on the meaning of quantum waves, but he has also designed experiments to probe their ‘existence’. My purpose, in what follows, shall be to contrast his ‘realistic’ interpretation of the wave function with local realism. Such a contrast illuminates the relational character of quantum physical properties and leads to recognize the ‘existence’ of quantum waves by their interference effects
The Complex Route
No full textThe essays collected in Part I trace the origins of linear perspective in the Renaissance culture and explore its impact from painting to mathematical thought. Is there any meaningful link between linear perspective, which gave painting a new dimension, and complex numbers, which seem to be doing the same to physics? Part II is focused on exploring this link and helping the reader see the “art” involved in connecting complex numbers and probability through the notion of “quantum probability amplitude.
A Silk Road from Leibniz to Quantum Information
No full textAt the roots of quantum physics we find two “specular” principles: a principle of distinguishability, which arranges the “uncertainty” of quantum
measurement, and a principle of continuity, which drives the evolution of quantum systems. The vital link between the two principles was sharply
captured by Leibniz. In his Theodicy, the issue of continuity (and indivisibles) is presented as one of “the two famous labyrinths” in which our
reason can lose its way; the other concerns “the great question of freedom and necessity.” According to Leibniz, only geometry can provide a
thread for the labyrinth of continuity. However, escaping from the labyrinth of Fate, Fortune, and Freedom requires a different perspective on nature. Leibniz’s perspective on the mechanism of nature is significantly different from Descartes’ or Newton’s. It reveals notable affinities with the Chinese correlative thinking. Surprisingly, the rationale behind these affinities may find a coherent setting in quantum information
Proof as a path of light
No full textAccording to certain medieval philosophers, perspective is a “demonstrative science” as it reveals the connection between sensible and intelligible visions by means of the mathematical rules of geometric optics. Carrying over concepts and methods of the medieval "perspectiva naturalis" into a plane surface, the Renaissance "perspectiva artificialis" unfurls a new “pictorial” space. To appreciate the impact of quantum theory on determinism and computation issues this paper will adopt a “perspectival” approach: the architecture of the theory, first captured in the ‘real’ three dimensional space, will lead us into a new ‘imaginary’ space. Here the bilateral symmetry coupling any possibility with its negation, sized by “complex probability amplitudes”, may dissolve the ‘ignorance’ of classical probabilities as well as the ‘blindness’ of finite mechanical procedures
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