Papers by Luigi Borzacchini
Proceedings of 6th International Workshop on Computer-Aided Software Engineering
System design recovery and impact analysis are critical phases in the process of sojhare maintena... more System design recovery and impact analysis are critical phases in the process of sojhare maintenance. We believe that the interdependence of abstraction recovmy tools and application domains plays a key role to provide intelligent assistance in sojhvare maintenance and advocate the use of a blackboard model to address the issue of knowledge sources integration A Software Maintenance Expert Sysrem (SLUES) based on a blackboard archirerture has been dewlooped to ewrluate the approach Explicit links between speci3cation, design and implementation objects have been represented in the blackboard working memory, as well as the design decisions in the abstraction recovery process. These links are then used in the sofnyare evolution process to support impact analysis.
Stone Age, Bronze Age, Iron Age, … and now we live in the Age of Signs. Today, everything in our ... more Stone Age, Bronze Age, Iron Age, … and now we live in the Age of Signs. Today, everything in our worldcars and planes, computers and iPhones, buildings and roads, games and wars, money and credit cards …is made with plastic, iron, rare earths, concrete, … but, before, it was only made with 'signs': algorithms, formulas and calculations, pins and passwords, codes (fiscal, banking, electoral, genetic, health, social security, ...). The age of Signs is very old. Its roots may have been in the alphabetic revolution, during the Greek Dark Ages, three thousand years ago, but its real emergence was symbolic algebra in the Scientific Revolution. Everyone remembers the 'x', the algebraic nightmare of his/her own youth. Students usually dislike algebra: a dry, mnemonic, incomprehensible, meaningless discipline. Geometry is fascinating, algebra is disturbing. Igor Šafarevič [1] saw algebra as a Cinderella of mathematics-perhaps the first Islamic clandestine-, while geometry, which had retained its Greek and Euclidean charm, was its favorite daughter: More than once I have met a person who said, "I didn't choose math as my profession, but I'll never forget the beauty of the elegant edifice built in geometry with its strict deduction of more and more complicated propositions, all beginning from the very simplest, most obvious statements!" Unfortunately, I have never heard a similar assessment concerning algebra.
Calcolo, 1980
In this paper we proof a theorem concerning lattice constants and hence three matricial equations... more In this paper we proof a theorem concerning lattice constants and hence three matricial equations for conversion matricesR: if H=ΔRT we have: i)H3=I; ii) HT Σ H= Σ; iii)(DH)2=I; where Δ,D, and ε are known when we can enumerate all non-isomorphic graphs withn vertices, we know (for Δ and ε) their edge-number and (for ε) the order of their automorphism group.
Calcolo, 1981
In this paper we study the connexion among graph enumerating polynomials and then we prove for th... more In this paper we study the connexion among graph enumerating polynomials and then we prove for these polynomials theorems concerning the reconstruction of a graph from its point-deleted subgraphs or edge-deleted partial graphs.
Academia Letters
Astrology (with alchemy) was the bridge between Aristotelian science and modern science, preservi... more Astrology (with alchemy) was the bridge between Aristotelian science and modern science, preserving the general Aristotelian framework, but opening the road to a science of the 'individuals'; beyond the classical science of the 'universals';. In Dante's Divina Commedia we can find also the beginning of an 'astral science'; which introduced the idea of science as a 'system'.
Academia Letters, 2021
Historicism and metaphysics seem incompatible ... However, history is an old mole, which, digging... more Historicism and metaphysics seem incompatible ... However, history is an old mole, which, digging, can appear in unpredictable places ...
In the last thirty years, I wrote many books about the history of mathematics and logic. However, I think they were about the philosophy of mathematics but reduced to just history because based on a weak skepticism: often I wrote “Truth about mathematics cannot be said or shown, but can be historically told”. For that reason, I avoided in my historical ‘tales’ any ‘philosophical’ background.
This way I followed the most genuine cultural south-Italian tradition: we could call it the
‘absolute historicism’,i.e. the ‘radical’ idea of the cultural history as the very primitive,
neither the object of a philosophy of history nor the empirical tool to support a philosophy of science. It was a minor branch of Western Culture, in comparison with the empirical-analytical Anglo-American and to the rationalist-philosophical continental ones
If G1 and G2 are two simple graphs, following C. Domb [l], let us denote*with (G,, G2) the number... more If G1 and G2 are two simple graphs, following C. Domb [l], let us denote*with (G,, G2) the number of subgraphs of G2 isomorphic with G1 (weak lattice constants). Let us denote with B, the set of non-isomorphic simple graphs, with n vertices, with B,, the set of simple graphs with n vertices and r edges (in both definitions isolated vertices are allowed). Let’s write G, < G2 if and only if G1 is subgraph of G2, with G1 E -El, and G2 E B,, ; then (B,,, -c ) is a partially ordered set and (G,, G2) is an incidence function on B,. Let’s write B,, = (gl, . . . , gA4}, with M = (B, I; where the total order defined on B,, satisfies the following conditions: (i) s iS a total extension Of 4, i.e. gi < gj =$ i Sj. (ii) If gi is not self-complementary, and gi is the complementary graph of gi, then gi = &+1-i ; self-complementary graphs must be set in centrall posirions in any order. (iii) If mj is the number of edges of gj, then mi < mj 3 i <i. Obviously such 2 total extension is al...
It is known that Information and Communication Technology are becoming increasingly useful in tea... more It is known that Information and Communication Technology are becoming increasingly useful in teaching mathematics, but so far it has not been considered as well that this technology, and symbolic calculus languages as Mathematica or Maple in particular, can pave the road to new connections between mathematics, computer and humanities. Music and Mathematics have always been deeply connected. At the beginning of our mathematics there is Pythagorean mathematics that was first and foremost music-driven, and the end of ancient music was connected to the equal-temperament, linked to the idea of real number. We give an outline of a Mathematica notebook about ancient Greek Music, that includes a short introduction to mathematical music theory, a part about Greek mathematical music theory and the difference between Pythagorean and equal-tempered tuning, a system to play ancient music, in which we can choose the system (enharmonic, chromatic, diatonic) and the mode (Doric, Lydian, Phrygian, ...
Sapere aprile 2019, 2019
Su Leonardo artista i giudizi sono abbastanza univoci, ma su Leonardo scienziato le opinioni sono... more Su Leonardo artista i giudizi sono abbastanza univoci, ma su Leonardo scienziato le opinioni sono molto diverse: c'è chi ne fa il 'nonno' della scienza moderna e c'è chi dalla storia della scienza lo esclude quasi del tutto. Quale è la sua collocazione nella storia della scienza?
Atti Accademia Pugliese delle Scienze, 2020
scienza per Leonardo non era un hobby, direi che doveva diventare il suo principale interesse. Un... more scienza per Leonardo non era un hobby, direi che doveva diventare il suo principale interesse. Un interesse non solo legato alla sua attività di pittore (anatomia e scienze naturali), ma centrato su matematica, meccanica, ottica, idraulica, astronomia e ingegneria-soprattutto il volo. E anche la pittura per lui era scienza. La scienza di Leonardo è il passaggio stretto-perché solitario e senza eredi diretti-tra la scienza medievale e quella moderna, ma ce ne dà il senso più profondo.
MUSIC EDUCATION, 2011
Today music is just entertainment. Perhaps it is also a cultural fact, but in an aesthetic realm,... more Today music is just entertainment. Perhaps it is also a cultural fact, but in an aesthetic realm, foreign to science. Indeed this is the view that has prevailed in the last three centuries. Until the 17 th century however, music was regarded either as a practice or as a mathematical theory, that every good mathematician and every good musicologist knew. In ancient Greece music played an even greater role, in the platonic polis it was part of both basic education (with gymnastics and grammar) and higher education (with arithmetic, geometry and astronomy).
Today music is just entertainment. Perhaps it is also a cultural fact, but in an aesthetic realm,... more Today music is just entertainment. Perhaps it is also a cultural fact, but in an aesthetic realm, foreign to science. Indeed this is the view that has prevailed in the last three centuries. Until the 17 th century however, music was regarded either as a practice or as a mathematical theory, that every good mathematician and every good musicologist knew. In ancient Greece music played an even greater role, in the platonic polis it was part of both basic education (with gymnastics and grammar) and higher education (with arithmetic, geometry and astronomy). In a previous paper I analyzed the crucial role played by musical mathematical theory in shaping ancient Greek mathematics. In this chapter my aim is to outline the role played by musical themes and musical mathematical theory between the Middle Ages and the early Modern Ages in the evolution of several ideas:. 1) Quantification of qualities: musical notation by neumes and tetragrams was the first example of the representation of a quality by a geometrical metaphor. 2) Symbolic representation 3) Opposition between the discrete and the continuous: time is continuous but music as a mathematical science is subordinated to arithmetic. 4) The correlation between arithmetic and geometric progressions: on the chord the sum of intervals tallies with the product of the corresponding ratios. 5) The numerical nature of ratios. In this way the evolution of musical mathematical theory in the 17 th century, from the arithmetic theory of harmonic consonances to the physical-mathematical theory of sound, can be interpreted as the emergence of a new view about the relation between mathematics and nature that shaped the Scientific Revolution: from mathematics as made up of immanent and immediately perceivable real attributes to mathematics as a representation language of a real, but concealed, structure of nature, attainable only by artificial experiment and measure. Such a dislocation explains the profound reasons for which the old connection between Muses and Science vanished in modern times, when musical harmony suddenly moved from the Scholastic Quadrivium to the Realm of Beauty, in perhaps the most astonishing turnaround in the history of human culture. This also hints at the role musical education could play in framing human cognition between the manipulation of signs and holistic perception.
Academia Letters
In a recent paper on Academia Letters (Borzacchini), I wrote about the 'historical scale' of the ... more In a recent paper on Academia Letters (Borzacchini), I wrote about the 'historical scale' of the breathtaking evolution of mathematics (sometimes an epochal and diffused revolution in a few decades!): it was difficult to ascribe it to an evolution of the brain, the hardwareevolution would be too slow!-, and it was just as difficult to read it as a simple evolution of ideas, the software-the changes would be too ephemeral and personal! My thesis was that the scale of such historical evolution could be successfully explained as the evolution of a midware, the actual architectural arrangement of the brain as it is constituted in the first years of the child's life. Furthermore, this hypothesis could be refined to read the evolution of mathematics as related to the evolution of the Brain Lateralization (BL) and the relationship between the two hemispheres: Left (LH: linguistic, analytic, syntactic) and Right (RH: holistic, iconic, spatial) (Springer and Deutsch). A premise: I fear that, with regard to the characteristics of the BL, it is also difficult to find a lexicon to describe it. My option is simple: to delineate it only in reference to mathematical knowledge, so that the two hemispheres' working essentially reduced to iconic (RH) and syntactic (LH) representations. These are the two classic types of representation: the first based on the similarity between the things and their icons, the second based on syntactic rules, connecting arbitrary signs and simulating the relationships between the represented things. These are the ancient Aristotelian aspects in the category of quantity: discrete (order, integers, writing) and continuous (figures, flow, speaking). Usually in the literature I have found references to the prevalence of one hemisphere over the other. In my opinion, however, the most relevant aspect in the history of mathematics
Academia Letters, 2021
In a recent paper on Academia Letters (Borzacchini), I wrote about the 'historical scale' of the ... more In a recent paper on Academia Letters (Borzacchini), I wrote about the 'historical scale' of the breathtaking evolution of mathematics (sometimes an epochal and diffused revolution in a few decades!): it was difficult to ascribe it to an evolution of the brain, the hardwareevolution would be too slow!-, and it was just as difficult to read it as a simple evolution of ideas, the software-the changes would be too ephemeral and personal! My thesis was that the scale of such historical evolution could be successfully explained as the evolution of a midware, the actual architectural arrangement of the brain as it is constituted in the first years of the child's life. Furthermore, this hypothesis could be refined to read the evolution of mathematics as related to the evolution of the Brain Lateralization (BL) and the relationship between the two hemispheres: Left (LH: linguistic, analytic, syntactic) and Right (RH: holistic, iconic, spatial) (Springer and Deutsch). A premise: I fear that, with regard to the characteristics of the BL, it is also difficult to find a lexicon to describe it. My option is simple: to delineate it only in reference to mathematical knowledge, so that the two hemispheres' working essentially reduced to iconic (RH) and syntactic (LH) representations. These are the two classic types of representation: the first based on the similarity between the things and their icons, the second based on syntactic rules, connecting arbitrary signs and simulating the relationships between the represented things. These are the ancient Aristotelian aspects in the category of quantity: discrete (order, integers, writing) and continuous (figures, flow, speaking). Usually in the literature I have found references to the prevalence of one hemisphere over the other. In my opinion, however, the most relevant aspect in the history of mathematics
Signs and Light. Light in the places of discourse. A.Ponzio and S. Petrilli edts.Semiotica, 136 ¼ (2001) 151-172, 2001
Haec qui bene conciliaverit, is totani nane difficultatem non sane laevem sustulerit, et lumini l... more Haec qui bene conciliaverit, is totani nane difficultatem non sane laevem sustulerit, et lumini lucem addens, de illius claritate apud plerosque multis falsitatum tenebris obvoluta bene merebitur. Who will solve all these not little problems, shedding light on light, will deserve well for such a brightness, which today for many people is surrounded by the great darkness of falsity.
Academia Letters, 2021
The emergence of modern science is maybe the main problem for the history of science. There is a ... more The emergence of modern science is maybe the main problem for the history of science. There is a fairly shared idea about its characters (experiments, mechanicism, mathematization), and many hypotheses about its roots: institutional (universities, modern state), economical (technological development, capitalism), cultural (crisis of Aristotelianism, craftsmanship, translations from Greek and Arabic), religious (autonomy of the state from religion, influences of classical philosophy on Christian religion), historical (immigration of scholars from Byzantium, oriental technology arrived by invasions). However, it is not a 'patchwork': there were strong interconnections among such historical phenomena. The father of the history of medieval science was Pierre Duhem (1861-1916), author of the celebrated Le Système du Monde. Perhaps his best known 'discovery' was that in the 13th and 14th century a tradition of Scholastic authors flourished, which he called Galileo's forerunners-Roger Bacon, John Duns Scotus, William Ockham, John Buridan, Nicholas Oresme, Albert of Saxonybecause they had developed a series of anti-Aristotelian criticisms, in which we can recognize many 'Galilean' theses. Furthermore, Duhem asserted that the Scientific Revolution could have occurred two centuries before the 17th century if it had not been delayed, due to the unscientific Renaissance culture, which produced a "vacuum" of nearly two centuries! Today, this thesis doesn't find many supporters, but the question is still alive: why such a gap between Galileo's forerunners and the beginning of the Scientific Revolution? A delay too long and deep to be explained only by wars and epidemics. And was the delay just a void? Here, too, we should rather be content with a historical 'tapestry', wherein there is, however, a theme usually almost completely ignored: the 'literary form' of science, its 'style'.
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Papers by Luigi Borzacchini
In the last thirty years, I wrote many books about the history of mathematics and logic. However, I think they were about the philosophy of mathematics but reduced to just history because based on a weak skepticism: often I wrote “Truth about mathematics cannot be said or shown, but can be historically told”. For that reason, I avoided in my historical ‘tales’ any ‘philosophical’ background.
This way I followed the most genuine cultural south-Italian tradition: we could call it the
‘absolute historicism’,i.e. the ‘radical’ idea of the cultural history as the very primitive,
neither the object of a philosophy of history nor the empirical tool to support a philosophy of science. It was a minor branch of Western Culture, in comparison with the empirical-analytical Anglo-American and to the rationalist-philosophical continental ones
In the last thirty years, I wrote many books about the history of mathematics and logic. However, I think they were about the philosophy of mathematics but reduced to just history because based on a weak skepticism: often I wrote “Truth about mathematics cannot be said or shown, but can be historically told”. For that reason, I avoided in my historical ‘tales’ any ‘philosophical’ background.
This way I followed the most genuine cultural south-Italian tradition: we could call it the
‘absolute historicism’,i.e. the ‘radical’ idea of the cultural history as the very primitive,
neither the object of a philosophy of history nor the empirical tool to support a philosophy of science. It was a minor branch of Western Culture, in comparison with the empirical-analytical Anglo-American and to the rationalist-philosophical continental ones
In ricordo del prof. Aldo Cossu, rettore dell'università di Bari, e come manifesto del 'mathematical pride', l'orgoglio matematico, l'onere e l'onore di produrre e diffondere quella conoscenza su cui si fonda la nostra civiltà.