Abstract
Quantum physics does not describe the world as the ultimate “artefact“ of God acting as a creator and ruler. If God played a role in the quantum world it would be as a gambler.
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Notes
- 1.
Letter to Max Born , 7 September 1944.
- 2.
According to Hacking, “What happened to signs, in becoming evidence, is largely responsible for our concept of probability ” (1975, p. 35).
- 3.
See Ekert’s essay in this volume.
- 4.
“The artists were self-taught and learned through practice. Fragments of Greek knowledge filtered down to them, but on the whole they sensed rather than grasped the Greek ideas and intellectual outlook. To an extent this was an advantage because, lacking formal schooling, they were free of indoctrination. Also, they enjoyed freedom of expression because their work was deemed ‘harmless’” (Kline 1972, p. 231).
- 5.
As in Alberti’s dedicatory letter of De pictura to Brunelleschi: “What man, however hard of heart or jealous, would not praise Filippo the architect when he sees here such an enormous construction towering above the skies, vast enough to cover the entire Tuscan population with its shadow, and done without the aid of beams or elaborate wooden supports?” (1436; 2004, p. 35). [“Chi mai sì duro o sì invido non lodasse Pippo architetto vedendo qui struttura sì grande, erta sopra e’ cieli, ampla da coprire con sua ombra tutti e’ popoli toscani, fatta senza alcuno aiuto di travamenti o di copia di legname, quale artificio certo, se io ben iudico, come a questi tempi era incredibile potersi, così forse appresso gli antichi fu non saputo nè conosciuto?” (1436; 1975, pp. 7–8).]
- 6.
“Una est ergo ratio generalis, ut consideremus totum circuitum, et ictus illos, quot modis contingere possunt, eorumque numerum, et ad residuum circuitus, eum numerum comparentur, et iuxta proportionem erit commutatio pignorum, ut aequali conditione certent” (Cardano 1663a, 2006, p. 63).
- 7.
A cubic equation such as \(x^{3} = 3px + 2q\) is solved by the rule:
$$\displaystyle{ x = \root{3}\of{q + \sqrt{q^{2 } - p^{3}}} + \root{3}\of{q -\sqrt{q^{2 } - p^{3}}.} }$$(6.1) - 8.
For a more detailed presentation, see Gavagna’s essay.
- 9.
Such as \((x + a)^{3} = x^{3} + 3x^{2}a + 3xa^{2} + a^{3}\).
- 10.
Namely when: p > q 2.
- 11.
“Un’altra sorte di R.c. L molto diverse dalle altre nasce dal capitolo di cubo uguale a tanti e numero quando il cubo di un terzo delli tanti è maggiore del quadrato della metà del numero, come in esso Capitolo si dimostrerà, la qual sorte di radici quadrate ha nel suo Algoritmo diversa operazione dalle altre e diverso nome; perché quando il cubato del terzo delli tanti è maggiore della metà del numero, lo eccesso loro non si può chiamare né più né meno, però lo chiamarò più di meno quando egli si doverà aggiongere, e quando si doverà cavare lo chiamerò men di meno, e questa operatione è necessarissima […] che molto più sono li casi dell’agguagliare dove ne nasce questa sorte di radici […] la quale parerà a molti più tosto sofistica che reale, e tale opinione ho tenuto anch’io, sin che ho trovato la sua dimostrazione in linee (come si dimostrerà nella dimostrazione del detto Capitolo in superficie piana)” (Bombelli 1572; 1929, pp. 133–134).
- 12.
After studying and translating the first five books of the Diophantine manuscript, Bombelli changed the form of his 1550 manuscript (Bortolotti 1929).
- 13.
“Tal proporzione è dalla immaginazione all’effetto, qual è dall’ombra al corpo ombroso, e la medesima proporzione è dalla poesia alla pittura, perché la poesia pone le sue cose nella immaginazione di lettere, e la pittura le dà realmente fuori dell’occhio, dal quale occhio riceve le similitudini non altrimenti che s’elle fossero naturali” (Trattato, I.2).
- 14.
For more details, see Vesely’s essay.
- 15.
Although the polyhedra drawn by Leonardo are in twenty-eight tables, De Divina Proportione does not put an upper limit on their number.
- 16.
“Primum quidem, quia generaliter duplus est labor, inde comparatione, si quid artifex delituit in magnitudine, numero, forma, colore, lituris, rugis, cavitatibus, aliisque innumeris, quae in unius medietatis figura lacebant, manifesta facta, operis turpitudinem declarant. Qui igitur fingere aliquid volunt, formam eius primum visam mente, quasi memoria concipere debent, inde typum quendam seorsum delineare subtiliu, post praesente eo quod signis singula animadvertendo ad amussim perficere, latet enim in unoquoque partium quaedam symmetria, quam si non mente conceperis oculorum vero praesidio tantum innixus tentes exprimere, operam luseris” (Cardano 1550, pp. 609–610).
- 17.
- 18.
“Le opere che l’occhio comanda alle mani sono infinite, come dimostra il pittore nelle finzioni d’infinite forme” (Trattato, I.24).
- 19.
“La scultura non è scienza ma arte meccanicissima, perché … in sé finisce dimostrando all’occhio quel che quello è, e non dà di sé alcuna ammirazione al suo contemplante, come fa la pittura, che in una piana superficie per forza di scienza dimostra le grandissime campagne co’ lontani orizzonti” (Trattato, I.31).
- 20.
See also El-Bizri’s essay in this book.
- 21.
As Underweysung der Messung (1525) documents, and as the style of the artist confirms, Dürer conceived a science of measure that obtained the most original results when it applied procedures typical of an artist’s workshop (bottega) to abstract mathematical objects. Applying the method of double projections, in use by carpenters and architects, to conic sections, Dürer obtained a construction that Gaspard Monge would theoretically codify at the end of the eighteenth century in his “descriptive” geometry.
- 22.
“First of all, on the surface on which I am going to paint, I draw a rectangle of whatever size I want, which I regard as an open window through which the subject to be painted is seen; and I decide how large I wish the human figures in the painting to be. I divide the height of this man into three parts, which will be proportional to the measure commonly called a ‘braccio’; for, as may be seen from the relationship of his limbs, three ‘braccia’ is just about the average height of a man’s body. With this measure I divide the bottom line of my rectangle into as many parts as it will hold; and this bottom line of the rectangle is for me proportional to the nearest transverse equidistant quantity seen on the pavement. Then I establish a point in the rectangle wherever I wish; and as it occupies the place where the centric ray strikes, I shall call this the centric point” (Alberti 1436, 2004, p. 54).
- 23.
“Perspective is by nature a two-edged sword”, Panofsky wrote, because it “subjects artistic phenomenon to stable and even mathematically exact rules”, but “the way [these rules] take effect is determined by the freely chosen position of a subjective ‘point of view’” (Panofsky 1924–25, 1997, p. 67).
- 24.
For more details on this construction, see Stillwell’s essay (chap. 1).
- 25.
For more on the magic of mirrors and symmetries, see Altmann’s essay (Chap. 5).
- 26.
The total over the events must be 1.
- 27.
Because the probability of each alternative is 1/4, the probability of a measurement is again 1/2.
- 28.
David Deutsch (1997) coined the suggestive term “photon-shadow”.
- 29.
Looking at Fig. 6.6, imagine a detector set on the path T, after the first STM. A photon on the path R cannot be measured.
- 30.
A device which does not detect the photon along T informs that the photon has been reflected.
- 31.
There are four possible results of a measurement.
- 32.
Therefore, the probability of each alternative pair of measurements is 1/4, and the probability of measuring the same observable is 1/2.
- 33.
Each particle can give its observables T and R either the same value or opposite values. As to the instructions of A, they must be either T A = R A or \(T_{A} = -R_{A}\). In the first case, S = 2T B , in the second, S = 2R B . Combining the instructions of A with the instructions of B, the value of S must be + 2 or − 2.
- 34.
For more details see Scarani (2003).
- 35.
- 36.
For more details, see Wheeler (1990), cf. also Wheeler & Zureck (1983) .
- 37.
The Argand plane is a two-dimensional plane where we can visualize any complex number c as a point and locate it by means of Cartesian coordinates \(\left (x,y\right )\) such that: \(c = \left (x + iy\right )\) or in polar form as \(c = \left \vert c\right \vert e^{i\theta } = \left \vert c\right \vert \left (\cos \theta +i\sin \theta \right )\).
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Angelini, A., Lupacchini, R. (2014). Artists and Gamblers on the Way to Quantum Physics. In: Lupacchini, R., Angelini, A. (eds) The Art of Science. Springer, Cham. https://doi.org/10.1007/978-3-319-02111-9_6
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