Abstract
Geometry and mathematics constitute a kind of knowledge, which is directly founded on these two pillars that support our “thinking thought”.
“Si omnia a veteribus inventa sunt,
hoc semper novum nobis erit,
usus et inventorum ab aliis
scientia ac dispositio.” \(^{1}\) (L. A. Seneca, ad Lucilium VII, II)
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Notes
- 1.
“Even if everything had been discovered by our ancestors, this will always remain new for us: the application as well as the understanding and ordering of what was invented by others”
- 2.
We must here emphasise that our perception of the world is first defined by our sight, i.e., by the detection of photons emitted by surrounding objects. The eye is itself a simple array detector which reacts to electromagnetic waves with wavelengths somewhere between approximately 390 and 750 nm, but it is the task of the brain to elaborate the perception of space and distance. This extremely complex elaboration results from a combination of images in movement. We can, therefore, understand why in our perception of reality space and time are essentially interrelated.
- 3.
In spite of the importance and transmission of his work, which has come down to us almost in its entirety, we know very little about his life. Even his place and time of birth are unknown. From different quotations by later authors we infer that he flourished around 300 B.C., sometime between Aristotle and Archimedes.
- 4.
Aristotle repeatedly address this problem and he solved it by accepting infinity as a potency, but not as an act. In his physical model the universe is, however, finite and, for instance, an infinite straight line cannot exist as a real entity.
- 5.
One should recall that Plato’s first authenticated definition of a straight line is “the line of which the middle always covers both ends” (Parmenides, 137 E), whereby he appeals here to our sense, implying that the line of sight is straight. Euclid’s definition is as follows: “a straight line is that which lies evenly with the points on itself”. This can be interpreted that among the lines having the same extremities the straight line is the least.
- 6.
It is worthwhile reporting Proclus’ criticism on the fifth Postulate. “It is impossible to assert without some limitation that two straight lines produced from angles less than two right angles do not meet. On the contrary, it is evident that some straight lines do meet, although the argument must be proven that this property belongs to all straight lines. For one might say that the lessening of the two right angles being subject to no limitation, with such and such an amount of lessening, in excess of this they meet.” [91], p. 37.
- 7.
It is known that, in order to completely liberate theory from a visual concretisation of geometrical curves, in 1847 Karl Georg Christian von Staudt, a German geometer, published a treaty of projective geometry [2] in which he did not use any figures.
- 8.
Note that the succession of squares is completed with one This involves the introduction of zero in the succession of the odd number. For the Pythagoreans zero represented the assertion of limits and the perfection of units, i.e., of the monad, whose shape is reproduced by successive applications of gnomons.
- 9.
Aristotle here means that, whenever one conceives of a number, its order of magnitude (decade) is given and, hence, the number is first expressed by a definite integer \(1, 2 \ldots 9\) of the highest decade and then by those of the lower decades. Actually, in modern mathematics infinity is not indicated as a number, but as a symbol (\(\infty \)) the use of which presupposes a complex background.
- 10.
The mediaeval Latin translation is conciser and clearer: “ annulos infinitos dicunt non habentes palam quoniam semper aliquid extra est accipere.” This is only a property of straight lines and circles. Aristotle indicates the analogy between an infinite straight line and a circle. Actually straight line and circle are considered as the same object in modern projective geometry: the straight line corresponds to a circle passing through a point P and having an infinite radius; whereas the circle centred in P and having an infinite radius corresponds to the straight line defined by all infinite’s points (i.e., all directions in the plane).
- 11.
Pierre de Fermat (1601–1665) formulated one of the most famous theorems of mathematics, known as “Last Theorem of Fermat”, which asserts that the above-mentioned equation does not have integer solutions for values of \(n\) greater than 2. The demonstration of this theorem occupied mathematicians until now and represented a source of the most important collateral results. The last theorem of Fermat was demonstrated in 1995 by using an interesting new type of number called \( p\)-adic, which will be presented in Chap. 4.
- 12.
Karl Popper [9], examining these ideas, also explained in Timaeus, concludes that Plato believed the field of continuum could be expressed as a linear combination of a countable base of irrational numbers with integer coefficients. It is not sure that this was really Plato’s precise idea, but, even if not, this hypothesis clearly indicates in which direction the speculations of the Pythagorean and the following platonic schools were moving.
- 13.
The exhaustion method is based on the following procedure: for example, to find the volume \(V\) of a solid whose shape, \(S\), can be increasingly approximated by a sequence of suitable simpler shapes \(S_{n}'\), whose volumes \(V_{n}' \) we can calculate. Then one moves on to demonstrate that, if the approximation of \(S \) by \( S_{n}'\) has been improved as much as needed, it is impossible that for sufficiently large \(n\) either \( V\) can be greater than \(V_{n}'\) or that \(V\) can be smaller than \(V_{n}'\). It must, moreover, be noted that the inequalities between quantities (volumes in our case) are here founded on visual relations between different figures.
- 14.
The book of Archimedes, in which he explains his “Method” of integration, was recovered only in 1906 in Constantinople, where it was buried under the dust of twenty centuries.
- 15.
Anaximander of Miletus (610–546 B.C), a philosopher-naturalist educated in the school of Thales, a fellow citizen of his, proposed a cosmological theory, according to which from the limitless One (ἄ\(\pi \varepsilon \iota \rho o\nu \), \(\acute{a}\) peiron), the principle of Creation (ἀ\(\rho \chi \acute{\eta }\), arch \( \acute{e}\)) is the centre of a vortex (\(\pi \varepsilon \rho \acute{\iota }\chi \rho \varepsilon \sigma \iota \varsigma \), per \( \acute{i}\) chresis) that, in dilating itself, creates at its periphery smaller vortices and thus differentiates all existing things through a separation of contrary elements. The unique quality of all beings is the fruit of the embezzlement of one of two contrary terms (injustice); this, however, will be recomposed from the justice in the original One. Therefore, infinity of universes are born and die continuously. This concept presents amazing similarities to some modern cosmological models. It is necessary to note that what remains of the work of Anaximander are only a few fragments quoted and commented on by Aristotle, but the interpretation of these extant passages is contested [4]. What, however, is pertinent here is the echo of his theories, which resounded in subsequent generations. Even today, Ilya Prigogine (1917–2003) and René Thom (1923–2002), two great experts in the theory of structures of complex thermodynamic systems, had a particular predilection for the ideas of Anaximander.
- 16.
The Pythagorean School investigated harmonic relations in music by comparing sound intervals with the length of the cither strings producing them. The main intervals of the octave, fifth and fourth intervals were elements of harmony of sounds and related to the fractions 1/2, 3/2 and 3/4. Ancient astronomy recognised five relationships based on the twelve divisions of the zodiac. Ptolomaeus taught that their significance came from an analogy with the ratios of the musical scale. The conjunction corresponds to unison; the opposition divides the circle in a 1:2 ratio (octave); the sextile (5:6) corresponds to a minor third; the square (3:4) corresponds to a perfect fourth; and the trine (2:3) to a perfect fifth.
- 17.
One may recall that the master of the great Plotinus was Ammonius Sacca, a philosopher of Indian origin, perhaps, as asserted by Jean Daniélou [69], a Buddhist monk, who naturally tended toward mystical and magical practices, which would become the carriers of a pernicious infection that affected Western thought for centuries. In metaphorical terms, science returned from Athens to Babylon.
- 18.
In order to transform Constantinople into the new cultural centre of the Roman Empire, Theodosius II founded there a university with 31 chairs. In 866 A.D. the school was reorganised and transferred to the Magnaura Palace by Bardas the learned and munificent Caesar, uncle of Michael III. The University of Constantinople consisted of four faculties: Grammar, Physics, Mathematics and Medicine. Excellent teachers were hired and generously remunerated by the State. The lessons and courses, held in Greek as well as in Latin, were free. While this process was taking place in Constantinople, Athens had decayed to the rank of a peripheral provincial town. However, pagan culture was still very much alive there, so that Justinian’s edict suppressing the Athens School was not merely symbolic, but rich in historical consequences.
- 19.
“Philosophers are searching for peace, but are engineers to secure peace with the launching precision of their catapults” (Philo of Byzantium , “Poliorcetica”).
- 20.
In the East, the Persian Empire was separated from Asia by the deserts of Kavir and Lut which extend from north to south for approximately 1000 km. The coastal road to Asia along the gulf of Oman crosses the desert-like, practically uninhabited and impracticable territories of Gedrosia. In the past, Persian armies had never succeeded in reaching the Indus valley via this road (Arrianus reports how Alexander, who decided to cross it with half of his army upon returning from his war campaign at the Indus mouth, risked perishing in those sandy solitudes, alternately battered by barren winds and terrible monsoonal hurricanes). The only easy access to India was to the north, following a long path that started from the Caspian Sea and, along the southern versant of the Elburz chain, until Herat; at that point a difficult mountain road towards the south-east led to Kandahar and, through the pass of Bolan, to arrive in western Pakistan. The main road to India, however, proceeded from Herat towards the north of Afghanistan and, following the border of Turkmenistan, reached the upper course of the Amu Darya (the ancient Oxus) up to Balkh, after which it turned south, towards the valley of Kabul and, finally, through the Khyber Pass, reached Peshawar, at the upper Indus water way. The spread of Islam to the East took place following this tortuous path, on which splendid towns flourished at that time, such as Marv, Termez, Bukhara, Balkh and, further east, Samarcand.
- 21.
Sarton decided to learn Arabic and then undertook a series of lengthy travels to the Middle East in order to rediscover and translate original manuscripts.
- 22.
What Mohammed knew of the Christian religion he had learnt from gnostic texts, which circulated in Christian communities of Nefud.
- 23.
The eventual evolution of Islam has been marked by an increase in the influence of Gnostic-Manichean ideas, which flowed into the political movement of Mahdism and into modern Islamic radicalism. The ominous aspects of this evolution have been thoroughly examined by Laurent Murawiec [95], who showed how these dark undercurrents are also present in European culture, where, however, they found barriers which impeded fantastic intellectual constructions and pseudo-scientific knowledge from becoming influential in Western society.
- 24.
On this subject it is interesting to note how Protestantism, which is nearer to gnostic Manichaeism than Catholicism, has encountered less difficulty than the latter in avoiding conflicts involving scientific theories apparently in contradiction to theological pronouncements.
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Ronchi, C. (2014). Time of Growth. In: The Tree of Knowledge. Springer, Cham. https://doi.org/10.1007/978-3-319-01484-5_1
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