Principle of perfection

Also called the principle of the best.

Principle of German philosopher and mathematician Gottfried Wilhelm Leibniz (1646-1716) that the actual world is the best of all possible worlds.

Bertrand Russell (1872-1970) argued that Leibniz did not fully distinguish this principle from that of sufficient reason.

Source:
B Russell, A Critical Exposition of the Philosophy of Leibniz (1900), §§14-15

Term and concept

Aristotle

The form of the word long fluctuated in various languages. The English language had the alternates, “perfection” and the Biblical “perfectness.”^[2] The word “perfection” derives from the Latin “perfectio“, and “perfect” — from “perfectus“. These expressions in turn come from “perficio” — “to finish”, “to bring to an end.” “Perfectio(n)” thus literally means “a finishing”, and “perfect(us)” — “finished”, much as in grammatical parlance (“perfect”).^[2]

Many modern languages have adopted their terms for the concept of “perfection” from the Latin: the French “parfait” and “perfection“; the Italian “perfetto” and “perfezione“; the Spanish “perfecto” and “perfección“; the English “perfect” and “perfection”; the Russian “совершенный” (sovyershenniy) and “совершенcтво” (sovyershenstvo); the Croatian and Serbian “savršen” and “savršenstvo“; the Czech “dokonalost“; the Slovak “dokonaly” and “dokonalost“; the Polish “doskonały” and “doskonałość.”^[2]

The genealogy of the concept of “perfection” reaches back beyond Latin, to Greek. The Greek equivalent of the Latin “perfectus” was “teleos.” The latter Greek expression generally had concrete referents, such as a perfect physician or flutist, a perfect comedy or a perfect social system. Hence the Greek “teleiotes” was not yet so fraught with abstract and superlative associations as would be the Latin “perfectio” or the modern “perfection.” To avoid the latter associations, the Greek term has generally been translated as “completeness” rather than “perfection.”^[3]

The oldest definition of “perfection”, fairly precise and distinguishing the shades of the concept, goes back to Aristotle. In Book Delta of the Metaphysics, he distinguishes three meanings of the term, or rather three shades of one meaning, but in any case three different concepts. That is perfect:

1. which is complete — which contains all the requisite parts;

2. which is so good that nothing of the kind could be better;

3. which has attained its purpose.^[4]

The first of these concepts is fairly well subsumed within the second. Between those two and the third, however, there arises a duality in concept. This duality was expressed by Thomas Aquinas, in the Summa Theologica, when he distinguished a twofold perfection: when a thing is perfect in itself — as he put it, in its substance; and when it perfectly serves its purpose.^[4]

The variants on the concept of perfection would have been quite of a piece for two thousand years, had they not been confused with other, kindred concepts. The chief of these was the concept of that which is the best: in Latin, “excellentia” (“excellence”). In antiquity, “excellentia” and “perfectio” made a pair; thus, for example, dignitaries were called “perfectissime“, just as they are now called “excellency.” Nevertheless, these two expression of high regard differ fundamentally: “excellentia” is a distinction among many, and implies comparison; while “perfectio” involves no comparison, and if something is deemed perfect, then it is deemed so in itself, without comparison to other things. Gottfried Wilhelm Leibniz, who thought much about perfection and held the world to be the best of possible worlds, did not claim that it was perfect.^[5]

Paradoxes

Vanini

The parallel existence of two concepts of perfection, one strict (“perfection,” as such) and the other loose (“excellence”), has given rise, perhaps since antiquity but certainly since the Renaissance, to a singular paradox: that the greatest perfection is imperfection. This was formulated by Lucilio Vanini (1585–1619), who had a precursor in the 16th-century writer Joseph Juste Scaliger, and they in turn referred to the ancient philosopher Empedocles. Their argument, as given by the first two, was that if the world were perfect, it could not improve and so would lack “true perfection,” which depends on progress. To Aristotle, “perfect” meant “complete” (“nothing to add or subtract”). To Empedocles, according to Vanini, perfection depends on incompleteness (“perfectio propter imperfectionem“), since the latter possesses a potential for development and for complementing with new characteristics (“perfectio complementii“). This view relates to the baroque esthetic of Vanini and Marin Mersenne: the perfection of an art work consists in its forcing the recipient to be active—to complement the art work by an effort of mind and imagination.^[6]

The paradox of perfection—that imperfection is perfect—applies not only to human affairs, but to technology. Thus, irregularity in semiconductor crystals (an imperfection, in the form of contaminants) is requisite for the production of semiconductors. The solution to the apparent paradox lies in a distinction between two concepts of “perfection”: that of regularity, and that of utility. Imperfection is perfect in technology, in the sense that irregularity is useful.^[7]

Perfect numbers

Perfect numbers have been distinguished ever since the ancient Greeks called them “teleioi.” There was, however, no consensus among the Greeks as to which numbers were “perfect” or why. A view that was shared by Plato held that 10 was a perfect number.^[8] Mathematicians, including the mathematician-philosopher Pythagoreans, proposed as a perfect number, the number 6.^[8]

The number 10 was thought perfect because there are 10 fingers to the two hands. The number 6 was believed perfect for being divisible in a special way: a sixth part of that number constitutes unity; a third is two; a half — three; two-thirds (Greek: dimoiron) is four; five-sixths (pentamoiron) is five; six is the perfect whole. The ancients also considered 6 a perfect number because the human foot constituted one-sixth the height of a man, hence the number 6 determined the height of the human body.^[8]

Euclid

Thus both numbers, 6 and 10, were credited with perfection, both on purely mathematical grounds and on grounds of their relevance in nature.^[8] Belief in the “perfection” of certain numbers survived antiquity, but this quality came to be ascribed to other numbers as well. The perfection of the number 3 actually became proverbial: “omne trinum perfectum” (Latin: all threes are perfect). Another number, 7, found a devotee in the 6th-century Pope Gregory I (Gregory the Great), who favored it on grounds similar to those of the Greek mathematicians who had seen 6 as a perfect number, and in addition for some reason he associated the number 7 with the concept of “eternity.”^[8]

The Middle Ages, however, championed the perfection of 6: Augustine and Alcuin wrote that God had created the world in 6 days because that was the perfect number.^[8]

The Greek mathematicians had regarded as perfect that number which equals the sum of its divisors that are smaller than itself. Such a number is neither 3 nor 7 nor 10, but 6, for 1 + 2 + 3 = 6.^[8]

But there are more numbers that show this property, such as 28, which = 1 + 2 + 4 + 7 + 14. It became customary to call such numbers “perfect.” Euclid gave a formula for (even) “perfect” numbers:

N_p = 2^p−1 (2^p − 1)

where p and 2^p − 1 are prime numbers.^[8]

Euclid had listed the first four perfect numbers: 6; 28; 496; and 8128. A manuscript of 1456 gave the fifth perfect number: 33,550,336. Gradually mathematicians found further perfect numbers (which are very rare). In 1652 the Polish polymath Jan Brożek noted that there was no perfect number between 10⁴ and 10⁷.^[9]

Despite over 2,000 years of study, it still is not known whether there exist infinitely many perfect numbers; or whether there are any odd ones.^[9]

Today the term “perfect number” is merely historic in nature, used for the sake of tradition. These peculiar numbers had received the name on account of their analogy to the construction of man, who was held to be nature’s most perfect creation, and above all on account of their own peculiar regularity. Thus, they had been so named on the same grounds as perfect objects in nature, and perfectly proportioned edifices and statues created by man; the numbers had come to be called “perfect” in order to emphasize their special regularity.^[9]

The Greek mathematicians had named these numbers “perfect” in the same sense in which philosophers and artists used the word. Jamblich (In Nicomachi arithmeticam, Leipzig, 1894) states that the Pythagoreans had called the number 6 “marriage,” “health,” and “beauty,” on account of the harmony and accord of that number.^[9]

The perfect numbers early on came to be treated as the measure of other numbers: those in which the sum of the divisors is greater than the number itself, as in 12, have — since as early as Theon of Smyrna, ca. 130 A.D. — been called “redundant” (Latin: redundantio), “more than perfect” (plus quam perfecti), or “abundant numbers”, and those the sum of whose divisors is smaller, as in 8, have been called “deficient numbers” (deficientes).^[9]

As of 7 December 2018, 51 perfect numbers had been identified.