Leibniz's Definition of Monad

N euroQ uantology | September 2006 | Vol. 4 | Issue 3 | Page 249-251 Kutateladze, SS. Leibniz’s definition of monad 249 Perspectives Lei bni z's Def i ni t i on of Monad Semen S. Kut at el adze 1 Abstract T his is a shor t discussion of t he definit ion of monad which was given by G. W . Leibniz in his Monadology. Key W ords: point, monad, Leibniz, micr oscope, nonstandar d analysis N euroQ uantology 2006; 4: 249-241 Acquired traits are never inherited. T his law of genetics determines many aspects of public life. Mankind creates and supports complicated social institutions for transferring to the young generations the experience of their ancestors. As biological species, we differ little from our paleolithic predecessors. So we may hope to comprehend the thoughts and ideas that are bequeathed to us by the greatest minds of the past epochs. T he outlook of Leibniz, proliferating with his works, occupies a unique place in human culture. W e can hardly find in the philosophical treatises of his predecessors and later thinkers something comparable with the phantasmagoric conceptions of monads, the special and stunning constructs of the world and mind which precede, comprise, and incor por ate all the infinite advents of the eter nity. M onadolody (Leibniz, 1992) is usually dated as of 1714. This article was never published during Leibniz's life. Moreover, it is generally accepted that the ver y term "monad" had appeared in his writings since 1690 when he was already an established and prominent scholar. T he special attention to the origin of the term "monad" and the particular investigation into the date of its first appearance in the works by Leibniz are in fact the present day products. There are now a few if any cultivated persons who never got acquaintance with the basics of planimetr y and heard nothing of Euclid. H owever, no one has ever met the concept of "monad" on the school bench. N either the contemporar y translations of Euclid's Elements nor the popular school textbooks contain this seemingly exotic term. H owever, the concept of "monad" is fundamental not only for Corresponding author :1Semen S. Kutateladze Address: Sobolev Institute O f Mathematics, Siberian D ivision of The Russian Academy of Sciences, N ovosibirsk, RUSSIA e-mail: sskut@ member.ams.org ISSN 1303 5150 www.neuroquantology.com N euroQ uantology | September 2006 | Vol. 4 | Issue 3 | Page 249-251 Kutateladze, SS. Leibniz’s definition of monad Euclidean geometr y but also for the whole science of the Ancient H ellada. By D efinition I of Book VII of Euclid's Elements (Euclid, 1949) a monad is "that by vir tue of which each of the things that exist is called one." Euclid proceeds with D efinition 2: "A number is a multitude composed of monads." N ote that the present day translations of the Euclid treatise substitute "unit" for "monad." A contemporar y reader can hardly understand why Sextus Empiricus, an outstanding scepticist of the second centur y, wrote when presenting the mathematical views of his predecessors as follows (Sextus Empiricus, 1976): "Pythagor as said that the or igin of the things that exist is a monad by vir tue of which each of the things that exist is called one." And furthermore: "A point is structured as a monad; indeed, a monad is a cer tain origin of numbers and likewise a point is a certain origin of lines." N ow some place is in order for the excerpt which can easily be misconceived as a citation from Monadology: "A whole as such is indivisible and a monad, since it is a monad, is not divisible. O r, if it splits into many pieces it becomes a union of many monads rather than a [simple] monad." It is worth obser ving that the ancients sharply perceived an exceptional status of the star t of counting. In order to count, one should firstly par ticularize the entities to count and only then to proceed with putting these entities into correspondence with some symbolic series of numerals. W e begin counting with making "each of the things one." The especial role of the start of counting is reflected in the almost millennium long dispute about whether or not the unit (read, monad) is a natural number. W e feel today that it is excessive to distinguish the key role of the unit or monad which signifies the start of counting. H owever, this was not always so. From the times of Euclid, all serious scientists knew about existence of the two basic concepts of mathematics: a point and a monad. By D efinition 1 of Book 1 of Euclid's Elements: "A point is that which has no ISSN 1303 5150 250 par ts." C learly this definition differs drastically from the definition of monad as that which makes one from many. T he cornerstone of geometr y is other than that of arithmetic. W ithout clear understanding of this circumstance it is impossible to comprehend the essence of the views of Leibniz. By the way, the modern set theor y refers to "that which has no par ts" as the empty set, the star ting cardinal of the von N eumann universe. T he present day mathematics seems to have no concept that is vocalized as "that which many make into one." W e will return to the modern mathematical definition of monad shor tly. Attempting to pursue the way of Leibniz's thought, we must always keep in mind that he was a mathematician by belief. From his earliest childhood, Leibniz dreamed of "some sor t of calculus" that operates in the "alphabet of human thoughts" and possesses the same beauty, strength, and integrity as mathematics in solving arithmetical and geometrical problems. Leibniz devoted many ar ticles to invention of this universal logical calculus. T he diversity and even polarity of the views of these writings proceed along with the universally accepted appraisal of Leibniz as a key figure of the prehistor y of the modern mathematical logic. Monadology is listed alongside the classical achievements of Leibniz which we express with the words culculamus and differentia. Leibniz always emphasized his love and devotion to mathematics. H e stressed constantly that his general methodological views base on "study into the methods of analysis in mathematics which I was engrossed in with such an eager that I do not know whether it is possible to find many who ser ved it with more toil. As a top mathematician of his age, Leibniz was in full command of Euclidean geometr y. T herefore, we are upmost bewildered already to read Item 1 of his Monadolody where he gave the first impression about his monad: "The Monad, of which we shall here speak, is nothing but a simple substance, which enters into compounds. By 'simple' is meant www.neuroquantology.com N euroQ uantology | September 2006 | Vol. 4 | Issue 3 | Page 249-251 Kutateladze, SS. Leibniz’s definition of monad 'without par ts."’ T his definition of monad as a "simple" substance without parts coincides with the Euclidean definition of point. At the same time the reference to compounds consisting of monads reminds us the structure of the definition of number which belongs to Euclid. The synthesis of both primar y definitions of Euclid in the Leibnizian monad is not accidental. W e must always bear in mind that the seventeenth centur y is the epoch of microscope. It was already in the 1610s that microscopes were massproduced in many European countries. From the 1660s Europe was enchanted by Antony van Leeuwenhoek's microscope. Let us make a mental experiment and aim a strong microscope at a region about a point at a mathematical line. W e will see in the eyepiece a blurred and dispersed cloud with unclear frontiers which is a visualization of the point under investigation. Under greater magnification, the portion of the "point-monad" we are looking at will enlarge, revealing extra details whereas disappearing partially from sight. H owever, we are still inspecting the same standard real number which you might prefer to percept as descr ibed by this process of "studying the microstructure of a physical straight line." Visualizing a point by microscope reveals its monadic essence. Leibniz could reason so or approximately so. In any case, the view of the monad of a standard real number as the collection of all infinitely close points is generally adopted in the contemporar y infinitesimal analysis resurrected under the name of nonstandard analysis in the works by Abraham Robinson in 1961. ISSN 1303 5150 251 References Leibniz G W . C ollected W orks. M oscow: Mysl, 1992;1:413-428. Euclid. Elements, In T hree Volumes. Moscow and Leningrad: Gostekhizdat. 1949. Sextus Empiricus. C ollected W orks. Vol. 1. M oscow: Mysl. 1976. Robinson A. N on-Standard Analysis. Pr inceton: Pr inceton U niver sity Press. 1996. www.neuroquantology.com