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MATHEMATICS OF THE ENLIGHTMENT

Lomonosov is the Russian titan of the epoch of scientific giants. Lomonosov was not a mathematician, but without mathematicians there would be no Lomonosov as the first and foremost Russian scientist at all.

The mathematical ideas of the second half of the seventeenth century and the first half of the eighteenth century greatly affected the scientific views of Lomonosov. His outlook was formed under the influence of his contemporaries Newton, Leibniz, Wolff, and Euler—the intellectual leaders of the Enlightenment.

**Foundation of the Russian Academy.**
Science in Russia had started with the foundation of the Academy of Sciences and Arts
which then evolved into the Russian Academy of Sciences of these days.
The turn of the sixteenth and seventeenth is a signpost of the
history of the mankind, the onset of the organized science.
The time of the birth of scientific societies and academies
accompanied the revolution in the natural sciences which
rested upon the discovery of differential and integral calculus.
The new language of mathematics brought about
an opportunity to make impeccably precise predictions of
future events.

To the patriotism of Peter the Great and the cosmopolitanism of Leibniz
we owe the foundation of the Saint Petersburg Academy of Sciences as
the center of Russian science. Peter and Leibniz stood at the cradle
of Russian science in much the same way as Euler and Catherine I
are the persons from whom we count the history of the national mathematical school
in Russia. We must also acclaim the outstanding role
of Leibniz who prepared for Peter a detailed plan of organizing academies
in Russia.
Leibniz viewed Russia as a bridge for connecting Europe with China whose
Confucianism would inoculate some necessary ethical principles for
bringing moral health to Europe.
Peter wanted to see Leibniz as an active organizer of the Saint Petersburg Academy,
he persuaded Leibniz in person and appointed Leibniz a *Justizrat* with a lavish salary.

It is worth observing that Peter visited the Royal Mint in London
in 1698 during the so-called “Grand Embassy.”
At that time Newton was Warden of the Mint and we can hardly imagine that
he ignored Peter's visits. Nevertheless there is no evidence that Peter met
Newton. It is certain that Jacob Bruce, one of the closest associates of Peter,
had discussions with Newton.
In 1714, two years after Peter made Leibniz a *Justizrat*,
Aleksandr Menshikov applied for membership in the London Royal Society, which was
an extraordinary and unpredictable event. What is more mysterious, Menshikov's
application was approved and he was notified of his new status by a letter from Newton
himself.

**Newton, Leibniz, and Revolution in Mathematics.**
The genius of Newton has revealed to the human race the mathematical
laws of nature and disclosed to a mathematician
a universal language for describing the ever-changing world.
The genius of Leibniz has pointed out to the mankind the opportunities
of mathematics as a reliable method of reasoning, the genuine logic
of human knowledge.
Leibnizian ideas of *mathesis universalis* and *calculemus*
arose once and forever as a dream and instrument of
science.

The influence of the ideas of Newton and Leibniz resulted in the scientific outlook of the epoch. The revolt of the natural sciences at the turn of the seventeenth and eighteenth centuries was determined by the invention of differential and integral calculus. The competing ideas of the common mathematics of Newton and Leibniz determined all principal trails of thought of the intellectual search of the epoch. The contribution of Lomonosov exhibits a convincing example of the general trends. To grasp the scientific approaches of Lomonosov and understand his creative revelations and naive delusions is impossible without deep analysis of and thorough comparison between the views of Newton and Leibniz.

The monads of Leibniz as well as the fluxions and fluents of Newton are products of the heroic epoch of the telescope and microscope. The independence of the discoveries of Leibniz and Newton is obvious, since their approaches, intellectual backgrounds, and intentions were radically different. Nevertheless, the groundless priority quarrel between Leibniz and Newton has become the behavioral pattern for many generations of scientists. Leibniz and Newton discovered the same formulas, part of which had already been known. Leibniz, as well as Newton, had his own priority in the invention of differential and integral calculus. Indeed, these scientists suggested the versions of mathematical analysis which were based on different grounds. Leibniz founded analysis on actual infinitesimals, erecting the tower of his perfect philosophical system known as monadology. The key technique of Newton was his method of “prime and ultimate ratios”—a kinetic predecessor of the modern limit theory.

The Leibnizian stationary vision of mathematical objects counterpoises the
Newtonian dynamical perception of ever-changing variable quantities.
The source of the ideas by Leibniz was the geometrical views
of antiquity which he was enchanted with from his earliest infantry.
The monad of Euclid is the mathematical tool of calculus, presenting a twin
to the point, the atom of geometry. Mathematics of Euclid is the product
of the human spirit. The monads of Leibniz, nurtured by his dream of *calculemus*,
are the universal instrument of creation whose understanding
brings a man to the divine providence in creating the best of all possible worlds.
Newton got acquaintance with Euclid only in his ripe years and so he traveled in his own way,
perceiving universal motion as something done at the creation of the world that
could thus never be reduced to any sum of states of rest.

Newton was the last scientific magician, and Leibniz was the first mathematical dreamer.

**Leibniz's Monadology.**
The outlook of Leibniz, proliferating with his works,
occupies a unique place in human culture.
We 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
incorporate all infinite advents of the eternity.

It is worth emphasizing that mathematics was the true source of the philosophical ideas of Leibniz.

As a top mathematician of his epoch, Leibniz
was in full command of Euclidean geometry.
Therefore, rather bewildering is Item 1 of Leibniz's *La Monadologie*
where Leibniz gave the first idea of what his monad actually is:
“The Monad, of which we shall here speak, is nothing but a
simple substance, which enters into compounds. By ‘simple’ is meant
‘without parts.’”
This definition of monad as a “simple” substance
without parts coincides with the Euclidean definition of
point. At the same time the reference to the compounds consisting of monads
reminds us the structure of the definition of number which belongs to Euclid.

To understand the outlook of Leibniz and the attraction of his ideas for natural sciences, we must bear in mind that Leibniz was a mathematician by belief. From his earliest childhood, Leibniz dreamed of “some sort 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 articles to invention of this universal logical calculus.

**Wolff, the Teacher of Lomonosov.**
The teacher of Lomonosov was Christian Wolff, an ardent propagator of the ideas of monadism
and the mathematical method. Wolff was considered by his contemporaries as
the second figure after Leibniz in the continental science.
The first figure of Misty Albion was Newton.
It is impossible to forget that the intellectual life of that epoch was
heavily contaminated with the nasty controversy about priority between
Newton and Leibniz.
The deplorable consequence of the confrontation was the stagnation
and isolation of the mathematical life of England.
As regards the continent, the slight but perceptible neglect
to the contribution by Newton led to dogmatization and canonization
of the teaching of Leibniz which was often understood with distortions.

Wolff was an epigone rather than a follower of Leibniz. The true disciples of Leibniz were Jacob, Jean, and Jacques Bernoulli as well as Euler who was a self-taught prodigy close to Bernoulli by the vogue and understanding of life. Euler devoted much time to opposing Wolff and Wolffians.

Note that Wolff was the trendsetter in matheatical education on the continental Europe of
the beginning of the eighteenth century.
After Leibniz's refusal to transfer to Saint Petersburg
for organizing the Academy, Peter considered Wolff as its possible leader.
Wolff's treatise *Der Anfangsgrüunde aller mathematischen Wissenshaft*
was published in four parts in 1710, abridged later for a wider readership,
and reprinted many times.

**Lomonosov and Wolffianism.**
The educational views of Wolff were
well accepted by Lomonosov, since Wolff and he were connected
with the warm feelings of mutual respect.
Wolff's mathematical method was a basis of
Lomonosov's scientific articles during many years of his creativity.
It should be observed that , unlike Wolff who had excellent
mathematical training, Lomonosov was not sufficiently well acquainted
even with Euclid's * Elements*, and he never possessed a working knowledge
of differential and integral calculus.
We must emphasize that Lomonosov never met
Euler. These circumstances explain to us why we can hardly find any practical
applications of mathematics in the papers of Lomonosov and why
some of his thoughts about the nature of mathematical knowledge
are naive and incorrect.

The attitude of Lomonosov to monads deserves a thorough examination.
Developing the atomistic ideas of corpuscular physics in his papers
of 1743 and 1744, as well as in his extensive correspondence, Lomonosov
sparingly use the concept of monad, especially distinguishing
*monades physicae*.
The physical monads of Lomonosov are closer to the conception of atoms
rather than mathematical monads or Leibnizian ideal monads.
Long-term personal contemplations over the structure of the matter led Lomonosov
to inventing his “corpuscular philosophy” close to molecular theory.

In February of 1754 Lomonosov wrote to Euler: “I am absolutely convinced that this mystical teaching must be completely destroyed by my arguments, I am afraid to spoil the elder years of the scholar whose benefactions to me I could not forget; otherwise I would not be scared to tease hornet-monadists throughout the whole of Germany.”

The man of a practical inclination, Lomonosov could not remain within the narrow limits of Wolffianism for ever. The real-world, sensory, and instructive experience drastically shifted aside the ideas of mathematical rationality, harmony, and beauty of the universal primitive cause in the methodological views and approaches of Lomonosov.

**Lomonosov and the Present Day.**
The scientific outlook of Lomonosov was based
on the mathematical ideas of the Enlightenment which
stemmed from the antique atomism.
The new mathematics was born as differential and integral calculus.
Differentiation discovers trends, and integration forecasts the future from trends.
The Christian ontology together with the microscope and the telescope
became the source of the scientific revolution in understanding the universe.
Leibniz's *La Monadologie* and Newton's *Philosophiae Naturalis Principia Mathematica*
changed the antique views of the atom—the indivisible material particle and the
monad—the original act of rigorous thought.

The physical outlook of the twenty first century has little in common with the atomism of the ancients. We grasp the laws of the microcosm within the quantum-mechanical conceptions and the uncertainty principle which are not reflected adequately in the Aristotle logic. Mathematics undergoes the revolutionary refusal from conservatism and categoricity. The freedom of modern mathematics does not reduce to the absence of exogenous limitations of the objects and methods of research. Mathematics and physics have grasped the new frontiers of their competence and demarcated the zones of mutual responsibility and the spheres of independent interests. The realities of the contemporary science cast new light on the Lomonosov contribution to the world's culture.

Characterizing Lomonosov as a “great champion of the great Peter,” Alexander Pushkin—the paragon and shrine of the Russian spirit—noticed: “Combining his extraordinary force of will with his extraordinary force of understanding, Lomonosov embraced all areas of the age of Enlightment. Craving for science constituted the strongest passion of his soul fraught with many passions. As a historian, rhetorician, mechanician, chemist, mineralogist, artist, and poet, he tried everything and travelled everywhere: He was the first who plunged into the history of our fatheland; he propounded the rules of public language; he gave the rules and examples of classical eloquence; he foresighted the discoveries of Franklin with the unfortunate Richman; and he founded an factory, made sculptures, and enriched arts with mosaic images. He crowned all these with opening to us the genuine sources of our poetical language.”

More than two dozens of decades elapsed from the death of Mikhailo Lomonosov, but his creative contribution still inspires thought in connection with the most topical and brand-new areas of mathematics and natural sciences. His enviable fate gives a supreme example for drafting life.

April 17, 2011

J. Appl. Indust. Math., 2011, V. 5, No. 2, 155–162.

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© Kutateladze S. S. 2011