A Gottfried of All Trades:
Leibniz and His Trans-Field Exploits
Andrew Nauffts, Benjamin Miller, and Paige Pendleton
History of Physical Sciences
April 21st, 2015
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Gottfried Wilhelm Leibniz was a man of immense intellect only mirrored by his thirst for
knowledge. His entire life’s works all stemmed from the same logic he used to explain the way
the world itself worked. Even though his works ranged from philosophy to mathematics to
politics and beyond, his same personal logic was the birthplace of it all. He saw reality through
his belief in everything falling into either absence or existence, zero or one. Everything had a
purpose and a logical means of process to Leibniz, it was discovering the process itself using his
logic which would make him famous. His works would later be the origins of Binary Code,
Calculus, metaphysics, and monadology. Even more things in modern day life can in some way
trace their origins to the works of Leibniz, so much so the list would be nearly impossible to
create. This case study will discuss the controversy surrounding his most notable achievement,
the discovery of the differential calculus, as well as his famous philosophies which mark him as
a great philosopher of the ages. All great men must start off as children however, and the journey
of their maturation often holds clues to how they became who they were. Leibniz was no
exception.
A Lifetime of Work: A Biography of Leibniz
Born on July 1st, 1646, Gottfried Wilhelm Leibniz would find himself a Philosopher,
Mathematician, a Doctor of Law, and Political Adviser before his death in 1716. Successful in
each of these professions, he would often juggle several of them at once giving opportunity to
fall directly in to the time’s equivalent of the conceptualized old-boy network. His exploits
would range in fields stretching even farther beyond these professions, making Leibniz a
Renaissance man of sorts. A man of great intellect, Leibniz pushed himself constantly to refine
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anything he was working on to a point of perfection through a highly systematic process before
ever publishing (Jolley, 9). Surprisingly, school wasn’t Leibniz’s path to genius. He was mostly
self-taught within his father’s library. In an odd twist of fate, the untimely death of Leibniz’s
father when Leibniz was only 6 years old may be responsible for the development of one of the
greatest minds in history. This is because Leibniz would only gain access to his father’s library
after he passed.
Prior to his demise, Leibniz’s father had been teaching Leibniz material well beyond his
age, mostly in the fields of history and philosophy. Leibniz would be brought to the University of
Leipzig with his father to learn almost every day, and after his father’s death, he was cut off from
learning this advanced material. Soon after he would be reprimanded by a school teacher for
reading books he had taken from his father’s library, as they were not suitable for a child. His
mother, in response, decided to keep him out of the library left by her husband. However, when
he was 8, a family friend took interest in the sheer intelligence shown by Leibniz, and convinced
his mother to allow him full access to the library. From age 8 on, he was exposed to Latin
classics, metaphysical works, and theological manuscripts. The works of Cicero, Pliny, Seneca,
Herodotus, Xenophon, Plato and others were at his fingertips. Leibniz would learn much from
this, and his entire world view would be shaped by the time spent in the library, and if it weren’t
for events transpiring in the way they did, he may never have developed the Leibnizian logic
which spawned calculus and binary code alike
By 1664, Leibniz would hold a degree of Master of Philosophy from the University of
Leipzig, and in 1666 he was rejected for a degree of doctor of law and decided to leave his
hometown of Leipzig, Saxony forever. He soon received his doctor of law degree in Nürnberg,
from the University of Altdorf. Receiving this degree would be the beginning of a series of
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events landing Leibniz in Paris by 1672 as a political diplomat (Duigan, 165). However, his
financier died, and Leibniz found himself without any financial support. In response to this,
Leibniz came up with a plan to both gain him financial support but also a better foothold within
the academic world (Ibid). He constructed a calculating machine to present to the Royal Society
in England, who made him an external member of the society in 1673 as a result. The Leibniz
wheel, as it was called, would later become a key component of the Arithmometer, the first
mass-produced mechanical calculator in history.
Following this came his discovery of the differential calculus in 1675. Between this time
and publishing his work on the differential calculus in 1684, Leibniz worked on theories in
regards to metaphysics and motion which would become further sources of controversy because
much of this work would not be published (Duigan, 165-6). His published philosophical works
caused some controversy as well. His published philosophical writings outlining the concept of
this world being the best possible of all worlds was an example of this. Stemming from his work
Principles of Sufficient Reason, Leibniz mapped out a series of constructs which, when
combined with his belief in God as being necessary, led him to create this idea of an imperfect
and yet best possible world. Even after his passing, this work was still receiving criticism,
including Voltaire’s Candide in 1759. Candide critiqued several different aspects of society as
well as several different philosophies and ideas, but the overall scheme of the book revolved
around attacking this notion of the world being the best it could possibly be through the events
experience by the main character and his love interest throughout the book.
An interdisciplinary thinker, his various philosophical theories have heavy influences
from his works in mathematics, motion, and metaphysics. The relationship shared between them
created a cohesive Leibniz model of the world in such a way which caught the eye of more than
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just the Royal Society. By 1700, Leibniz became a foreign member of the Academy of Science
of Paris as well. For the next and final 16 years of his life, much of his work was political.
Before his death in 1716, Leibniz earned the title of baron and became an official advisor to the
Austrian empire. However, in 1714 he would manage to publish 2 philosophical works,
Monadology and “Principles of Nature and Grace.” These would be his last works ever
published; Monadology would become one of the most recognizable works of his later
philosophy. Monadology would not only go on to influence metaphysics but it also was the first
monad based theorem, even though the word monad was never used(Duigan, 127; Jolley, 66-8).
The final 2 years of his life leading up to his death in 1716 would not be pleasant. His health
began to deteriorate rapidly from severe gout and he found himself less mobile by the day until
he was eventually bed ridden. These last 2 years would be spent in Hanover, and his death would
not be as momentous as it most likely should have been.
During his lifetime Leibniz made friend with many people or nobility and royalty from
all over Europe. He found himself tutoring the children Queen Sophie Charlotte of Prussia at her
personal request, a political advisor to multiple governments including the Austrian and Prussian,
he would be visited by men such as Peter the Great in his own home, and even was made a baron
out of shear gratitude from the Austrian emperor. However the various controversies surrounding
Leibniz eventually caught up to him, and by the time of his death many of those who once
praised him no longer viewed him a great man, excluding the more local Germanic royalty. It
made sense why the northern German and other nearby kingdoms wouldn’t turn their back on
Leibniz, due to him being a correspondent and friend to much of the noble families within the
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region. Despite his quiet death, Leibniz still went on to be considered a member of the “pantheon
of seven great philosophers.”1
A Feud between Giants: Who Discovered the Differential Calculus?
Leibniz was a fairly popular man, often rubbing elbows with royalty and intellectuals
alike, but one man, Isaac Newton, despised him. Subsequently, a feud spanning several decades
ensued over who truly discovered the differential calculus. It began with Leibniz’s time in Paris
from 1672-1676. He had no proper mathematical training and was practicing law at the time.
However, he was extremely drawn to the ideas of mathematicians. In this short time span, he
applied his logic and intellect to the compiled works of his predecessors to create calculus. His
notation system was “completely original and ingenious (Bardi, 9).” The decade following the
initial discovery was spent working out his system and symbols through the refinement of his
work. Leibniz himself devised the name calculus, stemming from the Latin word for a pebble
used in various roman tools with the purpose of counting (Bardi 9). The most notable of which
being the small stones on Roman abacuses known as calculi and a tool used to measure distance
in a similar fashion to an odometer which would drop a calculus (pebble) after each rotation of a
wheel across a distance. Leibniz had two scholarly papers of his calculus works published in
1684 and 1686. These two papers serve as the primary evidence in the argument for Leibniz
being the true creator of calculus. The following 2 decades were spent refining his work through
examination, consulting his equals, and studying the math of others – all in the name of
improving his calculus.
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Ross, page 1
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It was during the time he spent examining the math of others where Leibniz would
eventually come across Newton’s Opticks. It was here, in Opticks, where Newton attempted to
lay claim to the title of discoverer of the differential calculus. In 1704, when Newton published
“On the Quadrature of Curves” in the back of Opticks, it was his attempt to take back the
ownership of his invention. For 20 years Leibniz had been acknowledged as the king of calculus,
enjoying the glory he was receiving while Newton quietly stewed with anger. Newton was even
deemed a plagiarizer of Leibniz by some (Bardi, 9).
Jason Bardi, in his work The Calculus Wars, says:
This short treatise in the back of Opticks marked the quiet beginning of the calculus wars
because it was the light that revealed the long-hidden feelings of jealousy and resentment
between Leibniz and Newton. Newton had suffered in quiet humiliation for years with
the knowledge that he was the first inventor of calculus, and he was a smoldering fire
ready to be released into the flames. (Bardi, 10)
Newton felt this way because he allegedly came up with the idea of calculus in 1666 when he
was isolated at his parent’s estate during the outbreak of the Bubonic Plague. He planned on
publishing his work on calculus in De Analysi around the same time of his work in the field of
optics but Newton swore off publishing after the rejection he received from a letter he wrote
titled “New Theory about Light and Colors.” He didn't realize the effects the printing of this
would have…the controversy was immense. He didn't account for the time his
contemporaries would need to grasp the concept, especially considering it took years to grapple
with it himself. There would also be incredible resistance from those whose theories he would be
replacing. (Bardi, 4). Essentially, his letter “opened up a new dialogue, and he became
embroiled in bitter fights with [Robert] Hooke and others over his new theories-so much so that
he swore off publishing for decades. He even once told one of his colleagues that he would rather
wait until he died for his works to be published (Bardi, 5).” These feuds would lead Newton to
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some dark places, including taking and hiding Hooke’s work and notes immediately following
his death, delaying physics’ knowledge regarding springs for several years until these notes were
later discovered.
Newton didn’t publish his work on calculus because of personal reasons, but there were
also physical barriers to him printing De Analysi. The London fire of September 1666 destroyed
373 acres of land. Thousands of houses and buildings were scorched and the printing industry
took a massive hit. London was rebuilt in 3 years, which is commendable but the industry was
shaky. Producing a big book was a risk and very expensive for publishers, so authors of
mathematics (especially for such a complicated treatise) had almost no chance of being
published (Bardi, 38). Because of this De Analysi was tucked away. This turn of events lead to
Leibniz beating him to the punch.
The wars began with Newton’s publishing in the back of Opticks when he wanted to
claim the glory of the initial creation. In 1705, a European journal (closely associated with
Leibniz) printed an unnamed review of Newton’s essay. This escalated the conflict because this
anonymous person claimed that Newton had borrowed his ideas from Leibniz. Leibniz
continually denied this authorship of the claim, but it was later proved by a biographer that he
indeed wrote it (Bardi, 10). From the time of this attack to after Leibniz’s death in 1716, Newton
waged a war to claim ownership over calculus. Newton said, "Whether Mr. Leibniz invented it
after me, or had it from, is a question of no consequence, for second inventors have no rights
(Bardi, 10)." Leibniz took this extremely seriously. The group of masterminds he worked with in
Europe soon began writing letters in support of Leibniz. The following years for Leibniz would
be spent writing several anonymous attacks against Newton himself, alongside publishing his
other papers as well (Bardi, 11). Leibniz died in 1716 during the height of the wars. At this point
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"he and Newton were old men fighting openly about which of them deserved credit
and whether one had plagiarized the other. Their letters and their private writings are
bitter testimonies to their respective brilliance and rival's deceit” (Bardi, 11).
An Inter-disciplinary Construct: The Logic behind Leibniz’s Philosophies
Leibniz isn’t easily categorized in the history of thought, as he was a master of many
trades. He was a leading and influential thinker in the both the histories of mathematics and
philosophy. Leibniz would be credited with the discovery of binary (or the revision of it),
calculus, Boolean algebra, and formal logic. When one takes into account all of his musings and
meditations on these subjects, a clear picture emerges of the type of thinker Leibniz was, and
what his beliefs were. In his work there are clear overlaps between these disciplines.
Leibniz is widely held within the “pantheon of seven great philosophers,” (MacDonald,
1) and is considered a continental rationalist, alongside Descartes and Spinoza. This sort of
rational philosophy boiled down to a combination of analytic and symbolic representation,
wherein anything-from math to language to philosophy to the spiritual, and everything in
between-can be broken down into its most simple parts that comprise the greater concept. For
instance, Leibniz helped to refine the binary system, which today is used in cell phones and
computers, and many other computer-based technologies and devices. The significance of the
binary system, in addition to its practicality and technological uses, is the impact it had on
Leibniz’ philosophy. In mathematics, binary is a numeral system that represents numeric values
using two symbols: zero (0) and one (1), these being of a set that designates empty or full. This
helps us better understand Leibniz’s philosophy, or at least the context for his thought.
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Because he viewed numbers this way, he also viewed logic in this way, whereas other
philosophies and languages, even, focus on empirical evidence within a sentence, Leibniz’s
approach shifted focus to “causal relationships between sentences” (MacDonald 55) Just as
mathematicians use symbols like x and y in algebra, Leibniz believed words were just hollow
symbols that are given meaning, so instead of thinking about the content (as rhetoricians are
wont to do), Leibniz focused on the relationships between an x and a y. He is often considered
the most important thinker in the history of logic since Aristotle, and he even employed a sort of
scholastic and traditional method to his system, known as Genus and Differentia, or the “method
of division,” which stemmed from the Greek philosophers themselves (MacDonald, 49-51). This
method involved dividing things into branching classes, so that you are left with categories, and
subsets of categories, and at any given time there will be a complex concept that is defined by
simpler concepts, which in turn are defined by simpler concepts, and so on. One example is
“’Paris is the lover of Helen’... (which) asserts a relationship between two subjects...each is true
only because the other is true… (and) Ultimately, ‘God made Paris a lover because he was
making Helen loved,’” and vice-versa (MacDonald, 55). An easier example is perhaps the
concept ‘man,’ which can be understood by the terms ‘rational’ and ‘animal.’ These terms are
simpler than the thing defined, and the relationship is of a subject and predicate, where the
subject identifies a sentence, and the predicate attributes a certain property to that substance.
Another resultant in Leibniz’s numerical approach to thought is his view of the world (in the
universe) as God created it.
While working on the binary system at the same time as his notations on infinitesimal
calculus, it was conceivable to Leibniz to be able to trace or anticipate the path of an unfinished
line from just a rate of change-otherwise known as integral calculus. He discovered differential
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and integral calculus by observing quantities, or functions, and deriving a general principle for
constructing trajectories, planetary orbits, etc. However, while important to mathematics,
according to Leibniz, “the greatest significance of his discovery was metaphysical, or mystical,
as showing how the whole universe could be seen as constructed out of a Number.” Leibniz
believed in the existence of One and Zero, or nothing, and consequently that something could be
created out of nothing, i.e., the universe.
Leibniz’s Principle of Sufficient Reason states that nothing is without reason, and his
Principle of Identity states that a principle is necessarily true if it is either itself an identical
proposition, or can be reduced to one (identical predicate and subject). These two definitions,
and Leibniz’s belief in God, inform us that Leibniz believed God is the necessary truth, and we,
his creation, are made in his image (the identical proposition). In this way, Leibniz infamously
said “we live in the best of all possible worlds” (MacDonald, 103) I say infamous because this is
an Optimistic view of the universe, one that wasn’t shared as wholeheartedly by peer
philosophers, and in fact, Leibniz was the cause for and center of the 18th century, FrenchEnlightenment thinker Voltaire’s satirical novel Candide. So furthermore, things might seem
bad-for instance, natural disasters, disease etc. (criticisms which were often used against this
infamous statement of Leibniz)-but only relatively so, or due to our perspective, whereas they
might seem worse by another’s perspective. God wouldn’t create the world another way. So,
where in binary everything can be broken down to a 0 or a 1, the creation of the Earth reflects
this symbology, too; pure being (God, or a 1), and nothingness (0). God’s creative act was
therefore a “voluntary dilution of his own essence, and a mathematical computation” reflecting
perfect numbers (MacDonald, 101).
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Final Thoughts
Gottfried Wilhelm Leibniz was a man of many talents and great potential. He applied that
potential to a life of discovery and thought. From his work came calculus, binary code, the
theory of monads, etc. However, Leibniz is not a man heavily discussed in schools. He does not
get the attention Newton received in physics classrooms, nor does he receive acknowledgments
in the history classrooms. He does get discussed in the calculus classrooms, but not much beyond
his notation style vs. Newton’s. Leibniz was a man who works all stemmed from a similar logic,
and his logic was in turn often influenced by his own discoveries as well. An example of this
being his discovery of integrals through concepts he mentally constructed through his logic led
him to actually change his mental construct of the world to being an entirely number based
universal order. Evidence of the spiritual part of his logic becomes evidential here as well, as
many of his explanations often fall in to the metaphysical realm. He then moved from this point
to viewing the world as a binary construct, morphing his ideas in to one slowly evolving view of
the universe through numerical pattern identification.
In several ways Leibniz created something from nothing. He discovered calculus where
there was no calculus before. It took a specific type of understanding of the relationship shared
between trajectories, planes, dimensions, and numerical powers to discover the primitive
differential and integral calculi, an understanding no one else at the time possessed. However,
Leibnitz was careful about making his discovery accessible to others through its meticulously
designed notation system. Even Newton didn’t discover calculus through the same mental path,
which is most likely why his notation is in many ways nonsensical to most people, because it
only made sense to him. Leibniz was a giant among men, and a man who contributed more to the
human race than he took from it. It is a crime to not have him be studied in high school history
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classes at any point, and a shame he is only known for his calculus to most people who have
heard of them. However, to those who learn about the man himself. Those who learn about who
he was, and what he accomplished, can truly appreciate Leibnitz as the giant he is. In many ways
he was a renaissance man among boys, and a giant among men.
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Duigan, Brian. The 100 Most Influential Philosophers of All Time. 29 East 21st St., New
York, NY: Britannica Educational Publishing, 2010.
Jolley, Nicholas. Leibniz. 270 Madison Ave, New York, NY: Routledge, 2005.
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Mar. 2015. <http://faculty.humanities.uci.edu/bjbecker/RevoltingIdeas/leibniz.html>.
Ross, G. MacDonald. Past Masters: Leibniz. New York: Oxford University Press, 1984.